Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 231

This is a feedback system since the I/O equation includes a delayed version of
the output itself.

Q2: Find the I/O (input-output) relation of the system shown below.

Delay, T o

x ( t ) y ( t )

Ans. y(t) = x(t)+x(t - T o ) - y(t - T o ).

Tutorial 3

Q: Determine whether the analog time-delay T o (T o is constant) is:
1. Memoryless, 2. causal, 3. linear, 4. time-invariant, 5. stable.

Time−delay, T o

x ( t ) y ( t )

Solution:
1. Since y(t) = x(t - T 0 ), the output equals the input at a past time instant, t - T o ,
hence it is a memory system.
2. The output y(t) is not a function of x(t + t 0 ), t 0 [ 0, hence it is causal (does not
depend on future values of the input).
3. Let T represents the system transformation.
For the input x(t) = ap(t)+br(t) we have:

y ð t Þ¼ T f x ð t Þg¼ x ð t T 0 Þ¼ a p ð t T 0 Þþ b r ð t T 0 Þ
¼ a T f p ð t Þgþ b T f r ð t Þg

Hence, the system is linear.

4 : T f x ð t t 0 Þg¼ x ð t t 0 T 0 Þ ð 1 Þ

We have: y(t) = x(t - T 0 ), hence,
y ð t t 0 Þ¼ x ð t t 0 T 0 Þ ð 2 Þ

From Eqs. 1 and 2 we get:
y ð t t 0 Þ¼ T f x ð t t 0 Þg:

Hence, T is time-invariant.