SherbyNet - /page/wikibook/entry.mu

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Pulse compression

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Pulse compression is a <a href="Signal_processing" title="Signal processing">signal processing</a> technique commonly used by <a href="Radar" title="Radar">radar</a>, <a href="Sonar" title="Sonar">sonar</a> and <a href="Ultrasound" title="Ultrasound">echography</a> to either increase the range <a href="Angular_resolution" title="Angular resolution">resolution</a> when pulse length is constrained or increase the <a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">signal to noise</a> ratio when the <a href="Peak_power" title="Peak power">peak power</a> and the <a href="Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a> (or equivalently range resolution) of the transmitted signal are constrained. This is achieved by <a href="Modulation" class="mw-redirect" title="Modulation">modulating</a> the transmitted pulse and then <a href="Cross-correlation" title="Cross-correlation">correlating</a> the received signal with the transmitted pulse.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Simple_pulse">Simple pulse</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Signal_description">Signal description</h3></div>

The ideal model for the simplest, and historically first type of signals a pulse <a href="Radar" title="Radar">radar</a> or <a href="Sonar" title="Sonar">sonar</a> can transmit is a truncated sinusoidal pulse (also called a CW—carrier wave—pulse), of amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and <a href="Carrier_frequency" class="mw-redirect" title="Carrier frequency">carrier frequency</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}}">

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</math><img src="./6423b30a4c5770c59b5ab92dcb4ce378755440ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{0}}" loading="lazy">, truncated by a <a href="Rectangular_function" title="Rectangular function">rectangular function</a> of width, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">. The pulse is transmitted periodically, but that is not the main topic of this article; we will consider only a single pulse, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">. If we assume the pulse to start at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">, the signal can be written the following way, using the <a href="Sine_and_cosine#Relationship_to_complex_numbers" title="Sine and cosine">complex notation</a>:

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<mtext>otherwise</mtext>

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<annotation encoding="application/x-tex">{\displaystyle s(t)={\begin{cases}e^{2i\pi f_{0}t}&{\text{if}}\;0\leq t<T\\0&{\text{otherwise}}\end{cases}}}</annotation>

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</math><img src="./5e9172e3ab0068dccf46a7ad30411f2c28499262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.372ex; height:6.176ex;" alt="{\displaystyle s(t)={\begin{cases}e^{2i\pi f_{0}t}&{\text{if}}\;0\leq t<T\\0&{\text{otherwise}}\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Range_resolution">Range resolution</h3></div>

Let us determine the range resolution which can be obtained with such a signal. The return signal, written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(t)}">

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</math><img src="./ec653b463c709d42cb85133595bf0da29801f6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.698ex; height:2.843ex;" alt="{\displaystyle r(t)}" loading="lazy">, is an attenuated and time-shifted copy of the original transmitted signal (in reality, <a href="Doppler_effect" title="Doppler effect">Doppler effect</a> can play a role too, but this is not important here). There is also noise in the incoming signal, both on the imaginary and the real channel. The noise is assumed to be band-limited, that is to have frequencies only in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

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<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

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</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy"> (this generally holds in reality, where a <a href="Bandpass" class="mw-redirect" title="Bandpass">bandpass</a> filter is generally used as one of the first stages in the reception chain); we write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(t)}">

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<annotation encoding="application/x-tex">{\displaystyle N(t)}</annotation>

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</math><img src="./4a8aa64e349180ee714adfb6620d3bb4610d97d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle N(t)}" loading="lazy"> to denote that noise. To detect the incoming signal, a <a href="Matched_filter" title="Matched filter">matched filter</a> is commonly used. This method is optimal when a known signal is to be detected among additive noise having a <a href="Normal_distribution" title="Normal distribution">normal distribution</a>.

In other words, the <a href="Cross-correlation" title="Cross-correlation">cross-correlation</a> of the received signal with the transmitted signal is computed. This is achieved by <a href="Convolution" title="Convolution">convolving</a> the incoming signal with a <a href="Complex_conjugate" title="Complex conjugate">conjugated</a> and time-reversed version of the transmitted signal. This operation can be done either in software or with hardware. We write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle s,r\rangle (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \langle s,r\rangle (t)}</annotation>

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</math><img src="./0a23c288284fafd29f1cf3e2eb1b71534c816c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.631ex; height:2.843ex;" alt="{\displaystyle \langle s,r\rangle (t)}" loading="lazy"> for this cross-correlation. We have:

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<annotation encoding="application/x-tex">{\displaystyle \langle s,r\rangle (t)=\int _{t'\,=\,0}^{+\infty }s^{\star }(t')r(t+t')dt'}</annotation>

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</math><img src="./46d2f6e6b4bd55cb4ce2cf8b7aa8867e8e85b6df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.016ex; height:6.009ex;" alt="{\displaystyle \langle s,r\rangle (t)=\int _{t'\,=\,0}^{+\infty }s^{\star }(t')r(t+t')dt'}" loading="lazy"></dd></dl>

If the reflected signal comes back to the receiver at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{r}}">

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</math><img src="./a50aad1c6e0103f6f74a56aff0561f45714161a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.813ex; height:2.343ex;" alt="{\displaystyle t_{r}}" loading="lazy"> and is attenuated by factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, this yields:

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<annotation encoding="application/x-tex">{\displaystyle r(t)=\left\{{\begin{array}{ll}Ae^{2i\pi f_{0}(t\,-\,t_{r})}+N(t)&{\mbox{if}}\;t_{r}\leq t<t_{r}+T\\N(t)&{\mbox{otherwise}}\end{array}}\right.}</annotation>

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</math><img src="./0f42fc5893316e7c225b7d50c304ccda2dda384f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:48.623ex; height:6.509ex;" alt="{\displaystyle r(t)=\left\{{\begin{array}{ll}Ae^{2i\pi f_{0}(t\,-\,t_{r})}+N(t)&{\mbox{if}}\;t_{r}\leq t<t_{r}+T\\N(t)&{\mbox{otherwise}}\end{array}}\right.}" loading="lazy"></dd></dl>

Since we know the transmitted signal, we obtain:

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<annotation encoding="application/x-tex">{\displaystyle \langle s,r\rangle (t)=A\Lambda \left({\frac {t-t_{r}}{T}}\right)e^{2i\pi f_{0}(t\,-\,t_{r})}+N'(t)}</annotation>

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</math><img src="./866c37244a0e6888a64dd7e97ab6dc0547083442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.478ex; height:6.176ex;" alt="{\displaystyle \langle s,r\rangle (t)=A\Lambda \left({\frac {t-t_{r}}{T}}\right)e^{2i\pi f_{0}(t\,-\,t_{r})}+N'(t)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N'(t)}">

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</math><img src="./519056f81d53fe11ad54f63f7a4e2d9c3b970557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.457ex; height:3.009ex;" alt="{\displaystyle N'(t)}" loading="lazy">, is the result of the intercorrelation between the noise and the transmitted signal. Function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> is the triangle function, its value is 0 on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [-\infty ,-{\frac {1}{2}}]\cup [{\frac {1}{2}},+\infty ]}">

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</math><img src="./5f948e0e927638dad6950e50f50da5c9d1a5601a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.148ex; height:3.509ex;" alt="{\textstyle [0,{\frac {1}{2}}]}" loading="lazy"> until it reaches 0 again. Figures at the end of this paragraph show the shape of the intercorrelation for a sample signal (in red), in this case a real truncated sine, of duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=1}">

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</math><img src="./0c3daa0b80c432b6f5bf8e70bef2a3ce6dbc3567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.617ex; height:2.509ex;" alt="{\textstyle f_{0}=10}" loading="lazy"> hertz. Two echoes (in blue) come back with delays of 3 and 5 seconds and amplitudes equal to 0.5 and 0.3 times the amplitude of the transmitted pulse, respectively; these are just random values for the sake of the example. Since the signal is real, the intercorrelation is weighted by an additional <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style>1⁄2 factor.

If two pulses come back (nearly) at the same time, the intercorrelation is equal to the sum of the intercorrelations of the two elementary signals. To distinguish one "triangular" envelope from that of the other pulse, it is clearly visible that the times of arrival of the two pulses must be separated by at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> so that the maxima of both pulses can be separated. If this condition is not met, both triangles will be mixed together and impossible to separate.

Since the distance travelled by a wave during <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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</math><img src="./bd72d0d7b8ae7824511bebf90618f49d4e884dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.643ex; height:2.176ex;" alt="{\displaystyle cT}" loading="lazy"> (where c is the speed of the wave in the medium), and since this distance corresponds to a round-trip time, we get:

<th>Result 1

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<td>The range resolution with a sinusoidal pulse is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}cT}">

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Conclusion: to increase the resolution, the pulse length must be reduced.

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<caption>Example (simple impulsion): transmitted signal in red (carrier 10 hertz, amplitude 1, duration 1 second) and two echoes (in blue).

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<th>Before matched filtering

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<th>After matched filtering

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<div class="mw-heading mw-heading3"><h3 id="Energy_and_signal-to-noise_ratio_of_the_received_signal">Energy and signal-to-noise ratio of the received signal</h3></div>

The instantaneous power of the received pulse is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(t)=|r|^{2}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle P(t)=|r|^{2}(t)}</annotation>

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</math><img src="./d469a884fedfaff8c7583be2972e13ca65a203e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.538ex; height:3.343ex;" alt="{\displaystyle P(t)=|r|^{2}(t)}" loading="lazy">. The energy put into that signal is:

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<annotation encoding="application/x-tex">{\displaystyle E=\int _{0}^{T}P(t)dt=A^{2}T}</annotation>

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</math><img src="./12456d32b5bb9d46be377fbd4c4e6db5925835cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.097ex; height:6.176ex;" alt="{\displaystyle E=\int _{0}^{T}P(t)dt=A^{2}T}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is the standard deviation of the noise which is assumed to have the same bandwidth as the signal, the signal-to-noise ratio (SNR) at the receiver is:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle SNR={\frac {E_{r}}{\sigma ^{2}}}={\frac {A^{2}T}{\sigma ^{2}}}}</annotation>

</semantics>

</math><img src="./d204e24cc2f505056276ecef23a0d62f342eba27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.319ex; height:6.009ex;" alt="{\displaystyle SNR={\frac {E_{r}}{\sigma ^{2}}}={\frac {A^{2}T}{\sigma ^{2}}}}" loading="lazy"></dd></dl>

The SNR is proportional to pulse duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, if other parameters are held constant. This introduces a tradeoff: increasing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> improves the SNR, but reduces the resolution, and vice versa.

<div class="mw-heading mw-heading2"><h2 id="Pulse_compression_by_linear_frequency_modulation_(or_chirping)">Pulse compression by linear frequency modulation (or chirping)</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Basic_principles">Basic principles</h3></div>

How can one have a large enough pulse (to still have a good SNR at the receiver) without poor resolution? This is where pulse compression enters the picture. The basic principle is the following:

<ul><li>a signal is transmitted, with a long enough length so that the energy budget is correct</li>

<li>this signal is designed so that after matched filtering, the width of the intercorrelated signals is smaller than the width obtained by the standard sinusoidal pulse, as explained above (hence the name of the technique: pulse compression).</li></ul>

In <a href="Radar" title="Radar">radar</a> or <a href="Sonar" title="Sonar">sonar</a> applications, linear <a href="Chirp" title="Chirp">chirps</a> are the most typically used signals to achieve pulse compression. The pulse being of finite length, the amplitude is a <a href="Rectangle_function" class="mw-redirect" title="Rectangle function">rectangle function</a>. If the transmitted signal has a duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, begins at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> and linearly sweeps the frequency band <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

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</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy"> centered on carrier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f_{0}}</annotation>

</semantics>

</math><img src="./6423b30a4c5770c59b5ab92dcb4ce378755440ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{0}}" loading="lazy">, it can be written:

<msub>

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<mtext>otherwise</mtext>

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<annotation encoding="application/x-tex">{\displaystyle s_{c}(t)=\left\{{\begin{array}{ll}e^{i2\pi \left(\left(f_{0}\,-\,{\frac {\Delta f}{2}}\right)t\,+\,{\frac {\Delta f}{2T}}t^{2}\,\right)}&{\mbox{if}}\;0\leq t<T\\0&{\mbox{otherwise}}\end{array}}\right.}</annotation>

</semantics>

</math><img src="./960ef240cdcd08e03e2d4b1969217e3dc3e758c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:46.758ex; height:8.509ex;" alt="{\displaystyle s_{c}(t)=\left\{{\begin{array}{ll}e^{i2\pi \left(\left(f_{0}\,-\,{\frac {\Delta f}{2}}\right)t\,+\,{\frac {\Delta f}{2T}}t^{2}\,\right)}&{\mbox{if}}\;0\leq t<T\\0&{\mbox{otherwise}}\end{array}}\right.}" loading="lazy"></dd></dl>

The chirp definition above means that the phase of the chirped signal (that is, the argument of the complex exponential), is the quadratic:

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \phi (t)=2\pi \left(\left(f_{0}\,-\,{\frac {\Delta f}{2}}\right)t\,+\,{\frac {\Delta f}{2T}}t^{2}\,\right)}</annotation>

</semantics>

</math><img src="./40191bcc45d9fd3605ba6a819ba16fc1b1db5baa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.888ex; height:6.176ex;" alt="{\displaystyle \phi (t)=2\pi \left(\left(f_{0}\,-\,{\frac {\Delta f}{2}}\right)t\,+\,{\frac {\Delta f}{2T}}t^{2}\,\right)}" loading="lazy"></dd></dl>

thus the instantaneous frequency is (by definition):

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<annotation encoding="application/x-tex">{\displaystyle f(t)={\frac {1}{2\pi }}\left[{\frac {d\phi }{dt}}\right]_{t}=f_{0}-{\frac {\Delta f}{2}}+{\frac {\Delta f}{T}}t}</annotation>

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</math><img src="./7e18de051c6912fa2332f597c9ae65c306a71614.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.988ex; height:6.176ex;" alt="{\displaystyle f(t)={\frac {1}{2\pi }}\left[{\frac {d\phi }{dt}}\right]_{t}=f_{0}-{\frac {\Delta f}{2}}+{\frac {\Delta f}{T}}t}" loading="lazy"></dd></dl>

which is the intended linear ramp going from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{0}-{\frac {\Delta f}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{0}-{\frac {\Delta f}{2}}}</annotation>

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</math><img src="./30ca02fbea61eede6a448960cbe5236e224932ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.084ex; height:5.509ex;" alt="{\displaystyle f_{0}-{\frac {\Delta f}{2}}}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

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</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f_{0}+{\frac {\Delta f}{2}}}">

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<annotation encoding="application/x-tex">{\textstyle f_{0}+{\frac {\Delta f}{2}}}</annotation>

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</math><img src="./03f80ca408ff5ff6fb966a4ad759268322775754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.143ex; height:4.009ex;" alt="{\textstyle f_{0}+{\frac {\Delta f}{2}}}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=T}">

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<annotation encoding="application/x-tex">{\displaystyle t=T}</annotation>

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</math><img src="./5c6b2eabe8e275c2da71dbc61ca0ede73a418051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.574ex; height:2.176ex;" alt="{\displaystyle t=T}" loading="lazy">.

The relation of phase to frequency is often used in the other direction, starting with the desired <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

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</math><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"> and writing the chirp phase via the integration of frequency:

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<annotation encoding="application/x-tex">{\displaystyle \phi (t)=2\pi \int _{0}^{t}f(u)\,du}</annotation>

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</math><img src="./8fec1c85e4e0b9138236057f9283df50901c03f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.042ex; height:6.176ex;" alt="{\displaystyle \phi (t)=2\pi \int _{0}^{t}f(u)\,du}" loading="lazy"></dd></dl>

This transmitted signal is typically reflected by the target and undergoes attenuation due to various causes, so the received signal is a time-delayed, attenuated version of the transmitted signal plus an additive noise of constant power spectral density on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

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<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

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</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy">, and zero everywhere else:

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<mtext>otherwise</mtext>

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<annotation encoding="application/x-tex">{\displaystyle r(t)=\left\{{\begin{array}{ll}Ae^{i2\pi \left(\left(f_{0}\,-\,{\frac {\Delta f}{2}}\right)(t-t_{r})\,+\,{\frac {\Delta f}{2T}}(t-t_{r})^{2}\,\right)}+N(t)&{\mbox{if}}\;t_{r}\leq t<t_{r}+T\\N(t)&{\mbox{otherwise}}\end{array}}\right.}</annotation>

</semantics>

</math><img src="./dfa4dd5a8b84f448bb760a98d8ace5a60258fd88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:68.208ex; height:8.509ex;" alt="{\displaystyle r(t)=\left\{{\begin{array}{ll}Ae^{i2\pi \left(\left(f_{0}\,-\,{\frac {\Delta f}{2}}\right)(t-t_{r})\,+\,{\frac {\Delta f}{2T}}(t-t_{r})^{2}\,\right)}+N(t)&{\mbox{if}}\;t_{r}\leq t<t_{r}+T\\N(t)&{\mbox{otherwise}}\end{array}}\right.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Cross-correlation_between_the_transmitted_and_the_received_signal">Cross-correlation between the transmitted and the received signal</h3></div>

We now endeavor to compute the correlation of the received signal with the transmitted signals. Two actions are going to be taken to do this:

- The first action is a simplification. Instead of computing the cross-correlation we are going to compute an auto-correlation which amounts to assuming that the autocorrelation peak is centered at zero. This will not change the resolution and the amplitudes but will simplify the math:

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<annotation encoding="application/x-tex">{\displaystyle r'(t)={\begin{cases}Ae^{2i\pi \left(f_{0}\,+\,{\frac {\Delta f}{2T}}t\right)t}+N(t)&{\mbox{if}}\;-{\frac {T}{2}}\leq t<{\frac {T}{2}}\\N(t)&{\mbox{otherwise}}\end{cases}}}</annotation>

</semantics>

</math><img src="./3c60386b46c5b4bb3bb7edf1799352e74dedbd16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:51.409ex; height:8.509ex;" alt="{\displaystyle r'(t)={\begin{cases}Ae^{2i\pi \left(f_{0}\,+\,{\frac {\Delta f}{2T}}t\right)t}+N(t)&{\mbox{if}}\;-{\frac {T}{2}}\leq t<{\frac {T}{2}}\\N(t)&{\mbox{otherwise}}\end{cases}}}" loading="lazy"></dd></dl>

- The second action is, as shown below, is to set an amplitude for the reference signal which is not one, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \neq 1}">

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<annotation encoding="application/x-tex">{\displaystyle \rho \neq 1}</annotation>

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</math><img src="./41eaf8a5d8b6ec6c41b4392112a4433327ba6782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho \neq 1}" loading="lazy">. Constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is to be determined so that energy is conserved through correlation.

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<annotation encoding="application/x-tex">{\displaystyle s_{c}'(t)={\begin{cases}\rho e^{2i\pi \left(f_{0}\,+\,{\frac {\Delta f}{2T}}t\right)t}&{\mbox{if}}\;-{\frac {T}{2}}\leq t<{\frac {T}{2}}\\0&{\mbox{otherwise}}\end{cases}}}</annotation>

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</math><img src="./ac9bf343b4c641d95e15d2fecae61b6fd919151e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:43.616ex; height:8.509ex;" alt="{\displaystyle s_{c}'(t)={\begin{cases}\rho e^{2i\pi \left(f_{0}\,+\,{\frac {\Delta f}{2T}}t\right)t}&{\mbox{if}}\;-{\frac {T}{2}}\leq t<{\frac {T}{2}}\\0&{\mbox{otherwise}}\end{cases}}}" loading="lazy"></dd></dl>

Now, it can be shown<a href="#cite_note-2">[2]</a> that the correlation function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{c}'}">

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<annotation encoding="application/x-tex">{\displaystyle s_{c}'}</annotation>

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</math><img src="./81756cc8fc14d44cbca3ad920fa1dc9f018e36a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.035ex; height:2.509ex;" alt="{\displaystyle s_{c}'}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r'}">

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<annotation encoding="application/x-tex">{\displaystyle r'}</annotation>

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</math><img src="./20f4d06078c3550f2cd0812005ba6301d12cc4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.733ex; height:2.509ex;" alt="{\displaystyle r'}" loading="lazy"> is:

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<annotation encoding="application/x-tex">{\displaystyle \langle s_{c}',r'\rangle (t)=\rho A{\sqrt {T}}\Lambda \left({\frac {t}{T}}\right)\mathrm {sinc} \left[\Delta ft\Lambda \left({\frac {t}{T}}\right)\right]e^{2i\pi f_{0}t}+N'(t)}</annotation>

</semantics>

</math><img src="./6cec7bc53ea50cdb02fc039a4316454959ab935b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.396ex; height:6.176ex;" alt="{\displaystyle \langle s_{c}',r'\rangle (t)=\rho A{\sqrt {T}}\Lambda \left({\frac {t}{T}}\right)\mathrm {sinc} \left[\Delta ft\Lambda \left({\frac {t}{T}}\right)\right]e^{2i\pi f_{0}t}+N'(t)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N'(t)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N'(t)}</annotation>

</semantics>

</math><img src="./519056f81d53fe11ad54f63f7a4e2d9c3b970557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.457ex; height:3.009ex;" alt="{\displaystyle N'(t)}" loading="lazy"> is the correlation of the reference signal with the received noise.

<div class="mw-heading mw-heading4"><h4 id="Width_of_the_signal_after_correlation">Width of the signal after correlation</h4></div>

Assuming noise is zero, the maximum of the autocorrelation function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{c'}}">

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{c'}}</annotation>

</semantics>

</math><img src="./e2ca361812e02316483109e67ff82baab936ecc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.566ex; height:2.009ex;" alt="{\displaystyle s_{c'}}" loading="lazy"> is reached at 0. Around 0, this function behaves as the <a href="Sinc" class="mw-redirect" title="Sinc">sinc</a> (or cardinal sine) term, defined here as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sinc(x)=sin(\pi x)/(\pi x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle sinc(x)=sin(\pi x)/(\pi x)}</annotation>

</semantics>

</math><img src="./9085a194281fbe2a40d6fb2a0071c9f729ad41a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.924ex; height:2.843ex;" alt="{\displaystyle sinc(x)=sin(\pi x)/(\pi x)}" loading="lazy">. The −3 dB temporal width of that cardinal sine is more or less equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T'={\frac {1}{\Delta f}}}">

<msup>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle T'={\frac {1}{\Delta f}}}</annotation>

</semantics>

</math><img src="./7bff42d75d7175bd2a279a53c55e18ff8ac18088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.612ex; height:4.009ex;" alt="{\textstyle T'={\frac {1}{\Delta f}}}" loading="lazy">. Everything happens as if, after matched filtering, we had the resolution that would have been reached with a simple pulse of duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T'}</annotation>

</semantics>

</math><img src="./4b32d735f4fbb6eff3b35ed3dc1005a069d0b2e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle T'}" loading="lazy">. For the common values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T'}</annotation>

</semantics>

</math><img src="./4b32d735f4fbb6eff3b35ed3dc1005a069d0b2e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle T'}" loading="lazy"> is smaller than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, hence the pulse compression name.

Since the cardinal sine can have annoying <a href="Sidelobe" class="mw-redirect" title="Sidelobe">sidelobes</a>, a common practice is to filter the result by a window (<a href="Hamming_window" class="mw-redirect" title="Hamming window">Hamming</a>, <a href="Hann_function" title="Hann function">Hann</a>, etc.). In practice, this can be done at the same time as the adapted filtering by multiplying the reference chirp with the filter. The result will be a signal with a slightly lower maximum amplitude, but the sidelobes will be filtered out, which is more important.

<th>Result 2

</th></tr>

<tr>

<td>The distance resolution reachable with a linear frequency modulation of a pulse on a bandwidth <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy"> is: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {c}{2\Delta f}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {c}{2\Delta f}}}</annotation>

</semantics>

</math><img src="./b4a458485b1e3b5aea3c7a84c1aa90a8b574b48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:3.931ex; height:3.676ex;" alt="{\textstyle {\frac {c}{2\Delta f}}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is the speed of the wave.

</td></tr></tbody></table>

<th>Definition

</th></tr>

<tr>

<td>Ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {T}{T'}}=T\Delta f}">

<mfrac>

<msup>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {T}{T'}}=T\Delta f}</annotation>

</semantics>

</math><img src="./397a80163284ba3f7142cfc3a0105517de128e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:10.533ex; height:3.843ex;" alt="{\textstyle {\frac {T}{T'}}=T\Delta f}" loading="lazy"> is the pulse compression ratio. It is generally greater than 1 (usually, its value is 20 to 30).

</td></tr></tbody></table>

<caption>Example (chirped pulse): transmitted signal in red (carrier 10 hertz, modulation on 16 hertz, amplitude 1, duration 1 second) and two echoes (in blue).

</caption>

<td>

</td>

<td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading4"><h4 id="Energy_and_peak_power_after_correlation">Energy and peak power after correlation</h4></div>

When the reference signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{c}'}">

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{c}'}</annotation>

</semantics>

</math><img src="./81756cc8fc14d44cbca3ad920fa1dc9f018e36a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.035ex; height:2.509ex;" alt="{\displaystyle s_{c}'}" loading="lazy"> is correctly scaled using term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">, then it is possible to conserve the energy before and after correlation. The peak (and average) power before correlation is:

<msub>

<msup>

</msup>

</mrow>

</msub>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{r'}=|r'(t)|^{2}=P_{r'}^{peak}=A^{2}}</annotation>

</semantics>

</math><img src="./0b779a0f0a94b5fdecfa57c84080fd31293e2cd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.193ex; height:3.676ex;" alt="{\displaystyle P_{r'}=|r'(t)|^{2}=P_{r'}^{peak}=A^{2}}" loading="lazy"></dd></dl>

Since, before compression, the pulse is box-shaped, the energy before correlation is:

<msub>

<msup>

</msup>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{r'}=\int _{-T/2}^{T/2}|r'(t)|^{2}dt=A^{2}T}</annotation>

</semantics>

</math><img src="./03073229f955dd684da37723449e82622e1ad373.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.628ex; height:6.676ex;" alt="{\displaystyle E_{r'}=\int _{-T/2}^{T/2}|r'(t)|^{2}dt=A^{2}T}" loading="lazy"></dd></dl>

The peak power after correlation is reached at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy">:

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{<s_{c}',r'>}^{peak}=|<s_{c}',r'>(0)|^{2}=\rho ^{2}A^{2}T}</annotation>

</semantics>

</math><img src="./299936eaab48266e52b9c2c993adf3e55731b638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:36.731ex; height:4.009ex;" alt="{\displaystyle P_{<s_{c}',r'>}^{peak}=|<s_{c}',r'>(0)|^{2}=\rho ^{2}A^{2}T}" loading="lazy"></dd></dl>

Note that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =1}</annotation>

</semantics>

</math><img src="./6be1a852fccaddc2582e41b89a02b41b1ff4ffe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =1}" loading="lazy"> this peak power is the energy of the received signal before correlation, which is as expected.

After compression, the pulse is approximal by a box having a width equal to the typical width of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sinc}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle sinc}</annotation>

</semantics>

</math><img src="./58bb379a499bcc49bc73337668800d149a58130e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.294ex; height:2.176ex;" alt="{\displaystyle sinc}" loading="lazy"> function, that is, a width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'=1/\Delta f}">

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T'=1/\Delta f}</annotation>

</semantics>

</math><img src="./0e39e919db2ee892042439541c413390a8bdd424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.043ex; height:3.009ex;" alt="{\displaystyle T'=1/\Delta f}" loading="lazy">, so the energy after correlation is:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{<s_{c}',r'>}=\int _{-\infty }^{+\infty }|<s_{c}',r'>(t)|^{2}dt\approx P_{<s_{c}',r'>}^{peak}\times T'=\rho ^{2}{\frac {A^{2}T}{\Delta f}}}">

<msub>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{<s_{c}',r'>}=\int _{-\infty }^{+\infty }|<s_{c}',r'>(t)|^{2}dt\approx P_{<s_{c}',r'>}^{peak}\times T'=\rho ^{2}{\frac {A^{2}T}{\Delta f}}}</annotation>

</semantics>

</math><img src="./1d096431017fefdea3b00502c6229ae03a7a5e40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.397ex; height:6.343ex;" alt="{\displaystyle E_{<s_{c}',r'>}=\int _{-\infty }^{+\infty }|<s_{c}',r'>(t)|^{2}dt\approx P_{<s_{c}',r'>}^{peak}\times T'=\rho ^{2}{\frac {A^{2}T}{\Delta f}}}" loading="lazy"></dd></dl>

If energy is conserved:

<msub>

<msup>

</msup>

</mrow>

</msub>

<msub>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{r'}=E_{<s_{c}',r'>}}</annotation>

</semantics>

</math><img src="./cea43ba8e9061eda958b10d26962e9606581643d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.067ex; height:2.843ex;" alt="{\displaystyle E_{r'}=E_{<s_{c}',r'>}}" loading="lazy"></dd></dl>

... it comes that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\sqrt {\Delta f}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt {\Delta f}}}</annotation>

</semantics>

</math><img src="./22cf603669c72725b742c9f4b08e0c8371ec5a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.839ex; height:3.509ex;" alt="{\displaystyle \rho ={\sqrt {\Delta f}}}" loading="lazy"> so that the peak power after correlation is:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{<s_{c}',r'>}^{peak}=\rho ^{2}A^{2}T=P_{r'}\times \Delta f\times T}">

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{<s_{c}',r'>}^{peak}=\rho ^{2}A^{2}T=P_{r'}\times \Delta f\times T}</annotation>

</semantics>

</math><img src="./1a90dac5dc9128057a9233df9f5ad8b2e4ef0bd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:33.941ex; height:4.009ex;" alt="{\displaystyle P_{<s_{c}',r'>}^{peak}=\rho ^{2}A^{2}T=P_{r'}\times \Delta f\times T}" loading="lazy"></dd></dl>

As a conclusion, the peak power of the pulse-compressed signal is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f\times T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f\times T}</annotation>

</semantics>

</math><img src="./215e5db0975c68b5ab7faab0f60a16ee40665938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.691ex; height:2.509ex;" alt="{\displaystyle \Delta f\times T}" loading="lazy"> that of the raw received signal (assuming that the template <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{c}'}">

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{c}'}</annotation>

</semantics>

</math><img src="./81756cc8fc14d44cbca3ad920fa1dc9f018e36a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.035ex; height:2.509ex;" alt="{\displaystyle s_{c}'}" loading="lazy"> is correctly scaled to conserve energy through correlation).

<div class="mw-heading mw-heading3"><h3 id="Signal-to-noise_gain_after_correlation">Signal-to-noise gain after correlation</h3></div>

As we have seen above, things are written so that the energy of the signal does not vary during pulse compression. However, it is now located in the main lobe of the cardinal sine, whose width is approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T'\approx {\frac {1}{\Delta f}}}">

<msup>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle T'\approx {\frac {1}{\Delta f}}}</annotation>

</semantics>

</math><img src="./7b2fcfda9c7cdd52a82d7b7690877d0e9ac3b87f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:8.612ex; height:4.009ex;" alt="{\textstyle T'\approx {\frac {1}{\Delta f}}}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is the power of the signal before compression, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P'}</annotation>

</semantics>

</math><img src="./ef4efa52c8f47f06136f6ebfd1d68c1249aaca39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle P'}" loading="lazy"> the power of the signal after compression, energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is conserved and we have:

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=P\times T=P'\times T'}</annotation>

</semantics>

</math><img src="./88be32e233ce346d32c8d84a7879169cb5e53daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.946ex; height:2.509ex;" alt="{\displaystyle E=P\times T=P'\times T'}" loading="lazy"></dd></dl>

which yields an increase in power after pulse compression:

<msup>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle P'=P\times {\frac {T}{T'}}}</annotation>

</semantics>

</math><img src="./63e5f97669234a56011d63c125726dcc3fb3cdca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.431ex; height:5.176ex;" alt="{\displaystyle P'=P\times {\frac {T}{T'}}}" loading="lazy"></dd></dl>

In the spectral domain, the power spectrum of the chirp has a nearly constant spectral density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=P/\Delta f}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle D=P/\Delta f}</annotation>

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</math><img src="./61e2a508bf035c7dcac871274b2d4f04f2f30466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.145ex; height:2.843ex;" alt="{\displaystyle D=P/\Delta f}" loading="lazy"> in interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

<msub>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

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</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy"> and zero elsewhere, so that energy is equivalently expressed as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=P\times T=D.\Delta f.T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=P\times T=D.\Delta f.T}</annotation>

</semantics>

</math><img src="./f56d0e8c7212af37ff01a3a1b0e02406c52c6f97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.038ex; height:2.509ex;" alt="{\displaystyle E=P\times T=D.\Delta f.T}" loading="lazy">. This spectral density remains the same after matched filtering.

Imagining now an equivalent sinusoidal (CW) pulse of duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'=1/\Delta f}">

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle T'=1/\Delta f}</annotation>

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</math><img src="./0e39e919db2ee892042439541c413390a8bdd424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.043ex; height:3.009ex;" alt="{\displaystyle T'=1/\Delta f}" loading="lazy"> and identical input power, this equivalent sinusoidal pulse has an energy:

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<msup>

</msup>

<mfrac>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle E'=P\times T'=E{\frac {T'}{T}}}</annotation>

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</math><img src="./8e8080fad6565db14edcf55fe8fa083ddab35d32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.682ex; height:5.509ex;" alt="{\displaystyle E'=P\times T'=E{\frac {T'}{T}}}" loading="lazy"></dd></dl>

After matched filtering, the equivalent sinusoidal pulse turns into a triangular-shaped signal of twice its original width but the same peak power. Energy is conserved. The spectral domain is approximated by a nearly constant spectral density <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D'}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle D'}</annotation>

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</math><img src="./3c3bf8caca74bc346fa19acded4fc1a79e3ec114.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.609ex; height:2.509ex;" alt="{\displaystyle D'}" loading="lazy"> in interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

</semantics>

</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f\approx 1/T'}">

</mrow>

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \Delta f\approx 1/T'}</annotation>

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</math><img src="./33aa347ca90b4df970db98d86e16552544facdbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.043ex; height:3.009ex;" alt="{\displaystyle \Delta f\approx 1/T'}" loading="lazy">. Through conservation of energy, we have:

<msup>

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<mfrac>

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</mrow>

<mfrac>

<msup>

</msup>

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<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle E'=E{\frac {T'}{T}}=D\Delta fT{\frac {T'}{T}}=D\Delta fT'}</annotation>

</semantics>

</math><img src="./4003e8268cc0dd96fda1105e7dc9209f8c78071e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.35ex; height:5.509ex;" alt="{\displaystyle E'=E{\frac {T'}{T}}=D\Delta fT{\frac {T'}{T}}=D\Delta fT'}" loading="lazy"></dd></dl>

Since by definition we also have: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'=D'\Delta fT'}">

<msup>

</msup>

<msup>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle E'=D'\Delta fT'}</annotation>

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</math><img src="./b9a5fccf6f580ffffc8e7d773edac6b08b4f46e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.805ex; height:2.843ex;" alt="{\displaystyle E'=D'\Delta fT'}" loading="lazy"> it comes that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D'=D}">

<msup>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle D'=D}</annotation>

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</math><img src="./be385b9caa387e15469052a78324549d03267afe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.632ex; height:2.509ex;" alt="{\displaystyle D'=D}" loading="lazy"> meaning that the spectral densities of the chirped pulse, and the equivalent CW pulse are very nearly identical, and are equivalent to that of a bandpass filter on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

<msub>

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<msub>

</mrow>

</msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

</semantics>

</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy">. The filtering effect of correlation also acts on the noise, meaning that the reference band for the noise is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy"> and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=D'}">

<msup>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle D=D'}</annotation>

</semantics>

</math><img src="./a03ffb368187c30e67132effa059c4f4bb1783c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.632ex; height:2.509ex;" alt="{\displaystyle D=D'}" loading="lazy">, the same filtering effect is obtained on the noise in both cases after correlation. This means that the net effect of pulse compression is that, compared to the equivalent CW pulse, the <a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> (SNR) has improved by a factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T/T'}">

</mrow>

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle T/T'}</annotation>

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</math><img src="./cdafc57d3b01b4c5acd88800f6c3e84862e7d3f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:3.009ex;" alt="{\displaystyle T/T'}" loading="lazy"> because the signal is amplified but not the noise.

As a consequence:

<th>Result 3

</th></tr>

<tr>

<td>After pulse compression, the signal-to-noise ratio can be considered as being amplified by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\Delta f}</annotation>

</semantics>

</math><img src="./3670467d162aa79b83693707d2e1e08ad48a37b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.851ex; height:2.509ex;" alt="{\displaystyle T\Delta f}" loading="lazy"> as compared to the baseline situation of a continuous-wave pulse of duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'=1/\Delta f}">

<msup>

</msup>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T'=1/\Delta f}</annotation>

</semantics>

</math><img src="./0e39e919db2ee892042439541c413390a8bdd424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.043ex; height:3.009ex;" alt="{\displaystyle T'=1/\Delta f}" loading="lazy"> and the same amplitude as the chirp-modulated signal before compression, where the received signal and noise have (implicitly) undergone a bandpass filtering on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

<msub>

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<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

</semantics>

</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy">. This additional gain can be injected into the <a href="Radar_equation" class="mw-redirect" title="Radar equation">radar equation</a>.

</td></tr></tbody></table>

<caption>Example: same signals as above, plus an additive white Gaussian having undergone bandpass filtering (standard deviation of real part: 0.125 after filtering). After correlation, the power of the noise is unchanged. The signal itself is amplified by a factor four (or 16 for the power, as predicted by theory).

</caption>

<td>

</td>

<td>

</td></tr></tbody></table>

For technical reasons, correlation is not necessarily done for actual received CW pulses as for chirped pulses. However during <a href="Baseband" title="Baseband">baseband</a> shifting the signal undergoes a <a href="Bandpass" class="mw-redirect" title="Bandpass">bandpass</a> filtering on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}</annotation>

</semantics>

</math><img src="./0dbc58eea80c0bbb8441cb977506db96f88a16b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:2.843ex;" alt="{\displaystyle [f_{0}-\Delta f/2,f_{0}+\Delta f/2]}" loading="lazy"> which has the same net effect on the noise as the correlation, so the overall reasoning remains the same (that is, the SNR makes only sense for noise defined on a given bandwidth, here being that of the signal).

This gain in the SNR seems magical, but remember that the power spectral density does not represent the phase of the signal. In reality the phases are different for the equivalent CW pulse, the CW pulse after correlation, the original chirped pulse and the correlated chirped pulse, which explains the different shapes of the signals (especially the varying lengths) despite having (nearly) the same power spectrum in all cases. If the peak transmitting power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> and the bandwidth <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

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</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy"> are constrained, pulse compression thus achieves a better peak power (but same resolution) by transmitting a longer pulse (that is, more energy), compared to an equivalent CW pulse of same peak power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> and bandwidth <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy">, and squeezing the pulse by correlation. This works best only for a limited number of signal types which, after correlation, have a narrower peak than the original signal, and low sidelobes.

<div class="mw-heading mw-heading3"><h3 id="Stretch_processing">Stretch processing</h3></div>

While pulse compression can ensure good SNR and fine range resolution in the same time, digital signal processing in such a system can be difficult to implement because of the high instantaneous bandwidth of the waveform (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

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</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy"> can be hundreds of megahertz or even exceed 1 GHz.)

Stretch Processing is a technique for matched filtering of wideband chirping waveform and is suitable for applications seeking very fine range resolution over relatively short range intervals.<a href="#cite_note-3">[3]</a>

Picture above shows the scenario for analyzing stretch processing. The central reference point(CRP) is in the middle of the range window of interest at range of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation>

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</math><img src="./9b8916196f182fcbaaca54f931176a4a4f5769cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}" loading="lazy">, corresponding to a time delay of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

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</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy">.

If the transmitted waveform is the chirp waveform:

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle x(t)=\exp \left(j\pi {\frac {\Delta f}{T}}(t)^{2}\right)\exp(j2\pi f_{0}(t)),0\leq t\leq T}</annotation>

</semantics>

</math><img src="./f7e39bf4dab89b69a42732ea87f45a0aca505925.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.007ex; height:6.176ex;" alt="{\displaystyle x(t)=\exp \left(j\pi {\frac {\Delta f}{T}}(t)^{2}\right)\exp(j2\pi f_{0}(t)),0\leq t\leq T}" loading="lazy"></dd></dl>

then the echo from the target at distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{b}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{b}}</annotation>

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</math><img src="./565df947af6e95bcd8e1f201ac4988a286e84a84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.702ex; height:2.509ex;" alt="{\displaystyle R_{b}}" loading="lazy">can be expressed as:

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}(t)=\rho \exp \left(j\pi {\frac {\Delta f}{T}}(t-t_{b})^{2}\right)\exp(j2\pi f_{0}(t-t_{b})),0\leq t-t_{b}\leq T}</annotation>

</semantics>

</math><img src="./955476dec91994a1e6c62e1d9a5b2a214b6165b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.449ex; height:6.176ex;" alt="{\displaystyle {\bar {x}}(t)=\rho \exp \left(j\pi {\frac {\Delta f}{T}}(t-t_{b})^{2}\right)\exp(j2\pi f_{0}(t-t_{b})),0\leq t-t_{b}\leq T}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is proportional to the scatterer reflectivity.

We then multiply the echo by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \exp(-j2\pi f_{0}t)\exp \left(-j\pi {\frac {\Delta f}{T}}(t-t_{0})^{2}\right)}">

<msub>

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<annotation encoding="application/x-tex">{\textstyle \exp(-j2\pi f_{0}t)\exp \left(-j\pi {\frac {\Delta f}{T}}(t-t_{0})^{2}\right)}</annotation>

</semantics>

</math><img src="./185be97a7b69a173421627fb18f8e51758084745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.015ex; height:4.843ex;" alt="{\textstyle \exp(-j2\pi f_{0}t)\exp \left(-j\pi {\frac {\Delta f}{T}}(t-t_{0})^{2}\right)}" loading="lazy"> and the echo will become:

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

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</msub>

<msub>

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</msub>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(t)=\rho \exp \left(-j{\frac {4\pi R_{b}}{\lambda }}\right)\exp \left(-j2\pi {\frac {\Delta f}{T}}\delta t_{b}(t-t_{0})\right)\exp \left(j\pi {\frac {\Delta f}{T}}(\delta t_{b})^{2}\right),t_{0}\leq t-\delta t_{b}\leq t_{0}+T}</annotation>

</semantics>

</math><img src="./5fbfff9990d95f8b9ff1b9315016c380023d5c12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:92.537ex; height:6.176ex;" alt="{\displaystyle y(t)=\rho \exp \left(-j{\frac {4\pi R_{b}}{\lambda }}\right)\exp \left(-j2\pi {\frac {\Delta f}{T}}\delta t_{b}(t-t_{0})\right)\exp \left(j\pi {\frac {\Delta f}{T}}(\delta t_{b})^{2}\right),t_{0}\leq t-\delta t_{b}\leq t_{0}+T}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is the wavelength of electromagnetic wave in air.

After conducting sampling and discrete Fourier transform on y(t) the sinusoid frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{b}}</annotation>

</semantics>

</math><img src="./b151118e130ddb3d3eea064479c152bd16738f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle F_{b}}" loading="lazy"> can be solved:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{b}=-\delta t_{b}{\frac {\Delta f}{T}}(Hz)}</annotation>

</semantics>

</math><img src="./def1ffe8b00ef77411abc80ef2814b1572a302a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.176ex; height:5.509ex;" alt="{\displaystyle F_{b}=-\delta t_{b}{\frac {\Delta f}{T}}(Hz)}" loading="lazy"></dd></dl>

and the differential range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta R_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta R_{b}}</annotation>

</semantics>

</math><img src="./bff9d4b417c7f06e0ff1babaee561988e4dfe39d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.75ex; height:2.676ex;" alt="{\displaystyle \delta R_{b}}" loading="lazy"> can be obtained:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta R_{b}=-{\frac {cTF_{b}}{2\Delta f}}}</annotation>

</semantics>

</math><img src="./f7adda866d0b862840f4f386ecc5cda02d5cc633.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.568ex; height:5.843ex;" alt="{\displaystyle \delta R_{b}=-{\frac {cTF_{b}}{2\Delta f}}}" loading="lazy"></dd></dl>

To show that the bandwidth of y(t) is less than the original signal bandwidth <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy">, we suppose that the range window is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{w}={\frac {cT_{w}}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{w}={\frac {cT_{w}}{2}}}</annotation>

</semantics>

</math><img src="./60c61ed00991b5f548196aa9068617ecae2ba68b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.881ex; height:5.176ex;" alt="{\displaystyle R_{w}={\frac {cT_{w}}{2}}}" loading="lazy"> long. If the target is at the lower bound of the range window, the echo will arrive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}-T_{w}/2}">

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}-T_{w}/2}</annotation>

</semantics>

</math><img src="./604cab973e8ca32a59cd175a0023f317c63a26b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.826ex; height:2.843ex;" alt="{\displaystyle t_{0}-T_{w}/2}" loading="lazy"> seconds after transmission; similarly, If the target is at the upper bound of the range window, the echo will arrive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}+T_{w}/2}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}+T_{w}/2}</annotation>

</semantics>

</math><img src="./c67a864895d9c237523b32dd9ecc3c468f85b9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.826ex; height:2.843ex;" alt="{\displaystyle t_{0}+T_{w}/2}" loading="lazy"> seconds after transmission.

The differential arrive time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta t_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta t_{b}}</annotation>

</semantics>

</math><img src="./41dc50fecce071f22e5447992056dab7407e71cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.826ex; height:2.676ex;" alt="{\displaystyle \delta t_{b}}" loading="lazy"> for each case is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -T_{w}/2}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -T_{w}/2}</annotation>

</semantics>

</math><img src="./836f4ff27d3767c64dc1af7f574ce8b5a94ba825.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.9ex; height:2.843ex;" alt="{\displaystyle -T_{w}/2}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{w}/2}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{w}/2}</annotation>

</semantics>

</math><img src="./8fae267260bbac357f328ea22c848fec4e65e182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.091ex; height:2.843ex;" alt="{\displaystyle T_{w}/2}" loading="lazy">, respectively.

We can then obtain the bandwidth by considering the difference in sinusoid frequency for targets at the lower and upper bound of the range window:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f_{s}=F_{b,{\text{near}}}-F_{b,{\text{far}}}=-{\frac {\Delta f}{T}}(-T_{w}/2-T_{w}/2)={\frac {T_{w}}{T}}\Delta f}</annotation>

</semantics>

</math>

As a consequence:

<th>Result 4

</th></tr>

<tr>

<td>Through stretch processing, the bandwidth at the receiver output is less than the original signal bandwidth if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{w}<T}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{w}<T}</annotation>

</semantics>

</math><img src="./f980780bc8b54a672227d8ce6f2e31f908e660e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.501ex; height:2.509ex;" alt="{\displaystyle T_{w}<T}" loading="lazy">, thereby facilitating the implementation of DSP system in a linear-frequency-modulation radar system.

</td></tr></tbody></table>

To demonstrate that stretch processing preserves range resolution, we need to understand that y(t) is actually an impulse train with pulse duration T and period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{trans}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{trans}}</annotation>

</semantics>

</math><img src="./41a5532e0b5508dcb5d5a922bac70a8959b60d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.552ex; height:2.509ex;" alt="{\displaystyle T_{trans}}" loading="lazy">, which is equal to the period of the transmitted impulse train. As a result, the Fourier transform of y(t) is actually a sinc function with <a href="Rayleigh_resolution" class="mw-redirect" title="Rayleigh resolution">Rayleigh resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{T}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{T}}}</annotation>

</semantics>

</math><img src="./1447724b082b317d5abbf39692587197afcc7862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.993ex; height:3.509ex;" alt="{\textstyle {\frac {1}{T}}}" loading="lazy">. That is, the processor will be able to resolve scatterers whose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{b}}</annotation>

</semantics>

</math><img src="./b151118e130ddb3d3eea064479c152bd16738f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.432ex; height:2.509ex;" alt="{\displaystyle F_{b}}" loading="lazy"> are at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta F_{b}=1/T}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta F_{b}=1/T}</annotation>

</semantics>

</math><img src="./c047c79e6760fe478b249f3ee82101ec8c8d9136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.428ex; height:2.843ex;" alt="{\displaystyle \Delta F_{b}=1/T}" loading="lazy"> apart.

Consequently,

<mfrac>

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</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

<mrow>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T}}=\left\vert {\frac {\Delta f}{T}}\Delta (\delta t_{b})\right\vert \Rightarrow \left\vert \Delta (\delta t_{b})\right\vert ={\frac {1}{\Delta f}}}</annotation>

</semantics>

</math><img src="./efefd764dcd3adebeaa06e3dcd21cf0df78504b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.114ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{T}}=\left\vert {\frac {\Delta f}{T}}\Delta (\delta t_{b})\right\vert \Rightarrow \left\vert \Delta (\delta t_{b})\right\vert ={\frac {1}{\Delta f}}}" loading="lazy"></dd></dl>

and,

<msub>

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</msub>

<mfrac>

<mrow>

<msub>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (\delta R_{b})={\frac {c\Delta (\delta t_{b})}{2}}={\frac {c}{2\Delta f}}}</annotation>

</semantics>

</math><img src="./a4d435552148d21b41e9474b202735f6dabb02af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.319ex; height:6.343ex;" alt="{\displaystyle \Delta (\delta R_{b})={\frac {c\Delta (\delta t_{b})}{2}}={\frac {c}{2\Delta f}}}" loading="lazy"></dd></dl>

which is the same as the resolution of the original linear frequency modulation waveform.

<div class="mw-heading mw-heading3"><h3 id="Stepped-frequency_waveform">Stepped-frequency waveform</h3></div>

Although stretch processing can reduce the bandwidth of received baseband signal, all of the analog components in RF front-end circuitry still must be able to support an instantaneous bandwidth of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy">. In addition, the effective wavelength of the electromagnetic wave changes during the frequency sweep of a chirp signal, and therefore the antenna look direction will be inevitably changed in a <a href="Phased_array" title="Phased array">Phased array</a> system.

Stepped-frequency waveforms are an alternative technique that can preserve fine range resolution and SNR of the received signal without large instantaneous bandwidth. Unlike the chirping waveform, which sweeps linearly across a total bandwidth of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation>

</semantics>

</math><img src="./ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" loading="lazy"> in a single pulse, stepped-frequency waveform employs an impulse train where the frequency of each pulse is increased by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta F}</annotation>

</semantics>

</math><img src="./bcde5b83b29cc20f808fb4f349b838b82ed99a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.677ex; height:2.176ex;" alt="{\displaystyle \Delta F}" loading="lazy"> from the preceding pulse. The baseband signal can be expressed as:

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</mrow>

</munderover>

<msub>

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</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)=\sum _{m=0}^{M-1}x_{p}(t-mT)e^{j2\pi m\Delta F(t-mT)}}</annotation>

</semantics>

</math><img src="./6930ad9e392e5b061fbc8a9dfb6b4c58825cd17c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.397ex; height:7.343ex;" alt="{\displaystyle x(t)=\sum _{m=0}^{M-1}x_{p}(t-mT)e^{j2\pi m\Delta F(t-mT)}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{p}(t)}</annotation>

</semantics>

</math><img src="./39ef429104f7a85c88c8f42ef6b3de2c1f48e798.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.038ex; height:3.009ex;" alt="{\displaystyle x_{p}(t)}" loading="lazy"> is a rectangular impulse of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

</semantics>

</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy"> and M is the number of pulses in a single pulse train. The total bandwidth of the waveform is still equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f=M\Delta F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f=M\Delta F}</annotation>

</semantics>

</math><img src="./2f23a50fc8274f46b93bc584cf93c22ed624a985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.432ex; height:2.509ex;" alt="{\displaystyle \Delta f=M\Delta F}" loading="lazy">, but the analog components can be reset to support the frequency of the following pulse during the time between pulses. As a result, the problem mentioned above can be avoided.

To calculate the distance of the target corresponding to a delay <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{l}+\delta t}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{l}+\delta t}</annotation>

</semantics>

</math><img src="./6ddaf3a12ebb70edfba091cf7057acb4b7e5f8ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.291ex; height:2.676ex;" alt="{\displaystyle t_{l}+\delta t}" loading="lazy">, individual pulses are processed through the simple pulse matched filter:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{p}(t)=x_{p}^{*}(-t)}</annotation>

</semantics>

</math><img src="./b070777008c96d9bb7f70cb5987c8fa11d2e3020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.991ex; height:3.009ex;" alt="{\displaystyle h_{p}(t)=x_{p}^{*}(-t)}" loading="lazy"></dd></dl>

and the output of the matched filter is:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

<msub>

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</msub>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{m}(t)=s_{p}^{*}(t-(t_{l}+\delta t)-mT)e^{j2\pi m\Delta F(t-(t_{l}+\delta t)-mT)}}</annotation>

</semantics>

</math><img src="./d7fb572d23ed30e9d0cb6f254738cafcf8f617b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:50.696ex; height:3.509ex;" alt="{\displaystyle y_{m}(t)=s_{p}^{*}(t-(t_{l}+\delta t)-mT)e^{j2\pi m\Delta F(t-(t_{l}+\delta t)-mT)}}" loading="lazy"></dd></dl>

where

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{p}^{*}(t-(t_{l}+\delta t)-mT)=x_{p}(t-(t_{l}+\delta t)-mT)*h_{p}(t)}</annotation>

</semantics>

</math><img src="./874f01986190181da8982cb7b45d311efc31e933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.092ex; height:3.009ex;" alt="{\displaystyle s_{p}^{*}(t-(t_{l}+\delta t)-mT)=x_{p}(t-(t_{l}+\delta t)-mT)*h_{p}(t)}" loading="lazy"></dd></dl>

If we sample <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{m}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{m}(t)}</annotation>

</semantics>

</math><img src="./da222222725987d9ca1193b5f59a943fd735baaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.843ex;" alt="{\displaystyle y_{m}(t)}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{l}+mT}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=t_{l}+mT}</annotation>

</semantics>

</math><img src="./33b0de398ceaa41146621f292a9fe768e48b175d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.017ex; height:2.509ex;" alt="{\displaystyle t=t_{l}+mT}" loading="lazy">, we can get:

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y[l,m]=s_{p}^{*}(\delta t)e^{j2\pi m\Delta F\delta t}}</annotation>

</semantics>

</math><img src="./40dec0255d3be54095a8aab6e8c9657011501684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.297ex; height:3.343ex;" alt="{\displaystyle y[l,m]=s_{p}^{*}(\delta t)e^{j2\pi m\Delta F\delta t}}" loading="lazy"></dd></dl>

where l means the range bin l.

Conduct DTFT (m is served as time here) and we can get:

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y[l,\omega ]=\sum _{m=0}^{M-1}y[l,m]e^{-j\omega m}=s_{p}^{*}(\delta t)\sum _{m=0}^{M-1}e^{j(\omega -2\pi \Delta F\delta t)m}}</annotation>

</semantics>

</math><img src="./f192cd9de2d8f245cd0e425a6c01bce5dcc96659.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.769ex; height:7.343ex;" alt="{\displaystyle Y[l,\omega ]=\sum _{m=0}^{M-1}y[l,m]e^{-j\omega m}=s_{p}^{*}(\delta t)\sum _{m=0}^{M-1}e^{j(\omega -2\pi \Delta F\delta t)m}}" loading="lazy"></dd></dl>

, and the peak of the summation occurs when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi \Delta F\delta t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =2\pi \Delta F\delta t}</annotation>

</semantics>

</math><img src="./7f25f531d9d7ba4c18669322666e8c3fc84f20c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.604ex; height:2.343ex;" alt="{\displaystyle \omega =2\pi \Delta F\delta t}" loading="lazy">.

Consequently, the DTFT of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[l,m]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y[l,m]}</annotation>

</semantics>

</math><img src="./18f14c83106113538d8a6f6d24c1adba673f1d5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.217ex; height:2.843ex;" alt="{\displaystyle y[l,m]}" loading="lazy"> provides a measure of the delay of the target relative to the range bin delay <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{l}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{l}}</annotation>

</semantics>

</math><img src="./a8d4f1d4329575b7559de3830aaea50518788a47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.562ex; height:2.343ex;" alt="{\displaystyle t_{l}}" loading="lazy">:

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta t={\frac {\omega _{p}}{2\pi \Delta F}}={\frac {f_{p}}{\Delta F}}}</annotation>

</semantics>

</math>

and the differential range can be obtained:

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta R={\frac {cf_{p}}{2\Delta F}}}</annotation>

</semantics>

</math><img src="./855b891a11e3c487fbc65e69c061c6006f37b8a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.586ex; height:5.843ex;" alt="{\displaystyle \delta R={\frac {cf_{p}}{2\Delta F}}}" loading="lazy"></dd></dl>

where c is the speed of light.

To demonstrate stepped-frequency waveform preserves range resolution, it should be noticed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y[l,\omega ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y[l,\omega ]}</annotation>

</semantics>

</math><img src="./fe0b7c9a4c8753bd911bdc096a3202e0b1a3ff49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle Y[l,\omega ]}" loading="lazy"> is a sinc-like function, and therefore it has a Rayleigh resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f_{p}=1/M}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta f_{p}=1/M}</annotation>

</semantics>

</math><img src="./4de13d2595b418044e5ba2397e40d6a0c41dd748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12ex; height:3.009ex;" alt="{\displaystyle \Delta f_{p}=1/M}" loading="lazy">. As a result:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (\delta t)={\frac {1}{M\Delta F}}={\frac {1}{\Delta f}}}</annotation>

</semantics>

</math><img src="./c0be223db75a26c59528a687b36758ab76c5e534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.836ex; height:5.843ex;" alt="{\displaystyle \Delta (\delta t)={\frac {1}{M\Delta F}}={\frac {1}{\Delta f}}}" loading="lazy"></dd></dl>

and therefore the differential range resolution is :

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta (\delta R)={\frac {c}{2\Delta f}}}</annotation>

</semantics>

</math><img src="./8aa83dedd905a6f71dbceff0612601ffa8f6bed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.869ex; height:5.343ex;" alt="{\displaystyle \Delta (\delta R)={\frac {c}{2\Delta f}}}" loading="lazy"></dd></dl>

which is the same of the resolution of the original linear-frequency-modulation waveform.

<div class="mw-heading mw-heading2"><h2 id="Pulse_compression_by_phase_coding">Pulse compression by phase coding</h2></div>

There are other means to modulate the signal. <a href="Phase_modulation" title="Phase modulation">Phase modulation</a> is a commonly used technique; in this case, the pulse is divided in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> time slots of duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {T}{N}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {T}{N}}}</annotation>

</semantics>

</math><img src="./7cc5c1b624b9f2c0f51733aa9d67ee41f35ae7b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.295ex; height:3.509ex;" alt="{\textstyle {\frac {T}{N}}}" loading="lazy"> for which the phase at the origin is chosen according to a pre-established convention. For instance, it is possible to not change the phase for some time slots (which comes down to just leaving the signal as it is, in those slots) and de-phase the signal in the other slots by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> (which is equivalent of changing the sign of the signal); this is known as <a href="BPSK" class="mw-redirect" title="BPSK">binary phase-shift keying</a>. The precise way of choosing the sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,\pi \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0,\pi \}}</annotation>

</semantics>

</math><img src="./f09ef7701749dd136da9029e8b18f7f30a1427b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.853ex; height:2.843ex;" alt="{\displaystyle \{0,\pi \}}" loading="lazy"> phases can be done according to a technique known as <a href="Barker_code" title="Barker code">Barker codes</a>.

The advantages<a href="#cite_note-4">[4]</a> of the Barker codes are their simplicity (as indicated above, a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> de-phasing is a simple sign change), but the pulse compression ratio is lower than in the chirp case and the compression is very sensitive to frequency changes due to the <a href="Doppler_effect" title="Doppler effect">Doppler effect</a> if that change is larger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{T}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{T}}}</annotation>

</semantics>

</math><img src="./1447724b082b317d5abbf39692587197afcc7862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.993ex; height:3.509ex;" alt="{\textstyle {\frac {1}{T}}}" loading="lazy">.

Other <a href="Pseudorandom_binary_sequence" title="Pseudorandom binary sequence">pseudorandom binary sequences</a> have nearly optimal pulse compression properties, such as <a href="Gold_code" title="Gold code">Gold codes</a>, <a href="JPL_code" class="mw-redirect" title="JPL code">JPL codes</a> or <a href="Kasami_code" title="Kasami code">Kasami codes</a>, because their autocorrelation peak is very narrow. These sequences have other interesting properties making them suitable for <a href="GNSS" class="mw-redirect" title="GNSS">GNSS</a> positioning, for instance.

It is possible to code the sequence on more than two phases (polyphase coding). As with a linear chirp, pulse compression is achieved through intercorrelation.

<ul><li><a href="Continuous-wave_radar" title="Continuous-wave radar">Continuous-wave radar</a></li>

<li><a href="Spread_spectrum" title="Spread spectrum">Spread spectrum</a></li>

<li><a href="Chirp_compression" title="Chirp compression">Chirp compression</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> J. R. Klauder, A. C, Price, S. Darlington and W. J. Albersheim, ‘The Theory and Design of Chirp Radars,” Bell System Technical Journal 39, 745 (1960).

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Achim Hein, Processing of SAR Data: Fundamentals, Signal Processing, Interferometry, Springer, 2004, <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-05043-4</bdi>, pages 38 to 44. Very rigorous demonstration of the autocorrelation function of a chirp. The author works with real chirps, hence the factor of 1⁄2 in his book, which is not used here.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Richards, Mark A. 2014. Fundamentals of radar signal processing. New York [etc.]: McGraw-Hill Education.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> J.-P. Hardange, P. Lacomme, J.-C. Marchais, Radars aéroportés et spatiaux, Masson, Paris, 1995, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>2-225-84802-5</bdi>, page 104. Available in English: Air and Spaceborne Radar Systems: an introduction, Institute of Electrical Engineers, 2001, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-85296-981-3</bdi>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Nadav Levanon, and Eli Mozeson. Radar signals. Wiley. com, 2004.</li>

<li>Hao He, <a href="Jian_Li_(engineer)" title="Jian Li (engineer)">Jian Li</a>, and <a href="Peter_Stoica" title="Peter Stoica">Petre Stoica</a>. <a rel="nofollow" class="external text" href="http://www.sal.ufl.edu/book/">Waveform design for active sensing systems: a computational approach</a>. Cambridge University Press, 2012.</li>

<li>M. Soltanalian. <a rel="nofollow" class="external text" href="http://theses.eurasip.org/theses/573/signal-design-for-active-sensing-and/download/">Signal Design for Active Sensing and Communications</a>. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.</li>

<li>Solomon W. Golomb, and <a href="Guang_Gong" title="Guang Gong">Guang Gong</a>. <a rel="nofollow" class="external text" href="http://www.cambridge.org/us/academic/subjects/computer-science/cryptography-cryptology-and-coding/signal-design-good-correlation-wireless-communication-cryptography-and-radar">Signal design for good correlation: for wireless communication, cryptography, and radar</a>. Cambridge University Press, 2005.</li>

<li>Fulvio Gini, Antonio De Maio, and Lee Patton, eds. Waveform design and diversity for advanced radar systems. Institution of engineering and technology, 2012.</li>

<li>John J. Benedetto, Ioannis Konstantinidis, and Muralidhar Rangaswamy. "<a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/4775877/">Phase-coded waveforms and their design</a>." IEEE Signal Processing Magazine, 26.1 (2009): 22-31.</li>

<li>Ducoff, Michael R., and Byron W. Tietjen. "Pulse compression radar." Radar Handbook (2008): 8-3.</li></ul></div><div><div class="zim-footer">

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