Signals and Systems - Electrical Engineering

340 CHAPTER 5: Frequency Analysis: The Fourier Transform

n Zeros create “valleys” at the frequencies in the jaxis in front of the zeros imaginary parts. The closer the

zero is to the jaxis (from its left or its right, as the zeros are not restricted by stability to be in the open

left-hand s-plane) the closer the frequency response is to zero. If the zeros are on the jaxis, the frequency

response at the frequency of the zeros is zero. Thus, poles produce frequency responses that look like hills (or

like the main pole in a circus) around the frequencies of the poles, and zeros make the frequency response

go to zero in the form of valleys around the frequencies of the zeros.

nExample 5.18

Use MATLAB to find and plot the poles and zeros and the corresponding magnitude and phase

frequency responses of:

(a) A second-order band-pass filter and a high-pass filter realized using a series connection of a

resistor, an inductor, and a capacitor, each with unit resistance, inductance, and capacitance.

Let the input be a voltage sourcevs(t)and initial conditions be zero.

(b) An all-pass filter with a transfer function

H(s)=

s^2 −2.5s+ 1

s^2 +2.5s+ 1

Solution

Our functionfreq respscomputes and plots the poles and the zeros of the filter transfer function

and the corresponding frequency response (the function requests the coefficients of its numerator

and denominator in decreasing order of powers ofs).

(a) As from a Example 5.16, the transfer functions of the band-pass and high-pass second-order

filters are

Hbp(s)=

s

s^2 +s+ 1

Hhp(s)=

s^2

s^2 +s+ 1

The denominator in the two cases is exactly the same since the values ofR,L, andCremain

the same for the two filters—the only difference is in the numerator.

To compute the frequency response of these filters and to plot their poles and zeros, we

used the following script, which uses two functions:freqresps, which we give below, and

splane, which plots the poles and zeros. The coefficients of the numerator and the denomi-

nator correspond to the coefficients, from the highest to the lowest order ofs, of the transfer

function.

%%%%%%%%%%%%%%%%%%%%%

% Example 5.18---Frequency response

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n = [0 1 0]; % numerator coefficients -- bandpass

% n = [1 0 0]; % numerator coefficients -- highpass

d = [1 1 1]; % denominator coefficients