10.2 Discrete-Time Fourier Transform 595
H1up = unwrap(H1p); % unwrapped phase of H1(z)
H2m = abs(H2(1:128)); H2p = angle(H2(1:128)); % magnitude/phase of H2(z)
H2up = unwrap(H2p); % unwrapped phase of H1(z)
n
10.2.8 Convolution Sum
The computation of the convolution sum, just like the convolution integral in the continuous-time
domain, is simplified in the Fourier domain.
Ifh[n]is the impulse response of a stable LTI system, its outputy[n]can be computed by means of the
convolution sum
y[n]=
∑
k
x[k]h[n−k]
wherex[n]is the input. The Z-transform ofy[n]is the product
Y(z)=H(z)X(z) ROC:RY=RH∩RX
If the unit circle is included inRY, then
Y(ejω)=H(ejω)X(ejω) or
|Y(ejω)|=|H(ejω)||X(ejω)|
∠Y(ejω)=∠H(ejω)+∠X(ejω) (10.25)
Remarks
n Since the system is stable, the ROC of H(z)includes the unit circle, and so if the ROC of X(z)includes
the unit circle, the intersection of these regions will also include the unit circle.
n We will see later that it is still possible for y[n]to have a DTFT when the input x[n]does not have a
Z-transform with a region of convergence including the unit circle, as when the input is periodic. In this
case the output is also periodic. These signals are not finite energy, but finite power, and can be represented
by DTFT containing analog delta functions.
nExample 10.12
LetH(z)be the cascade of first-order systems with transfer functions
Hi(z)=Ki
z− 1 /αi
z−αi∗
|z|>|αi|, i=1,...,N− 1
where|αi|<1 andKi>0. Such a system is called anall-pass systembecause its magnitude response
is a constant for all frequencies. If the DTFT of the filter inputx[n] isX(ejω), determine the gains{Ki}
so that the magnitude of the DTFT of the outputy[n] of the system coincides with the magnitude
ofX(ejω).
