1 1. time domain representation of signals and systems 1.1 discrete-time signals 1.2 operations on...

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1. Time Domain Representation of Signals and Systems

1.1 Discrete-Time Signals

1.2 Operations on Sequences

1.3 Classification of Sequences

1.4 Some Basic Sequences

1.5 The Sampling Process

1.6 Discrete-Time Systems

1.7 Classification of Discrete-Time Systems

1.8 Time-Domain Characterization of LTI Systems

1.9 Correlation

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1.1 Discrete-Time Signals

• There are basically two types of discrete time

signals:

– Sampled-data signals in which samples

are continuous-valued

– digital signals in which samples are

discrete-valued

• Digital signals are obtained by quantizing the sample values either by rounding or truncation

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Discrete-Time Signals

• Signals are represented as sequences of

numbers, called samples

• A sample value of a typical signal or

sequence is denoted as x[n] with n being an

integer in the range

• x[n] is defined only for integer values of n and

is undefined for non-integer values of n

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

n

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• Discrete-time signal may also be written as a

sequence of numbers inside braces:

x[n]={…,-0.2, 2.2,1.1,0.2,-0.7,2.9,…}

• In the above, x[-1]=-0.2, x[0]=2.2 x[1]=1.1 etc.

• The arrow is placed under the sample at

time index n = 0

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• The graphical representation of a discrete time

signal with real-valued samples is as shown below:

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• In some applications, a discrete-time sequence

{x[n]} may be generated by periodically

sampling a continuous-time signal xa(t) at

uniform time intervals

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• 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

samples is called the sampling interval or

sampling period

• Reciprocal of sampling interval T, denoted

as FT is called sampling frequency:

FT=1/T

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• 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

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• A complex sequence {x[n]} can be written

as {x[n]}={xre[n]+jxim[n]} where xre[n] and

xim[n] are the real and imaginary parts of x[n]

• The complex conjugate sequence of {x[n]}

is given by {x*[n]}={xre[n]- jxim[n]}

• Often the braces are ignored to denote a

sequence if there is no ambiguity

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• 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: where:

with

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

1 2N n N N

2N 1 2N N

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• Examples:

x[n]= n2,

is a finite-length sequence of length 8

y[n]=cos(0.4n)

is an infinite-length sequence

3 4n

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• 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, ie: by

appending it with zeros

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• Example:

is a finite-length sequence of length-12

obtained by zero-padding the sequence

with 4 zero-valued samples

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Discrete-Time Signals• A right-sided sequence x[n] has zero valued samples for

• If a right-sided sequence is called a causal sequence

1n N

1 0N

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Discrete-Time Signals• A left-sided sequence x[n] has zero-valued

samples for 2n N

• If a left-sided sequence is called a anti-causal sequence

2 0N

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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

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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 that we describe

next:

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Basic Operations• Product (modulation) operation:

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

Modulator:

• An application is in forming a finite-length

sequence from an infinite-length sequence

by multiplying with a window sequence

• This process is usually called windowing

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Basic Operations

• Addition operation: y[n]=x[n]+w[n]

Adder:

• Multiplication operation: y[n] = A.x[n]

Multiplier:

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Basic Operations

• Time-shifting operation: y[n] = x[n − N] , where N is an integer

• If N > 0, it is delaying operation e.g. unit delay: y[n] = x[n −1]

• If N < 0, it is an advance operation, e.g. unit advance: y[n] = x[n +1]

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Basic Operations

• Time-reversal operation:

y[n] = x[−n]

• Branching operation: Used to provide

multiple copies of a sequence

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Basic Operations

• Example: Consider the two following

sequences of length 5 defined for 0 ≤ n ≤ 4:

{a[n]}={3 4 6 − 90}

{b[n]}={2 −1 4 5 −3}

• New sequences generated from the above two

sequences by applying the basic operations are

as follows:

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Basic Operations

{c[n]}= {a[n]⋅b[n]}= {6 − 4 24 − 450}

{d[n]}= {a[n]+ b[n]}= {5 3 10 − 4 −3}

{e[n]}={4.5 6 9 13.50}

• As pointed out by the above examples,

operations 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

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Basic Operations

• 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 0 ≤ n ≤ 2 :{f [n]}= {− 2 1 −3}

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Basic Operations

• 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]}+{fe[n]} ={1 5 3 − 9 0}

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Combinations of Basic Operations

• Example:

y[n] =α1x[n]+α 2x[n −1]+α3x[n − 2]+α4x[n − 3]

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1.3 Classification of Sequences

Based on Symmetry

• Conjugate-symmetric sequence:

x[n] = x*[−n]

• If x[n] is real, then it is an even sequence

An Even Sequence

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Classification of Sequences

Based on Symmetry

• Conjugate-antisymmetric sequence:

x[n] = −x*[−n]

• If x[n] is real, then it is an odd sequence

An Odd Sequence

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Classification of Sequence

Based on Symmetry

• 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-antisymmetric 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

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Classification of SequencesBased on Symmetry

• Any complex sequence can be expressed as a sum of its conjugate-symmetric part and its conjugate-antisymmetric part, if the

parent sequence is of odd length defined for a symmetric interval, −M ≤ 0 ≤ M:

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

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

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

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Based on Symmetry

• Example: Consider the complex length-7

sequence defined for − 3 ≤ n ≤ 3:

{g[n]} = {0, 1+ j4, −2+ j3, 4− j2, −5− j6, −j2,3}

• Its conjugate sequence is then given by:

{g*[n]} = {0, 1− j4, −2− j3, 4+ j2, −5+ j6, j2,3}

• The time-reversed version of the above is:

{g*[−n]} = {3, j2, −5+ j6, 4+ j2, −2− j3,1−j4,0}

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Based on Symmetry

• Therefore {gcs[n]}=1/2{g[n]+ g *[−n]}

={1.5, 0.5+ j3, −3.5+ j4.5, 4, −3.5− j4.5, 0.5− j3, 1.5}

• Likewise {gca[n]}=1/2 {g[n]− g *[−n]}

={−1.5, 0.5+ j, 1.5− j1.5, − j2, −1.5− j1.5, −0.5− j, 1.5}

• It can be easily verified that gcs[n]= gcs*[-n]= and gca[n] = − gca*[-n]

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Classification of Sequences:

Periodic and Aperiodic Signals

• A sequence x[n] satisfying:

x[n]=x[n+kN]

is called a periodic sequence with a

period N where N is a positive integer

and k is any integer

• Smallest value of N satisfying x[n]=x[n+kN]

is called the fundamental period

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Periodic and Aperiodic Signals

• Example: Periodic sequence with period N=7

•A sequence not satisfying the periodicity condition is called an aperiodic sequence

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Energy and Power Signals

• The total energy of a sequence x[n] is defined by:

• An infinite length sequence with finite sample values may or may not have finite energy

• A finite length sequence with finite sample

values has finite energy

2

| [ ] |xn

E x n

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• The average power of an aperiodic

sequence is defined by:

• Now, we define the energy of a sequence

x[n] over a finite interval − K ≤ n ≤ K as:

2lim

1| [ ] |

2 1

K

xk n K

P x nK

2

, | [ ] |K

x Kn K

X x n

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Classification of Sequences:Energy and Power Signals

• Then, the average power is:

• The average power of a periodic sequence x[n] with a period N is given by:

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

,lim1

2 1x x Kk

P EK

21

0

1| [ ] |

N

xn

P x nN

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• Example: Consider the causal sequence

defined by:3( 1) , 0

[ ]0 , 0

n nx n

n

• x[n] has infinite energy and its average power is given by:

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Energy and Power Signals• 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:

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Other Types of Classifications

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

each of its samples is of magnitude less than or

equal to a finite positive number Bx, i.e.,

• Example: The sequence x[n]=cos(0.3πn) is a

bounded sequence as:

|x[n]| = |cos0.3πn| ≤1

| [ ] | xx n B

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Other Types of Classifications

• A sequence x[n] is said to be absolutely

summable if:

Example:

is an absolutely summable sequence as:

| [ ] |n

x n

0.3 , 0[ ]

0 , 0

n ny n

n

0

1| 0.3 | 1.42857

1 0.3n

n

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1.4 Some Basic Sequences

• Unit sample sequence:

• Unit step sequence:

1 0[ ]

0 0

nn

n

1 0[ ]

0 0

for nn

for n

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Some Basic Sequences

• Unit impulse and unit step sequence shifted by k samples:

• Relations between the unit sample and the

step sequence:

1, 1, [ ] , [ ]=

0, 0,

n k n kn k n k

n k n k

0[ ] [ ]

[ ] [ ] [ 1]k

n n k

n n n

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• Real sinusoidal sequence:

x[n] = Asin(ωon + φ)

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

• Example:

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Some Basic Sequences• Exponential sequence:

x[n] = Aαn , − ∞ < n < ∞

where A and α are real or complex numbers• If we write:

then we can express:

where:

0 0( ) , | |j je A A e

0 0( )[ ] | | [ ] [ ]j njre imx n A e e x n jx n

0

0

0

0

[ ] | | cos( ),

[ ] | | sin( )

nre

nim

x n A e n

x n A e n

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• x [n] of a complex exponential

sequence are real sinusoidal sequences with constant (σo= 0), growing (σo > 0) or

decaying (σo < 0) amplitudes for n > 0

• Example: x[n] exp(-1/12+jπ/6)n

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• Real exponential sequence:

x[n] =Aαn, −∞ < n < ∞

where A and α are real numbers. Example:

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• The sinusoidal sequence Asin(ωon + φ) and the complex exponential sequence

Bexp( jωon) are periodic sequences of period N as long as ωoN = 2πr where N and r are positive integers

• The smallest possible value of N satisfying ωoN = 2πr is the fundamental period

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• If 2π/ωo is a noninteger rational number,

then the period will be a multiple of 2π/ωo

• Otherwise, the sequence is aperiodic

• Example: x[n] = sin( 3n + φ) is aperiodic

even though it has a sinusoidal envelope

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Some Basic Sequences• Example: Period of Acos(ωon + φ) ωo = 0.1π

Period N=2πr/0.1π=20 for r=1

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Some Basic Sequences• Representation of an arbitrary sequence:• An arbitrary sequence can be represented as a

weighted sum of some basic sequence and its delayed (advanced) versions.

• Example:

[ ] 0.5 [ 2] 1.5 [ 1] [ 2] [ 4] 0.75 [ 6]x n n n n n n

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1.5 The Sampling Process

• Often, a sequence x[n] is developed by sampling a continuous-time signal xa(t)

• The relation between the two signals is:

[ ] ( ) | ( ), ..., 2, 1,0,1,2,...a t nT ax n x t x nT n

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The Sampling Process

• The time variable t of the continuous-time

signal is related to the time variable n of the discrete-time signal x[n] only at discrete-time instants tn given by:

( )ax t

2n

T T

n nt nT

F

With FT =1/T denoting the sampling frequency and ΩT = 2π FT denoting the sampling angular frequency

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• Consider the continuous-time signal

x(t) = Acos(2πfot + φ) = Acos(Ωot + φ)

• The corresponding discrete-time signal is:

• Where: ωo = 2πΩo /ΩT = ΩoT is the

normalized digital angular frequency

of x[n]

00

0

2[ ] cos( ) cos( )

cos( )T

x n A nT A n

A n

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• If the unit of the sampling period T is in

seconds, then:

-the unit of the analog frequency f0 is hertz

-the unit of the normalized analog angular

frequency Ω0 is radians/second

-the unit of the normalized digital angular

frequency ωo is radians/sample

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• Example: Determine the discrete-time

signal v[n] obtained by uniformly sampling

at a sampling rate of 200 Hz the continuous

time signal:

v(t)=6cos(60πt)+2cos(100 πt)+10sin(140 πt)

composed of a weighted sum of 3 sinusoidal

signals of frequencies 30Hz, 50 Hz, and 70Hz

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1.6 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:

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Discrete-Time Systems: Examples

• Examples of 2-input, 1-output discrete-time systems are e.g. the modulator and the adder

• Examples of 1-input, 1-output discrete-time systems are e.g. the multiplier, the unit delay, the unit advance and the discrete-time system as shown earlier:

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Discrete-Time Systems: Examples• Up-sampling– process of increasing the

sampling rate of a signal. Eg Application: Up-sampling image data such

as photograph means increase the resolution of the photograph.

• Down-Sampling-also known as sub-sampling means process of reducing the sampling rate of a signal

Eg Application: Down-sampling to reduce the data rate or the size of the data

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1.7 Classification of Discrete-Time Systems

• Classification of discrete-time systems:

-Linear Systems

-Shift-Invariant Systems

-Causal Systems

-Stable Systems

-Passive and Lossless Systems

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Linear Systems

• Linear systems have the property that if:

x1[n]y1[n] and x2[n]y2[n] , then:

x[n]=αx1[n]+ßx2[n] y[n]= αy1[n]+ßy2[n]

• If input consists of a sum of scaled sequences,

then the corresponding output is a sum of

scaled outputs corresponding to the individual

input sequences

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α x1[n] y1[n] w[n]

x2[n] y2[n]

ß

x1[n] α

x[n] y[n]

x2[n]

ß

System

System

X

X+

X

X

+ System

w[n]=y[n] when system is linear!

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Shift-Invariant Systems• A discrete time system is said to be time invariance if an input is delayed(shifted) by

n0, the output is delayed(shifted) by the same amount:

x[n-n0]y[n-n0]Eg: Test the sequence below if they are time invariant or not:1) y[n]=(x[n])2

2) y[n]=x[-n]3) y[n]=nx[n]

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Shift-Invariant Systems

• Illustration of the up-sampling operation:

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Linear Time-Invariant Systems• Linear Time-Invariant (LTI) System:• A system satisfying both the linearity and• the time-invariance property is called a LTI system• 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

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Causal Systems

• In a causal system, the n-th output sample

y[n0] depends only on input samples x[n] for n

≤ no and does not depend on input samples for

n > no

• Simply speaking, for a causal system,

changes in output samples do not precede

changes in the input samples

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Causal Systems

• Examples of causal systems:

1) discrete-time system given earlier:

• Example of a non-causal system:

factor-of-2 interpolator:

y[n]=xu[n]+1/2{xu[n-1]+xu[n+1]}

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Stable Systems

• There are various definitions of stability

• We consider here the bounded-input,

bounded-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, where Bx and By are finite

constants

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Convolution and LTI System• From LTI system below, impulse response h[n] is

simply the output when the input is unit impulse sequence δ[n]

• Any signal x[n] can be represented as sum of scaled and shifted impulse signals:

[ ] [ ] [ ]

... [ 2] [ 2] [ 1] [ 1] [0] [ ]

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

lx n x n n k

x n x n x n

x n x n

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Convolution and LTI System• x[n] is a representation of linear combination

of scaled, shifted impulses.• Since LTI systems respond in simple and

predictable ways to sum of signals and to shifted signals, this representation is particularly useful for deriving a general formula of LTI system

• From figure, response to input δ[n] is by definition the impulse response h[n]

• Time Invariance gives information response due to δ[n-1] is h[n-1]

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Convolution and LTI System

• Therefore we can write the whole family of input-output pairs:

δ[n]h[n]

δ[n-1]h[n-1]

δ[n-2]h[n-2]

δ[n-l]h[n-l]

• Now we are in position to use linearity, because x[n] express a general input signal as a linear combination of shifted impulse signals

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Convolution and LTI Systemx[0]δ[n]x[0]h[n]

x[1]δ[n-1]x[1]h[n-1]x[2]δ[n-2]x[2]h[n-2]x[l]δ[n-l]x[l]h[n-l]

• The we use superposition to put it all together:

CONVOLUTION SUM

OR

[ ] [ ] [ ]l

x n x n n l

[ ] [ ] [ ]l

y n h n x n l [ ] [ ] [ ]l

y n x n h n l

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• Properties of the Convolution Sum:

1) Commutative property:

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

2) Associative property:

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

3) Distributive property:

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

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Computation of the Convolution Sum:

• 1) Time-reverse h[k] to form h[−k]

• 2) Shift h[−k] to the right by n sampling

periods 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[n] of the convolution

sum

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• Schematic Representation of Convolution:

• 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

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• 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

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• If both the input sequence and the impulse

response sequence are of infinite length, the 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

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• Example: Develop the sequence y[n]

generated by the convolution of the sequences x[n] and h[n] shown below

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Time-Domain Characterization

of LTI Discrete-Time Systems

• The sequence {y[n]} generated by the

convolution sum is shown below:

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• 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

• Convolution Using MATLAB

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

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• Convolution Using MATLAB

Example:

a = [− 2 0 1 −1 3];

b = [1 2 0 -1];

Then conv(a,b) yields:

[−2 −4 1 3 1 5 1 −3]

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Simple Interconnection

Schemes of LTI Systems

• Two simple interconnection schemes of LTI systems are:

1) Cascade Connection

2) Parallel Connection

• They are widely used for developing

complex LTI systems from simple LTI

system sections

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Simple InterconnectionSchemes of LTI Systems

• 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:

h[n] = h1[n] * h2[n]

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• 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

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• Parallel Connection

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

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

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• Consider the LTI discrete-time system where:

1

2

3

4

1[ ] [ ] [ 1]

21 1

[ ] [ ] [ 1]2 4

[ ] 2 [ ]

1[ ] 1( ) [ ]

2n

h n n n

h n n n

h n n

h n n

88



• Simplifying the block-diagram we obtain:

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• Overall impulse response h[n] is given by:

h[n] = h1[n]+ h2[n] * (h3[n]+h4[n])

= h1[n]+ h2[n]*h3[n]+h2[n]*h4[n]

• Now:

2 3

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

2 41

[ ] [ 1]2

h n h n n n n

n n

90



2 4

1

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

2 4 2

1 1 1[ ] [ 1]

2 2 2

1 1[ ] [ 1]

2 2

11 [ ] [ ]

2

:

1 1[ ] [ ] [ 1] [ ] [ 1] [ ] [ ]

2 2

n

n n

n n

n

h n h n n n n

n n

n n

n n

Therefore

h n n n n n n n

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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 output of the system

• {pk} and {xk} are constants characterizing the system

0 0[ ] [ ]

N M

k kk k

d y n k p x n k

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• The order of the LTI 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 even

though such a system, in general, has an

impulse response of infinite length

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

provided d0 ≠0

1 00 0

[ ] . [ ] . [ ]N Mk k

k k

d py n y n k x n k

d d

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Classification of LTI


Based on Impulse Response Length

• If h[n] is of finite length,Eg:

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

then it is known as a finite impulse

response (FIR) discrete-time system

• The convolution sum reduces to:2

1

[ ] [ ] [ ]N

k Ny n h k x n k

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• 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

96



• If the impulse response is of infinite length,

then it is known as an infinite impulse

response (IIR) discrete-time system

• For a causal IIR system with a causal input x[n], the convolution sum can be expressed in the form:

• This convolution sum can be used to compute

the output samples

0[ ] [ ] [ ]

n

ky n h k x n k

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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

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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

99


• Example: In radar and sonar applications, the

received signal reflected from the target is the

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

100


Definitions

• A measure of similarity between a pair of

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

l=0,+1, +2,…

• • The parameter l called lag, indicates the

• time-shift between the pair of signals

[ ] [ ] [ ], xyn

r l x n y n l

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• 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 l

• 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]

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• The autocorrelation sequence of x[n] is

given by

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

[ ] [ ] [ ], xyn

r l x n y n l

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An examination of the expression for the

cross-correlation

reveals that it looks quite similar to that of

the linear convolution

• This similarity is much clearer if we rewrite

the expression for the cross-correlation as:

[ ] [ ] [ ], xyn

r l x n y n l

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

r l x n y l n x l y l

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• 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]

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Correlation Computation

Using MATLAB

• The cross-correlation and autocorrelation

sequences can easily be computed using

MATLAB, xcorr

1 1. time domain representation of signals and systems 1.1 discrete-time signals 1.2 operations on...

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