Digital Signal Processing - Analysis of linear time invariant systems

inuous-time system
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 4
Discrete-time system
Digital system
 A discrete-time system is digital if it 
operates on discrete-time signals whose amplitudes 
are quantized
 Quantization maps each continuous 
amplitude level into a number
 The digital system employs digital hardware
1. explicitly in the form of logic circuits
2. implicitly when the operations on the signals are 
executed by writing a computer program
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 5
Analysis and design
 Analysis of a system is investigation of the 
properties and the behavior (response) of an 
existing system
 Design of a system is the choice and 
arrangement of systems components to perform 
a specific task
 Design by analysis is accomplished by modifying 
the characteristics of an existing system
 Design by synthesis: we define the form of the 
system directly from its specifications
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 6
Block diagram
 Block diagram is a pictorial representation of a system 
that provides a method for characterizing the 
relationships among the components
 Single block with one input and one output is the 
simplest form of the block diagram
 Interior of the rectangle representing the block contains 
(a) component name, 
(b) component description, or 
(c) the symbol for the mathematical operation to be 
performed on input to yield output
 Arrows represent the 
direction of signal flow
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 7
Basic elements of block diagram
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 8
Summing point
Takeoff point
Interconnections of blocks
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 9
Blocks connected 
in cascade
Blocks connected 
in feedback
Blocks connected 
in parallel
State
 For some systems, the output at time t0
depends not only on the input applied at t0, 
but also on the input applied before t0
 The state is the information at t0 that, 
together with input for t ≥ t0, determines 
uniquely output for t ≥ t0
 Dynamical equation is the set of equations 
that describes unique relations between the 
input, output, and state
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 10
Relaxed system
 A system is said to be relaxed at time t0 if the 
output for t ≥ t0 is solely and uniquely 
determined by the input for t ≥ t0
 If the concept of energy is applicable, the 
system is said to be relaxed at t0 if no energy 
is stored in the system at t0
 A system is said to be zero-input if the output 
for t ≥ t0 is solely and uniquely determined by 
the state
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 11
Some basic concepts and terminologies 
in discrete time systems
• Linear Time Invariant systems - LTI. 
• Input and output signals with relationship of 
discrete-time convolution via impulse response of 
system.
• LTI system can be separated into FIR (Finite 
Impulse Response) and IIR (Infinite Impulse 
Response). 
• FIR system can be modeled in the block or 
sample-by sample processing
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 12
The representation of signals 
in term of impulses
Example 
2. Input/Output Rules.
Sample-to-sample processing method - I/O rule
Block processing method
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 14
    ,,,,,,,,,, 210210 n
H
n yyyyxxxx   etc. ,,, 221100 yxyxyx
HHH 
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Example 1:
Example 2:
Hx
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DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 15
    ,2,2,2,2,2,,,,, 4321043210 xxxxxxxxxx
H
3. Linearity and Time-Invariance LTI
 Consider a relaxed system in which there is 
one independent variable t
 A linear system is a system which has the 
property that if
 input x1(n) produces an output y1(n) and
 input x2(n) produces an output y2(n), then
 input c1 x1(n) + c2 x2(n) produces an 
output c1 y1(n) + c2 y2(n) for any x1(n), 
x2(n) and arbitrary constants c1 and c2
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 16
* Linearity
(1)
(2)
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 17
     nxanaxnx 21 
     nyanyany 2211 
Testing linearity
Time-invariant system
 A relaxed system is time-invariant if a time shift 
in the input signal causes a 
time shift in the output signal
 In the case of discrete-time digital systems, we 
often use the term 
shift-invariant instead of time-invariant
 Characteristics and parameters of a time-invariant 
system do not change with time
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 18
* Time invariance
D is delay operator
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 19
Testing Time-Invariance with delay by D samples
Linear time-invariant (LTI) system
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 20
Continuous-time system is LTI if its input-output relationship 
can be described by the ordinary linear constant coefficient 
differential equation
Discrete-time system is LTI if its input-output relationship 
can be described by the linear constant coefficients 
difference equation
4. Impulse response
Dirac Delta function
   nhn H    ,,,,,0,0,0,1 3210 hhhhH
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 21
 






0n if0
0n if1
n
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 22
Delayed impulse response of an LTI system
(LTI form)
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 23
                   3322|110 nhxnhxnhxnhxny
      
m
mnhmxny
Convolution
Response to linear combination of inputs
5. FIR and IIR Filters
An FIR filter has impulse response h(n) that 
extends only over a finite time interval 0 ≤ n ≤ M
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 24
  ,0,0,0,,,,, 210 Mhhhh
• M is filter order. Length of impulse response 
• h = {h0, h1, h2, , hM} is : LH = M + 1
{h0, h1, h2, , hM} named filter coefficients, or 
filter weights, or filter taps.
Example: third order FIR filter h = [h0,h1, h2, h3] has 
I/O equation:
y(n) = h0x(n) + h1x(n-1) + h2x(n-2) + h3x(n-3)
Example: Find impulse response of the following FIR 
filter:
(a) y(n) = 2x(n) + 3x(n-1) + 5x(n-2) + 2x(n-3)
(b) y(n) = x(n) - 4x(n-4)
Solution: 
(a) h = [2, 3, 5, 2]
(b) h = [1, 0, 0, 0, -4]
if input is x(n) = (n), then output y(n) = h(n):
(a) h(n) = (n) + 3(n – 1) + 5(n – 2) + 2(n – 3)
(b) h(n) = (n) – (n – 4) 
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 25
FIR FILTERING EQUATION
An IIR filter has an impulse response h(n) of infinite 
duration, defined over the infinite interval. 
IIR equation
Example: Find the convolution form and impulse response of 
the following IIR filter: y(n) = 0,25y(n – 2) + x(n)
Solve: impulse response h(n) = 0,25h(n – 2) + (n)
h(–2) = h(–1) = 0; h(0) = 0,25h(–2) + (0) = 1
h(1) = 0,25h(–1) + (1) = 0; h(2) = 0,25h(0) + (2) = 0,25 = 0,52
h(3) = 0,25h(1) + (3) = 0; h(4) = 0,25h(2) + (4) = 0,252 = 0,54
n ≥0
   
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odd 
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DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien
26
h ={1, 0, 0.52, 0, 0.54, 0,. . .}; y(n) = x(n) + 0.52x(n – 2) + 0.252x(n – 4)
     



0m
mnxmhny
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 27
Example: Find the difference equation of the following 
IIR filter: h ={2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, . . .}
with the cycle lasts for 4 samples
Solve: h(n – 4) ={0, 0, 0, 0, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, . . .}
h(n) – h(n – 4) = {2, 3, 4, 5, 0, 0, 0, 0,. . .}
h(n) – h(n – 4) = 2d(n) + 3d(n – 1) + 4d(n – 2) + 5d(n – 3)
h(n) = h(n – 4) + 2d(n) + 3d(n – 1) + 4d(n – 2) + 5d(n – 3) 
Or yn = yn – 4 + 2xn + 3xn-1 + 4xn-2 + 5xn-3
IIR Filter has the impulse response h(n)
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 28
     

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     
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6. Causality and stability
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 29
Delay D to produce the causal,
hD(n) = h(n – D) 
I/O equation for causal filter hD(n)
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 30
     



Dm
mnxmhny
     



0m
DD mnxmhny
Example: Consider the typical 5-tap smoothing filter having 
filter coefficient h(n) = 1/5 for -2 ≤ n ≤2. The corresponding 
I/O convolutional equation 
It is called a smoother or average because at each n it 
replaces the current sample x(n) by its average with the two 
samples ahead and two samples behind it. Its anti-causal 
part has duration D=2 and can be made causal wit the time 
delay of two units 
              4321
5
1
22  nxnxnxnxnxnyny
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 31
       


2
2
2
2 5
1
mm
mnxmnxmhny
          2112
5
1
 nxnxnxnxnx
Stability:
A stable LTI system is one whose impulse 
response h(n) goes to zero sufficiently fast as n 
to be infinitive, so that the output y(n) never 
diverges and |y(n)| ≤ B if input is limited 
|x(n)| ≤ A. 
A necessary and sufficient condition for an LTI 
system to be stable is 
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 32
  

n
nh
DSP Lectured by Assoc. Prof. Dr. Thuong Le-Tien 33
LINEAR CONSTANT COEFFICIENT DIFFERENCE EQUATION
Example: y[n] – y[ n-1] = x[n]
A recursive difference equation
Example: a recursive form of a moving average system