1. CEN352 Dr. Nassim Ammour King Saud University
Chapter 3
Digital Signals and Systems
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2. CEN352 Dr. Nassim Ammour King Saud University
Introduction
This chapter introduces notations for digital signals and special digital sequences.
The chapter continues to study some properties of linear systems such as time invariance, causality, impulse response, difference equations, and digital convolution.
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3. Graphical representation of a discrete-time signal 𝒙(𝒏)
Digital Signals
A discrete-time signal 𝑥(𝑛) is a function of an integer variable 𝑛, where 𝑛∈ℤ.
Graphical representation of a discrete-time signal 𝒙(𝒏)
For floating point DS processor, the amplitudes can be floating points.
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Digital Signals Contd.
Besides the graphical representation, there are some alternative representations.
𝑥 𝑛 = 1, 𝑓𝑜𝑟 𝑛=1,3 4, 𝑓𝑜𝑟 𝑛=2 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
1. Functional representation
2. Tabular representation
3. Sequence representation (bold or arrow for origin n=0)
𝑥 𝑛 = …,0 ,𝟎 ,1 ,4 ,1, 0, 0,… 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒−𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑔𝑛𝑎𝑙
𝑥 𝑛 = 0 ,−2 ,1 ,4 ,−1, 𝑓𝑖𝑛𝑖𝑡𝑒−𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑔𝑛𝑎𝑙
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Common Digital Sequences
1. Unit-impulse sequence:
2. Unit-step sequence:
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Shifted Sequences
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Sinusoidal and Exponential Sequences
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Example 1
The sum
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Generation of Digital Signals
To generate the digital sequence 𝑥 𝑛 from the analog signal 𝑥 𝑡 :
uniformly sampling at the time interval of ∆𝑡=𝑇
𝑥 𝑛 = 𝑥(𝑡) 𝑡=𝑛𝑇 =𝑥(𝑛𝑇)
Example 2
Convert analog signal 𝑥 𝑡 into digital signal 𝑥 𝑛 , when sampling period is 125 microsecond, also plot sample values.
𝑥 𝑡 =10 𝑒 −5000𝑡 𝑢(𝑡
Solution:
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Example 2 (contd.)
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Classification of Discrete-Time Signals
𝐸= 𝑛=−∞ ∞ 𝑥 𝑛 2
Energy signals: The energy E of a signal 𝑥 𝑛
Periodic signals and aperiodic signals: A signal 𝑥 𝑛 is periodic with period N (N>0) if
𝑥 𝑛+𝑁 =𝑥 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛
Symmetric (even) and anti-symmetric (odd) signals
Anti-symmetric
𝑥 −𝑛 =−𝑥 𝑛
symmetric
𝑥 −𝑛 =𝑥 𝑛
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Digital Systems
A digital system is a device or an algorithm that performs prescribed operations (or transformation) on a digital signal 𝑥 𝑛 and produces an output signal 𝑦 𝑛 .
Digital System
Input signal
(Excitation)
Output signal
(Response)
𝑥 𝑛
𝑦 𝑛
Example 3
𝑥 𝑛 = 𝑛 , −3≤𝑛≤3 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Determine the response of the following system to the input signal
𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒 𝑚 ′ 𝑠 𝑜𝑢𝑡𝑝𝑢𝑡 𝑦 𝑛 = 𝑥 𝑛+1 +𝑥 𝑛 +𝑥(𝑛−1
Solution:
The output of this system is the mean value of the present, immediate past, and the immediate future samples.
For 𝑛= 𝑦 0 = 1 3 𝑥 −1 +𝑥 0 +𝑥(1) = = 2 3
𝑦 𝑛 = …, 0,1, 5 3 , 2, 1, 2 3 , 1, 2, 5 3 , 1, 0, …
Repeating this computation for every value of 𝑛
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Block Diagram of Discrete-Time Systems
Constant multiplier
Unit delay element
Adder
Unit advance element
Signal multiplier
Example 4
Solution:
Sketch the block diagram representation of the discrete-time system described by the input-output relation.
𝑦 𝑛 = 1 4 𝑦 𝑛− 𝑥 𝑛 𝑥(𝑛−1)
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14. Classification of Discrete-Time Systems
Static and dynamic systems
𝑦 𝑛 =𝑥 𝑛 +3𝑥(𝑛−1
𝑦 𝑛 =𝑛 𝑥 𝑛 +𝑏 𝑥 3 (𝑛
Dynamic (have memory)
past or future samples of the input
Static or memory-less
no past or future samples of the input
Time-invariant System (Time Shift Invariance)
The system input-output characteristics do not change with time (a shifted input signal will produce a shifted output signal, with the same of shifting amount).
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Linear System
A digital system is linear if and only if it satisfy the superposition principle:
𝑆 𝑎 1 𝑥 1 𝑛 + 𝑎 2 𝑥 2 𝑛 = 𝑎 1 𝑆 𝑥 1 𝑛 + 𝑎 2 𝑆 𝑥 2 𝑛
Homogeneity (deals with amplitude)
Additivity
Homogeneity & Additivity
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Example 5 (a)
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Example 5 (b)
System
𝐿𝑖𝑛𝑒𝑎𝑟 ???
4 𝑦 1 𝑛 +2 𝑦 2 𝑛
𝑦(𝑛)≠4 𝑦 1 𝑛 +2 𝑦 2 𝑛
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Example 6 (a)
Given the linear system y 𝑛 =2𝑥(𝑛−5), find whether the system is time invariant or not.
Solution:
Let the shifted input be: 𝑥 2 𝑛 = 𝑥 1 (𝑛− 𝑛 0 ),
Therefore system output: 𝑦 2 𝑛 =2 𝑥 2 𝑛−5 =2 𝑥 1 𝑛− 𝑛 0 −5 ,
Shifting
𝑦 1 𝑛 =2 𝑥 1 𝑛−5 by 𝑛 0 samples leads to 𝑦 1 𝑛− 𝑛 0 =2 𝑥 1 𝑛−5− 𝑛 0
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Example 6 (b)
Given the linear system y 𝑛 =𝑛 𝑥(𝑛), find whether the system is time invariant or not.
𝑦 1 𝑛 =𝑛 𝑥 1 (𝑛)
Solution:
Let the shifted input be: 𝑥 2 𝑛 = 𝑥 1 (𝑛− 𝑛 0 )
Therefore system output
due to the shifted input
𝑦 2 𝑛 =𝑛 𝑥 2 𝑛 =𝑛 𝑥 1 𝑛− 𝑛 0
Shifting
𝑦 1 𝑛 =𝑛 𝑥 1 𝑛 by 𝑛 0 samples
(replace 𝑛 by 𝑛−𝑛 0 ) leads to
𝑦 1 𝑛− 𝑛 0 =(𝑛− 𝑛 0 ) 𝑥 1 𝑛− 𝑛 0
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Causality
Causal System:
Output y(n) at time n depends only on current input 𝑥 𝑛 , at time n or previous inputs, such as 𝑥 𝑛 , 𝑥 𝑛−1 , 𝑥 𝑛−2 , etc.
Example:
Non Causal System:
Output y(n) at time n depends on future inputs, such as 𝑥 𝑛+1 , 𝑥 𝑛+2 , etc.
Example:
The non causal system cannot be realized in real time.
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Difference Equation
A causal, linear, time-invariant system (LTI) can be described by a difference equation as follow:
Outputs
Inputs
After rearranging:
Finally:
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Example 7
Identify non zero system coefficients of the following difference equations.
Solution:
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System Representation
Using Impulse Response
A linear time-invariant system (LTI system) can be completely described by its unit-impulse response ℎ(𝑛) due to the impulse input 𝛿(𝑛) with zero initial conditions.
Impulse input with
zero initial conditions
Impulse Response
𝑦 𝑛 =ℎ 𝑛 ⊗𝑥 𝑛
Any input
Convolution
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Example 8 (a)
Given the linear time-invariant system:
Solution:
a. let 𝑥 𝑛 =𝛿(𝑛), then
Therefore,
c. The system output
b. The block diagram of the LTI system
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Example 8 (b)
Solution:
a. Let 𝑥 𝑛 =𝛿(𝑛), then
With the calculated results, we can predict the impulse response as
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Example 8 (b) – contd.
b. The system block diagram
c. The output sequence
Finite Impulse Response (FIR) system:
When the difference equation contains no previous outputs, i.e. ‘a’ coefficients are zero ( See example 8 (a) )
Infinite Impulse Response (IIR) system:
When the difference equation contains previous outputs, i.e. ‘a’ coefficients are not all zero. ( See example 8 (b) )
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BIBO Stability
BIBO: Bounded In and Bounded Out
A stable system is one for which every bounded input produces a bounded output.
We have:
Let, in the worst case, every input value reaches to maximum value M.
Using absolute values of the impulse responses,
If the impulse responses are finite number, then output is also finite.
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Stable system.

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BIBO Stability –contd.
To determine whether a system is stable, we apply the following equation:
Impulse response is decreasing to zero.
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29. The summation is finite, so the system is stable.
Example 9
Given a linear system given by:
Which is described by the unit-impulse response:
Determine whether the system is stable or not.
Solution:
Using definition of step function:
For a< 1, we know
The summation is finite, so the system is stable.
Therefore
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Digital Convolution
A LTI system can be represented using a digital convolution sum.
The unit-impulse response ℎ(𝑛) relates the system input and output.
Commutative
The sequences are interchangeable.
Convolution sum requires ℎ(𝑛) to be reversed and shifted.
If ℎ(𝑛) is the given sequence, ℎ(−𝑛) is the reversed sequence.
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Reversed Sequence
Given a sequence
Sketch the sequence ℎ(𝑘) and reversed sequence ℎ(−𝑘).
Solution:
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32. Convolution Using Table Method
Example 9
Solution:
Length =3
Length =3
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Convolution length = 3 +3 –1 = 5

33. Convolution Using Table Method
Example 10
Solution:
Convolution length = –1 = 4
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Convolution Properties
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