◀ Prev · Parts · Full · Next ▶

For rect ⁡ ⁡ ( x / a ) {\displaystyle \operatorname {rect} (x/a)} , its Fourier transform is ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t a ) ⋅ ⋅ e − − i 2 π π f t d t = a sin ⁡ ⁡ ( π π a f ) π π a f = a sinc π π ⁡ ⁡ ( a f ) . {\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=a{\frac {\sin(\pi af)}{\pi af}}=a\ \operatorname {sinc} _{\pi }{(af)}.}

We can define the triangular function as the convolution of two rectangular functions:

t r i ( t / T ) = r e c t ( 2 t / T ) ∗ ∗ r e c t ( 2 t / T ) . {\displaystyle \operatorname {tri(t/T)} =\operatorname {rect(2t/T)} *\operatorname {rect(2t/T)} .\,}

Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with a = − − 1 / 2 , b = 1 / 2. {\displaystyle a=-1/2,b=1/2.} The characteristic function is

φ φ ( k ) = sin ⁡ ⁡ ( k / 2 ) k / 2 , {\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}},}

and its moment-generating function is

◀ Prev · Parts · Full · Next ▶