Signals and Systems - Electrical Engineering

Problems 701

11.7 What Have We Accomplished? Where Do We Go from Here?....................

In Chapter 6 and in this chapter you have been introduced to the most important application of lin-
ear time-invariant systems: filtering. The design and realization of analog and discrete filters gathers
many practical issues in signals and systems. If you pursue this topic, you will see the significance,
for instance, of passive and active elements, feedback and operational amplifiers, reactance functions,
and frequency transformation in analog filtering. The design and realization of discrete filters brings
together interesting topics such as quantization error and its effect on the filters, optimization meth-
ods for filter design, stabilization of unstable filters, etc. If you pursue filtering deeper, you will find
that there is a lot more on filter design than what we have provided you in this chapter. A lot more.
Also remember that MATLAB has a large number of tools to design and implement filters.

Problems............................................................................................

11.1. FIR filters: causality and phase—MATLAB

A three-point moving-average filter is of the form:

y[n]=β

(

αx[n−1]+x[n]+αx[n+1]

)

whereαandβare constants, andx[n]is the input andy[n]is the output of the filter.

(a) Determine the transfer functionH(z)=Y(z)/X(z)of the filter, and from it determine the frequency

responseH(ejω)of the filter in terms ofαandβ.

(b) Find the values ofαandβso that the dc gain of the filter is unity, and the filter has a zero phase. For

α=0.5and the corresponding value ofβ, sketchH(ejω)and find the poles and zeros ofH(z)and plot

them in thez-plane. Verify your results using MATLAB.

(c) Suppose we letv[n]=y[n−1]be the output of a second filter. Is this filter causal? Find its transfer

functionG(z)=V(z)/X(z). Use MATLAB to compute the unwrapped phase ofG(z)and to plot the

poles and zeros ofG(z)and explain the relation betweenG(z)andH(z).

11.2. FIR and IIR filters: causality and zero phase—MATLAB

Let the filterH(z)be the cascade of a causal filter with transfer functionG(z)and an anti-causal filter with

transfer functionG(z−^1 ), so that

H(z)=G(z)G(z−^1 )

(a) Suppose thatG(z)is an FIR filter with transfer function

G(z)=

1

3

( 1 + 2 z−^1 +z−^2 )

Find the frequency responseH(ejω)and determine its phase.

(b) Determine the impulse response of the filterH(z). IsH(z)a causal filter? If not, would delaying its

impulse response make it causal? Explain. What would be the transfer function of the causal filter.

(c) Use MATLAB to verify the unwrapped phase ofH(z)you obtained analytically, and to plot the poles

and zeros ofH(z).

(d) How would you use the MATLAB functionconvto find the impulse response ofH(z).

(e) Suppose then thatG(z)= 1 /( 1 −0.5z−^1 ). Find the filterH(z)=G(z)G(z−^1 ). Is this filter zero phase?

If so, where are its poles and zeros? If you think of filterH(z)as causal, is it BIBO stable?