Discrete Time Signals and Systems

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DISCRETE TIME SIGNALS AND SYSTEMS

BY: DR.AZURA HAMZAH

PROF.SHOZO KOMAKI PROF.NOZOMU HAMADA


DISCRETE TIME SIGNALS


Discrete-time signals:

● analog signal

– t represents any physical quantity, time in sec.

● Discrete signal: discrete-time signal

– N is integer valued, represents discrete instances in times

)(txa

)(nx

}),1(),0(),1(,{)}({)( xxxnxnx

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Discrete-time signals:

Time: defined only at discrete values of time: ,

Descriptions: sequences of real or complex

numbers ,

Note A.: it take on values in the continuous

interval ,

Note B.: sampling of analogue signals:

• sampling interval, period: ,

• sampling rate: number of samples per

second,

• sampling frequency (Hz): .

( ) ( )f nT f n

T

1/Sf T

( ) ( , ) ,f n a b for a b

t nT


SEQUENCES


Representation by a Sequence

Concerned with processing signals that are represented by sequences.

nnxx )},({

1 2

3 4 5 6 7

8 9 10 -1 -2 -3 -4 -5 -6 -7 -8

n

x(n)


Important Sequences

● Unit-sample sequence (n)

1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8

n

(n)

a discrete-time impulse; or

an impulse ,0,0,1,0,0,0,0

0,1)(

n

nn


Important Sequences

● Unit-step sequence u(n)

1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8

n

u(n)

Fact:

)1()()( nunun

,1,1,1,0,0,0,0

0,1)(

n

nnu


Important Sequences

●Real exponential sequence

nAnx )(

1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8

n

x(n)

. . .

. . .


Important Sequences

● The exponential sequence

x[n] = Aαn

– Properties:

• If A and α are real then x[n] is real.

• Moreover if 0≤α<1 & A>0, then the sequence x[n] is positive and decreasing with increasing n.

• If -1≤α≤0, then the decreasing sequence values alternate in sign.


Important Sequences

●Sinusoidal sequence

)cos()( 0 nAnx

n

x(n)


– The sinusoidal sequence

x[n] = A cos(ωn+Φ)

• ω frequency of the sequence (in radians).

• A is magnitude of the sequence.

• Φ is the phase offset (in radians).

Important Sequences


Important Sequences

●Complex exponential sequence

njenx )( 0)(


Important Sequences

● A sequence x(n) is defined to be periodic with period N if

NNnxnx allfor )()(

Example: consider njenx 0)(

)()( 0000 )( Nnxeeeenx njNjNnjnj

kN 20 0

2

kN

0

2

must be a rational

number


Energy of a Sequence

● Energy of a sequence is defined by

|)(| 2

n

n

nxE


1

0

2|)(|1 N

n

x nxN

P

Power of a Sequence

● Power of a sequence is defined by


Operations on Sequences

1)Sum

2)Product

3)Multiplication

)}()({ nynxyx

)}()({ nynxyx

)}({ nxx


4) Time-Shift

● Delay by n0 samples

– n→n-n0

● Advance by n0 samples

– n→n+n0

● Example: δ[n] δ[n-2]

)()( 0nnxny


● Any sequence (digital signal) can be expressed as a weighted sum of unit sample shifted in time; e.g.,:

● In General:

● On the other hand:

...]3[]2[]1[

][

]1[]2[]3[...][

321

0

123

nanana

na

nanananp

k

knkxnx ][][][

])[(][ nxTny


Sequence Representation Using delay unit

k

knkxnx )()()(

1

2

3 4 5 6

7

8 9 10 -1 -2 -3 -4 -5 -6 -7 -8

n

x(n)

a1

a2 a7

a-3

)7()3()1()3()( 7213 nananananx


0 0 , an integern n n n Time shifting


5) Time Reversal

● Time reversal corresponds to reflection of the signal along n=0, the time axis (n → -n)

● Time folding


6) Time Scaling

x[n]=u[n] n=0 y[0]=x[0]=1 y[n]=x[n/2] n=1 y[1]=x[1/2] – undefined y[1]=0 n=2 y[2]=x[1]=1 …

x[n] y[n]

0 0

1a t t t t

a


Scaling; Signal Compression

n Kn K an integer > 1


CLASSIFICATION OF SYSTEMS


Types of Systems

● Causal & Non-causal

● Linear & Non Linear

● Time Variant &Time-invariant

● Stable & Unstable

● Static & Dynamic

● Recursive & Non-recursive Systems


1) Causal vs. Non-causal Systems.

Definition.

A system is said to be causal if the output of the system at any time

n (i.e., y(n)) depends only on present and past inputs (i.e., x(n), x(n-

1), x(n-2), … ). Output does not depend on the future values of the

input. In mathematical terms, the output of a causal system satisfies

an equation of the form

where is some arbitrary function. If a system does not satisfy

this definition, it is called non-causal.

( ) ( ), ( 1), ( 2),y n F x n x n x n

[.]F


28

Examples:

The causal system:

The non-causal system:

0

( ) ( ) ( )N

k

y n h k x n k

2( ) ( ) ( )y n x n bx n k

3( ) ( 1) ( 1)y n nx n bx n 10

10

( ) ( ) ( )k

y n h k x n k


2) Linear vs. Non-linear Systems.

A discrete-time system is called linear if only if it satisfies the linear

superposition principle. In the other case, the system is called non-

linear.

Definition. A relaxed system is linear if only if

for any arbitrary input sequences and , and any

arbitrary constants and .

The multiplicative (scaling) property of a linear system:

The additivity property of a linear system:

[.]H

1 1 2 2 1 1 2 2( ) ( ) ( ) ( )H a x n a x n a H x n a H x n

1( )x n2( )x n

1a 2a

1 1 1 1( ) ( )H a x n a H x n

1 2 1 2( ) ( ) ( ) ( )H x n x n H x n H x n


30

Examples:

The linear systems:

The non-linear systems:

0

( ) ( ) ( )N

k

y n h k x n k

2( ) ( ) ( )y n x n bx n k

3( ) ( ) ( 1)y n nx n bx n 0

( ) ( ) ( ) ( 1)N

k

y n h k x n k x n k


3) Time-Invariant vs. Time-Variant Systems.

A discrete-time system is called time-invariant if its input-output

characteristics do not change with time. In the other case, the

system is called time-variant.

Definition. A relaxed system is time- or shift-invariant if

only if

implies that

for every input signal and every time shift k .

[.]H

( ) ( )Hx n y n

( ) ( )Hx n k y n k

( )x n

( ) ( )y n H x n

( ) ( )y n k H x n k


Examples:

The time-invariant systems:

The time-variable systems:

3( ) ( ) ( )y n x n bx n

0

( ) ( ) ( )N

k

y n h k x n k

3( ) ( ) ( 1)y n nx n bx n

0

( ) ( ) ( )N

N n

k

y n h k x n k

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4) Stable vs. Unstable of Systems.

An arbitrary relaxed system is said to be bounded input - bounded

output (BIBO) stable if and only if every bounded input produces

the bounded output. It means, that there exist some finite numbers

say and , such that

for all n. If for some bounded input sequence x(n) , the output y(n)

is unbounded (infinite), the system is classified as unstable.

xM yM

( ) ( )x yx n M y n M


34

Examples:

The stable systems:

The unstable system:

0

( ) ( ) ( )N

k

y n h k x n k

2( ) ( ) 3 ( )y n x n x n k

3( ) 3 ( 1)ny n x n


5) Static vs. Dynamic Systems.

A discrete-time system is called static or memoryless if its output

at any time instant n depends on the input sample at the same time,

but not on the past or future samples of the input. In the other case,

the system is said to be dynamic or to have memory.

If the output of a system at time n is completely determined by the

input samples in the interval from n-N to n ( ), the system is

said to have memory of duration N.

If , the system is static or memoryless.

If , the system is said to have finite memory.

If , the system is said to have infinite memory.

0N

0N

0 N

N


Examples:

The static (memoryless) systems:

The dynamic systems with finite memory:

The dynamic system with infinite memory:

3( ) ( ) ( )y n nx n bx n

0

( ) ( ) ( )N

k

y n h k x n k

0

( ) ( ) ( )k

y n h k x n k


1. 6) Recursive vs. Non-recursive Systems.

A system whose output y(n) at time n depends on any number of the

past outputs values ( e.g. y(n-1), y(n-2), … ), is called a recursive

system. Then, the output of a causal recursive system can be

expressed in general as

where F[.] is some arbitrary function. In contrast, if y(n) at time n

depends only on the present and past inputs

then such a system is called nonrecursive.

( ) ( 1), ( 2), , ( ), ( ), ( 1), , ( )y n F y n y n y n N x n x n x n M

( ) ( ), ( 1), , ( )y n F x n x n x n M


38

Examples:

The nonrecursive system:

The recursive system:

0

( ) ( ) ( )N

k

y n h k x n k

0 1

( ) ( ) ( ) ( ) ( )N N

k k

y n b k x n k a k y n k


DISCRETE TIME SYSTEMS


Discrete-Time System

● A discrete-time system is a device or algorithm that operates on a discrete-time signal called the input or excitation (e.g. x(n)), according to some rule (e.g. T{.}) to produce another discrete-time signal called the output or response (e.g. y(n)).

● A discrete–time system can be thought of as a transformation T(x) of an input sequence to an output sequence:

y[n] = T{x[n]}

T{●} x[n] y[n]



● If T{x} is restricted to have properties of

– Linearity, and

– Time invariance,

Then the system is referred to as linear time-invariant (LTI) system.



● Definition of LTI systems.

– x1[n] and x2[n] inputs to a discrete-time system.

– a & b, arbitrary constants, then

– The system is linear if and only if:

T{ax1[n] + bx2[n]} = aT{x1[n]} + bT{x2[n]}

Principle of superposition.


Principle of Superposition.

T{ax1[n] + bx2[n]} = aT{x1[n]} + bT{x2[n]}

x1

x2

a

b

T{} y= T{ax1[n] + bx2[n]} ax1[n] + bx2[n]

x1

x2

T{}

T{}

a

b

bT{x2[n]}

y= aT{x1[n]}+bT{x2[n]}

aT{x1[n]}


Properties of LTI Systems .

h1[n] h2[n] x[n] y[n]

h2[n] h1[n] x[n] y[n]

h1[n]*h2[n] x[n] y[n]

h2[n]*h1[n] x[n] y[n]

h1[n]

h2[n]

x[n] y[n]

h1[n]+h2[n] x[n] y[n]

h2[n]+h1[n] x[n] y[n]


CONVOLUTION


– LTI system is completely characterized by its impulse response h[n]:

• Impulse response is defined as the system’s response to a unit sample (or impulse).

• “*” denotes the convolution operator.

][*][][

][][][

nhnxny

knhkxnyk

Convolution


.H

LTI system

unit impulse

( )n ( ) ( )h n H n

impulse response

LTI system description by convolution (convolution sum):

( ) ( ) ( ) ( ) ( ) ( )* ( ) ( )* ( )k k

y n h k x n k x k h n k h n x n x n h n

Impulse Response and Convolution


● By applying Superposition Principle and Linear Time Invariance we can derive expression that defines convolution:

k k

kk

knhkxknTkxny

knkxTknkxTnxTny

][][])[(][][

][][)][][(])[(][

Convolution


Convolution Sum

T [ ]

(n) h(n)

x(n) y(n)

)(*)()()()( nhnxknhkxnyk

convolution A linear shift-invariant system is completely

characterized by its impulse response.


Properties of Convolution Math

)(*)()()()( nhnxknhkxnyk

)(*)()()()( nxnhknxkhnyk

)(*)()(*)( nxnhnhnx



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 systems are identical.



h1(n)+h2(n) x(n) y(n)

These two systems are identical.

h1(n)

x(n)

h2(n)

y(n) +


Properties of Convolution

Commutative Property

][*][][*][ nxnhnhnx

Distributive Property

])[*][(])[*][(

])[][(*][

21

21

nhnxnhnx

nhnhnx

Associative Property

][*])[*][(

][*])[*][(

][*][*][

12

21

21

nhnhnx

nhnhnx

nhnhnx

http://stellar.mit.edu/S/course/6/sp08/6.003/courseMaterial/topics/topic1/lectureNotes/Lecture__3/Lecture__3.pdf




Example

0 1 2 3 4 5 6

)()()( Nnununx

00

0)(

n

nanh

n

y(n)=? 0 1 2 3 4 5 6


Example

)()()(*)()( knhkxnhnxnyk

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(0k)


Example


0 1 2 3 4 5 6

k x(k)

0 1 2 3 4 5 6

k h(0k)

0 1 2 3 4 5 6

k h(1k)

compute y(0)

compute y(1)

How to compute y(n)?


Example


0 1 2 3 4 5 6

k x(k)

0 1 2 3 4 5 6

k h(0k)

0 1 2 3 4 5 6

k h(1k)

compute y(0)

compute y(1)

How to computer y(n)?

Two conditions have to be considered.

n<N and nN.


Example


1

1

1

)1(

00 11

1)(

a

aa

a

aaaaany

nnn

n

k

knn

k

kn

n < N

n N

11

1

0

1

0 11

1)(

a

aa

a

aaaaany

NnnNn

N

k

knN

k

kn

Discrete Time Signals and Systems

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