Post on 05-Apr-2018
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Chapter 2: Discrete-Time Signals and
Systems
Dr. Deepa Kundur
Texas A&M
January 17, 2008
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Some Elementary Discrete-Time Signals
1. unit sample sequence (a.k.a. Kronecker delta function):
(n) =
1, for n = 00, for n = 0
2. unit step signal:
u(n) = 1, for n 0
0,
forn 12 > 19 0 if 3n2 + 1 is an integer; undefined otherwise
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Simple Manipulation of Discrete-Time Signals
Graph of x( 32 n + 1).
n
-1
2
-2
3
-3
1
3
1 2 3 40 6 7 8 95
1211
10
-1-2-3 13 14
1
2 2
-2
-1
This signal is undefined for values ofn
that are not even integers and zero for
even integers not shown on this sketch.
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Input-Output Description of Dst-Time Systems
Discrete-timeSystem
x(n)
Discrete-time
signal
y(n)
Discrete-time
signal
input/excitation
output/response
Input-output description (exact structure of system isunknown or ignored):
y(n) = T[x(n)]
black box representation:
x(n)T
y(n)
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Classification of Discrete-Time Systems
Why is this so important?
mathematical techniques developed to analyze systemsare often contingent upon the general characteristics of
the systems being considered for a system to possess a given property, the property
must hold for every possible input to the system to disprove a property, need a single counter-example
to prove a property, need to prove for the general case
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Static vs. Dynamic Systems
Static (Memoryless): output at any time instant n
depends at most on the input sample at the same time,but not on past or future samples of the input; examples:
y(n) = ax(n) y(n) = nx(n) + bx3(n)
Dynamic (Memory): a system that is not static;examples: y(n) = x(n) 5x(n 6) y(n) =
nk= x(n k)
General description of a static system:y(n) = T[x(n), n]
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Time-invariant vs. Time-variant Systems
Time-invariant system: input-output characteristics donot change with time
a system is time-invarariant iff
x(n)T
y(n) = x(n k)T
y(n k)
for every input x(n) and every time shift k.
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Time-invariant vs. Time-variant Systems
Examples: time-invariant or not?
y(n) = A x(n)
y(n) = n x(n)
y(n) = x(n) + x2(n 2) y(n) = x(n)
y(n) = x(n + 1)
y(n) = 11x(n+2)
y(n) = e3x(n)
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Linear vs. Nonlinear Systems
Linear system: obeys superposition principle
a system is linear iff
T[a1 x1(n) + a2 x2(n)] = a1 T[x1(n)] + a2 T[x2(n)]
for any arbitrary input sequences x1(n) and x2(n), andany arbitrary constants a1 and a2
L N l S
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Linear vs. Nonlinear Systems
Examples: linear or not?
y(n) = A x(n)
y(n) = n x(n)
y(n) = x(n) + x2(n 2) y(n) = x(n)
y(n) = x(n + 1)
y(n) = 11x(n+2)
y(n) = e3x(n)
C l N l S
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Causal vs. Noncausal Systems
Causal system: output of system at any time n dependsonly on present and past inputs
a system is causal iff
y(n) = F [x(n), x(n 1), x(n 2), . . .]
for any arbitrary input sequences x1(n) and x2(n), and
any arbitrary constantsa
1 anda
2
C l N l S
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Causal vs. Noncausal Systems
Examples: causal or not?
y(n) = A x(n)
y(n) = n x(n)
y(n) = x(n) + x2(n 2) y(n) = x(n)
y(n) = x(n + 1)
y(n) = 11x(n+2)
y(n) = e3x(n)
S bl U bl S
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Stable vs. Unstable Systems
Bounded Input-Bounded output (BIBO) Stable: everybounded input produces a bounded output
a system is BIBO stable iff
|x(n)| Mx < = |y(n)| My <
for all n.
St bl U t bl S t
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Stable vs. Unstable Systems
Examples: causal or not?
y(n) = A x(n)
y(n) = n x(n)
y(n) = x(n) + x2(n 2) y(n) = x(n)
y(n) = x(n + 1)
y(n) = 11x(n+2)
y(n) = e3x(n)
Bl k Di R ti
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Block Diagram Represenation
Adder:
Constant multiplier:
Signal multiplier:
+
+
Unit delay:
Unit advance:
The Con ol tion S m
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The Convolution Sum
Recall:
x(n) =
k=
x(k)(n k)
See Figure 2.3.1 of text .
The Convolution Sum
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The Convolution Sum
Let the response of a linear time-invariant (LTI) system to the
unit sample input
(n
) beh
(n
).
(n)T
h(n)
(n k)T
h(n k)
(n k)T
h(n k)
x(k) (n k)T
x(k) h(n k)
k= x(k)(n k)
T
k= x(k)h(n k)
x(n)T
y(n)
The Convolution Sum
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The Convolution Sum
Therefore,
y(n) =
k=
x(k)h(n k) = x(n) h(n)
for any LTI system.
Properties of Convolution
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Properties of Convolution
Associative and Commutative Laws:
x(n) h(n) = h(n) x(n)
[x(n) h1(n)] h2(n) = x(n) [h1(n) h2(n)]
x(n)h (n)
1h (n)
2
h (n) * h (n)1 2 h (n) * h (n)2 1=
y(n)
x(n) h (n)2 h (n)1 y(n)
Properties of Convolution
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Properties of Convolution
Distributive Law:
x(n) [h1(n) + h2(n)] = x(n) h1(n) + x(n) h2(n)
h (n) + h (n)1 2
h (n)1
h (n)2
x(n) y(n)+
Causality and Convolution
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Causality and Convolution
For a causal system, y(n) only depends on present and pastinputs values. Therefore, for a causal system, we have:
y(n) =
k=h(k)x(n k)
=1
k=
h(k)x(n k) +k=0
h(k)x(n k)
=
k=0
h(k)x(n k)
Finite-Impulse Reponse vs Infinite Impulse
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Finite-Impulse Reponse vs. Infinite Impulse
Response
Finite impulse response (FIR):
y(n) =M1k=0
h(k)x(n k)
Infinite impulse response (IIR):
y(n) =
k=0 h(k)x(n k)
How would one realize these systems? Two classes: recursiveand nonrecursive.
Infinite Impulse Response System Realization
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Infinite Impulse Response System Realization
There is a practical and computationally efficient means ofimplementing a family of IIR systems that makes use of . . .
. . . difference equations.
Infinite Impulse Response System Realization
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Infinite Impulse Response System Realization
Consider an accumulator:
y(n) =
n
k=0
x(k) n = 0, 1, 2, . . .
Memory requirements grow with increasing n!
Infinite Impulse Response System Realization
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Infinite Impulse Response System Realization
y(n) =n
k=0
x(k)
=
n1k=0
x(k) + x(n)
= y(n 1) + x(n)
y(n) = y(n 1) + x(n)
+
recursive implementation
Linear Constant-Coefficient Difference Equations
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Linear Constant Coefficient Difference Equations
Example for a linear constant-coefficient difference
equation (LCCDE):
y(n) = y(n 1) + x(n)
Initial conditions: at rest for n < 0; i.e., y(1) = 0
General expression for Nth-order LCCDE:
N
k=0
aky(n k) =
M
k=0
bkx(n k) a0 1
Initial conditions: y(1),y(2),y(3), . . . ,y(N)
Direct Form I vs. Direct Form II Realizations
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ect o s ect o ea at o s
y(n) = N
k=1
aky(n k) +Mk=0
bkx(n k)
is equivalent to the cascade of the following systems:
v(n)output 1
=Mk=1
bk x(n k) input 1
nonrecursive
y(n)output 2
=
Nk=1
aky(n k) + v(n)input 2
recursive
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