Discrete Time Signals & Systems

1. EC533: Digital Signal Processing
Lecture 4
Discrete-Time Signals &
Systems

2. x(n) y(n)
4.1 – Discrete-Time Signals DSP
By tradition, a discrete-time signal is represented as a sequence of numbers:
x(n)=5,3,4,3,6,…. for n=0,1,2,… x(n)
x(nT)=5,3,4,3,6,…. for n=0,1,2,… 6
5
4
xn=5,3,4,3,6,…., for n=0,1,2,… 3 3
T 2T 3T 4T nT
where x(n) indicates the value of the signal at a discrete time n(nT), also, it may indicates
the sequence itself.
• In DSP, it is common to omit T as the sampling frequency is assumed to be unity.

3. 4.1.1 – Important Discrete Signals
As we are considering processing signals that are represented by sequences,
we shall introduce the following basic signals:
a)Unit Impulse (Unit Sample)
1
0 n
b)Unit Step
1
0 1 2 3 4 n

4. 4.1.1 – Important Discrete Signals – cont.
Delayed & Advanced Sequences
1 delayed
For f(±n ±m);
If n & m have the same sign, the sequence will be 0 2 n
advanced by m samples (shifted left). If n & m
have the opposite signs, the sequence will be
delayed by m samples (shifted right).
1 advanced
‐1 0 n
c) Unit Ramp
r (n)
⎧n for n > 0⎫
r ( n) = ⎨ ⎬
⎩0 otherwise⎭
0 1 2 3 4 n

5. 4.1.1 – Important Discrete Signals – cont.
d) exponential signal:
x(n) = a n for ∀n
a = r.e jθ
When |a|>1 The signal converges to 0 at ∞

6. 4.1.1 – Important Discrete Signals – cont.
Unit Impulse & Unit Step Relationship
1
0 1 2 3 4 n
1
1 2 3 4 n
1
1
0 1 2 3 4 n
0 n
Hence, the unit impulse signal can be used as a basic building block for the construction
& representation of other signals. 3
2
1 0.5
‐1 0 1 3 n

7. 4.2 – Discrete-Time Systems
• A discrete-time system is essentially a mathematical algorithm that takes an
input sequence, x(n), & produces an output sequence y(n). e.g., digital controllers,
digital spectrum analyzers, & digital filters.
x(n) y(n)
DT system
• The discrete-time system is described by its impulse (unit sample) response h(n)
if h(n)
then h(n)
Impulse Response
Characteristics of discrete-time systems
Linearity
Shift Invariance (Time Invariance)
Stability
Causality

8. 4.3 – Characteristics of discrete-time systems
• A discrete system is linear if it satisfies the superposition principle, that is:
If y1(n) is the o/p for the i/p x1(n) , y2(n) is the o/p for the i/p x2(n)
then the o/p for the i/p α x1 (n) + β x2 (n) is α y1 (n) + β y2 (n)
h(n)
h(n)
h(n)

9. Linear System: Example 1
n
Accumulator y ( n ) = ∑ x (l )
l = −∞
if x ( n) = α x1 (n) + β x2 (n)
Then
n
y (n ) = ∑ α x1 (l ) + β x 2 (l )
l = −∞
n n
= ∑ α x1 (l ) + ∑ β x 2 (l )
l = −∞ l = −∞
n n
= α ∑ x1 (l ) + β ∑ x 2 ( l )
l = −∞ l = −∞
= α y1 ( n ) + β y 2 ( n )

10. Linear System: Example 2
y [ n ] = ( x [ n ]) 2
if x(n) = α x1 (n) + β x2 (n)
Then
y [ n ] = ( α x 1 [ n ] + β x 2 [ n ]) 2
≠ α y1[ n ] + β y 2 [ n ]

11. 4.3.2 – Shift Invariance (Time Invariance)
A system is shift invariant if any delay in the i/p produces a similar
delay in the o/p.
x1(n) y1(n)
i.e if x1 ( n ) ⎯
⎯→ y1 ( n ) h(n)
then if x ( n ) = x1 ( n − n 0 )
y ( n ) = y1 ( n − n 0 ) x1(n-k) y1(n-k)
h(n)
i.e. process doesn’t depend on absolute value of n

12. Shift Invariant system (Example 1)
• Upsampler x(n) y (n)
L
Not shift‐invariant

13. Shift Invariant system (Example 2)
Scaling by the time index
Hence,
if
then
Not shift-invariant parameters depend on n

14. Linear Shift Invariance (LSI) Systems
• Systems which are both linear and shift invariant are easily
manipulated mathematically
• If discrete index corresponds to time, called Linear Time
Invariant (LTI)
• There is a wide and useful class of DSP systems as digital
filters.

15. Linear Time-Invariant (LTI) systems
• LTI Systems can be fully characterized by the convolution sun
•Since
•Due to linearity and shift-invariance:
•Then
Convolution sum
•The convolution describes how the I/p to a system interacts with the
system to produce the O/p.

16. Properties of Convolution Sum
1- Commutative:
2- Distributive:
x(n) h1(n) y(n)
h2(n)
Parallel connection
3- Associative:
x(n) y(n) x(n) y(n)
h1(n) h2(n)
Cascade connection

17. LTI System Representation
• LTI systems can be represented by 2 forms:
1. Convolution Sum:
2. Difference Equation:
If b’s=0 FIR (Finite Impulse Response) system
else IIR (Infinite Impulse Response) system

18. 4.3.3 –Stability
• A system is stable if each Bounded Input produces a Bounded
Output (BIBO).
Bounded i/p :
Bounded o/p :
for
Condition for Stability
The system is stable if it is absolutely summable.

19. 4.3.4 – Causality
• A system is causal if there is no output when there is no input,
The o/p of a causal system depends only on the present & past
values of the i/p to the system & doesn’t predict future.

20. Causality (Example)
Moving Average
depends on Causal
‘Centered’ Moving Average
looks forward in time noncausal can be made causal by
delaying