This document contains 7 problems related to analyzing discrete-time systems. Problem 1 involves determining the input-output relationship of a system formed by connecting two other systems in series. Problem 2 analyzes properties like linearity and time-invariance for a given system. Problems 3-4 check these properties for additional systems and statements. Problem 5 calculates the step response and response to other inputs for a given system. Problem 6 determines the steady state periodic response of a system to a periodic input using MATLAB code. Problem 7 involves finding controller parameters to place closed loop poles and determining the steady state output for a feedback system.

1. 4th
Year Biomedical Eng. Systems Engineering 2020 Assignment#1
Solve the following backgroundproblems:
1- Consider a system S with input x(n) and output y(n). This system is obtained through a
series interconnection of a system S1 followed by a system S2. The input-output
relationships for S1 and S2 are:
S1: y1(n) = 2 x1(n) + 4 x1(n-1)
S2: y2(n) = x2(n-2) + 0.5 x2(n-3)
where x1(n) and x2(n) denote input signals.
(a) Determine the input-output relationship for system S.
(b) Does the input-output relationship of system S change if the order in which S1 and S2
are connected in series is reversed (i.e., if S2 follows S1)?
2- Consider a discrete-time system with input x(n) and output y9n) related by
𝑦( 𝑛) = ∑ 𝑥( 𝑘),
𝑘=𝑛+4
𝑘=𝑛−4
(a) Is the system linear?
(b) Is the system time-invariant?
(c) Is the system causal?
3- Check the general properties of (Linearity, Time-invariance and Causality) for the
following systems
(a) y(n) = x(-n)
(b) y(n) = x(n-2) – 2 x(n-8)
(c) y(n) = n x(n)
(d) 𝑦( 𝑛) = {
𝑥( 𝑛), 𝑛 ≥ 1
0, 𝑛 = 0
𝑥( 𝑛 + 1) 𝑛 ≤ −1
4- (a) Is the following statement true or false?
The series interconnection of two linear time-invariant systems is itself a linear, time-
invariant system (Justify your answer).
(b) Is the following statement true or false?
The series interconnection of two nonlinear systems is itself nonlinear system (Justify
your answer).

2. (d) Consider three systems with the following input-output relationships:
System 1 : 𝑦(𝑛) = {
𝑥 (
𝑛
2
) 𝑛 𝑒𝑣𝑒𝑛
0 𝑛 𝑜𝑑𝑑
System 2 : y(n) = x(n) + 0.5 x(n-1) + 0.25 x(n-2)
System 3 : y(n) = x(2n)
Suppose that these systems are connected in series System 1 then System 2 followed
by System 3. Find the input-output relationship for the overall connected system. Is
this system linear? Is it time-invariant?
5- a) Find the step response y(k) of a discrete system whose pulse transfer function H(z) is
given by
H(z) = 1 / (1 – 0.8 z-1)
b) What is the steady state value of y(k)?
c) If the input the system of part a) is u(k) – u(k-2), where u(k) is a unit step sequence,
can you find the output due to this input in terms of y(k) (the response of part a)).
d) Determine the response of this system to an input sequence which is given by {1, 2, 3,
0, 0 ,0 ,0 ………0}.
6- Given that the response of a linear and stable discrete system to a periodic input is
periodic in the steady state. Find the steady state response of a system whose pulse
transfer function is given by H(z) = 2 / (1 – 0.6 z-1) to an input and periodic sequence x(k)
= {1, 1, -1, 1, 1, -1, ……}.
Hint: The output y(k) will look like { y0, y1, y2, ………., α, β, λ, α, β, λ, α, β, λ, …}. Use
the input-output relationship to find α, β, λ.
Use MATLAB to generate the periodic input sequence and the corresponding output
sequence. The following MATLAB code will help you in doing so.
Sysd = tf ( Num, Den, Ts); This generate a discrete system with sampling period Ts
and Num is the numerator of the pulse transfer function and Den is the denominator of
the pulse transfer function. In this problem Sysd = tf ( [2], [ 1 -1], 0.1);
Next generate one period of the input sequence using the code:
OnePeriod = [1 1 -1];
To generate a twenty period of this sequence use
Uk = repmat(OnePeriod, [ 20]);
To find the response of this system to the periodic input sequence; use the code
t = 0 : 0.1 : 5.9; this generate a time sequence from 0 to 5.9 secs. With 0.1 sec
increment.
y = lsim(Sysd, Uk, t); this generates the output sequence y.
plot( t’, y) to plot the response of the system y against time sequence t.

3. 7- Given the system shown in Fig.
and G(z) = 1/ ((z -1) (z -0.6)) & H(z) = K (z – α) /( z – β )
a) Find the values of k, α and β such that the closed loop poles of the system are located
at z = 0.3 + 0.4 j, z = 0.3 – 0.4 j and z = 0.05
b) Determine the steady state response of Y(z).