itroduction to discrete time signals

8/12/2019 Itroduction to Discrete Time Signals

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Digital Signal Processing

B. Discrete time signals

Athanassios C. Iossifides

October 2012


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Athanassios Iossifides DIGITAL SIGNAL PROCESSING 2

.1 Representation of discrete time signals

.2 Fundamental discrete time signals

.3 Transformations on the independent variable

.4 Transformations on the dependent variable.5 Signal classification

.6 Signal analysis and synthesis with impulses

.7 Signal correlation

.8 Examples

. Discrete time signals


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.1 Representation of discrete

time signals



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.1 Representation of discrete time signals

Discrete time signals are functions (sequences) defined only for

integer values of the independent variable.

Functional representation

1, 1

5, 2

( ) 2, 3

0,

n

n

x n n

otherwise


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.2 Fundamental discrete time

signals



7/37Athanassios Iossifides DIGITAL SIGNAL PROCESSING 7


Impulse (delta) or unit sample sequence

Unit step sequence

Unit ramp sequence

1, 0( )

0, 0

n n

n

1, 0( )

0, 0

nu n

n

, 0( )

0, 0

n nr n

n




Exponential signals

a real number ( ) nx n a

0 1a

1a




Exponential signals

a real number ( ) nx n a

1 0a

1a




Exponential signal

a complex number

( )

( )

(cos sin )

( ) ( )

n

j

n jn

n

R I

x n a

a re

x n r e

r n j n

x n jx n

Real part

Imaginary part




Exponential signal

a complex number

( )

( )

( ) ( )

( )

( )

n

j

n jn

n

x n a

a re

x n r e

x n n

x n r

n n

Magnitude

Angle



.3 Transformations on the

independent variable





Shift

( ) ( )x n x n

( ) ( )x n x n

0

0

0

0

n delay

n precedence

( )x n

( 3)x n

( 2)x n

0n n n




Folding (reflection)

n n

( )x n

( )x n




Down sampling

The resulting sequence corresponds to the one that we would have if the

sampling frequency was M times smaller.

n Mn

( )x n

( )x n

(2 )x n



.4 Transformations on the

dependent variable (signal

operations)




.4 Signal operations

Addition, multiplication and scaling

AdditionMultiplication

Scaling

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

2( ) ( ),y n cx n n

1 2( ) ( ) ( )y n x n x n

1 2( ) ( ) ( )y n x n x n



.5 Signal classification



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Energy signals and power signals

Signal energy

Mean signal power

When the energy of a signal is finite (0 < < ), then the signal is called

energy signal.

When the mean power of a signal is finite (0 < P < ), then the signal is called

power signal.

A signal may be either

a power signal

or an energy signal

or none of the two.

2( )

n

E x n

21 1lim ( ) lim

2 1 2 1

N

NN N

n N

P x n E N N


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Periodic signals

A signal is periodic with fundamental period(> 0) if and only if

Let x1(n) x2(n) discrete time signals of period 1and 2respectively.

The signals

are periodic with period

where gcd(N1,N2) is the greatest common divisor of 1and 2.

Periodic signals are power signals (when they dont have infinite values in the duration

of one period). The mean power of a periodic signal can be calculated by

( ) ( ),x n N x n n

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

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

1 2

1 2gcd( , )

N NN

N N

12

0

1( )

N

n

P x nN


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Even and odd signals

A signal is said to be evenwhenA signal is said to be oddwhen

Every real valued signal can be written (decomposed) as a sum of an even and

an odd signal, that is

where

( ) ( )x n x n

( ) ( ) ( )e ox n x n x n 1

( ) [ ( ) ( )]2

1( ) [ ( ) ( )]

2

e

o

x n x n x n

x n x n x n

( ) ( )x n x n


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Even and odd signals

( ) ( ) ( 10)x n u n u n

1( ) [ ( ) ( )]

2ex n x n x n

1( ) [ ( ) ( )]

2ox n x n x n


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.6 Signal analysis and synthesis

with impulses



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Letx(n) a discrete time signal.

(n-k) is zero everywhere except at shift (position) n =k.

The productx(n)(n-k) equals to the value ofx(n) at the position n= k, that is

The above is true for any shift k. Repeating this procedure for all k we can

reproducex(n) as a sum of weighted unit sample sequences (impulses), as

follows

( ) ( ) ( ) ( )x n n k x k n k

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

k k

k k

k

x n n k x k n k

x n n k x k n k

x n x k n k


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.7 Correlation



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Auto correlation

Letx(n) a discrete time, real valued energy signal. The autocorrelationfunction is defined as

Auto correlation gives us information about the similarity of a signal and its

shifted (delayed) replicas.

For periodic signals

( ) ( ) ( ), 1, 2, 3,...xxn

r l x n x n l l

2

(0) ( )xx nr x n E

1

0

1

0

1( ) ( ) ( ),

1( ) ( ) ( ),

N

xx

n

N

xy

n

r l x n x n l N the period N

r l x n y n l N the period N


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Calculation of autocorrelation

( 6) ( ) ( 6)xxn

r x n x n

(0)xxr E

(2) ( ) ( 2)xxn

r x n x n


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Cross correlation

Letx(n) and y(n) two real valued discrete time signals. The cross correlationfunction of the two signals is defined as

or

Cross correlation informs us about the similarity of two shifted signals

between each other.

When two sequences that are synchronized (l 0) have a cross correlation

value equal to zero, then these sequences are calledorthogonal

( ) ( ) ( ), 1, 2, 3,...xyn

r l x n y n l l

( ) ( ) ( ) ( ), 1, 2, 3,...yx xy n

r l x n l y n r l l

(0) ( ) ( ) 0xyn

r x n y n


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Signal correlation applications

Auto and cross correlation between signals are used in several applications oftelecommunications:

Distance estimation (.. radar)

Signal synchronization (Barker, Golay sequences, etc)

Spread spectrum signals for multipath mitigation.

CDMA systems (PN, Gold sequences, etc.)

In these applications specific sequences that present very low auto and cross

correlation values are used, so that the sequences and their shifted versions

can be easily separated (discriminated) among each other.


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.8 Examples

Signal representation Proakis 2.1(a,)

Representation of the above signal with the use of unit sample and unit step

signals

1 , 3 13

( ) 1, 0 3

0,

nn

x n n

1 2( ) {...0, , ,1,1,1,1,0...}

3 3x n

1 2( ) ( 2) ( 1) ( ) ( 4)

3 3x n n n u n u n


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.8 Examples

Signal transformations Proakis 2.2(a), 2.2()

Find y(n) x(4 n)

First I shift the signal and then fold it.

1 1( ) {...,0,1,1,1,1, , ,0,...}

2 2x n

( ) (4 )

1 1

{...,0, , ,1,1,1,1,0,...}2 2

y n x n


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.8 Examples

Signal transformations Proakis 2.2()

Find

First I cratex(n-1)

1 1( ) {...,0,1,1,1,1, , ,0,...}2 2

x n

( ) ( 1) ( 3)y n x n n

( 3) {...,0,0,0,0,0,1,0,0,...} n

1 1( 1) {...,0,0,1,1,1,1, , ,0,...}

2 2x n

( 1) ( 3) {...,0,0,0,0,0,1,0,0,...}x n n


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.8 Examples

Are these periodic signals?

( ) cos( / 8)cos( / 8)x n n n

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

( ) cos( / 2) sin( / 8)x n n n


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Autocorrelation application to distance estimation

Transmitted signalx(n)barker code

(very good autocorrelation

properties)

Received signal afterattenuation and noise

addition

Ddelay

w(n)Gaussian noise

Correlation of x(n)

and y(n)

Distance D estimation

.8 Examples

( ) ( ) ( )y n ax n D w n

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