LTI DISCRETE-TIME SYSTEMS IN THE FREQUENCY DOMAIN

LTI DISCRETE-TIME SYSTEMS IN THE FREQUENCY DOMAIN

Illustration Filters

The length of the filter is N + 1.

The following program, which uses the function sinc, can be used to compute the above

approximation.

% Program

% Impulse Response of Truncated Ideal Lowpass Filter

clf;

fc = 0.25;

n = [-6.5:1:6.5];

y = 2*fc*sinc(2*fc*n);k = n+6.5;

stem(k,y);title(’N = 13’);axis([0 13 -0.2 0.6]);

xlabel(’Time index n’);ylabel(’Amplitude’); grid

Stability Test

% Program

% Stability Test

clf;

den = input(’Denominator coefficients = ’);

ki = poly2rc(den);

disp(’Stability test parameters are’);

disp(ki);

DIGITAL PROCESSING OF CONTINUOUS-TIME SIGNALS

% Program

% Illustration of the Sampling Process

% in the Time Domain

clf;

t = 0:0.0005:1;

f = 13;

xa = cos(2*pi*f*t);

subplot(2,1,1)

plot(t,xa);grid

xlabel(’Time, msec’);ylabel(’Amplitude’);

title(’Continuous-time signal x_{a}(t)’);

axis([0 1 -1.2 1.2])

subplot(2,1,2);

T = 0.1;

n = 0:T:1;

xs = cos(2*pi*f*n);

k = 0:length(n)-1;

stem(k,xs); grid

xlabel(’Time index n’);ylabel(’Amplitude’);

title(’Discrete-time signal x[n]’);

axis([0 (length(n)-1) -1.2 1.2])

Illustration Aliasing Effect in the Time Domain

% Program

% Illustration of Aliasing Effect in the Time Domain

% Program adapted from [Kra94] with permission from

% The Mathworks, Inc., Natick, MA.

clf;

T = 0.1;f = 13;

n = (0:T:1)’;

xs = cos(2*pi*f*n);

t = linspace(-0.5,1.5,500)’;

ya = sinc((1/T)*t(:,ones(size(n))) - (1/T)*n(:,ones(size(t)))’)*xs;

plot(n,xs,’o’,t,ya);grid;

xlabel(’Time, msec’);ylabel(’Amplitude’);

title(’Reconstructed continuous-time signal y_{a}(t)’);

axis([0 1 -1.2 1.2]);

Illustration Design of Analog Lowpass Filters

Program P5_3 can be used for the design of the Butterworth lowpass filter.

% Program P5_3

% Design of Analog Lowpass Filter

clf;

Fp = 3500;Fs = 4500;

Wp = 2*pi*Fp; Ws = 2*pi*Fs;

[N, Wn] = buttord(Wp, Ws, 0.5, 30,’s’);

[b,a] = butter(N, Wn, ’s’);

wa = 0:(3*Ws)/511:3*Ws;

h = freqs(b,a,wa);

plot(wa/(2*pi), 20*log10(abs(h)));grid

xlabel(’Frequency, Hz’);ylabel(’Gain, dB’);

title(’Gain response’);

axis([0 3*Fs -60 5]);

Illustration Binary Equivalent of a Decimal Number

Program used to the generate the binary equivalent in sign-magnitude form of

a decimal fraction.

% Program

% Determines the binary equivalent of a

% decimal number in sign-magnitude form

d = input(’Type in the decimal fraction = ’);

b = input(’Type in the desired wordlength = ’);

d1 = abs(d);

beq = [zeros(1,b)];

for k = 1:b

int = fix(2*d1);

beq(k) = int;

d1 = 2*d1 - int;

end

if sign(d) == -1;

bin = [1 beq];

else

bin = [0 beq];

end

disp(’The binary equivalent is’);

disp(bin)

Illustration Decimal Equivalent of a Binary Number

Program to perform the reverse process and generates the decimal equivalent of a binary

fraction in sign-magnitude form.

% Program

% Determines the decimal equivalent of a

% binary number in sign-magnitude form

bin = input(’Type in the binary fraction = ’);

b = length(bin) - 1; d = 0;

for k = 1:b

d = d + bin(k+1)*2^(-k);

end

if sign(bin(1)) == 0;

dec = d;

else

dec = - d;

end

disp(’The decimal equivalent is’);

disp(dec);