Question

**Problem 1**

Consider the discrete-time LTI system characterized by the following difference equation with input and initial conditions specified:

*y*[n] - 2 *y*[n-1] – 3 *y*[n-2] =
*x*[n] , with *y*[0] = -1 and *y*[1] = 0,
*x*[*n*] = (-1/2)n u[n-2].

? Write a MATLAB program to simulate this difference
equation.

You may try the commands ‘*filter*’ or ‘*filtic*’ or
create a loop to compute the values recursively.

? **Printout and plot the values** of the input
signal, x[n] and the output signal, y[n] over the range 1 ? n ?
10.

? Solve this difference equation **by hand** using
the **classical method**.

? **Verify that the values** of the output signal,
y[n] produced by MATLAB are the same as those

calculated by hand for the values of n = 2, 3, 4.

Answer #1

**MATLAB CODE:**

y[n]+3y[n-1]-4y[n-2]=x[n]

y[1]=0 and y[2]=1,

and x[n]=3nu[n]

clc

n=3:5;

a=[1 3 -4];

b=[3 0 0];

yi=[1 0];

xi=[1 1];

x=3*n;

zi=filter(b,a,yi,xi);

y=filter(b,a,3*n,zi);

subplot(2,1,1)

plot(n,y)

xlabel('----n------>')

ylabel('----values---->')

title('plot of output signal y over the range from n=3 to
n=5')

subplot(2,1,2)

plot(n,x)

xlabel('----n---->')

ylabel('----values---->')

title('plot of input signal x over the range from n=3 to n=5')

**OUTPUT:**

CHAPTER 13: DISCRETE-TIME SIGNAL (TEXTBOOK SIGNALS AND SYSTEM BY
MAHMOOD NAHVI
11. In an LTI system, x(n) is the input and h(n) is the
unit-sample response. Find and sketch the
output y(n) for the following cases:
i) x(n) = 0.3nu(n) and h(n) = 0.4nu(n)
ii) x(n) = 0.5nu(n) and h(n) = 0.6nu(n)
iii) x(n) = 0.5|n|u(n) and h(n) = 0.6nu(n)

Consider a causal LTI system described by the difference
equation:
y[n] = 0.5 y[n-1] + x[n] – x[n-1]
(a) Determine the system function H(z) and plot a pole-zero pattern
in the complex z-plane.
(b) Find the system response using partial fraction expansion when
the input is x[n] = 2u[n]. Plot the result.

Solve this signal problem.
Suppose the output y[n] of a DT LTI system with input x[n] is
y[n-1] - 10/3y[n] + y[n+1] = x[n]
The system is stable and the impulse response of h[n] =
A1*(B1)^n*C1 + A2*(B2)^n*C2 is then,
What is A1?
What is B1?
What is C1?
What is A2?
What is B2?
What is C2?

System 3 : Consider the discrete time system represented by the
following difference equation:
y(n) ? x(n) ? x(n ? 2) ? 0.8y(n ?1) ? 0.64 y(n ? 2)
a) Draw the corresponding BLOCK DIAGRAM
b) Obtain the TRANSFER FUNCTION, H(z) , for this
system.
c) Calculate and plot the POLES and ZEROS of the transfer
function.
d) State the FREQUENCY RESPONSE Equation , H(ej? ) ,
for this system.
System 4 : Consider the discrete time system represented by...

The signal x[n] is the input of an LTI system with impulse
function of h[n]. x[n] = (0.4)^n u[n] and h[n] = (0.2)^n u[n].
(a) What is the DTFT of the output of the LTI system?
(b) What are the Energy density spectrums of the input and
output signals?
(c) What would be the inverse DTFT: X(w) =
1/(1-0.25e^-j(w-2))
(d) How would part (c) differ for the DTFT: X(w) =
1/(1-0.25e^-j(w-2)) + 1/(1-0.25e^-j(w+2))

Write a MATLAB function that implements the difference equation
for the system assuming null initial conditions. Include source
code .
y(n) = x(n) + 2x(n-1) + x(n-2) + 0.8 y(n-1) - 0.64 y(n-2)
Use Matlab to calculate the response of the system to p(n) = 0
for all n expect when p(10) = 1.
Using stem, plot the resulting output to n=100
What is the amplitude of the output squence? Was the signal
amplified? How much?

An LTI system has an impulse answer of h[n] = a^(n)H[n], H[n] is
the Heaviside step function. Obtain the output y[n] from the system
when the input is x[n]=H[n]. 2. Consider the discrete system
defined by> y[n] - ay[n-1] =x[n] Find the output when the input
is x[n] = Kb^(n)H[n], and y[-1]=y_(-1)\ Find the output when the
input is x[n] = K ẟ [n], and y[-1]=a Find the impulse response when
the system is initially at rest. Find the Heaviside...

Problem 1....... you can use Matlab
The following Scilab code generates a 10-second “chirp” with
discrete frequencies ranging from 0 to 0.2 with a sampling
frequency of 8 kHz.
clear;
Fs = 8000;
Nbits = 16;
tMax = 10;
N = Fs*tMax+1;
f = linspace(0.0,0.2,N);
x = zeros(f);
phi = 0;
for n=0:N-1 x(n+1) = 0.8*sin(phi);
phi = phi+2*%pi*f(n+1);
end sound(x,Fs,Nbits);
sleep(10000); //allows full sound to play
Add code that calculates and plays y (n)=h(n)?x (n) where h(n)
is the...

For the LTI system described by the following system functions,
determine (i) the impulse response (ii) the difference equation
representation (iii) the pole-zero plot, and (iv) the steady state
output y(n) if the input is x[n] = 3cos(πn/3)u[n].
a. H(z) = (z+1)/(z-0.5), causal system (Hint: you need to
express H(z) in z-1 to find the difference equation )
b. H(z) = (1 + z-1+ z-2)/(1-1.7z-1+0.6z-2), stable system
c. Is the system given in (a) stable? Is the system given in...

For an LTI system h[n], the output is given by
y[n] = 2δ[n-1],
given that
x[n] = δ[n]-2δ[n-1]+ 2δ[n-2].
a) Find the transfer function H(z) (7 Points).
b) Find the difference equation of the overall system (8
Points).
c) Given that the system is causal find h[n] (10 Points).
d) Given that the system does not have Fourier Transform, find h[n]
(10 Points).

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