Chapter 2 Handouts

8/2/2019 Chapter 2 Handouts

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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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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)



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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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There is a practical and computationally efficient means ofimplementing a family of IIR systems that makes use of . . .

. . . difference equations.



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Consider an accumulator:

y(n) =

n

k=0

x(k) n = 0, 1, 2, . . .

Memory requirements grow with increasing n!



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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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