discrete-time signal and systems

Chap 2Discrete-Time Signal and

Systems

2018/9/18 2

2.1 Discrete-Time Signals: Sequences

Continuous-Time Signal: xa(t)

Discrete-Time Signals are represented mathematically as sequences of numbers.

A sequence of number x is x = {x[n]}

Where x[n] is the n-th number in the sequence.

n is an integer.

Relationship between discrete-time signal and continuous-time signal is

T is called the sampling period, and its reciprocal is the sampling frequency.

( ),ax n x nT n

Figure 2.1 Graphic representation of a discrete-time signal.

2018/9/18 3

Figure 2.2 (a) Segment of a continuous-time speech signal xa(t ). (b)

Sequence of samples x[n] = xa(nT ) obtained from the signal in part (a)

with T = 125 µs.

2018/9/18 4

2018/9/18 5

Basic Sequences

Unit Sample Sequence:

Unit Step Sequence:

Exponential Sequence:

If A and a are real numbers, then the the sequence is real.

Sinusoidal Sequence: x[n]= Acos[w0n + f], for all n,

= Acos[(w0 + 2pr)n + f]

where A and f are real constants; n and r are integer.

w0 and f are frequency and phase of the complex sinusoid, respectively.

So, -p < w0 <= p or 0 <= w0 < 2p

0, 0

1, 0

nn

n

1, 0

0, 0

nu n

n

n

k

u n k

1 2 ...u n n n n 0k

u n n k

1n u n u n

nx n Aa

Figure 2.3 Some basic sequences. The sequences shown play

important roles in the analysis and representation of discrete-time

signals and systems.

2018/9/18 6

2018/9/18 7

Fig2.4 Sequence as a sum of scaled, delayed impulses

p[n] = a-3[n+3] + a1[n-1] + a2[n-2] + a7[n-7]

k

x n x k n k

Any sequence

2018/9/18 8

Sequence Operations

The product and sum of two sequences are defined as the sample-by-sample product and sum, respectively.

Multiplication of a sequence by a number is defined as multiplication of each sample value by this number.

A sequence y[n] is said to be a delayed or shifted version of a sequence x[n] ify[n] = x[n – nd], where nd is an integer.

Example 2.1 Combining Basic Sequences

x[n]= Aan, n >= 0,

= 0, n < 0,

or x[n] = Aanu[n].

2018/9/18 9

Differences between continuous-time and discrete-time concerns their periodicity

In the continuous-time case, a sinusoidal signal and a complex exponential signal are both periodic, with the period equal to 2p divided by the frequency.

In the discrete-time case, a periodic sequence is a sequence for which

x[n] = x[n + N], for all n,

where the period N is necessarily an integer.

For the discrete-time sinusoid

A cos[w0n + f] = A cos[w0n + w0N + f]

Which w0N = 2pk ,

where k is an integer.

Period of Continuous-Time Signals

2018/9/18 10

In physics, angular frequency ω

(also referred to by the terms

radian frequency) is a scalar

measure of rotation rate

One revolution is equal to 2π

radians

ω is the angular frequency

(radians per second),

T is the period (in seconds),

f is the ordinary frequency (in

hertz)

ω = 2π f = 2π/T

T=2π/ ω

2018/9/18 11

Example 2.1 Periodic and Aperiodic Discrete-Time Sinusoids

Increasing the frequency of a discrete-time sinusoid does not necessarily decrease the period of the signal.

Example: x1[n] = cos[pn /4] has a period of N = 8, since

x1[n + 8] = cos[p(n + 8)/4] = cos[pn /4] = x1[n].

But x2[n] = cos(3pn /8] has a period of N = 16, because

x2[n + 8] = cos[3p(n + 8)/8] = cos[3pn/8 + 3p] = -x2 [n]

For a continuous-time sinusoidal signal, as frequency increases, it oscillates more and more rapidly.

The integer restriction on n causes some sinusoidal signals not to be periodic at all.

Example: There is no integer N such that the signal x[n] = cos[n] satisfies the condition x[n + N] = x[n] for all n.

There are N distinguishable frequencies for which the corresponding sequences are periodic with period N.

One set of frequencies is wk = 2pk /N, k = 0, 1, …, N-1.

2018/9/18 12

Fig 2.5 For the discrete-time sinusoidal signal, as its frequency increases

from 0 toward p, it oscillates more and more rapidly. However, as its frequency increases from p to 2p, the oscillations become slower.

2018/9/18 13

2.2 Discrete-Time Systems

The discrete-time system is defined mathematically as a transformation or operator that maps an input sequence with values x[n] into an output sequence with values y[n].

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

Classes of systems are defined by placing constraints on the properties of the transformation T{.}.

Examples of some simple and useful systems

Ideal Delay System: y[n]= x[n – nd]

Moving Average System:

y[n]= (Sx[n – k])/(M1+M2+1)

Memoryless System: y[n]= (x[n])2 , nd = 0

Accumulator System: y[n]= Sx[k] , k = [-infinity,n]

Compressor System: y[n]= x[Mn] , M = + integer

Forward Difference System: y[n]= x[n + 1] – x[n]

Backward Difference System: y[n]= x[n] – x[n – 1]

2018/9/18 14

2018/9/18 15

Example 2.3 Moving Average

2

11 2

1 1

1 2

2

1[ ] [ ]

1

1{ [ ] [ 1] ... [ ]

1

[ 1] .. [ ]}

M

k M

y n x n kM M

x n M x n M x nM M

x n x n M

The general moving average system is defined by the equation

This system computes the nth sample of the output sequence as the average of (M1+M2+1) samples of the input sequence around the nth sample.

2018/9/18 16

Example 2.3 Moving Average

Figure 2.7 Sequence values involved in computing a moving average

with M1 = 0 and M2 = 5.

2018/9/18 17

Discrete-Time System

2.2.1 Memoryless System: if the output y[n] at every value of n depends only on the input x[n] at the same value of n.

2

for each value of .y n x n n


2.2.2 Linear System is defined by the principle of superposition:

Additivity Property:

Homogeneity (Scaling) Property:

Example 2.6 Nonlinear System: y[n]=log10(|x[n]|)2018/9/18 18

1 2 1 2 1 2T x n x n T x n T x n y n y n

T ax n aT x n ay n

1 2 1 2T ax n bx n aT x n bT x n

2018/9/18 19


2.2.3 Time-Invariant System is a system for which a time shift or delay of the input sequence causes a corresponding shift in the output sequence. Equivalent to Shift-Invariant System

Example 2.8: The compressor system

Consider the response

Delaying the output y[n] by n0 samples

NOT time-invariant:

[ ], -y n x Mn n

1 1 0[ ] [ ] [ ]y n x Mn x Mn n

0 0[ ] [ ( )]y n n x M n n

1 0 0[ ] [ ] y n y n n for all M and n


2.2.4 Causal System: for every n0, the output sequence value at the index n = n0 depends only on the input sequence values for n <= n0.

Example 2.9

The forward difference system– NOT causal

The backward difference system– causal

2018/9/18 20

[ ] [ 1] [ ]y n x n x n

[ ] [ ] [ 1]y n x n x n


2.2.5 Stable System: is stable in the BIBO sense iff every bounded input sequence produces a bounded output sequence.

, for allxx n B n.

, for allyy n B n.

Example 2.10 Testing for stability or instability

10[ ] log ( [ ] ) , when [ ] 0y n x n x n

2018/9/18 22

2.3 Linear Time-Invariant System

Defining the output for an unit sample input as the Unit Sample Response

Let hk[n] be the response of the system to [n-k], an impulse occurring at n=k. Then

System

0 n n0

x[n]=[n] y[n]=h[n]:unit sample response

k

y n T x k n k

2018/9/18 23


From principle of superposition

For linearity, hk[n] will depend on both n and k. If property of time invariance implies hk[n] will depend only on n, then

hk[n]=h[n – k]=T{[n – k]}

So, y[n]=Sx[k]h[n-k]

Convolution sum

k

k k

y n x k T n k x k h n

for allk

y n x k h n k n.

2018/9/18 24


By Linearity (Superposition Property)

The output for an arbitrary input signal is the superposition of a series of “shifted, scaled unit sample response”---- look at the index k

That is a linear time-invariant system is completely characterized by its impulse response h[n].

[ ] [ ] [ ]

[ ] [ ] [ ]

k

kk k

x n x k n k

y n x k h n k

Contribution of the k-th input x[k] to the output signal

2018/9/18 25

Fig 2.8 Representation of the output of an LTI system as the

superposition of responses to individual samples of the input

2018/9/18 26

Computation of Convolution Sum

k

knhkxny ][][][

Contribution to the output signal at time n

input signal

reflected-over version of h[k] located at k = n

A Different Way to visualize the convolution sum ---- looked at the index n

2018/9/18 27


Suppose h[k] is the sequence shown in Fig.2.9(a).

First consider h[-k] plotted against k; h[-k] is simply h[k]

reflected or “flipped” about k=0, as shown in Fig.2.9(b).

Replacing k by k-n, where n is a fixed integer, leads to a

shift of the origin of the sequence h[-k] to k=n , as shown in

Fig.2.9(c) for the case n=4.

Generalizing Example 2.12 , it should be clear that in

general the sequence h[n-k], - < k< , is

obtained by

1. Reflecting h[k] about the origin to obtain h[-k];

2. Shifting the origin of the reflected sequence to

k=n

2018/9/18 28


2018/9/18 29

Example 2.11

n

[ ] [ ] [ ]

1, 0 n N-1

0, otherwise

The input is x[n]=a [ ]

h n u n u n N

u n

( 0, 1 )

[ ] [ ] [ ]k

a a convergence

y n x k h n k

Consider a system with impulse response

2018/9/18 30

Example 2.11 Figure 2.10

2018/9/18 31

Example 2.11

:

1, n-(N-1) k n[ ]

0, otherwise

Sol

h n k

(1) n<0 => x[k] and h[n-k] do not overlap

=>y[n]=0, n<0

n

k=0

n+1

(2) 0 1 ( n-(N-1) 0, [ ] [ ] , 0 ~ )

=>y[n]= , 0 1

1-a = , 0 1

1

k

n N x k h n k k n

a n N

n Na

有值

2018/9/18 32

Example 2.11

n-N+1

n

k=n-N+1

n-N+1 1

N

(3) n>N-1

=>y[n]= , N-1<n

a =

1-a

1-a =a ( ), N-1<n

1-a

k

n

a

a

1

1

0, n<0

1[ ] , 0 1

1

1( ), N-1<n

1

n

Nn N

ay n n N

a

aa

a

2018/9/18 33

2.4 Properties of Linear Time-Invariant Systems

Properties of the convolution operation

Commutative property(交換律): x[n]*h[n] = h[n]*x[n]

Associative(結合律):

(x[n]*h1[n])*h2[n]=x[n]* (h1[n]*h2[n])

2018/9/18 34

2.4 Properties of Linear Time-Invariant

Systems

Properties of the convolution operation

Distributive property(分配律):

x[n]*(h1[n]+h2[n]) = x[n]* h1[n]+ x[n]*h2 [n]

2018/9/18 35

2.4 Properties of Linear Time-Invariant Systems

The impulse response of a cascadecombination of LTI systems is independent of the order in which they are cascaded.

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

The impulse response of a parallelconnection of LTI systems is the sum of the individual impulse responses.

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

2018/9/18 36

Stability

Linear Time Invariant Systems are stable iff the impulse response is absolutely summable

This can be shown as follows. If bounded input gives bounded output (BIBO). If |x[n]|<B, for all n

kkk

]k[hB]kn[x]k[h]kn[x]k[h]n[y

| [ ] |k

S h k

2018/9/18 37

Impulse Response of LTI Systems

To find the impulse responses of the systems, it can be done by simply computing the response of each system to [n]. Ideal Delay: h[n] = [n – nd] , nd a positive fixed integer.

Moving Average: h[n] =

Accumulator: h[n] = u[n].

Forward Difference: h[n] = [n + 1] – [n].

Backward Difference: h[n] = [n] – [n - 1].

The systems which their impulse responses have only a finite number of nonzero samples are called finite-duration impulse response (FIR) systems. But if their impulse responses are infinite duration, they will be called infinite-duration impulse response (IIR) systems.

If h[n] = 0 for n < 0, the system is causal.

2

11

1

21

M

M

]kn[MM

2018/9/18 38

Commutative property of convolution

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

= [ 1] ( [ 1] [ ])

= [ ] [ 1]

h n n n n

n n n

n n

Equivalent systems

2018/9/18 39

Identity system and Inverse system

[ ] [ ] ( [ ] [ 1])

= [ ] [ 1]

= [ ]

h n u n n n

u n u n

n

2018/9/18 40

2.5 Linear Constant-Coefficient Difference Equations

The Nth-order linear constant-coefficient difference equation:

The output y[n] = yp[n] + yh[n], where yh[n] is the homogeneous solution obtained from the homogeneous equation:

The sequence yh[n] is in a member of a family of solutions of the form

where complex number zm are roots of the polynomial

N M

mk mnxbknya0 0

][][

N

hk ]kn[ya0

0

1

[ ]N

n

h m my n A z

0

0N

k

kk

a z

2018/9/18 41

Linear Constant-Coefficient Difference Equations (cont.)

The recursive representation of the system is

If every sample of the input x[n], together with a set of auxiliary values, y[-1]; y[-2]; …; y[-N], is specified, then y[n] can be determined.

To generate values of y[n] for n < -N, we can rearrange in the form

N M

mk mnxa

bkny

a

any

1 0 00

][][][

1

0 0

][][][N M

N

m

N

k mnxa

bkny

a

aNny

2018/9/18 42

Example 2.12

[ ] [ 1] [ ] : First -ordery n ay n x n

n

n

x[n]= [n]

=> 1. y[n]=a [ ] : causal , a 1

2. y[n]=-a [ 1] : noncausal , a 1

if

u n stable

u n stable

Figure 2.15 Block diagram of a recursive difference equation

representing an accumulator.

2018/9/18 43

Example 2.12

: (1) Initial rest conditions: causal

Let x[n]= [n]

Then y[n]=0, n<0

y[0]=a y[-1]+1=1

y[1]=a y[0]=a

.

sol

.

n y[n]=a y[n-1]=a

n thus y[n] =a u[n]

(2) Assume y[n]=0, for n>0 : noncausal

Let x[n]= [n]

1 y[n-1]= ( [ ] [ ])

a

1 or y[n] = ( [ 1] [ 1])

a

Then y[n]=0 , n>0

1 y[0]= ( [1] [1]) 0

a

y[-

y n x n

y n x n

y x

1

Thus

11]= ( [0] [0])

a

.

.

1 y[n]= [ 1]

a

y[n]= [ 1]

n

n

y x a

y n a

a u n

2018/9/18 44

Example 2.13

2018/9/18 45

Example 2.13

2018/9/18 46

Example 2.13

For a given input, a linear constant-coefficient difference equation for discrete time systems does not provide a unique specification of the output

N

k=0

代入上式 =>

有

p

h

N

k=0

N

hm=1

[ ] [ ] [ ]

[ ] is determined by x [n]

y [n] is any solution to the equation with x[n]=0

[ ] 0 : Homogeneous equation

=> y [n]=

0

p h

p

k h

nm m

kk

y n y n y n

y n

a y n k

A z

a z

個根N

2018/9/18 47

Example 2.13

Since yh[n] has N undetermined coefficients , a set of N auxiliary conditions is required for unique specification of y[n] for a given x[n]

If the input x[n] together with a set of auxiliary values , say y[-1], y[-2],..,y[-N], is specified, then y[0] can be determined from.Eq(2.97).

With y[0],y[-1],..,y[-N+1] available, y[1] can then be calculated, and so on.

Using this procedure, y[n] is said to be computed recursively; i.e., the output computation involves not only the input sequence but also previous values of the output sequence.

1 00 0

[ ] [ ] [ ], (2.97)N M

k k

k k

a by n y n k x n k

a a

2018/9/18 48

Summarization for a system for which the input and output satisfy a linear constant-coefficient difference equation

The output for a given input is not uniquely specified. Auxiliary information or conditions are required.

If the auxiliary information is in the form of N sequential values of the output, later values can be obtained by rearranging the difference equation as a recursive relation running forward in n, and prior values can be obtained by rearranging the difference equation as a recursive relation running backward in n.

Linearity, time invariance, and causality of the system will depend on the auxiliary conditions. If an additional condition is that the system is initially at rest (called initial-rest conditions), then the system will be linear, time invariant, and causal.

2018/9/18 49

Example of Recursive Computation of Difference Equation

Given y[n] = ay[n-1] + x[n], if x[n] = K[n] and y[-1] = cy[0]= ac + Ky[1]= ay[0] + 0= a2c + aKy[2]= ay[1] + 0= a3c + a2Ky[3]= ay[2] + 0= a4c + a3K

… = … = …y[n]= ay[n - 1] + 0 = an+1c + anK, for n >= 0.

y[n –1]= a-1(y[n] – x[n]) or y[n] = a-1 (y[n + 1] – x[n + 1])y[-2]= a-1(y[-1] – x[-1])= a-1cy[-3]= a-1(y[-2] – x[-2])= a-2cy[-4]= a-1 (y[-3] – x[-3])= a-3c

… = … = …y[n]= a-1 (y[n + 1] – x[n + 1])= an+1c, for n <= -1.

In sum,

y[n]= an+1c+Kanu[n], for all n

2018/9/18 50

2.6 Frequency-Domain Representation ofDiscrete-Time Signals and Systems

Consider an input sequence

The corresponding output of a linear time-invariant system with impulse response h[n] is

If we define

The above equation becomes

[ ] ,j nx n e nw

( )[ ] [ ] ( [ ] )j n k j n j k

k k

y n h k e e h k ew w w

( ) [ ]j j k

k

H e h k ew w

[ ] ( )j j ny n H e ew w

2018/9/18 51


ejwn: Eigenfunction of the SystemThe associated eigenvalue H(ejw) describes

the change in complex amplitude of a complex exponential input signal as a function of the frequency w, and is called the frequency response of the system.

In general, H(ejw) is complex and can be expressed in terms of its real and imaginary parts as

H(ejw) = HR(ejw) + jHI(e

jw)

= |H(ejw)|ejARG{H(.)}

2018/9/18 52


Eigenfunction and eigenvalue

What is the eigenfunction of a system T{.}?

Cf[n]=T{f[n]} , where C is a complex constant, eigenvalue.

The output waveform has the same shape of the input waveform.

The complex exponential sequence is the eigenfunction of any LTI system.

2018/9/18 53

A broad class signals can be represented as a linear combination of complex exponentials in the form

From the principle of superposition, the corresponding output of a LTI system is

[ ] kj n

k

k

x n ewa


[ ] ( )k kj j n

k

k

y n H e ew wa

2018/9/18 54

2018/9/18 55


The concept of the frequency response of LTI systems is essentially the same for continuous-time and discrete-time systems. However, an important distinction arises because the frequency response of discrete-time linear time-invariant systems is always a periodic function of the frequency variable w with period 2p.

( 2 ) ( 2 )

2

( 2 )

( ) [ ]

1

( ) ( ), int

j j n

n

j n

j r j

H e h n e

e

H e H e for r an eger

w p w p

p

w p w

Figure 2.17 Ideal lowpass filter showing (a) periodicity of the frequency

response and (b) one period of the periodic frequency response.

Figure 2.18 Ideal frequency-selective filters. (a) Highpass filter. (b) Bandstop

filter. (c) Bandpass filter. In each case, the frequency response is periodic with

period 2π. Only one period is shown.

2018/9/18 58

Example 2.16 Frequency Response of the Moving-Average System

2

11 2

1eq. 2.121

1

Mj j n

n M

H e eM M

w w

2

1

02

10

1

Mj j n

n

M H e eM

w w

The impulse response of the moving average system of example 2.3 is

The frequency response is

1 2

1 2

1,

1[ ]

0, Otherwise

M n MM Mh n

Example 2.16 Frequency Response of the Moving-Average System

2

2 2 2

2

1

2

1 /2 1 /2 1 /2

/2 /2 /22

2 /2

2

1 1eq. 2.123

1 1

1 ( )

1

sin 1 / 21

1 sin / 2

j M

j

j

j M j M j M

j j j

j M

eH e

M e

e e e

M e e e

Me

M

ww

w

w w w

w w w

ww

w

2

2

sin 2 1 / 21eq. 2.124

2 1 sin / 2

jM

H eM

ww

w

If the moving average is symmetric, i.e., if M1=M2, then

Figure 2.19 (a) Magnitude and (b) phase of the frequency response of

the moving-average system for the case

M1 = 0 and M2 = 4.

2018/9/18 61

2.6.2 Suddenly Applied Complex Exponential Inputs

The above eigenfunction analysis is valid when the input is applied to the system at n=-infinity .

What happens if the input applies to a causal LTI system at n=0?

2018/9/18 62

2.6.2 Suddenly Applied Complex Exponential Inputs

Consider n>=0

Figure 2.20 Illustration of a real part of suddenly applied complex

exponential input with (a) FIR and (b) IIR.

The solid dots indicate the samples x[k] of the suddenly applied complex exponential input.

In the FIR case, the output would consist of steady-state component for n>=8

In the IIR case, the “missing” samples have less and less effect as n increases.

2018/9/18 64

2.7 Representation of Sequences byFourier Transform

Forward Fourier Transform: Analysis

, X(ejw) is periodic with period of 2p.

It may be called discrete-time Fourier transform (DTFT).

Inverse Fourier Transform: Synthesis

We can expressed X(ejw) in various forms.

Rectangular form: XR(ejw) + j XI(e

jw)

Polar form:

X(ejw) may be called the Fourier spectrum or, simply, the spectrum .

njj enxeX ww ][)(

p

p

ww wp

de)e(X]n[x njj

21

( )( ) | ( ) |jj j j X eX e X e eww w

2018/9/1865

The phase is not uniquely specified, since any integer multiple of may be added to at any value of

without affecting the result of the complex exponentiation

Principle value

Denoted as

If we want to refer to a continuous phase function for , we use Continuous phase function

( )jX e w

2p

( )jX e w

( )jX e wp p

[ ( )]jARG X e w

arg[ ( )]jX e w

0 w p

w

2018/9/18 66

2018/9/18 67

2018/9/18 68

Example 2.18 Square-Summability for the Ideal Lowpass Filter

lp

lp

h [n] 0, for n<0 => noncausal

h [n] is not absolutely summable

2018/9/18 69


M

cM

( )

Consider H ( ) as the sum of a finite number of terms

sin w H ( )

n

1 = [ ]

2

1 =

2

c

c

jw

Mjw jwn

n M

wMj n jwn

n M w

j w n

n

e

ne e

e d e

e

p

p

p

1 sin[(2 1)( ) / 2] =

2 sin[( ) / 2]

c

c

c

c

w M

Mw

w

w

d

M wd

w

p

sin[ ] is not obsolutely summable

sin sin 1ln

n=-

sin

n

w nch n

lp n

w n w nc c dn dn n

n n n

w n nc

p

p p

不隨變大而變小

Figure 2.21 Convergence of the Fourier transform. The oscillatory

behavior at ω = ωc is often called the Gibbs phenomenon.

2018/9/18 71


M => oscillatory behavior at w=w (Gibbs phenomenon)c

is more rapid

M => oscillations converge in location toward the point w=wc

=>no uniform convergence

h [n] is sqlp

uare summable, and

2 lim ( ) ( ) 0

M

jw jwH e H e dw

lp Mpp

2018/9/18 72

Special Functions

2018/9/18 73

Special Functions

2018/9/18 74

2.8 Symmetry Properties ofthe Fourier Transform

They are often useful for simplifying the solution of problems.

A conjugate-symmetric sequence xe[n] is defined as a sequence for which xe[n] = xe*[-n].

A conjugate-antisymmetric sequence xo[n] is defined as a sequence for which xo[n] = -xo*[-n].

Any sequence x[n] = xe[n] + xo[n]

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

xo[n] = (x[n] - x*[-n])/2 = -xo*[-n]

A real sequence that is conjugate symmetric is called an even sequence xe[n] = xe[-n]

A real sequence that is conjugate antisymmetric is called an odd sequence xo[n] = -xo[-n].

Table 2.1 SYMMETRY PROPERTIES OF THE FOURIER

TRANSFORM

2018/9/18 76

Example 2.21 Illustration of Symmetry Properties

Figure 2.22 Frequency response for a system with impulse response h[n] = anu[n]. (a)

Real part. a > 0; a = 0.75 (solid curve) and a = 0.5 (dashed curve). (b) Imaginary part. (c)

Magnitude. a > 0; a = 0.75 (solid curve) and a = 0.5 (dashed curve). (d) Phase.

2018/9/18 78

2.9 Fourier Transform Theorem

-1

-1

:

( ) { [ ]},

[ ] { ( )},

[ ] ( )

That is , F denotes the operation of

"taking the Fourier transform of x[n],"

and F is the inverse of that operation.

Operator notationjw

jw

F jw

X e F x n

x n F X e

x n X e

2018/9/18 79


Linearity Theoremax[n] + by[n] aX(ejw) + bY(ejw)

Time Shifting and Frequency Shiftingx[n – nd] e-jwnd X(ejw)

ejw0nx[n] X(ej(ww0)

Time Reversalx[-n] X(e-jw),

if x[n] real: x[-n] X*(ejw)

Differentiation in Frequencynx[n] jdX(ejw)/dw

2018/9/18 80


Parseval’s Theorem, |X(ejw)|2 is called energy density spectrum.

The Convolution Theoremy[n]= x[n]*h[n] Y(ejw)=X(ejw)H(ejw),

The Modulation or Windowing Theorem

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

22 1[ ] ( )

2

jE x n X e dp

w

p

wp

p

p

ww wp

deYeXnynx jj )(*)(21

][*][

( )1( ) ( ) ( )

2

j j jY e X e W e dp

w w

p

p

2018/9/18 81

2018/9/18 82

2.10 Discrete-Time Random Signals

A stochastic signal is considered to be a member of an ensemble of discrete-time signals that is characterized by a set of probability density functions.

While stochastic signals are not absolutely summable or square summable and, consequently, do not directly have Fourier transforms, many (but not all) of the properties of such signals can be summarized in terms of averages such as the autocorrelation or autocovariance sequence, for which the Fourier transform often exists.

The Fourier transform of the autocovariance sequence has a useful interpretation in terms of the frequency distribution of the power in the signal.

The effect of processing stochastic signals with a discrete-time linear system can be conveniently described in terms of the effect of the system on the autocovariance sequence.

2018/9/18 83

Discrete-Time Random Signals (cont.)

If sequence x[n] is stationary, it may be characterized by its means mx and its autocorrelation function fxx[m], or we may also have additional information about first or second order probability distributions.

For many applications, it is sufficient to characterize both the input and output in terms of simple averages, such as the mean, variance, and autocorrelation.

The cross-correlation between input and output is the convolution of the impulse response with the input autocorrelation sequence. Then:

Fxy(e jw) = H(e jw)Fxx(e jw) (see the next page)

If the input is zero mean white noise for which fxx[m] = sx

2 [m]:

Fxy[m] = sx2 H(e jw)

2018/9/18 84

( ) [ ] [ ]

[ ] [ ] [ ]

[ ] [ ]

F

j

xy

k

xxk

e Ε x n y n m

=Ε x n h k x n m k

= h k m k

w

f

Project of Chapter 2

2018/9/18 85

discrete-time signal and systems

Documents