Mohammed Almathami
x(t)*q(t) = T*SUM{x[k]*d(t-k*T)} .------.
x(t)--->(*)------------------------------------>| H(f) |---> x(t)
^ '------'
|
| +inf
'------- q(t) = T*SUM{ d(t - k*T) }
k=-inf
where d(t) = 'dirac' impulse function
and T = 1/Fs = sampling period
Fs = sampling frequency
+inf
q(t) = T * SUM{ d(t - k*T) } is the "sampling function", is periodic
k=-inf with period T, and can be expressed as a
Fourier series. It turns out that ALL of
the Fourier coefficients are equal to 1.
+inf
q(t) = SUM{ c[n]*exp(j*2*pi*n/T*t) }
n=-inf
where c[n] is the Fourier series coefficient.
t0+T
c[n] = 1/T * integral{q(t) * exp(-j*2*pi*n/T*t) dt}
t0
and t0 can be any time. We choose t0 to be -T/2.
T/2 +inf
c[n] = 1/T * integral{T * SUM{ d(t - k*T) } * exp(-j*2*pi*n/T*t) dt}
-T/2 k=-inf
T/2
= integral{ d(t - 0*T) * exp(-j*2*pi*n/T*t) dt}
-T/2
T/2
= integral{ d(t) * exp(-j*2*pi*n/T*t) dt}
-T/2
= exp(-j*2*pi*n/T*0) = 1
