chapter 2 discrete-time signals and systems. §2.1.1 discrete-time signals: time-domain...

Chapter 2

Discrete-Time Discrete-Time Signals and SystemsSignals and Systems

§2.1.1 Discrete-Time Signals:Time-Domain Representation

Signals represented as sequences of numbers, called samples

Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range - n

x[n] defined only for integer values of n and undefined for noninteger values of n

Discrete-time signal represented by {x[n]}


Discrete-time signal may also be written as a sequence of numbers inside braces:

{x[n]}={…,-0.2,2.2,1.1,0.2,-3.7,2.9,…}

In the above, x[-1]= -0.2, x[0]=2.2, x[1]=1.1, et

c. The arrow is placed under the sample at tim

e index n = 0


Graphical representation of a discrete-time signal with real-valued samples is as shown below:


In some applications, a discrete-time sequence {x[n]} may be generated by periodically sampling a continuous-time signal xa(t) at uniform intervals of time


Here, n-th sample is given by

x[n]=xa(t) |t=nT=xa(nT), n=…,-2,-1,0,1,… The spacing T between two consecutive sam

ples is called the sampling interval or sampling period

Reciprocal of sampling interval T, denoted as FT , is called the sampling frequency:

TFT

1

TFT

1


Unit of sampling frequency is cycles per second, or hertz (Hz) , if T is in seconds

Whether or not the sequence {x[n]} has been obtained by sampling, the quantity x[n] is called the n-th sample of the sequence

{x[n]} is a real sequence, if the n-th sample x[n] is real for all values of n

Otherwise, {x[n]} is a complex sequence


A complex sequence {x[n]} can be written as {x[n]}={xre[n]}+j{xim[n]} where xre and xim are the real and imaginary parts of x[n]

The complex conjugate sequence of {x[n]} is given by {x*[n]}={xre[n]}-j{xim[n]}

Often the braces are ignored to denote a sequence if there is no ambiguity


Example - {x[n]}={cos0.25n} is a real sequence

{y[n]}={ej0.3n} is a complex sequence We can write {y[n]}={cos0.3n + jsin0.3n} ={cos0.3n} + j{sin0.3n}

where {yre[n]}={cos0.3n}

{yim[n]}={sin0.3n}


Example –

{w[n]}={cos0.3n}- j{sin0.3n}={e-j0.3n} is the complex conjugate sequence of {y[n]}

That is,

{w[n]}= {y*[n]}


Two types of discrete-time signals: - Sampled-data signals in which samples are continuous-valued- Digital signals in which samples are discrete-valued

Signals in a practical digital signal processing system are digital signals obtained by quantizing the sample values either by rounding or truncation


Example –

Am

pli

tud

eDigital signal

Am

pli

tud

e

Boxedcar signal

Time,t Time,t


A discrete-time signal may be a finite-length or an infinite-length sequence

Finite-length (also called finite-duration or finite-extent) sequence is defined only for a finite time interval: N1 n N2

where - < N1 and N2 < with N1 N2

Length or duration of the above finite-length sequence is N= N2 - N1+ 1


Example – x[n]=n2, -3 n 4 is a finite-length sequence of length 4 -(-3)+1=8

y[n]=cos0.4n is an infinite-length sequence


A length- N sequence is often referred to as an N-point sequence

The length of a finite-length sequence can be increased by zero-padding, i.e., by appending it with zeros


Example –

is a finite-length sequence of length 12 obtained by zero-padding x[n] =n2, -3≤n≤4 with 4 zero-valued samples

85,0

43,][

2

n

nnnxe


A right-sided sequence x[n] has zero-valued samples for n < N1

nN1

A right-sided sequence

If N1 0, a right-sided sequence is called a causal sequence


A left-sided sequence x[n] has zero-valued samples for n ＞ N2

If N2≤0, a left-sided sequence is called a anti-causal sequence

N 2

n

A left-sided sequence


Size of a Signal

Given by the norm of the signal

Lp - norm

where p is a positive integer

p

n

p

pnxx

/1

][


The value of p is typically 1 or 2 or ∞

L2 –norm

2x

is the root-mean-squared (rms) value of {x[n]}


is the peak absolute value of {x[n]}

is the peak absolute value of {x[n]}, i.e.

1xL1 - norm

xL∞ - norm

maxxx


Example – Let {y[n]}, 0≤n≤N-1, be an approximation of {x

[n]}, 0≤n≤N-1 An estimate of the relative error is given by the

ratio of the L2 -norm of the difference signal and the L2 -norm of {x[n]}:

p

N

n

N

nrel

nx

nxnyE

/1

1

0

2

1

0

2

][

][][

§2.1.2 Operations on Sequences

A single-input, single-output discrete-time system operates on a sequence, called the input sequence, according some prescribed rules and develops another sequence, called the output sequence, with more desirable properties

x[n] y[n]

Input sequence Output sequence

Discrete-timesystem

§2.1.2 Operations on Sequences

For example, the input may be a signal corrupted with additive noise

Discrete-time system is designed to generate an output by removing the noise component from the input

In most cases, the operation defining a particular discrete-time system is composed of some basic operations

§2.2.1 Basic Operations Product (modulation) operation:

x[n] y[n]

w[n]y[n]=x[n].w[n]-Modulator

An application is in forming a finite-length sequence from an infinite-length sequence by multiplying the latter with a finite-length sequence called an window sequence

Process called windowing

§2.2.1 Basic Operations Addition operation:

Ax[n] y[n] y[n]=A.x[n]–Multiplier

Multiplication operation

y[n]=x[n]+w[n]–Adderx[n] y[n]

w[n]

§2.2.1 Basic Operations Time-shifting operation: y[n]=x[n-N] where N

is an integer If N>0, it is delaying operation

1z y[n]x[n] y[n]=x[n-1]

–Unit delay

y[n]x[n] z y[n]=x[n-1]

–Unit advance

If N<0, it is an advance operation

§2.2.1 Basic Operations

Time-reversal (folding) operation:

y[n]=x[n-1] Branching operation: Used to provide multipl

e copies of a sequence

x[n] x[n]

x[n]


Example - Consider the two following sequences of length 5 defined for 0n4 :

{a[n]}={3 4 6 –9 0}

{b[n]}={2 –1 4 5 –3} New sequences generated from the above t

wo sequences by applying the basic operations are as follows:


{c[n]}={a[n].b[n]}={6 –4 24 –45 0}

{d[n]}={a[n]+b[n]}={5 3 10 –4 -3}

{e[n]}=(3/2){a[n]}={4.5 6 9 –13.5 0} As pointed out by the above example, opera

tions on two or more sequences can be carried out if all sequences involved are of same length and defined for the same range of the time index n


However if the sequences are not of same length, in some situations, this problem can be circumvented by appending zero-valued samples to the sequence(s) of smaller lengths to make all sequences have the same range of the time index

Example - Consider the sequence of length 3 defined for 0n 2: {f[n]}={-2, 1, -3}


We cannot add the length-3 sequence to the length-5 sequence {a[n]} defined earlier

We therefore first append {f[n]} with 2 zero-valued samples resulting in a length-5 sequence {fe[n]}={-2 1 –3 0 0}

Then

{g[n]}={a[n]}+{f[n]}={1 5 3 –9 0}


Ensemble Averaging A very simple application of the addition ope

ration in improving the quality of measured data corrupted by an additive random noise

In some cases, actual uncorrupted data vector s remains essentially the same from one measurement to next


While the additive noise vector is random and not reproducible

Let di denote the noise vector corrupting the i-th measurement of the uncorrupted date vector s:

xi=s+di


The average data vector, called the ensemble average, obtained after K measurements is given by

For large values of K, xave is usually reasonable replica of the desired data vector s

k

ii

k

ii

k

iiave d

ksds

kx

kx

111

1)(

11

§2.2.1 Basic Operations Example


We cannot add the length-3 sequence {f[n]} to the length-5 sequence {a[n]} defined earlier

We therefore first append {f[n]} with 2 zero-valued samples resulting in a length-5 sequence {fe[n]}={−2 1 -3 0 0}

Then

{g[n]}= {g[n]}+{fe[n]}={1 5 3 -9 0}

§2.2.1 Combinations of Basic Operations

Example -

y[n]=1x[n]+ 2x[n-1]+ 3[n-2]+ 4x[n-3]

§2.2.2 Sampling Rate Alteration

Employed to generate a new sequence y[n] with a sampling rate F’T higher or lower than that of the sampling rate FT of a given sequence x[n]

Sampling rate alteration ratio is

R= F’T / FT If R>1, the process called interpolation If R<1, the process called decimation


In up-sampling by an integer factor L>1, L−1 equidistant zero-valued samples are inserted by the up-sampler between each two consecutive samples of the input sequence x[n]:

otherwise,0

,2,,0],/[][

LLnLnxnxu

x[n] xu[n]L


An example of the up-sampling


In down-sampling by an integer factor M>1, every M-th samples of the input sequence are kept and M-1 in-between samples areremoved:

x[n]=x[nM]

x[n] y[n]M


An example of the down-sampling operation

Classification of SequencesBased on Symmetry

Conjugate-symmetric sequence:

x[n]=-x*[n] If x[n] is real, then it is an even sequence


Conjugate-antisymmetric sequence:

x[n]=-x*[-n] If x[n] is real, then it is an odd sequence


It follows from the definition that for a conjugate-symmetric sequence {x[n]}, x[0] must be a real number

Likewise, it follows from the definition that for a conjugate anti-symmetric sequence {y[n]}, y[0] must be an imaginary number

From the above, it also follows that for an odd sequence {w[n]}, w[0]=0


Any complex sequence can be expressed as a sum of its conjugate-symmetric part and its conjugate-antisymmetric part:

x[n]=xcs[n]+ xca[n]where

xcs[n]=1/2(x[n]+x*[-n])

xca[n]=1/2(x[n]-x*[-n])


Example – Consider the length-7 sequence defined for -3≤n≤3

}3,2,65,24,32,41,0{]}[{ jjjjjng

Its conjugate sequence is then given

The time-reversed version of the above

}3,2,65,24,32,41,0{]}[*{ jjjjjng

}0,41,32,24,65,2,3{]}[*{ jjjjjng


Likewise

}5.1,5.0,5.15.1,2,5.15.1,5.0,5.1{

]}[*][{2

1]}[{

jjjjj

ngngngca

Therefore

}5.1,35.0,5.45.3,4,5.45.3,35.0,5.1{

]}[*][{2

1]}[{

jjjj

ngngngcs

It can be easily verified that

and ][][ * ngng caca ][][ * ngng cscs


Any real sequence can be expressed as a sum of its even part and its odd part:

x[n]=xev[n]+ xod[n]

where

xev[n]=1/2(x[n]+x[-n])

xod[n]=1/2(x[n]-x[-n])


A length-N sequence x[n], 0≤n≤N-1, can be expressed as x[n]=xpcs[n]+ xpca[n]

where

xpcs[n]=1/2(x[n]+x*[<-n>N]), 0≤n≤N-1,

is the periodic conjugate-symmetric part and

xpca[n]=1/2(x[n]-x*[<-n>N]), 0≤n≤N-1,

is the periodic conjugate-antisymmetric part


For a real sequence, the periodic conjugate- symmetric part, is a real sequence and is called the periodic even part xpe[n]

For a real sequence, the periodic conjugate- antisymmetric part, is a real sequence and is called the periodic odd part xpo[n]


A length-N sequence x[n] is called a periodic conjugate-symmetric sequence, if

x[n]=x*[<-n>N ]=x*[<N-n>N ] and is called a periodic conjugate-antisymmetric sequence if

x [n]=-x*[<-n>N])=- x*[<N-n>N ]


A finite-length real periodic conjugate- symmetric sequence is called a symmetric sequence

A finite-length real periodic conjugate- antisymmetric sequence is called a antisymmetric sequence


Example - Consider the length-4 sequence defined for 0≤n≤3: {u[n]}={1+j4, -2+j3, 4-j2, -5-j6}

Its conjugate sequence is given by {u*[n]}={1-j4, -2-j3, 4+j2, -5+j6}

To determine the modulo-4 time-reversed version {u*[<-n>4]} observe the following:


u*[<-0>4]=u*[0]=1-j4

u*[<-1>4]=u*[3]=5+j6

u*[<-2>4]=u*[2]=4+j2

u*[<-3>4]=u*[1]=-2-j3 Hence

{u*[<-n>4]}={1-j4, -5+j6, 4+j2, -2-j3}


Therefore

{upcs[n]}=1/2(u[n]+u*[<-n>4])

={1, -3.5+j4.5, 4, -3.5-j4.5} Likewise

{upca[n]}=1/2(u[n]-u*[<-n>4])

={j4, 1.5-j1.5, -2, -1.5-j1.5}

A sequence satisfying][~ nx ][~][~ kNnxnx


Smallest value of N satisfying is called the fundamental period

][~][~ kNnxnx

is called a periodic sequence with a period N where N is a positive integer and k is any integer

§2.2.3 Classification of Sequences based on periodicity

Example –

A sequence satisfying the periodicity condition is called an periodic sequence

§2.2.4 Classification of Sequences Energy and Power Signals

Total energy of a sequence x[n] is defined by

nnx 2

x ][ An infinite length sequence with finite sampl

e values may or may not have finite energy A finite length sequence with finite sample v

alues has finite energy


The average power of an aperiodic sequence is defined by

K

KnKK

nxP 2

121

x ][lim

K

KnKx nx 2

, ][

Define the energy of a sequence x[n] over a finite interval -K n K as


ThenKx

K KP ,x 12

1lim

The average power of an infinite-length sequence may be finite

1

0

2][~1 N

nx nx

NP

The average power of a periodic sequence

with a period N is given by][~ nx


Example –Consider the causal sequence defined by

5.412

)1(9lim19

12

1lim

0x

K

K

KP

K

K

nK

Note: x[n] has infinite energy Its average power is given by

0,0

0,)1(3][

n

nnx

n


An infinite energy signal with finite average power is called a power signal

Example - A periodic sequence which has a finite average power but infinite energy

A finite energy signal with zero average power is called an energy signal

Example - A finite-length sequence which has finite energy but zero average power

Other Types of Classiffications

A sequence x[n] is said to be bounded if

xBnx ][

13.0cos][ nnx

Example - The sequence x[n]=cos(0.3n) is a bounded sequence as

Other Types of Classiffications A sequence x[n] is said to be absolutely sum

mable if

nnx ][

00030

nnny

n

,,.][

is an absolutely summable sequence as

428571

3011

300

..

.n

n

Example - The sequence

Other Types of Classiffications

A sequence x[n] is said to be square-summable if

nnx 2][

nnnh 4.0sin][

is square-summable but not absolutely summable

Example - The sequence

§2.3 Basic Sequences

Unit sample sequence -

0,0

0,1][

n

nn

1

–4 –3 –2 –1 0 1 2 3 4 5 6n

0,0

0,1][

n

nn

–4 –3 –2 –1 0 1 2 3 4 5 6

1

n

Unit step sequence -


Real sinusoidal sequence -

x[n]=Acos(0n+)

where A is the amplitude, 0 is the angular frequency, and is the phase of x[n]

Example -

§2.3 Basic Sequences Exponential sequence -

,][ nAnx n

,)( oo je , jeAA

],[][][ )( nxjnxeeAnx imrenjj oo

),cos(][ neAnx on

reo

)sin(][ neAnx on

imo

where

then we can express

where A and are real or complex numbers

If we write

§2.3 Basic Sequences xre[n] and xim[n] of a complex exponential seque

nce are real sinusoidal sequences with constant (0=0), growing (0>0) , and decaying (0<0) amplitudes for n > 0

njnx )exp(][612

1

Real part Imaginary part


Real exponential sequence -

x[n]=An, -< n < where A and are real numbers

=1.2 =0.9

§2.3 Basic Sequences Sinusoidal sequence Acos(0n + ) and comp

lex exponential sequence Bexp(j0n) are periodic sequences of period N if 0N=2rwhere N and r are positive integers

Smallest value of N satisfying 0N=2ris the fundamental period of the sequence

To verify the above fact, consider

x1[n]= Acos(0n + )

x2[n]= Acos(0 ( n+N) + )


Now

x2[n]= cos(0 ( n+N) + )

= cos(0n+)cos0N - sin(0n+)sin0N

which will be equal to cos(0n+)=x1[n] only if sin0N= 0 and cos0N = 1

These two conditions are met if and only if 0N= 2r or 2/0 = N/r


If 2/0 is a noninteger rational number, then the period will be a multiple of 2/0

Otherwise, the sequence is aperiodic

Example - is an aperiodic sequence

)3sin(][ nnx


Here 0=0

Hence period N=2r/0=20 for r=0


Here 0=0.1

Hence N=2r/0=20 for r=1

0 = 0.1


Property 1 - Consider x[n]=exp(j1n) and y[n]=exp(j2n) with 0≤ 1< and 2k≤ 2

<2(k +1) where k is any positive integer If 2= 1+2k, then x[n]= y[n] Thus, x[n] and y[n] are indistinguishable


Property 2 - The frequency of oscillation of Acos(0n) increases as 0 increases from 0 to ,and then decreases as 0 increases from

to 2 Thus, frequencies in the neighborhood of =

0 are called low frequencies, whereas, frequencies in the neighborhood of = are called high frequences


Because of Property 1, a frequency 0 in the neighborhood =2k is indistinguishable from a frequency 0-2k in the neighborhood of ω=0 and a frequency 0 in the neighborhood of =(2k+1) is indistinguishable from a frequency 0- (2k+1) in the neighborhood of ω=π


Frequencies in the neighborhood of ω=2πk are usually called low frequencies

Frequencies in the neighborhood of ω=π(2k+1) are usually called high frequencies

v1[n]=cos(0.1πn)= cos(0.9πn) is a low frequency signal

v2[n]=cos(0.8πn)= cos(1.2πn) is a high frequency signal

§2.3 Basic Sequences An arbitrary sequence can be represented in

the time-domain as a weighted sum of some basic sequence and its delayed (advanced) versions

]6[75.0]4[

]2[]1[5.1]2[5.0][

nn

nnnnx

§2.4 The Sampling Process Often, a discrete-time sequence x[n] is devel

oped by uniformly sampling a continuous-time signal xa(t) as indicated below

The relation between the two signals is x[n] =xa(t)|t=nT=xa (nT), n=…, -2, -1, 0, 1, 2, …

§2.4 The Sampling Process

Time variable t of xa(t) is related to the time variable n of x[n] only at discrete-time instants tn given by

with FT=1/T denoting the sampling frequency and T= 2πFT denoting the sampling angular frequency

TTn

n

F

nnTt

2

§2.4 The Sampling Process Consider the continuous-time signal

)cos()2cos()( tAtfAtx oo

is the normalized digital angular frequency of x[n]

ToToo /2where

The corresponding discrete-time signal is

)cos(

)2

cos()cos(][

0

nA

nAnTAnxT

oo


If the unit of sampling period T is in seconds The unit of normalized digital angular

frequency 0 is radians/sample The unit of normalized analog angular

frequency 0 is radians/second The unit of analog frequency f0 is hertz (Hz)

§2.4 The Sampling Process The three continuous-time signals

)6cos()(1 ttg )14cos()(2 ttg )26cos()(3 ttg

)6.0cos(][1 nng )4.1cos(][2 nng )6.2cos(][3 nng

of frequencies 3Hz, 7Hz, and 13Hz, are sampled at a sampling rate of 10Hz, i.e. with T = 0.1 sec. generating the three sequences

§2.4 The Sampling Process Plots of these sequences (shown with circles)

and their parent time functions are shown below:

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

time

Am

plitu

de

Note that each sequence has exactly the same sample value for any given n


This fact can also be verified by observing that

)6.0cos()6.02(cos)4.1cos(][2 nnnng

)6.0cos()6.02(cos)6.2cos(][3 nnnng

As a result, all three sequences are identical and it is difficult to associate a unique continuous-time function with each of these sequences


The above phenomenon of a continuous-time signal of higher frequency acquiring the identity of a sinusoidal sequence of lower frequency after sampling is called aliasing


Since there are an infinite number of continuous-time signals that can lead to the same sequence when sampled periodically, additional conditions need to imposed so that the sequence {x[n]}={xa[nT]} can uniquely represent the parent continuous-time signal xa(t)

In this case, xa(t) can be fully recovered from {x[n]}

§2.4 The Sampling Process Example - Determine the discrete-time sign

al v[n] obtained by uniformly sampling at a sampling rate of 200Hz the continuous- time signal

Note: va(t) is composed of 5 sinusoidal signals of frequencies 30Hz, 150Hz, 170Hz, 250Hz and 330Hz

)660sin(10)500cos(4)340cos(2)300sin()60cos(6)(

ttttttva


The sampling period is T=1/200=0.005 sec The generated discrete-time signal v[n] is th

us given by


Note: v[n] is composed of 3 discrete-time sinusoidal signals of normalized angular frequencies: 0.3π, 0.5π, and 0.7π

)7.0sin(10)6435.05.0cos(5)3.0cos(8)7.0sin(10)5.0cos(4

)3.0cos(2)5.0sin(3)3.0cos(6))7.04sin((10))5.02cos((4

))3.02cos((2))5.02sin((3)3.0cos(6)3.3sin(10)5.2cos(4

)7.1cos(2)5.1sin(3)3.0cos(6][

nnnnn

nnnnn

nnnnn

nnnnv


Note: An identical discrete-time signal is also generated by uniformly sampling at a 200-Hz sampling rate the following continuous-time signals:

)700sin(3)460cos(6)260sin(10)100cos(4)60cos(2)(

)140sin(10)6435.0100cos(5)60cos(8)(

ttttttg

ttttw

a

a


Recall 0=20/T

Thus if T>20, then the corresponding normalized digital angular frequency 0 of the discrete-time signal obtained by sampling the parent continuous-time sinusoidal signal will be in the range -<<

Conclusion: No aliasing


On the other hand, if T < 20 , the normalized digital angular frequency will foldover into a lower digital frequency

0=(20/T)2 in the range -<< because of aliasing

Hence, to prevent aliasing, the sampling frequency T should be greater than 2 times the frequency 0 of the sinusoidal signal being sampled

§2.4 The Sampling Process Generalization: Consider an arbitrary contin

uous-time signal xa(t) composed of a weighted sum of a number of sinusoidal signals

xa(t) can be represented uniquely by its sampled version {x[n]} if the sampling frequency T is chosen to be greater than 2 times the highest frequency contained in xa(t)


The condition to be satisfied by the sampling frequency to prevent aliasing is called the sampling theorem

A formal proof of this theorem will be presented later

§2.5 Discrete-Time Systems A discrete-time system processes a given

input sequence x[n] to generates an output sequence y[n] with more desirable properties

In most applications, the discrete-time system is a single-input, single-output system:

x[n] y[n]

Input sequence Output sequence

Discrete-TimeSystem

Discrete-Time Systems:Examples

2-input, 1-output discrete-time systems - Modulator, adder

1-input, 1-output discrete-time systems - Multiplier, unit delay, unit advance


Accumulator :

nxny

][][

][]1[][][1

nxnynxxn

The output y[n] at time instant n is the sum of the input sample x[n] at time instant n and the previous output y[n-1] at time instant n-1 which is the sum of all previous input sample values from - to n-1

The system accumulatively adds, i.e., it accumulates all input sample values


Accumulator - Input-output relation can also be written in the form

The second form is used for a causal input sequence, in which case y[-1] is called the initial condition

0,][]1[

][][][

0

0

1

nxy

xxny

n

n


Used in smoothing random variations in data

In most applications, the data x[n] is a bounded sequence

M-point moving-average system –

1

0

][1

][M

k

knxM

ny

M-point average y[n] is also a bounded sequence


If there is no bias in the measurements, an improved estimate of the noisy data is obtained by simply increasing M

A direct implementation of the M-point moving average system requires M−1 additions, 1 division, and storage of M−1 past input data samples

A more efficient implementation is developed next


Hence

1

0

1

1

1

0

][][]1[1

][][][1

][][][1

][

M

M

M

MnxnxnxM

MnxnxnxM

MnxMnxnxM

ny

])[][(1

]1[][ MnxnxM

nyny


Computation of the modified M-point moving average system using the recursive equation now requires 2 additions and 1 division

An application: Consider

x[n] = s[n] + d[n],

where s[n] is the signal corrupted by a noise d[n]


s[n]=2[n(0.9)n], d[n]-random signal

Discrete-Time Systems:Examples Exponentially Weighted Running Average Filter

Computation of the running average requires only 2 additions, 1 multiplication and storage of the previous running average

Does not require storage of past input data samples

10,][]1[][ nxnyny


For 0<α<1, the exponentially weighted average filter places more emphasis on current data samples and less emphasis on past data samples as illustrated below

][]1[]2[]3[][]1[])2[]3[(

][]1[]2[][])1[]2[(][

23

2

2

nxnxnxnynxnxnxny

nxnxnynxnxnyny

Discrete-Time Systems:Examples Linear interpolation - Employed to estimate sa

mple values between pairs of adjacent sample values of a discrete-time sequence

Factor-of-4 interpolation


Factor-of-2 interpolator –

])1[]1[(2

1][][ nxnxnxny uuu


])1[]2[(3

2

])2[]1[(3

1][][

nxnx

nxnxnxny

uu

uuu


Median Filter – The median of a set of (2K+1) numbers is th

e number such that K numbers from the set have values greater than this number an

d the other K numbers have values smaller Median can be determined by rank-ordering

the numbers in the set by their values and choosing the number at the middle


Median Filter – Example: Consider the set of numbers

{2, -3, 10, 5, -1} Rank-order set is given by

{-3, -1, 2, 5, 10} Hence,

Med{2, -3, 10, 5, -1}=2


Median Filter – Implemented by sliding a window of odd len

gth over the input sequence {x[n]} one sample at a time

Output y[n] at instant n is the median value of the samples inside the window centered at n


Median Filter – Finds applications in removing additive rand

om noise, which shows up as sudden large errors in the corrupted signal

Usually used for the smoothing of signals corrupted by impulse noise

Discrete-Time Systems:Examples Median Filtering Example –

§2.5 Discrete-Time Systems: Classification

Linear System Shift-Invariant System Causal System Stable System Passive and Lossless Systems

§2.5.1 Linear Discrete-Time Systems

Definition - If y1[n] is the output due to an input x1[n] and y2[n] is the output due to an input x2[n] then for an input

x[n] =αx1[n] +βx2[n] the output is given by

y[n] =αy1[n] +βy2[n] Above property must hold for any arbitrary cons

tants α and β and for all possible inputs x1[n] and x2[n]


][][][ 21 nxnxnx For an input

the output is

Hence, the above system is linear

nn

xnyxny

][][,][][ 2211 Accumulator –

nn

n

nynyxx

xxny

][][][][

])[][(][

2121

21


The outputs y1[n] and y2[n] for inputs x1[n] and x2[n] are given by

])[][(]1[][ 20

1

xxynyn

The output y[n] for an input αx1[n]+βx2[n] is given by

nn

xynyxyny0

2220

111 ][]1[][][]1[][


]1[]1[]1[ 21 yyy

Now

][][][ 21 nynyny if Thus

)][][(]1[]1[

)][]1[()][]1[(

])[][

02

0121

022

011

21

nn

nn

xxyy

xyxy

nyny

Now

)][][(]1[]1[

)][]1[()][]1[(

])[][

02

0121

022

011

21

nn

nn

xxyy

xyxy

nyny


For the causal accumulator to be linear the condition y[-1]=αy1[-1]+βy2[-1] must hold for all initial conditions y[-1], y1[-1], y2[-1], and all constants α and β

This condition cannot be satisfied unless the accumulator is initially at rest with zero initial condition

For nonzero initial condition, the system is nonlinear

Nonlinear Discrete-Time Systems

The median filter described earlier is a nonlinear discrete-time system

To show this, consider a median filter with a window of length 3

Output of the filter for an input

{x1[n]}={3, 4, 5}, 0≤n≤2is

{y1[n]}={3, 4, 4}, 0≤n≤2


Output for an input

{x2[n]}={2, -1, -1}, 0≤n≤2is

{y2[n]}={0, -1, -1}, 0≤n≤2 However, the output for an input

{x[n]}={x1[n]+ x2[n]}is

{y[n]}={3, 4, 3}


Note

{y1[n]+y2[n]}={3, 3, 3}≠{y [n]} Hence, the median filter is a nonlinear discre

te-time system

§2.5.1 Shift-Invariant System For a shift-invariant system, if y1[n] is the re

sponse to an input x1[n] , then the response to an input x[n]=x1[n-n0]

is simply y[n]=y1[n-n0]

where n0 is any positive or negative integer The above relation must hold for any arbitrar

y input and its corresponding output The above property is called time-invariance

property, or shift-invariant proterty

§2.5.1 Shift-Invariant System

In the case of sequences and systems with indices n related to discrete instants of time, the above property is called time-invariance property

Time-invariance property ensures that for a specified input, the output is independent of the time the input is being applied

§2.5.1 Shift-Invariant System Example - Consider the up-sampler with an i

nput-output relation given by

otherwise,0

,2,,0,]/[][

LLnLnxnxu

For an input x1[n]=x[n-n0] the output x1,u[n] is given by

otherwise,0

,2,,0,]/)[(otherwise,0

,2,,0,]/[][

0

1,1

LLnLLnnx

LLnLnxnx u

§2.5.1 Shift-Invariant System

However from the definition of the up-sampler

Hence, the up-sampler is a time-varying system

][otherwise,0

,2,,,/][][

,1

0000

0

nx

LnLnnnLnnxnnx

u

u

§2.5.2 Linear Time-Invariant system

Linear Time-Invariant (LTI) System - A system satisfying both the linearity and

the time-invariance property LTI systems are mathematically easy to

analyze and characterize, and consequently, easy to design

Highly useful signal processing algorithms have been developed utilizing this class of systems over the last several decades

Causal System

In a causal system, the n0-th output sampley[n0] depends only on input samples x[n] forn≤n0 and does not depend on input samples for n>n0

Let y1[n] and y2[n] be the responses of acausal discrete-time system to the inputs x1

[n] and x2[n], respectively

Causal System

Then

x1[n]= x2[n] for n<N

implies also that

y1[n]= y2[n] for n<N For a causal system, changes in output sam

ples do not precede changes in the input samples

Causal System Examples of causal systems:

y[n]=α1x[n]+α2x[n-1]+α3x[n-2]+α4x[n-3]

y[n]=b0x[n]+b1x[n-1]+b2x[n-2]

+a1y[n-1]+a2y[n-2]y[n]=y[n-1]+x[n]

Examples of noncausal systems:

y[n]=xu[n]+1/2(xu[n-1]+ xu[n+1])

y[n]=xu[n]+1/3(xu[n-1]+ xu[n+2])

+ 2/3(xu[n-2]+ xu[n+1])

Causal System

A noncausal system can be implemented as a causal system by delaying the output by an appropriate number of samples

For example a causal implementation of the factor-of-2 interpolator is given by

y[n]=xu[n-1]+1/2(xu[n-2]+ xu[n])

Stable System

There are various definitions of stability We consider here the bounded-input, bound

ed-output (BIBO) stability If y[n] is the response to an input x[n] and if

|x[n]|≤Bx for all values of n Then

|y[n]|≤By for all values of n

Stable System

Example - The M-point moving average filter is BIBO stable:

For a bounded input |x[n]|≤Bx we have

1

0

][1

][M

k

knxM

ny

xx

M

k

M

k

BMBM

knxM

knxM

ny

)(1

][1

][1

][1

0

1

0

§2.5.3 Passive and Lossless Systems

A discrete-time system is defined to be passive if, for every finite-energy input x[n], the output y[n] has, at most, the same energy, i.e.

nnnxny 22 ][][

For a lossless system, the above inequality is satisfied with an equal sign for every input

§2.5.3 Passive and Lossless Systems

Example - Consider the discrete-time system defined by y[n]=x[n-N] with N a positive integer

Its output energy is given by

nnnxny 222 ][][

Hence, it is a passive system if || 1 and is a lossless system if || =1

§2.5.4 Impulse and Step Responses

The response of a discrete-time system to a unit sample sequence {δ[n]} is called the unit impulse response or simply, the impulse response, and is denoted by {h[n]}

The response of a discrete-time system to a unit step sequence {μ[n]} is called the unit step response or simply, the step response, and is denoted by {s[n]}


Example - The impulse response of the system

y[n]=a1x[n]+a2x[n-1]+a3x[n-2]+a4x[n-3]

is obtained by setting x[n]=δ[n] resulting in

h[n]=a1δ[n]+a2δ[n-1]+a3δ[n-2]+a4δ[n-3] The impulse response is thus a finite-length

sequence of length 4 given by{h[n]={a1, a2, a3, a4}


Example - The impulse response of the discrete-time accumulator

n

xny

][][

][][][ nnhn

is obtained by setting x[n] = δ[n] resulting in


Example - The impulse response {h[n]} of the factor-of-2 interpolator

])[][(][][ 1121 nxnxnxny uuu

])[][(][][ 1121 nnnnh

}.,.{]}[{ 50150

nh

is obtained by setting xu[n]= [n] and is given by

The impulse response is thus a finite-length sequence of length 3:

§2.6 Time-Domain Characterization of LTI Discrete-Time System

Input-Output Relationship –

A consequence of the linear, time-invariance property is that an LTI discrete-time system is completely characterized by its impulse response

Knowing the impulse response one can compute the output of the system for any arbitrary input


Let h[n] denote the impulse response of a LTI discrete-time system

Compute its output y[n] for the input:

]5[75.0]2[]1[5.1]2[5.0][ nnnnnx

As the system is linear, we can compute its outputs for each member of the input separately and add the individual outputs to determine y[n]


Since the system is time-invariant

]5[]5[]2[]2[

]1[]1[]2[]2[

outputinput

nhnnhnnhnnhn


Likewise, as the system is linear

][.][.][ 151250 nhnhny][.][ 57502 nhnh

Hence because of the linearity property we get Likewise, as the system is linear

]5[75.0]5[75.0]2[]2[]1[5.1]1[5.1]2[5.0]2[5.0

outputinput

nhnnhn

nhnnhn


Now, any arbitrary input sequence x[n] can be expressed as a linear combination of delayed and advanced unit sample sequences in the form

kknkxnx ][][][

The response of the LTI system to an input x[k][n-k] will be x[k]h[n-k]


Hence, the response y[n] to an input

kknkxnx ][][][

kknhkxny ][][][

kkhknxny ][][][

which can be alternately written as

will be

§2.6.1 Convolution Sum

The summation

kknhknxknhkxny ][][][][][

y[n] = x[n] h[n]*

is called the convolution sum of the sequences x[n] and h[n] and represented compactly as

§2.6.1 Convolution SumProperties -

Commutative property:

x[n] h[n] = h[n] x[n]* *

(x[n] h[n]) y[n] = x[n] (h[n] y[n])****

x[n] (h[n] + y[n]) = x[n] h[n] + x[n] y[n]** *

Associative property :

Distributive property :


Interpretation - 1) Time-reverse h[k] to form h[-k] 2) Shift h[-k] to the right by n sampling perio

ds if n > 0 or shift to the left by n sampling periods if n < 0 to form h[n-k]

3) Form the product v[k]=x[k]h[n-k] 4) Sum all samples of v[k] to develop the

n-th sample of y[k] of the convolution sum


Schematic Representation -

nz][ knh

][ kh

][kx

][kv][ny

k

The computation of an output sample using the convolution sum is simply a sum of products

Involves fairly simple operations such as additions, multiplications, and delays

§2.6.1 Convolution Sum We illustrate the convolution operation for th

e following two sequences:

Figures on the next several slides the steps involved in the computation of

otherwise,0

50,1][

nnx

otherwise,0

50,3.08.1][

nnnh

y[n]= x[n] h[n]*

Time-Domain Characterization of LTI Discrete-Time System

In practice, if either the input or the impulse response is of finite length, the convolution sum can be used to compute the output sample as it involves a finite sum of products

If both the input sequence and the impulse response sequence are of finite length, the output sequence is also of finite length


If both the input sequence and the impulse response sequence are of infinite length, convolution sum cannot be used to compute the output

For systems characterized by an infinite impulse response sequence, an alternate time-domain description involving a finite sum of products will be considered


Example - Develop the sequence y[n] generated by the convolution of the sequences x[n] and h[n] shown below


As can be seen from the shifted time-reversed version {h[n-k]} for n<0, shown below for n =-3 , for any value of the sample index k, the k-th sample of either {x[k]} or is zero


As a result, for n<0, the product of the k-th samples of {x[k]} and {h[n-k]} is always zero, and hence y[n] = 0 for n < 0

Consider now the computation of y[0]

The sequence {h[n-k]} is shown on the right


The product sequence {x[k]h[-k]} is plotted below which has a single nonzero sample x[0]h[0] for k=0

Thus y[0]=x[0]x[0]=-2


For the computation of y[1], we shift {h[-k]} to the right by one sample period to form {h[1-k]} as shown below on the left

The product sequence {x[k]h[1-k]} is shown below on the right

Hence, y[1]=x[0]h[1]+x[1]h[0]=-4+0=-4


To calculate y[2], we form as shown below on the left -6

The product sequence {x[k]h[2-k]} is plotted below on the right

y[2]=x[0]h[2]+x[1]h[1]+x[2]h[0]=0+0+1=1


Continuing the process we get y[3]=x[0]h[3]+x[1]h[2]+x[2]h[1]+x[3]h[0] =2+0+0+1=3 y[4]=x[1]h[3]+x[2]h[2]+x[3]h[1]+x[4]h[0] =0+0-2+3=1 y[5]=x[2]h[3]+x[3]h[2]+x[4]h[1]=-1+0+6=5 y[6]=x[3]h[3]+x[4]h[2]=1+0=1 y[6]=x[4]h[3]=-3


From the plot of {h[n-k]} for n > 7 and theplot of {x[k]} as shown below, it can be seen that there is no overlap between these two sequences

As a result y[n] = 0 for n > 7


The sequence {y[n]} generated by the convolution sum is shown below


Note: The sum of indices of each sample product inside the convolution sum is equal to the index of the sample being generated by the convolution operation

For example, the computation of y[3] in the previous example involves the products x[0]h[3], x[1]h[2], x[2]h[1], and x[3]h[0]

The sum of indices in each of these products is equal to 3


In the example considered the convolution of a sequence {x[n]} of length 5 with a sequence {h[n]} of length 4 resulted in a sequence {y[n]} of length 8

In general, if the lengths of the two sequences being convolved are M and N, then the sequence generated by the convolution is of length M+N-1

Tabular Method ofConvolution Sum Computation

Can be used to convolve two finite-length sequences

Consider the convolution of {g[n]}, 0≤n≤3, with {h[n]}, 0≤n≤2, generating the

Samples of {g[n]} and {h[n]} are then multiplied using the conventional multiplication method without any carry operation

sequence y[n]= g[n] h[n]*


The samples y[n] generated by the convolution sum are obtained by adding the entries in the column above each sample

]5[]4[]3[]2[]1[]0[:][

]2[]3[]2[]2[]2[]1[]2[]0[]1[]3[]1[]2[]1[]1[]1[]0[

]0[]3[]0[]2[]0[]1[]0[]0[]2[]1[]0[:][

]3[]2[]1[]0[:][543210:

yyyyyyny

hghghghghghghghg

hghghghghhhnh

ggggngn


The samples of {y[n]} are given by y[0]=g[0]h[0] y[1]=g[1]h[0]+g[0]h[1] y[2]=g[2]h[0]+g[1]h[1]+g[0]h[2] y[3]=g[3]h[0]+g[2]h[1]+g[1]h[2] y[4]=g[3]h[1]+g[2]h[2] y[5]=g[3]h[2]


The method can also be applied to convolve two finite-length two-sided sequences

In this case, a decimal point is first placed to the right of the sample with the time index n = 0 for each sequence

Next, convolution is computed ignoring the location of the decimal point


Finally, the decimal point is inserted according to the rules of conventional multiplication

The sample immediately to the left of the decimal point is then located at the time index n = 0

Convolution Using MATLAB

The M-file conv implements the convolution sum of two finite-length sequences

If a=[-2 0 1 -1 3]

b=[1 2 0 -1]

then conv(a,b) yields

[-2 -4 1 3 1 5 1 -3]

Simple InterconnectionSchemes

Two simple interconnection schemes are: Cascade Connection Parallel Connection

Cascade Connection

Impulse response h[n] of the cascade of two LTI discrete-time systems with impulse responses h1[n] and h2[n] is given by

y[n]= h1[n] h2[n]*

][nh1][nh2][nh1 ][nh2

][][ nhnh 1 ][nh2][nh1 *

Cascade Connection

Note: The ordering of the systems in the cascade has no effect on the overall impulse response because of the commutative property of convolution

A cascade connection of two stable systems is stable

A cascade connection of two passive (lossless) systems is passive (lossless)

Cascade Connection

An application is in the development of an inverse system

If the cascade connection satisfies the relation

then the LTI system h1[n] is said to be the inverse of h2[n] and vice-versa

=[n]

h1[n] h2[n]*

If the impulse response of the channel is known, then x[n] can be recovered by designing an inverse system of the channel

Cascade Connection An application of the inverse system concept

is in the recovery of a signal x[n] from its distorted version appearing at the output of a transmission channel

][ˆ nx

=[n]

h1[n] h2[n]*

][nh2][nh1x[n]

x[n]

][ˆ nxchannel Inverse system

Cascade Connection

Example - Consider the discrete-time accumulator with an impulse response µ[n]

Its inverse system satisfy the condition It follows from the above that h2[n]=0 for n<0

and

1for0][

1]0[

02

2

nh

hn

Cascade Connection

Thus the impulse response of the inverse system of the discrete-time accumulator is given by

h2[n]=[n]- [n-1]

which is called a backward difference system

Parallel Connection

Impulse response h[n] of the parallel connection of two LTI h1[n] discrete-time systems with impulse responses and h2[n] is given by

h[n]=h1[n]+h2[n]

][nh2

][nh1 ][][ nhnh 1 ][nh2][nh1

§2.6.2 Simple Interconnection Schemes

h1[n]=[n]+0.5[n-1],

h2[n]=0.5[n]-0.25[n-1],

h3[n]=2[n],

h4[n]=- 2(0.5)n[n]

][nh2

][nh1

][nh4

][nh3

Consider the discrete-time system where


Simplifying the block-diagram we obtain

][nh2

][nh1

][][ 43 nhnh

][nh1

])[][(][ 432 nhnhnh *


Overall impulse response h[n] is given by

][][][][][ nhnhnhnhnh 42321 ])[][(][][][ nhnhnhnhnh 4321 *

* *

][2])1[][(][][41

21

32 nnnnhnh

]1[][21 nn

* *

Now,


Therefore][][]1[

2

1][]1[

2

1][][ nnnnnnnh

][)

2

1(2])1[

4

1][

2

1(][][ 42 nnnnhnh n* *

][][)2

1(

]1[)2

1(][)

2

1(

]1[)2

1(

2

1][)

2

1( 1

nn

nn

nn

n

nn

nn

Stability Condition of an LTI Discrete-Time System

BIBO Stability Condition - A discrete- time is BIBO stable if and only if the output sequence {y[n]} remains bounded for all bounded input sequence {x[n]}

An LTI discrete-time system is BIBO stable if and only if its impulse response sequence {h[n]} is absolutely summable, i.e.

n

nhS ][


Proof: Assume h[n] is a real sequence Since the input sequence x[n] is bounded w

e have xBnx ][

Therefore

SBkhB

knxkhknxkhny

xk

x

kk

][

][][][][][


Thus, S < ∞ implies |y[n]|≤By < ∞ indicating that y[n] is also bounded

To prove the converse, assume y[n] is bounded, i.e., |y[n]|≤By

Consider the input given by

0][if,

0][if}),{sgn(][

nhK

nhnhnx


where sgn(c) = +1 if c > 0 and sgn(c) =-1 if c < 0 and |K| ≤ 1

Note: Since |x[n]| ≤ 1, is obviously bounded For this input, y[n] at n = 0 is

Therefore, |y[n]|≤By implies S < ∞

k

yBSkhkhy ][])[sgn(]0[


Example - Consider a causal LTI discrete- time system with an impulse response h[n]=(α)nµ[n]

For this system

Therefore S < ∞ if |α|<1 for which the system is BIBO stable

If |α|<1, the system is not BIBO stable

1if1

1][

0

n

n

n

n nS

Causality Condition of an LTI Discrete-Time System

Let x1[n] and x2[n] be two input sequences with

x1[n]=x2[n] for n≤n0

The corresponding output samples at n=n0 of an LTI system with an impulse response {h[n]} are then given by


][][

][][][][][

01

1

010

0101

knxkh

knxkhknxkhny

k

kk

][][

][][][][][

02

1

020

0202

knxkh

knxkhknxkhny

k

kk


If the LTI system is also causal, then

y1[n0]=y2[n0]

As x1[n]=x2[n] for n≤n0

][][][][ 020

010

knxkhknxkhkk

This implies

][][][][ 02

1

01

1

knxkhknxkhkk


As x1[n] ≠x2[n] for n>n0 the only way the condition

will hold if both sums are equal to zero, which is satisfied if h[k] = 0 for k < 0

][][][][ 02

1

01

1

knxkhknxkhkk


]3[]2[]1[][][ 4321 nxnxnxnxny

An LTI discrete-time system is causal if and only if its impulse response {h[n]} is a causal sequence

Example - The discrete-time system defined by

is a causal system as it has a causal impulse response

}{]}[{ 4321 nh

Example - The discrete-time accumulator defined by


is a causal system as it has a causal impulse response given by

][][][ nnyn

][][][ nnhn


Example - The factor-of-2 interpolator defined by

])1[]1[(2

1][][ nxnxnxny uuu

is noncausal as it has a noncausal impulse response given by

}5.015.0{]}[{ nh


Note: A noncausal LTI discrete-time system with a finite-length impulse response can often be realized as a causal system by inserting an appropriate amount of delay

For example, a causal version of the factor- of-2 interpolator is obtained by delaying the input by one sample period:

])[]2[(2

1]1[][ nxnxnxny uuu

Finite-Dimensional LTI Discrete-Time Systems

An important subclass of LTI discrete-time systems is characterized by a linear constant coefficient difference equation of the form

x[n] and y[n] are, respectively, the input and the output of the system

{dk} and {pk} are constants aracterizing the system

M

kk

N

kk knxpknyd

00

][][


The order of the system is given by max(N,M), which is the order of the difference equation

It is possible to implement an LTI system characterized by a constant coefficient difference equation as here the computation involves two finite sums of products


If we assume the system to be causal, then the output y[n] can be recursively computed using

provided d0≠0

y[n] can be computed for all n≥n0, knowing x[n] and the initial conditions

y[n0-1], y[n0-2],…, y[n0-N]

M

k

kN

k

k knxd

pkny

d

dny

0 01 0

][][][

§2.7 Classification of LTI Discrete-Time Systems

Based on Impulse Response Length - If the impulse response h[n] is of finite length,

i.e.,

h[n]=0 for N1<n<N2 and N1<N2

then it is known as a finite impulse response (FIR) discrete-time system

The convolution sum description here is

2

1

][][][N

Nkknxkhny


The output y[n] of an FIR LTI discrete-time system can be computed directly from the convolution sum as it is a finite sum of products

Examples of FIR LTI discrete-time systems are the moving-average system and the linear interpolators


If the impulse response is of infinite length, then it is known as an infinite impulse response (IIR) discrete-time system

The class of IIR systems we are concerned with in this course are characterized by linear constant coefficient difference equations


Example - The discrete-time accumulator defined by

y[n]=y[n-1]+x[n]

is seen to be an IIR system


Example - The familiar numerical integration formulas that are used to numerically solve integrals of the form

t

dxty0

)()(

can be shown to be characterized by linear constant coefficient difference equations, and hence, are examples of IIR systems


If we divide the interval of integration into n equal parts of length T, then the previous integral can be rewritten as

nT

Tn

dxTnynTy)1(

)())1(()(

nT

dxnTy0

)()(

where we have set t = nT and used the notation


Using the trapezoidal method we can write

)}())1(({)(2

)1(

nTxTnxdx TnT

Tn

)}())1(({))1(()(2

nTxTnxTnynTy T

Hence, a numerical representation of the definite integral is given by


Let y[n] = y(nT) and x[n] = x(nT) Then

)}())1(({))1(()(2

nTxTnxTnynTy T

]}1[][{]1[][2

nxnxnyny T

which is recognized as the difference equation representation of a first-order IIR discrete-time system

reduces to


Based on the Output Calculation Process Nonrecursive System - Here the output can

be calculated sequentially, knowing only the present and past input samples

Recursive System - Here the output computation involves past output samples in addition to the present and past input samples


Based on the Coefficients - Real Discrete-Time System - The impulse re

sponse samples are real valued Complex Discrete-Time System - The impul

se response samples are complex valued

§2.8 Correlation of Signals

There are applications where it is necessary to compare one reference signal with one or more signals to determine the similarity between the pair and to determine additional information based on the similarity


For example, in digital communications, a set of data symbols are represented by a set of unique discrete-time sequences

If one of these sequences has been transmitted, the receiver has to determine which particular sequence has been received by comparing the received signal with every member of possible sequences from the set


Similarly, in radar and sonar applications, the received signal reflected from the target is a delayed version of the transmitted signal and by measuring the delay, one can determine the location of the target

The detection problem gets more complicated in practice, as often the received signal is corrupted by additive random noise


Definitions A measure of similarity between a pair of en

ergy signals, x[n] and y[n], is given by the cross-correlation sequence rxy [ℓ] defined by

...,,,],[][][ 210

nxy nynxr

The parameter ℓ called lag, indicates the time-shift between the pair of signals


y[n] is said to be shifted by ℓ samples to the right with respect to the reference sequence x[n] for positive values of ℓ, and shifted by ℓ samples to the left for negative values of

The ordering of the subscripts xy in the definition of rxy [ℓ] specifies that x[n] is the reference sequence which remains fixed in time while y[n] is being shifted with respect to x[n]

§2.8 Correlation of Signals If y[n] is made the reference signal and shift

x[n] with respect to y[n], then the corresponding cross-correlation sequence is given by

Thus, ryx [ℓ] is obtained by time-reversing rxy [ℓ]

][][][

][][][

xym

nyx

rmxmy

nxnyr


The autocorrelation sequence of x[n] is given by

nxx nxnxr ][][][

obtained by setting y[n] = x[n] in the definition of the cross-correlation sequence rxy [ℓ]

Note: , the energy of the signal x[n]

xnxx Enxr

][]0[ 2

§2.8 Correlation of Signals From the relation ryx [ℓ]= rxy [-ℓ] it follows that

rxx [ℓ]= rxx [-ℓ] implying that rxx [ℓ] is an even function for real x[n]

An examination of

reveals that the expression for the cross- correlation looks quite similar to that of the linear convolution

yxyyxxxy EErrr ]0[]0[][

§2.8 Correlation of Signals This similarity is much clearer if we rewrite t

he expression for the cross-correlation as

The cross-correlation of y[n] with the reference signal x[n] can be computed by processing x[n] with an LTI discrete-time

system of impulse response y[−n]

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

yxnynxr

nxy *

x[n] rxy[n]y[-n]


Likewise, the autocorrelation of x[n] can be computed by processing x[n] with an LTI discrete-time system of impulse response x[-n]

x[n] rxx[n]x[-n]

Properties of Autocorrelation andCross-correlation Sequences

Consider two finite-energy sequences x[n] and y[n]

The energy of the combined sequence ax[n]+y[n-ℓ] is also finite and nonnegative, i.e.,

0][][][2

][)][][(2

222

nn

nn

nynynxa

nxanynax


Thus

a2 rxx[0]+2arxy[ℓ]+ryy[0]≥0

where rxx[0]=Ex>0 and ryy[0]=Ey>0 We can rewrite the equation on the previous

slide as

for any finite value of a

01]0[][

][]0[1

a

rr

rra

yyxy

xyxx


or, equivalently,

]0[][

][]0[

yyxy

xyxx

rr

rr

Or, in other words, the matrix

is positive semidefinite 0][]0[]0[ 2 xyyyxx rrr



The last inequality on the previous slide provides an upper bound for the cross- correlation samples

If we set y[n] = x[n], then the inequality reduces to

xxxxy Err ]0[][


Thus, at zero lag (ℓ=0), the sample valuel of the autocorrelation sequence has its maxi

mum value Now consider the case

y[n] =±bx[n-N] where N is an integer and b > 0 is an arbitrary number

In this case Ey=b2Ex


Therefore

xxyx bEEbEE 22

Using the above result in

we get


]0[][]0[ xxxyxx brrbr

Correlation ComputationUsing MATLAB

The cross-correlation and autocorrelation sequences can easily be computed using MATLAB

Example - Consider the two finite-length sequences

x[n]=[1 3 -2 1 2 -1 4 4 2]

y[n]=[2 -1 4 1 -2 3]


The cross-correlation sequence rxy[n] computed using Program 2_7 of text is plotted below


The autocorrelation sequence rxx[ℓ] computed using Program 2_7 is shown below

Note: At zero lag, rxx[0] is the maximum


The plot below shows the cross-correlation of x[n] and y[n]=x[n-N] for N = 4

Note: The peak of the cross-correlation is precisely the value of the delay N


The plot below shows the autocorrelation of x[n] corrupted with an additive random noise generated using the function randn

Note: The autocorrelation still exhibits a peak at zero lag


The autocorrelation and the cross- correlation can also be computed using the function xcorr

However, the correlation sequences generated using this function are the time- reversed version of those generated using Programs 2_7 and 2_8

Normalized Forms ofCorrelation

Normalized forms of autocorrelation and cross-correlation are given by

They are often used for convenience in comparing and displaying

Note: |ρxx [ℓ]|≤1 and |ρxy [ℓ]|≤1 independent of the range of values of x[n] and y[n]

]0[]0[

][][,

]0[

][][

yyxx

xyxy

xx

xxxx

rr

r

r

r

Correlation Computation forPower Signals

The cross-correlation sequence for a pair of power signals, x[n] and y[n], is defined as

K

KnK

xx nxnxK

r ][][12

1lim][

The autocorrelation sequence of a power signal x[n] is given by

K

KnK

xy nynxK

r ][][12

1lim][

Correlation Computation forPeriodic Signals

1

0~~ ][~][~1

][N

nyx nynxN

r

1

0~~ ][~][~1

][N

nxx nxnxN

r

The autocorrelation sequence of a periodic signal of period N is given by][~ nx

The cross-correlation sequence for a pair of periodic signals of period N, and

is defined as

][~ nx ][~ ny

The periodicity property of the autocorrelation sequence can be exploited to determine the period of a periodic signal that may have been corrupted by an additive random disturbance


Note: Both and are also periodic signals with a period N

][~~ yxr][~~ xxr


Let be a periodic signal corrupted by the random noise d[n] resulting in the signal

][~ nx

which is observed for 0≤n≤M-1 where M>>N

][][~][ ndnxnw


The autocorrelation of w[n] is given by

][][][][

][~][1

][][~1

][][1

][~][~(1

])[][~])([][~(1

][][1

][

~~~~

1

0

1

0

1

0

1

0

1

0

1

0

xddxddxx

M

n

M

n

M

n

M

n

M

n

M

nww

rrrr

nxndM

ndnxM

ndndM

nxnxM

ndnxndnxM

nwnwM

r


In the last equation on the previous slide,

is a periodic sequence with a period N and hence will have peaks at ℓ=0, N, 2N,… with the same amplitudes as ℓ approaches M

][~~ xxr

As and d[n] are not correlated, samples of cross-correlation sequences

and are likely to be very small relative to the amplitudes of ][~~ xxr

][~ dxr ][~ xdr

][~ nx


The autocorrelation rdd [ℓ] of d[n] will show a peak at ℓ=0 with other samples having rapidly decreasing amplitudes with increasing values of |ℓ|

Hence, peaks of rww [ℓ] for ℓ>0 areessentially due to the peaks of and can be used to determine whether is a periodic sequence and also its period N if the peaks occur at periodic intervals

][~~ xxr][~ nx

Correlation Computation of aPeriodic Signal Using MATLAB

Example - We determine the period of the sinusoidal sequence x[n] =cos(0.25n), 0≤n≤95 corrupted by an additive uniformly distributed random noise of amplitude in the range [-0.5,0.5]

Using Program 2_8 of text we arrive at the plot of rww[ℓ] shown on the next slide


As can be seen from the plot given above, there is a strong peak at zero lag

However, there are distinct peaks at lags that are multiples of 8 indicating the period of the sinusoidal sequence to be 8 as expected


Figure below shows the plot of rdd[ℓ]

As can be seen rdd[ℓ] shows a very strongpeak at only zero lag

chapter 2 discrete-time signals and systems. §2.1.1 discrete-time signals: time-domain...

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