# signal and system(c) by fawadjatt-1

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```									  Signals and Systems

(Oppenheim & Willsky)
(Section 1.6 Basic System Properties)

Instructor
Waseem Khan
waseem.khan@mail.au.edu.pk

Department of Electrical Engineering
Air University
Systems with and without memory (1.6.1)
A system whose output at any time instant depends only
on the input at the same time, is a system without
memory. e.g. y[n] = (x[n])2
y(t) = R x(t)      → without memory
y([n] = abs(x[n]) → without memory
y([n] = x[n-1]     → with memory (output depends on
previous sample)
n
y[n] =   ∑ x[k ]   → with memory (output depends on all
k = −∞                previous samples)

→ with memory (voltage across capacitor
depends on history of current)
Invertibility and Invertible Systems (1.6.2)
A system is said to be invertible if distinct input to a
e.g. y(t) = 2 x(t). Its inverse system is w(t) = ½ y(t).

y(t) = [x(t)]2 is a non-invertible system.
Causality (1.6.3)
A system is causal if the output at any time depends only
on present and past input values.
Causal                     Non Causal
n
y[n] =   ∑ x[k ]
k = −∞

y([n] = abs(x[n])                    y[n] = x[n+1]

All memory less systems are causal.
Stability (1.6.4)
A system is said to be stable if every bounded input to
the system leads to a bounded output.
If there exists a finite value ‘A’ such that
|x(t)| < A      for all ‘t’
and there exists a finite value ‘B’ such that
|y(t)| < B      for all ‘t’
the system is stable.
Examples: y(t) = tan(x(t)) → unstable
y(t) = ex(t)     → stable              y(t) = tx(t) → unstable
y[n] = nx[n]   → unstable                    n
y[n] =   ∑ x[k ]
k = −∞
→ unstable
Time Invariance (1.6.5)
A system is time-invariant if the behaviour and
characteristics of the system are fixed and do not
change with time.

If input x(t) to a system leads to output y(t), and x(t-t0)
leads to output y(t-t0), the system is time invariant.

In DT, if output is y[n] for input x[n] and output is y[n-n0]
for x[n-n0], the system is TI.

See examples 1.14, 1.15
Linearity (1.6.6)
A system that satisfies principle of superposition, is
called linear system.
If x1[n] and x2[n] are two inputs of a system and y1[n]
and y2[n] are the respective outputs, then the system is
linear if and only if output of the system for input
ax1[n]+bx2[n]
is         ay1[n] + by2[n]
Linearity (Example)
y[n] = (x[n])2
Lets check for linearity.
y1[n] = (x1[n])2
y2[n] = (x2[n])2
Then output for input x1[n] + x2[n] is
y[n]= (x1[n] + x2[n])2
= (x1[n])2 + (x2[n])2 + 2x1[n] x2[n]
≠ (x1[n])2 + (x2[n])2

Hence, this system is not linear.

See examples 1.17~ 1.20 yourself.
Practice Problems

1.9~1.11
1.15~1.17
1.19-1.20
1.27-1.28
1.31

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