introduction to digital signal processing

Introduction to Digital Signal Processing
UNIT 1

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Contents
Introduction to DSP
Discrete time signals and sequences
Linear Shift Invariant systems
Stability and Causality
Linear constant coefficient Difference equations
Frequency domain representation of discrete
time signals and systems

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What is a Signal ?
Anything which carries information is a signal.
Examples
 Human voice, chirping of birds, smoke signals, gestures
(sign language), fragrances of the flowers.
 Many of our body functions are regulated by chemical
signals, blind people use sense of touch, ECG, EEG
 Bees communicate by their dancing pattern.

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Some examples of modern high speed signals
 Voltage charger in a telephone wire
 Electromagnetic field emanating from a transmitting
antenna,
 Variation of light intensity in an optical fiber.
In this course we will learn some of the mathematical
representations of the signals
A signal is a real (or complex) valued function of one or
more real variable(s)

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A continuous-time signal x (t) (discrete-time signal x[n])
is a function of an independent continuous variable t (discrete
variable n)

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Classification of signals
a) Based on no of independent variables
 1D : When the function depends on a single variable, the
signal is said to be one dimensional.
EX : A speech signal, daily maximum temperature, annual
rainfall at a place
 MULTIDIMENSIONAL : When the function depends on
two or more variables, the signal is said to be
multidimensional.

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Images (2D): light intensity as a function of 2D coordinates
Black and white or grey scale images (I=0-255)
Colour images: I=red(0-255), green(0-255), blue(0-255)
Video (3D) : Sequence of images, called frames
Is a function of 3 variables = 2 spatial coordinates and time
Our physical world is four dimensional ( Three spatial and one
temporal) EX: speed of wind , pressure etc.,

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Classification of signals
b) Based on nature of independent variables
 Continuous signals
 Discrete signals
Time
Amplitude
analog signals
continuous-time
signals
discrete-time
signals
digital signals
Continuous
Continuous
Discrete
Discrete
A signal is said to be continuous when it is defined for all
instants of time

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Digital signals that are discrete in
time and quantized in amplitude
Discrete time signals are defined
for discrete instances of time
Amplitude is continuous

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Classification of signals contd..
c) Based on nature of indeterminacy
 Deterministic signals
 Random signals
d) Based on periodicity
e) Based on causality
f) Based on energy content in signal
g) Natural or synthetic signals

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Need for processing
To obtain signal in more desirable form (transformation of
signal) or interpretation and manipulation of signals
Ex: Noise cancellation, Multiplexing and demultiplexing
Amplification of signals

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Signal Process Systems
analog
system
signal output
continuous-time signal continuous-time signal
discrete-
time
system
signal output
discrete-time signal discrete-time signal
digital
system
signal output
digital signal digital signal
ASP
DSP

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ASP : In Classical radio,
Telephone, Radar and
Television systems
For signals that are not
digitized

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 Digital Signal Processors (DSP) take real-world (natural)
signals like voice, audio, video, temperature, pressure, or
position that have been digitized and then mathematically
manipulate them.
 A DSP is designed for performing mathematical functions like
"add", "subtract", "multiply" and "divide" very quickly.
What is a Digital Signal Processing ?

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Advantages of Digital Signal Processing
(1) Flexibility : Same hardware can be used to do various kind of
signal processing operation.
while in the analog signal processing one has to design a
system for each kind of operation.
(2) Repeatability : The same signal processing operation can be
repeated again and again giving same results
while in analog systems there may be parameter variation due
to change in temperature or supply voltage.

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(3) High Accuracy : The accuracy of the analog filter is affected by
the tolerance of the circuit components used for design the
filter, but DSP has superior control of accuracy.
(4) Flexibility in Configuration : For reconfiguring an analog
system, we can only do it by redesign of system hardware,
where as a DSP System can be easily reconfigured only by
changing the program.
(5) Ease of Data Storage : On magnetic media, without the loss
of fidelity the digital signals can be stored and can be
processed off-line in a remote laboratory.

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(6) Time Sharing : The cost of the processing signal can be
reduced in DSP by the sharing of a given processor among a
number of signals.
(7) Cheaper : The digital realization is much cheaper than the
analog realization in many applications.
(8) Stability : Less sensitive environmental changes
(9) Applicable for very low frequency signal processing
e.g. seismic signals

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Disadvantages of Digital Signal Processing
(1) Power Consumption : The DSP chip consists of over 4 Lakh
transistors, which will yields to dissipate high power (1 Watt),
whereas the analog signal processing includes only passive
circuit elements like resistors, capacitors and inductors, which
will leads to only low power dissipation.
(2) Processing of signals beyond higher frequencies (beyond
GHz) and below lower frequencies (a few Hz) involves
limitations

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(3) Information is lost because we only take samples of the signal
at intervals .
(4) Speed is limited and increased complexity due to A/D and
D/A Converters
(5) Frequency range of operation is limited due to sampling
Disadvantages of Digital Signal Processing

Discrete-Time Signals---
Sequences
Discrete-Time Signals and Systems

Sampling Of Low Pass Signal
Let m(t) be a base band signal which is band limited
and its highest frequency component is fM.
Let the values of m(t) be determined at regular
intervals separated by Ts ≤ 1/2 fM,that is the signal is
periodically sampled for every Ts seconds .Then these
samples m(nTS),where n is an integer,uniquely
determine the signal and the signal may be
reconstructed from these samples with no distortion.
The time Ts is called Sampling Time
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Representation of sequences

 Tabular Representation Ggraph
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1 2
3 4 5 6 7
8 9 10
-1
-2
-3
-4
-5
-6
-7
-8
n
x[n]

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Elementary discrete-time signals / sequences
1. Unit sample sequence or Unit impulse or Kronecker
delta function (much simpler than the Dirac impulse)

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2. unit step sequence

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3. Unit Ramp sequence
4. Signum Function

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5. Exponential Function

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6. Sinusoidal sequence

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Periodic only if

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Classification of discrete-time signals
1.Periodic and Aperiodic sequence N, m must be integers
P1: Determine whether or not each of the following signals is periodic.
If a signal is periodic, specify its fundamental period
Always periodic
Not periodic always

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2.Symmetric (Even) and Anti Symmetric (odd) sequence
For an Even signal x(n) = x(-n)
For an odd signal x(-n) = -x(n)
P2.Find the odd and even components of x(n)
For even part

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3. Power or Energy sequence
The energy of a discrete-time signal is defined as
The average power of a signal is defined as
If E is finite, [0 ≤ E < ] and P = 0 then x(n) is called an energy signal
ꝏ
The average power of periodic sequence with period N is given by
If E is infinite and If P is finite and nonzero, then x[n] is called a
power signal.

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P4. Determine the power & energy of the unit step sequence.
P5. Test whether the given signal is an energy signal or a power signal.
x[n] = 5 (a constant signal)

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Operations on Sequences
1. Amplitude Scaling: (A Constant Multiplier)
2.Addition of two signals (An Adder)

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3. The product of two signals (A signal Multiplier)
4. Shifting
A unit delay element A unit advance element
Time shifting y[n] = x[n − k]. k can be positive (delayed signal) or
negative (advanced signal)

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5. Folding or reflection or time-reversal
y[n] = x[−n]

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x(n − 2)
x(−n) x(2 − n)
Try x(-2 - n)
Try x(-2 + n)

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6. Time-scaling
Down sampling /compression
Up sampling /expansion

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x(n) = u(n + 3) + 0.5 u(n − 1)
x(n) = δ(n + 3) + 0.5 δ(n − 1)
x(n) = 2^n · δ(n − 4)
x(n) = 2^n · u(−n − 2)
x(n) = (−1)^n . u(−n − 4)
x(n) = 2 δ(n + 4) − δ(n − 2) + u(n − 3)
Practice problems
Make an accurate sketch of each of the discrete-time signals

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Make an accurate sketch of each of the discrete-time signals

Linear Shift
Invariant systems

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 A device or an algorithm that performs some prescribed
operation on a discrete time signal (input or excitation) to
produce another discrete time signal (output or response)
Discrete-time system
y[n] = T {x[n]}
Accumulator system 



n
k
k
x
n
y )
(
)
(
Ideal Delay System )
(
)
( d
n
n
x
n
y 

Squarer  2
]
[
)
( n
x
n
y 
Compressor ]
[
)
( Mn
x
n
y 
Examples
T [ ]
x[n] y[n] = T
[ x[n] ]

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 Adder
 Constant multiplier
 Signal multiplier
 Unit delay element (why z −1 clear later)
How are delays implemented?.
Block diagram representation of discrete-time systems
With buffers / latches / flip-flops

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Classification of discrete-time systems
 Linear & non-linear systems
 Time-variant & Time-Invariant systems
 Static & Dynamic systems
 Causal & noncausal systems
 Stable & unstable systems
Time properties (causality, memory, time invariance)
Amplitude properties (stability, invertibility, linearity)

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1. Linear & non-linear systems
System is said to be linear if it obeys superposition principle
1. Homogeneity or scaling property
2. Additivity property
These two properties together comprise the principle of
superposition
T {a x[n] } = a
T{ x[n] }
If a = 0 we see that zero input signal implies zero output
signal for a linear system
T { x1 [n] + x2 [n] } = T { x1 [n] } + T { x2 [n] }

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A system T is linear iff
T { α x1[n] + β x2[n] } = α T { x1[n] } + β T { x2[n] }
x3[n] = α x1[n] + β x2[n]
= T { x3[n] }
α
α
β
β
T
T
T

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Determine whether the system given below are
linear or non-linear.
)
(
)
( n
x
n
y 
)
(
)
( 2
n
x
n
y 
)
(
)
( n
nx
n
y 
 
)
(
cos
)
( n
x
n
y 
)
(
)
( n
x
n
y 
 
)
(
log
)
( n
x
n
y 

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1.
2. )
(
)
( n
nx
n
y 
 
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
1
2
1
2
1
3
2
1
3
n
y
n
y
n
nx
n
nx
n
x
n
x
n
n
y
n
x
n
x
n
x

















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2. Time-variant & Time-Invariant systems
Systems whose input-output behavior does not change with time are
called Time-invariant
A relaxed system T is called Time Invariant or Shift Invariant iff
implies that
Graphically:

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)
(
)
(
)
(
)
,
(
k
n
x
n
x
at
n
y
k
n
y



k
n
n
n
y
k
n
y



 )
(
)
(
1. y(n) = x(-n)
2. y(n) = x(n) + x(n-1)
3. y(n) = n x(n)
4. y(n) = x(2n)
5. y(n) = x(n) sinω0n
If delayed output is equal to output due to delayed input then
system is said to be Shift-invariant

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1. y(n) = x(2n)
2. y(n) = x(n) sinω0n

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3. Static & Dynamic systems
 For a Static system or Memory less system, the output y[n]
depends only on the current input x[n], not on previous (n-1) or
future (n+1) inputs .
 Output depends on past, present & future samples of I/P -
Dynamic (with memory)
Ex

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4. Causal & NonCausal systems
 For a Causal system, the output y[n] at any time n depends
only on the “ present ” and “ past ” inputs i.e.,
y(n) = x(n)+u(n+1)
 Non causal system – The system does not satisfy the above
condition.
y(n) = x(-n)
y(n) = x(2n) For n < n0 ; i.e., n=1, n0=4
y(n) = ax(n)+b
Ex
Ex

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5. Stable & unstable systems
 A system is Bounded-Input Bounded-Output (BIBO)
stable iff every bounded input produces a bounded output
If ∃ Mx s.t. | x[n] | ≤ Mx < ∞ n, then there must exist an
∀
My s.t. | y[n] | ≤ My < ∞ n
∀
unstable
k
n
x
n
y
stable
k
n
x
m
n
y
unstable
n
n
x
n
y
k
n
k











0
1
0
2
)
(
)
(
)
(
1
)
(
)
(
)
(
Ex

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Analysis of Linear Time Invariant systems
Why analysis? So far we only have input-output relationships.
For a given system can compute y[n] for selected x[n]’s , but
very difficult to design filters etc
LTI h (n)

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 Decompose input signal x[n] into a weighted sum of elementary
Two particularly good choices for the elementary functions
• impulse functions δ [n − k]
• complex exponentials
 Determine response of system to each elementary
 Apply superposition property:
Techniques for the analysis of linear systems

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Resolution of discrete-time signal into impulses
This is the sifting property of the unit impulse function
x(n) = {0.5, 1.5, 0, -1, 1, 0.75, 2}
x(n) = 0.5 δ(n+2)+1.5 δ(n+1)-δ(n-1)+δ(n-2)+0.75 δ(n-3)+2 δ(n-4)
T [ ]
(n) h(n)
x(n) y(n)
Impulse Response

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Let the symbol h[n-k] to denote the system output when the input
is δ[n − k] , i.e.,
Because the system is Time Invariant system
Response of LTI systems to arbitrary inputs:
The Convolution Sum
T [ ]
x(n) y(n) = T [x(n)]
)
(
)
(
)
( k
n
k
x
n
x
k

 



 






 



)
(
)
(
)
( k
n
k
x
T
n
y
k

)
(
)]
(
[ k
n
h
k
n
T 




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74
)]
(
[
)
(
)
( k
n
T
k
x
n
y
k

 




)
(
)
( k
n
h
k
x
k

 



Delayed Impulse
as input
Delayed Impulse
Response







 



)
(
)
(
)
( k
n
k
x
T
n
y
k

)
(
*
)
( n
h
n
x

convolution
The convolution sum shows that with the impulse response h[n], you
can compute the output y[n] for any input signal x[n] . Thus
A LTI system is characterized completely by its impulse response h[n]

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Properties of Convolution
)
(
*
)
(
)
(
)
(
)
( n
h
n
x
k
n
h
k
x
n
y
k


 



)
(
*
)
(
)
(
)
(
)
( n
x
n
h
k
n
x
k
h
n
y
k


 



Convolution is Commutative
)
(
*
)
(
)
(
*
)
( n
x
n
h
n
h
n
x 
If x[n] has n = N1, . . . , N1 + L1 − 1 (length L1) and
h[n] has n = N2, . . . , N2 + L2 − 1 (length L2)
then y[n] = x[n] h[n] has L = L1 + L2 − 1
∗

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Convolution is Associative
h1(n)
x(n) h2(n) y(n)
h2(n)
x(n) h1(n) y(n)
h1(n)*h2(n)
x(n) y(n)
These three systems are identical.
Convolution is Associate

1. Given x[n] and h[n], choose an initial value of n, the
starting time for evaluating the output sequence y[n]. If
x[n] starts at n = n1 and h[n] starts at n = n2 then n = n1+n2
is a good choice.
2. Express both the sequences in terms of the index k to get
h[k] and x[k].
3. Tilt h[k] about k=0 to obtain h[-k] and shift by n to the
right if n is positive and left if n is negative to obtain h[n-
k].
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Convolution Process steps
)
(
)
(
)
( k
n
h
k
x
n
y
k

 




4. Multiply the two sequences x[k] and h[n-k]
element by element and sum the products to get
y[n].
5. Increment the index n, shift the sequence
h[n-k] from left to right by one sample and repeat
step-4.
6. Repeat step-5 until the sum of products is zero for
all remaining values of n.
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Convolution Process steps
)
(
)
(
)
( k
n
h
k
x
n
y
k

 




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79
Example.1
].
[
*
]
[
]
[
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(
n
h
n
x
n
y
find
n
h
and
n
x
Given




]
[n
h y(n)=?
0 1 2 3 4 5 6
n
0 1 2 3 4 5 6
n
-1
3 2
1 2
1
1
2
2

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Example.1
n1= 0, n2= -1  n = n1+n2= -1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x
3
1
.
3
)
1
(
)
(
)
1
( 




 



k
h
k
x
y
k
1 2 3 4 5 6 7
k
h[-1-k]
0
-2 -1
-3
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



3 2
1 2 1
1
2
2
1
1
2 2
2 1
1
2

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81
Example.1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
8
2
6
1
.
2
2
.
3
)
(
)
(
)
0
( 





 



k
h
k
x
y
k
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



3 2
1 2 1
1
2
2
n = -1+1=0
2 1
1
2
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x

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82
Example.1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
1 2 3 4 5 6 7
k
h[1-k]
0
-2 -1
8
1
.
1
2
.
2
1
.
3
)
1
(
)
(
)
1
( 




 



k
h
k
x
y
k
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



n = 0+1=1
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
2 1
1
2
3 2
1 2 1
1
2
2
2 1
1
2
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x

Dec 20, 2024
83
Example.1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
1 2 3 4 5 6 7
k
h[2-k]
0
-2 -1
12
1
.
2
2
.
1
1
.
2
2
.
3
)
2
(
)
(
)
2
( 





 



k
h
k
x
y
k
3 2
1 2 1
1
2
2
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
2 1
1
2
2 1
1
2
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



n = 1+1=2
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x

Dec 20, 2024
84
Example.1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
h[3-k]
1 2 3 4 5 6 7
k
0
-2 -1
9
2
.
2
1
.
1
2
.
2
)
3
(
)
(
)
3
( 




 



k
h
k
x
y
k
3 2
1 2 1
1
2
2
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
2 1
1
2
2 1
1
2
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



n = 2+1=3
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x

Dec 20, 2024
85
Example.1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
h[4-k]
1 2 3 4 5 6 7
k
0
-2 -1
4
1
.
2
2
.
1
)
4
(
)
(
)
4
( 



 



k
h
k
x
y
k
3 2
1 2 1
1
2
2
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
2 1
1
2
2 1
1
2
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



n = 3+1=4
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x

Dec 20, 2024
86
Example.1
0 1 2 3 4 5 6
k
x[k]
-1 0 1 2 3 4 5
k
h[k]
h[5-k]
1 2 3 4 5 6 7
k
0
-2 -1
4
2
.
2
)
5
(
)
(
)
5
( 


 



k
h
k
x
y
k
3 2
1 2 1
1
2
2
1 2 3 4 5 6 7
k
h[-k]
0
-2 -1
2 1
1
2
2 1
1
2
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



n = 4+1=5 n1= 3, n2= 2  n = n1+n2= 5
}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x

Dec 20, 2024
87
Example.1
0 1 2 3 4 5 6
n
y[n
]
-1
)
(
)
(
)
( k
n
h
k
x
n
y
k

 



}
2
,
1
,
2
,
1
{
]
[
}
2
,
1
,
2
,
3
{
)
(


 n
h
and
n
x
3
8 8
12
9
4 4
}
4
,
4
,
9
,
12
,
8
,
8
,
3
{
)
(


n
y

Stability and Causality
Linear Time Invariant systems

Dec 20, 2024
90
Stability
 Stable systems --- every bounded input produce a
bounded output (BIBO)
 Necessary and sufficient condition for a BIBO is







k
k
h
S |
)
(
|

Dec 20, 2024
91
Causality
 Causal systems --- output for y(n0) depends only on
x(n) with n n0.
 A causal system whose impulse response h(n) satisfies
0
for
0
)
( 
 n
n
h

Dec 20, 2024
92
Example:
 Show that the linear shift-invariant system with
impulse response h(n)=an
u(n) where |a|<1 is
stable.





 





 1
1
|
)
(
|
0
k
k
k a
a
k
h
S

Dec 20, 2024
93
Example:
(
)
( d
n
n
x
n
y 

Accumulator 



n
k
k
x
n
y )
(
)
(
Moving Average 






2
1
)
(
1
1
)
(
2
1
M
M
k
k
n
x
M
M
n
y
T [ ]
x(n
)
y(n)=T[x(n)
]
 Check the following systems for stability and
causality

Dec 20, 2024
94
Example:
Squarer  2
]
[
)
( n
x
n
y 
Forward difference ]
[
]
1
[
)
( n
x
n
x
n
y 


Compressor eger
ve
is
M
n
Mn
x
n
y int
,
],
[
)
( 






T [ ]
x(n
)
y(n)=T[x(n)
]
 Check the following systems for stability and
causality
Backward difference ]
1
[
]
[
)
( 

 n
x
n
x
n
y

Solution
Select x1[n]=x2[n] for n≤k
x1[n]≠x2[n] for n>k
Find T{x1[n]}=x1[n-nd] and T{x2[n]}=x2[n-nd]
At n=k T{x1[n]}= T{x2[n]} if nd is positive
So system is causal for nd≥0
System is noncausal for nd<0
If x[n]<∞ .i.e. M, bounded input y[n] also bounded .
So system is stable.
Dec 20, 2024
95
Ideal Delay System
)
(
)
( d
n
n
x
n
y 


Solution
Select x1[n]=x2[n] for n≤k x1[n]≠x2[n] for n>k
T{x1[n]}= and
T{x2[n]}=
T{x1[n]}= T{x2[n]} if M1≤0 and M2>-M1
Then the system is causal otherwise the system is noncausal
Dec 20, 2024
96
Moving Average system







2
1
)
(
1
1
)
(
2
1
M
M
k
k
n
x
M
M
n
y






2
1
)
(
1
1
1
2
1
M
M
k
k
n
x
M
M






2
1
)
(
1
1
2
2
1
M
M
k
k
n
x
M
M

Solutions
If x[n]<∞ .i.e. M bounded input then output y[n] also
bounded. So the system is stable.
Dec 20, 2024
97
Moving Average







2
1
)
(
1
1
)
(
2
1
M
M
k
k
n
x
M
M
n
y
M
M
M
M
M
M
M
M
M
n
y
then
M
n
x
If
M
M
k













2
1
1
)
1
(
1
1
]
[
]
[
2
1
2
1
2
1

Dec 20, 2024
98
Example.2
0 1 2 3 4 5 6
)
(
)
(
)
( N
n
u
n
u
n
x 








0
0
0
)
(
n
n
a
n
h
n
y(n)=?
0 1 2 3 4 5 6 n
n

Dec 20, 2024
99
Example
)
(
)
(
)
(
*
)
(
)
( k
n
h
k
x
n
h
n
x
n
y
k


 



0 1 2 3 4 5 6
k
x(k)
0 1 2 3 4 5 6
k
h(k)
0 1 2 3 4 5 6
k
h(0k)

Dec 20, 2024
10
Example
)
(
)
(
)
(
*
)
(
)
( k
n
h
k
x
n
h
n
x
n
y
k


 



0 1 2 3 4 5 6
k
x(k)
0 1 2 3 4 5 6
k
h(0k)
0 1 2 3 4 5 6
k
h(1k)
compute y(0)
compute y(1)
How to computer y(n)?

Dec 20, 2024
10
Example
)
(
)
(
)
(
*
)
(
)
( k
n
h
k
x
n
h
n
x
n
y
k


 



0 1 2 3 4 5 6
k
x(k)
0 1 2 3 4 5 6
k
h(0k)
0 1 2 3 4 5 6
k
h(1k)
compute y(0)
compute y(1)
How to computer y(n)?
Two conditions have to be considered.
n<N and nN.

Dec 20, 2024
10
Example
)
(
)
(
)
(
*
)
(
)
( k
n
h
k
x
n
h
n
x
n
y
k


 



1
1
1
)
1
(
0
0 1
1
1
)
( 















 
 a
a
a
a
a
a
a
a
a
n
y
n
n
n
n
k
k
n
n
k
k
n
n < N
n  N
1
1
1
0
1
0 1
1
1
)
( 
















 
 a
a
a
a
a
a
a
a
a
n
y
N
n
n
N
n
N
k
k
n
N
k
k
n

Dec 20, 2024
10
Example
)
(
)
(
)
(
*
)
(
)
( k
n
h
k
x
n
h
n
x
n
y
k


 



1
1
1
)
1
(
0
0 1
1
1
)
( 















 
 a
a
a
a
a
a
a
a
a
n
y
n
n
n
n
k
k
n
n
k
k
n
n < N
n  N
1
1
1
0
1
0 1
1
1
)
( 
















 
 a
a
a
a
a
a
a
a
a
n
y
N
n
n
N
n
N
k
k
n
N
k
k
n
0
2
4
6
0 5 10 15 20 25 30 35 40 45 50

Dec 20, 2024
10
Impulse Response of
the Ideal Delay System
(
)
( d
n
n
x
n
y 

)
(
)
( d
n
n
n
h 


By letting x(n)=(n) and y(n)=h(n),
(n nd)
0 1 2 3 4 5 6 nd
n
n

Dec 20, 2024
10
Impulse Response of
the Ideal Delay System
(n nd)
0 1 2 3 4 5 6 nd
)
(
)
(
*
)
( d
d n
n
x
n
n
n
x 



(n nd) ： Shifts input by nd
and gives that as output
n n

Dec 20, 2024
10
Impulse Response of
the Moving Average
Moving Average 






2
1
)
(
1
1
)
(
2
1
M
M
k
k
n
x
M
M
n
y







2
1
)
(
1
1
)
(
2
1
M
M
k
k
n
M
M
n
h 











otherwise
M
n
M
for
M
M
n
h
0
1
1
)
( 2
1
2
1
M1  0  M2
. . . . . .
Sequence of (n k)
n

Dec 20, 2024
10
Impulse Response of
the Accumulator
)
(
)
(
)
( n
u
k
n
h
n
k


 


Unit-step function
Accumulator 



n
k
k
x
n
y )
(
)
(
0
. . .
n

Solutions
Select x1[n]=x2[n] for n≤k x1[n]≠x2[n] for n>k
T{x1[n]}= and T{x2[n]}=
T{x1[n]}= T{x2[n]} so the system is causal.
So the system is unstable.
Dec 20, 2024
10
Accumulator 



n
k
k
x
n
y )
(
)
(



n
k
k
x )
(
1 


n
k
k
x )
(
2







n
k
M
n
y
then
M
n
x
If ]
[
]
[

Linear Constant-Coefficient
Difference Equations

Dec 20, 2024
11
Nth
Order Difference Equations

 




M
k
k
N
k
k k
n
x
b
k
n
y
a
0
0
)
(
)
(
Examples:
(
)
( d
n
n
x
n
y 

Accumulator )
(
)
1
(
)
( n
x
n
y
n
y 


Moving Average 




M
k
k
n
x
M
n
y
0
)
(
1
1
)
(

Dec 20, 2024
11
Compute y(n)

 




M
k
k
N
k
k k
n
x
b
k
n
y
a
0
0
)
(
)
(

 






M
k
k
N
k
k
k
n
x
a
b
k
n
y
a
a
n
y
0 0
1 0
)
(
)
(
)
(

Dec 20, 2024
11
The Ideal Delay System
)
(
)
( d
n
n
x
n
y 

Delay Delay Delay
. . .
x(n) y(n)
nd sample delays
x(n) y(n)=x(n-nd)
)
(
)
( d
n
n
n
h 



Dec 20, 2024
11
The Moving Average





M
k
k
n
x
M
n
y
0
)
(
1
1
)
(





M
k
k
n
M
n
h
0
)
(
1
1
)
( 
 
)
1
(
)
(
1
1




 M
n
u
n
u
M
  )
(
*
)
1
(
)
(
1
1
n
u
M
n
n
M




 


Dec 20, 2024
11
The Moving Average





M
k
k
n
x
M
n
y
0
)
(
1
1
)
(
  )
(
*
)
1
(
)
(
1
1
)
( n
u
M
n
n
M
n
h 



 

Attenuator
1
1

M
+
M+1 sample
delay
Accumulator
system
+
_
x(n
)
y(n
)

Dec 20, 2024
11
Frequency Response
n
j
e  n
j
j
e
e
H 

)
(
)
( 
j
e
H
Eigen value
Eigen function






k
k
j
j
e
k
h
e
H 

)
(
)
(

Dec 20, 2024
11
Frequency Response






k
k
j
j
e
k
h
e
H 

)
(
)
(
)
(
)
(
)
( 

 j
I
j
R
j
e
jH
e
H
e
H 

)
(
|
)
(
|
)
(


 j
e
H
j
j
e
e
H
e
H 

magnitude
Phase

Dec 20, 2024
11
Example:
)
(
)
( d
n
n
x
n
y 
 )
(
)
( d
n
n
n
h 


d
n
j
k
n
j
d
k
n
j
j
e
e
n
k
e
k
h
e
H 
















 
 )
(
)
(
)
(
1
|
)
(
| 

j
e
H
d
j
n
e
H 


 
)
(
Magnitude
Phase

Dec 20, 2024
11
Example:
d
n
j
j
e
e
H 



)
(
)
cos(
)
( 0 


 n
A
n
x
n
j
j
n
j
j
e
e
A
e
e
A
n
x 0
0
2
2
)
( 







d
d n
j
n
j
j
n
j
n
j
j
e
e
e
A
e
e
e
A
n
y 0
0
0
0
2
2
)
( 










)
(
)
( 0
0
2
2
d
d n
n
j
j
n
n
j
j
e
e
A
e
e
A 




 



]
)
(
cos[
)
( 0 



 d
n
n
A
n
y

Dec 20, 2024
11
Periodic Nature of
Frequency Response






k
k
j
j
e
k
h
e
H 

)
(
)
(












k
jk
j
e
k
h
e
H )
2
(
)
2
(
)
(
)
(







k
jk
e
k
h )
(
)
( 
 j
e
H

,
2
,
1
,
0
)
(
)
( )
2
(



 
m
e
H
e
H m
j
j 



Dec 20, 2024
12
Moving Average





M
k
k
n
x
M
n
y
0
)
(
1
1
)
(





M
k
k
n
M
n
h
0
)
(
1
1
)
( 








k
jk
j
e
k
h
e
H )
(
)
(
h(n)
0 0 M






M
k
jk
e
M 0
1
1











 




j
M
j
e
e
M 1
1
1
1 )
1
(











 












)
(
)
(
1
1
2
/
2
/
2
/
2
/
)
1
(
2
/
)
1
(
2
/
)
1
(
j
j
j
M
j
M
j
M
j
e
e
e
e
e
e
M











 









)
(
)
(
1
1
2
/
2
/
2
/
)
1
(
2
/
)
1
(
2
/
j
j
M
j
M
j
M
j
e
e
e
e
e
M












 

)
2
/
sin(
]
2
/
)
1
(
sin[
1
1 2
/ M
e
M
M
j
n n

Dec 20, 2024
12
Moving Average












 


)
2
/
sin(
]
2
/
)
1
(
sin[
1
1
)
( 2
/ M
e
M
e
H M
j
j
)
2
/
sin(
]
2
/
)
1
(
sin[
1
1
|
)
(
|





 M
M
e
H j

introduction to digital signal processing

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