Signals and Systems - Electrical Engineering

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

11.20. Down-sampling transformations—MATLAB

Consider down sampling the impulse responseh[n]of a filter with transfer function

H(z)=

1

1 −0.5z−^1

(a) Use MATLAB to ploth[n]and the down sampled impulse responseg[n]=h[2n].

(b) Plot the magnitude responses corresponding toh[n]andg[n]and comment on the effect of the down

sampling.

11.21. Modulation property transformation—MATLAB

Consider a moving-average, low-pass, FIR filter,

H(z)=

1 +z−^1 +z−^2

3

(a) Use the modulation property to convert the given filter into a high-pass filter.

(b) Use MATLAB to plot the magnitude responses of the low-pass and the high-pass filters.

11.22. Implementation of IIR rational transformation—MATLAB

Use MATLAB to design a Butterworth second-order low-pass discrete filterH(Z)with half-power fre-

quencyθhp=π/ 2 and a dc gain of 1. Consider this low-pass filter a prototype that can be used to

obtain other filters. Implement using MATLAB the frequency transformationsZ−^1 =N(z)/D(z)using the

convolution property to multiply polynomials to obtain:

(a) A high-pass filter with a half-power frequencyωhp=π/ 3 from the low-pass filter.

(b) A band-pass filter withω 1 =π/ 2 andω 2 = 3 π/ 4 from the low-pass filter.

(c) Plot the magnitude of the low-pass, high-pass, and band-pass filters.

Give the corresponding transfer functions for the low-pass as well as the high-pass and the band-pass

filters.

11.23. Parallel connection of IIR filters—MATLAB

Use MATLAB to design a Butterworth second-order low-pass discrete filter with half-power frequency

θhp=π/ 2 and a dc gain of 1; call itH(z). Use this filter as a prototype to obtain a filter composed of a

parallel combination of the following filters:

(a) Assume that we upsample byL= 2 the impulse responseh(n)ofH(z)to get a new filterH 1 (z)=

H(z^2 ). DetermineH 1 (z)and plot its magnitude using MATLAB.

(b) Assume then that we shiftH(z)byπ/ 2 to get a band-pass filterH 2 (z). Find the transfer function of

H 2 (z)fromH(z)and then plot its magnitude.

(c) If the filtersH 1 (z)andH 2 (z)are connected in parallel, what is the overall transfer functionG(z)of the

parallel connection? Plot the magnitude response corresponding toG(z).

11.24. Realization of IIR filters

Consider the following transfer function:

H(z)=

2 (z− 1 )(z^2 +

√

2 z+ 1 )

(z+0.5)(z^2 −0.9z+0.81)

(a) Develop a cascade realization ofH(z)using first- and second-order sections. Use direct form II to

realize each of the sections.

(b) Develop a parallel realization ofH(z)by considering first- and second-order sections, each realized

using direct form II.