Signals and Systems - Electrical Engineering

646 C H A P T E R 11: Introduction to the Design of Discrete Filters

That is, the outputy[n] is a combination of the inputx[n] and of a delayed and attenuated version

of the input. Determine the transfer function of the filter that gives the above input–output equa-

tion. Use MATLAB to plot its magnitude and phase. If the phase is nonlinear, how would you

recover the inputx[n] (which is the message)? Let the input bex[n]= 2 +cos(πn/ 4 )+cos(πn). In

practice, the delayN 0 and the attenuationαare not known at the receiver and need to be estimated.

What would happen if the delay is estimated to be 12 and the attenuation 0.79?

Solution

The transfer function of the filter with inputx[n] and outputy[n] is

H(z)=

Y(z)

X(z)

= 1 −0.8z−^11 =

z^11 −0.8

z^11

with a pole ofz=0 of multiplicity 11, and zeros the roots ofz^11 −0.8=0, or

zk=(0.8)^1 /^11 ej^2 πk/^11 k=0,..., 10

Using thefreqzfunction to plot its magnitude and phase responses (see Figure 11.5), we find that

the phase is nonlinear, and as such the output ofH(z),y[n], will not be a delayed version of the

input. To recover the input, we use aninverse filter G(z)such that cascaded withH(z)the overall

filter is an all-pass filter (i.e.,H(z)G(z)=1). Thus,

G(z)=

z^11

z^11 −0.8

The poles and zeros and the magnitude and the phase responses ofH(z) are shown in

Figure 11.5(a). The filters with transfer functionsH(z)andG(z)are calledcomb filtersgiven the

shape of their magnitude responses.

If the delay is estimated to be 11 and the attenuation 0.8, the input signalx[n] (the message) is

recovered exactly; however, if we have slight variations on these values the message might not be

recovered. When the delay is estimated to be 12 and the attenuation 0.79, the inverse filter is

G(z)=

z^12

z^12 −0.79

having the poles, zeros, and magnitude and phase responses shown in Figure 11.5(b). In

Figure 11.5(c), the effect of these changes are illustrated. The output of the inverse filterz[n] does

not resemble the sent signalx[n]. The signaly[n] is the output of the channel with a transfer

functionH(z). n