Chapter 3 Discrete-Time Fourier Transform. §3.1 The Continuous-Time Fourier Transform Definition...

Chapter 3

Discrete-Time FouriDiscrete-Time Fourier Transformer Transform

§3.1 The Continuous-Time FourierTransform

Definition – The CTFT of a continuous- time signal xa(t) is given by

dtetxjX tjaa

－

)()(

Often referred to as the Fourier spectrum or simply the spectrum of the continuous-time signal


Definition – The inverse CTFT of a Fourier transform Xa(jΩ) is given by

dejXtx tjaa

－

)(2

1)(

)()(CTFT

jXtx aa

Often referred to as the Fourier integral A CTFT pair will be denoted as


Ω is real and denotes the continuous-time angular frequency variable in radians

In general, the CTFT is a complex function of Ω in the range -∞<Ω< ∞

It can be expressed in the polar form as)()()( aj

aa ejXjX

)(arg)( jX aawhere


The quantity |Xa(jΩ)| is called the magnitude spectrum and the quantity θa(Ω) is called the phase spectrum

Both spectrums are real functions of Ω In general, the CTFT Xa(jΩ) exists if xa(t) sati

sfies the Dirichlet conditions given on the next slide


Dirichlet Conditions (a) The signal xa(t) has a finite number of

discontinuities and a finite number of maxima and minima in any finite interval

(b) The signal is absolutely integrable, i.e.,

dttxa

－

)(


If the Dirichlet conditions are satisfied, then

dejX tj

a

－

)(21

converges to xa(t) at values of t except at values of t where xa(t) has discontinuities

It can be shomn that if xa(t) is absolutely integrable,then |Xa(jΩ)|<∞ proving the existence of the CTFT

§3.1.2 Energy Density Spectrum

The total energy εx of a finite energy continuous-time complex signal xa(t) is given by

dttxtxdttx aaax )()()( *2

dtdejXtx tjaax

)(

21)( *

The above expression can be rewritten


Interchanging the order of the integration we get

djX

djXjX

ddtetxjX

a

aa

tjaax

－

－

－－

2

*

*

)(21

)()(21

)()(21


Hence

djXdttx aa

－－

22)(

21)(

The above relation is more commonly known as the Parseval’s relation for finite- energy continuous-time signals


The quantity |Xa(jΩ)|2 is called the energy density spectrum of xa(t) and usually denoted as

2)()( jXS axx

b

a

xxrx dS )(21

,

The energy over a specified range of frequencies Ωa≤Ω≤Ωb can be computed using

§3.1.3 Band-limited Continuous-Time Signals

A full-band, finite-energy, continuous-time signal has a spectrum occupying the whole frequency range -∞< Ω< ∞

A band-limited continuous-time signal has a spectrum that is limited to a portion of the frequency range -∞< Ω< ∞


An ideal band-limited signal has a spectrum that is zero outside a finite frequency range Ωa≤|Ω|≤Ωb, hat is

b

aa jX

,0

0,0)(

However, an ideal band-limited signal cannot be generated in practice


Band-limited signals are classified according to the frequency range where most of the signal’s is concentrated

A lowpass, continuous-time signal has a spectrum occupying the frequency range |Ω|≤ Ω

p<∞ where Ωp is called the bandwidth of the signal


A highpass, continuous-time signal has a spectrum occupying the frequency range 0< Ωp ≤|Ω|<∞ where the bandwidth of the signal is from Ωp to ∞

A bandpass, continuous-time signal has a spectrum occupying the frequency range 0< ΩL ≤|Ω| ≤ΩH<∞ where ΩH-ΩL is the bandwidth

§3.2 The Discrete-Time FourierTransform

Definition – The discrete-time Fourier transform (DTFT) X(ejω) of a sequence

x[n] is given by

n

njj enxeX ][)(

)()()( jim

jre

j ejXeXeX

In general, X(ejω) is a complex function of the real variable ω and can be written as


Xre(ejω) and Xim(ejω) are, respectively, the real and imaginary parts of X (ejω)， and are real function of ω

X (ejω) can alternately be expressed as )()()( jjj eeXeX

)(arg)( jeXwhere


|X (ejω)| is called the magnitude function θ(ω) is called the phase function Both quantities are again real functions of ω In many applications, the DTFT is called the

Fourier spectrum Likewise, |X (ejω)| and θ(ω) are called the ma

gnitude and phase spectra


For a real sequence x[n], |X (ejω)| and Xre(ejω) are even functions of ω, whereas, θ(ω) and Xim

(ejω) are odd function of ω Note:

）（

）＋（

）））

jj

2jjj

e| X(e | e| X(e | X(e

k

The phase function θ(ω) cannot be uniquely specified for any DTFT

for any integer k


Unless otherwise stated, we shall assume that the phase function θ(ω) is restricted to the following range of values:

-π≤θ(ω) ＜ π

called the principal value


The DTFTs of some sequences exhibit discontinuities of 2π in their phase responses

An alternate type of phase function that is a continuous function of ω is often used

It is derived from the original phase function by removing the discontinuities of 2π


The process of removing the discontinuities is called “unwrapping”

The continuous phase function generated

unwrapping is denoted as θc(ω) In some cases, discontinuities of π may be

present after unwrapping


Example – The DTFT of the unit sample sequence δ[n] is given by

1]0[][)(

nj

n

j ene

1],[][ nnx n

Example – Consider the causal sequence


Its DTFT is given by

1 je

0

0

11

][)(

nj

nnj

n

njn

n

njnj

ee

eeneX

）（

as


The magnitude and phase of the DTFT X (ejω)=1/(1-0.5 e-jω) are shown below


The DTFT X (ejω) of a sequence x(n) is a continuous function of ω

It is also a periodic function of ω with a period 2π:

)(][][

][)(

2

)2()2(

ooo

oo

j

n

njknj

n

nj

n

nkjkj

eXenxeenx

enxeX


represents the Fourier series representation of the periodic function

As a result, the Fourier coefficients x[n] can be computed from X (ejω) using the Fourier in

tegral

n

njj enxeX ][)(

deeXnx njj )(21

][

Therefore


Inverse discrete-time Fourier transform:

deeXnx njj )(21

][

deexnx njj

][21

][

Proof:


The order of integration and summation can be interchanged if the summation inside the brackets converges uniformly, i.e. X (ejω) exists

Then

)()(sin

][21][

][21

)(

nn

xdex

deex

nj

njj

－－


Now

][,0,1

)()(sin

nnn

nn

][][][)(

)(sin][ nxnx

nn

x

Hence


Convergence Condition – An infinite series of the form

n

njj enxeX ][)(

K

Kn

njjK enxeX ][)(

may or may not converge Let


Then for uniform convergence of X (ejω) ,

0)()(lim

jK

j

KeXeX

nnx ][

Now, if x[n] is an absolutely summable sequence, i.e., if


for all values of ω Thus, the absolute summability of x[n] is a s

ufficient condition for the existence of the DTFT X (ejω)

nn

njj nxenxeX ][][)(

Then


Example – The sequence x[n] = αnµ [n] for |α|<1 is absolutely summable as

11

][0n

n

n

n n

and its DTFT X (ejω) thereore converges to 1/(1-αe-jω) uniformly


an absolutely summable sequence has always a finite energy

However, a finite-energy sequence is not necessarily absolutely summable

,][][2

2

nnnxnx

Since


Example – The sequence

][nx 00

11

nnn

,,/

61 2

1

2

nx n

has a finite energy equal to

But, x[n] is not absolutely summable


To represent a finite energy sequence x[n]

that is not absolutely summable by a DTFT X

(ejω) , it is necessary to consider a mean-square convergence of X (ejω)：

0)()(lim 2deXeX jK

j

K

K

Kn

njjK enxeX ][)(

where


must approach zero at each value of ω as K goes to ∞

In such a case, the absolute value of the error |X (ejω)- XK (ejω)| may not go to zero as K goes to ∞ and the DTFT is no longer bounded

)()( jK

j eXeX

Here, the total energy of the error


Example – Consider the DTFT

c

cjLP eH

,0

0,1)(

-π π-ωc ωc0

HLP (ejω)

ω

1

shown below


The inverse DTFT of HLP (ejω) is given by

nn

njn

ejn

e

denh

cnjnj

njLP

cc

c

c

,sin

21

21][

hLP [n] is a finite-energy sequence, but it is not absolutely summable

The energy of hLP [n] is given by ωc/π


does not uniformly converge to HLP (ejω)

for all values of ω, but converges to HLP (ejω) in the mean-square sense

njK

Kn

cK

Kn

njLP e

nn

enh

sin][

As a result


The mean-square convergence property of the sequence hLP [n] can be further

illustrated by examining the plot of the function

njK

Kn

cjKLP e

nn

eH

sin

)(,

for various values of K as shown next


As can be seen from these plots, independent of the value of K there are ripples in the plot of HLP,K (ejω) around both sides of the point ω=ωc

The number of ripples increases as K increases with the height of the largest ripple remaining the same for all values of K


holds indicating the convergence of

HLP,K (ejω) to HLP (ejω) The oscillatory behavior of HLP,K (ejω) approxi

mating HLP (ejω) in the mean-square sense at a point of discontinuity is known as the Gibbs phenomenon

0)()(lim2

, deHeH jKLP

jLP

K

As K goes to infinity, the condition


The DTFT can also be defined for a certain class of sequences which are neither absolutely summable nor square summable

Examples of such sequences are the unit step sequence µ[n], the sinusoidal sequence cos(ω0n+φ) and the exponential sequence Aαn

For this type of sequences, a DTFT representation is possible using the Dirac delta function δ(ω)


A Dirac delta function δ(ω) is a function of ω with infinite height, zero width, and unit area

It is the limiting form of a unit area pulse function p∆(ω) as ∆ goes to zero satisfying

2

20

1

)(p

ddp )()(lim0


Example –Consider the complex exponential sequence

nj oenx ][

k

oj keX )2(2)(

where δ(ω) is an impulse function of ωand-π≤ω0≤π

Its DTFT is given by


is a periodic function of ω with a period 2πand is called a periodic impulse train

To verify that X(ejω) give above is indeed the DTFT of x[n]=ej0n we compute the inverse DTFT of X(ejω)

k

oj keX )2(2)(

The function


where we have used the sampling property of the impulse function δ(ω)

njnjo

k

njo

ede

deknx

0

)2(221][

＝）（

Thus

Commonly Used DTFT Pairs

j

kj

k

nj

k

en

ke

n

ke

k

n

11)1(,][

)2(1

1][

)2(2

)2(21

1][

00

Sequence DTFT

§3.3 DTFT Theorems

There are a number of important properties of the DTFT that are useful in signal processing applications

These are listed here without proof Their proofs are quite straightforward We illustrate the applications of some of the

DTFT properities

Table 3.1: DTFT Properties: Symmetry Relations

x[n]: A complex sequence

Table 3.2: DTFT Properties: Symmetry Relations

x[n]: A real sequence

Table 3.4:General Properties of DTFT

§3.3 DTFT Theorems

Example – Determine the DTFT Y(e jω) of y[n]=(n+1)n[n], ||<1

Let x[n]= n[n], ||<1 We can therefore write

y[n]=nx[n]+x[n] From Table 3.3, the DTFT of x[n] is given

by

jj

eeX

1

1)(

§3.3 DTFT Theorems Using the differentiation property of the

DTFT given in Table 3.2, we observe that the DTFT of nx[n] is given by

2)1(11)(

j

j

j

j

ee

eddj

dedX

j

22 )1(1

11

)1()(

jjj

jj

eeee

eY

Next using the linearity property of the DTFT given in Table 3.4 we arrive at

§3.3 DTFT Theorems Example – Determine the DTFT V(e jω) of the

sequence v[n] defined by

d0v[n]+d1v[n-1] = p0 [n] + p1 [n-1] From Table 3.3, the DTFT of δ[n] is 1 Using the time-shifting property of the

DTFT given in Table 3.4 we observe that the DTFT of δ[n-1] is e -jω and the DTFT of

v[n-1] is e− jω V(e jω)

§3.3 DTFT Theorems Using the linearity property of Table 3.4 we t

hen obtain the frequency-domain representation of

d0v[n]+d1v[n-1] = p0[n] + p1[n-1]

as

d0V(ej)+ d1e-jV(ej) = p0 + p1e-j

Solving the above equation we get

j

jj

edd

eppeV

10

10)(

§3.4 Energy Density Spectrum of a Dicrete-Time Sequence

The total energy of a finite-energy sequence g[n] is given by

n

g ng2

][

deGng j

ng

22)(

21][

From Parseval’s relation given in Table 3.4

we observe that


is called the energy density spectrum The area under this curve in the range

-π≤ω≤π divided by 2π is the energy of the sequence

2)()( j

gg eGS

The quantity


Example – Compute the energy of the

sequence

nn

nnh c

LP ,sin

][

deHnh jLP

nLP

22

)(21][

－

c

cjLP eH

,001

)(，

here

where


Hence, hLP [n] is a finite-energy lowpass sequence

c

nLP

c

c

dnh21][

2

－

Therefore

§3.5 Band-limited Discrete-timeSignals

Since the spectrum of a discrete-time signal is a periodic function of ω with a period 2π, a full-band signal has a spectrum occupying

the frequency range π≤ω≤π A band-limited discrete-time signal has a sp

ectrum that is limited to a portion of the frequency range π≤ω≤π


An ideal band-limited signal has a spectrum that is zero outside a frequency range

0<ωa≤|ω|≤ωb<π, this is

b

ajeX,0

0,0)(

An ideal band-limited discrete-time signal cannot be generated in practice


A classification of a band-limited discrete- time signal is based on the frequency range where most of the signal’s energy is concentrated

A lowpass discrete-time real signal has a spectrum occupying the frequency range 0<|ω|≤ωp<π and has a bandwidth of ωp


A highpass discrete-time real signal has a spectrum occupying the frequency range 0<ω

p ≤ |ω|<π and has a bandwidth of

π- ωp

A bandpass discrete-time real signal has a spectrum occupying the frequency range 0<ω

L≤|ω|≤ωH<π and has a bandwidth of ωH - ωL


Example – Consider the sequence

x[n]= (0.5)n[n] Its DTFT is given below on the left along wit

h its magnitude spectrum shown below on the right

jj

eeX

5.01

1)(


It can be shown that 80% of the energy of this lowpass signal is contained in the frequency range 0≤|ω|≤0.5081π

Hence, we can define the 80% bandwidth to be 0.5081π radians

§3.6 DTFT Computation UsingMATLAB

The function freqz can be used to compute the values of the DTFT of a sequence, described as a rational function in the form of

NjN

j

MjM

jj

ededdepepp

eX

........

)(10

10

at a prescribed set of discrete frequencypoints


For example, the statement

H = freqz(num,den,w)

returns the frequency response values as a vector H of a DTFT defined in terms of the vectors num and den containing the coefficients pi and di, respectively at a prescribed set of frequencies between 0 and 2π given by the vector w


There are several other forms of the

function freqz Program 3_1.m in the text can be used to co

mpute the values of the DTFT of a real sequence

It computes the real and imaginary parts, and the magnitude and phase of the DTFT


Example – Plots of the real and imaginary parts, and the magnitude and phase of the DTFT

43

2

43

2

41.06.17.237.21

008.0033.005.0033.0008.0

)(

jj

jj

jj

jj

j

eeee

eeee

eX

are shown on the next slide


Note: The phase spectrum displays a discontinuity of 2π at ω=0.72

This discontinuity

can be removed

using the function

unwrap as

indicated below

Linear Convolution UsingDTFT

An important property of the DTFT is given by the convolution theorem in Table 3.4

It states that if y[n] = x[n] h[n], then the DTFT Y(e jω) of y[n] is given by

Y(e jω)= X(e jω)H(e jω) An implication of this result is that the linear

convolution y[n] of the sequences x[n] and h[n] can be performed as follows:

＊

Linear Convolution UsingDTFT

1) Compute the DTFTs X(e jω) and H(e jω) of the sequences x[n] and h[n], respectively

2) Form the DTFT Y(e jω)=X(e jω)H(e jω) 3) Compute the IDFT y[n] of Y(e jω)

DTFT

DTFT

IDTFT

x[n]

h[n]y[n]

H(ejω)

X(ejω)Y(ejω)

§3.7 The Unwrapped PhaseFunction

In numerical computation, when the computed phase function is outside the range [-π,π], the phase is computed modulo 2π, to bring the computed value to this range

Thus the phase functions of some sequences exhibit discontinuities of 2π radians in the plot


For example, there is a discontinuity of 2π at ω = 0.72 in the phase response below

432

432

41.06.17.237.21008.0033.005.0033.0008.0)( jjjj

jjjjj

eeeeeeeeeX


In such cases, often an alternate type of phase function that is continuous function of ω is derived from the original phase function by removing the discontinuities of2π

Process of discontinuity removal is called unwrapping the phase

The unwrapped phase function will be denoted as θc(ω)


In MATLAB, the unwrapping can be implemented using the M-file unwrap

The unwrapped phase function of the DTFT of previous page is shown below


The conditions under which the phase function will be a continuous function of ω is next derived

Now

)()()(ln jeXeX jj

)(arg)( jeHwhere


If in X(e jω) exits,then its derivative with respect to ω also exists and is given by

dedX

jd

edXeX

dedX

eXdeXd

jim

jre

j

j

j

j

)()()(

1

)()(

1)(ln


From ln X(e jω)= |X(e jω)|+ jθ(ω),

d ln X(e jω)/dω is also given by

dd

jd

eXd

deXd

jj )()()(ln


Thus, dθ(ω)/dω is given by the imaginary part of

dedX

jd

edXeX

jim

jre

j

)()()(

1

])(

)(

)()([

)(

1)(2

dedX

eX

dedX

eXeXd

d

jrej

im

jimj

rej

Hence,


The phase function can thus be defined unequivocally by its derivative dθ(ω)/dω:

dd

d

0

)()(

with the constraintθ(0)=0


The phase function defined by

is called the unwrapped phase function of X(e jω) and it is a continuous function of ω

dd

d

0

)()( ＝

→ ln X(e jω) exits


Moreover, the phase function will be an odd function of ω if

If the above constraint is not satisfied, then the computed phase function will exhibit absolute jumps greater than π

0)(1

2

0

dd

d

§3.8 The Frequency Response Most discrete-time signals encountered in pr

actice can be represented as a linear combination of a very large, maybe infinite, number of sinusoidal discrete-time signals of different angular frequencies

Thus, knowing the response of the LTI system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superposition property

§3.8 The Frequency Response

An important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal multiplied by a complex constant

We consider here one such eigen function as the input


Consider the LTI discrete-time system with an impulse response h[n] shown below

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

k

knxkhny ][][][

Its input-output relationship in the time- domain is given by the convolution sum


If the input is of the form nenx nj ,][

nj

k

kj

k

knj eekhekhny

][][][ )(

k

kjj ekheH ][)(

then it follows that the output is given by

Let


Then we can

y[n]=H(ejω) ejωn Thus for a complex exponential input signal

ejωn, the output of an LTI discrete-time system is also a complex exponential signal of the same frequency multiplied by a complex constant H(ejω)

Thus ejωn is an eigen function of the system


The quantity H(e jω) is called the frequency response of the LTI discrete-time system

H(e jω) provides a frequency-domain description of the system

H(e jω) is precisely the DTFT of the impulse response h[n] of the system

§3.8 The Frequency Response H(e jω) , in general, is a complex function

of ω with a period 2π It can be expressed in terms of its real and i

maginary parts

）（ jjj eeHeH )()(

)(arg)( jeH

or, in terms of its magnitude and phase,

where

)()()( jim

jre

j ejHeHeH


The function |H(e jω)|is called the magnitude response and the function θ(ω) is called the phase response of the LTI discrete-time system

Design specifications for the LTI discrete- time system, in many applications, are given in terms of the magnitude response or the phase response or both


In some cases, the magnitude function is specified in decibels as

dBeHG j )(log20)( 10

)()( GA

where G(ω) is called the gain function The negative of the gain function

is called the attenuation or loss function


Note: Magnitude and phase functions are real functions of ω, whereas the frequency response is a complex function of ω

If the impulse response h[n] is real then it follows from Table 3.2 that the magnitude function is an even function of ω:

)()(

)()( jj eHeH －and the phase function is an odd function of ω:


Likewise, for a real impulse response h[n], H

re(e jω) is even and Him(e jω) is odd Example – Consider the M-point moving ave

rage filter with an impulse response given by

otherwise,010,/1

][MnM

nh

1

0

1)(M

n

njj eM

eH －

Its frequency response is then given by


与 44页重复


Or,

2/)1(

0

0

)2/sin()2/sin(1

111)1(1

1)(

Mj

j

jMjM

n

nj

Mn

nj

n

njj

eM

M

ee

Mee

M

eeM

eH

－

－

－－－

－－


Thus, the magnitude response of the M-point moving average filter is given by

)2/sin()2/sin(1)(

MM

eH j

2/

0

)2(2

)1()(

M

kM

kM

and the phase response is given by

§3.8.4 Frequency Response Computation Using MATLAB

The function freqz(h,1,w) can be used to determine the values of the frequency response vector h at a set of given frequency points w

From h, the real and imaginary parts can be computed using the functions real and imag, and the magnitude and phase functions using the functions abs and angle


Example – Program 3_2.m can be used to generate the magnitude and gain responses of an M-point moving average filter as shown below


The phase response of a discrete-time system when determined by a computer may exhibit jumps by an amount 2πcaused by the way the arctangent function is computed

The phase response can be made a continuous function of ω by unwrapping the phase response across the jumps


To this end the function unwrap can be used, provided the computed phase is in radians

The jumps by the amount of 2π should not be confused with the jumps caused by the zeros of the frequency response as indicated in the phase response of the moving average filter

§3.8.5 Steady-State and Transient Responses

Note that the frequency response also determines the steady-state response of an LTI discrete-time system to a sinusoidal input

Example – Determine the steady-state output y[n] of a real coefficient LTI discrete-time system with a frequency response H (e jω) for an input

nnAnx ),cos(][ 0


We can express the input x[n] as

][][][ ngngnx

njj eeH 00 )(

njj eAeng 0

21][

where

Now the output of the system for an input e jω0n is simply


Because of linearity, the response v[n] to an input g[n] is given by

njjj eeHAenv 00 )(2

1][

njjj eeHAenv 00 )(2

1][ －－－

Likewise, the output v*[n] to the input g*[n] is


Combining the last two equations we get

))(cos()(

)(21

)(21)(

21

][][][

00

)()(

0

00000

0000

neHA

eeeeeeeHA

eeHAeeeHAe

nvnvny

j

njjjnjjjj

njjjnjjj －－－


Thus, the output y[n] has the same sinusoidal waveform as the input with two differences:

(1) the amplitude is multiplied by |H (e jω0)|, the value of the magnitude function atω=ω0

(2) the output has a phase lag relative to the input by an amount θ(ω0), the value phase function at ω=ω0

§3.8.6 Response to a Causal Exponential Sequence

The expression for the steady-state response developed earlier assumes that the system is initially relaxed before the application of the input x[n]

In practice, excitation x[n] to a discrete-time system is usually a right-sided sequence applied at some sample index n=n0

We develop the expression for the output for such an input


Without any loss of generality, assume x[n]=0 for n<0

From the input-output relation

][][ nenx nj

－kknxkhny ][][][

][][][0

)( nekhnyn

k

knj

we observe that for an input

the output is given by


Or,][][][

0

neekhny njn

k

kj

nj

nk

kjnj

k

kj

njn

k

kj

eekheekh

eekhny

10

0

][][

][][

The output for n<0 is y[n]=0 The output for n≥0 is given by


Or,nj

nk

kjnjj eekheeHny

1

][)(][

njjtr eeHny )(][

The first term on the RHS is the same as that obtained when the input is applied at n=0 to an initially relaxed system and is the steady-state response:


The second term on the RHS is called the transient response:

nj

nk

kjtr eekhny

1

][][

1 011

)( ][][][][nk knk

nkjtr khkhekhny

To determine the effect of the above term on the total output response, we observe


For a causal, stable LTI IIR discrete-time system, h[n] is absolutely summable

As a result, the transient response ytr[n] is a bounded sequence

Moreover, as n→∞,

0][1

nkkh

and hence, the transient response decays to zero as n gets very large


For a causal FIR LTI discrete-time system with an impulse response h[n] of length

N+1, h[n]=0 for n>N Hence, ytr[n]=0 for n>N-1 Here the output reaches the steady-state val

ue ysr[n]= H (e jω)e jωn at n=N

§3.8.7 The Concept of Filtering One application of an LTI discrete-time syst

em is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components

Such systems are called digital filters and one of the main subjects of discussion in this course

§3.8.7 The Concept of Filtering

The key to the filtering process is

deeXnx njj )(][21

It expresses an arbitrary input as a linear weighted sum of an infinite number of exponential sequences, or equivalently, as a linear weighted sum of sinusoidal sequences

§3.8.7 The Concept of Filtering Thus, by appropriately choosing the values

of the magnitude function |H (e jω)| of the LTI digital filter at frequencies corresponding to the frequencies of the sinusoidal components of the input, some of these components can be selectively heavily attenuated or filtered with respect to the others


To understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient LTI discrete-time system characterized by a magnitude function

c

cjeH,0

,1)(


We apply an input

2121 0,cos][ cnBnAnx

)(cos)(

)(cos)(][

22

11

2

1

neHB

neHAnyj

j

to this system Because of linearity, the output of this syste

m is of the form


As

0)(,1)( 21 jj eHeH

)(cos)(][ 111 neHAny j

the output reduces to

Thus, the system acts like a lowpass filter In the following example, we consider the de

sign of a very simple digital filter


Example – The input consists of a sum of two sinusoidal sequences of angular frequencies

0.1 rad/sample and 0.4 rad/sample We need to design a highpass filter that will p

ass the high-frequency component of the input but block the low-frequency component

For simplicity, assume the filter to be an FIR filter of length 3 with an impulse response:

h[0]=h[2]=0 , h[1]= 1

§3.8.7 The Concept of Filtering The convolution sum description of this filter i

s then given by

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

=0x[n]+1x[n-1]+0x[n-2] y[n] and x[n] are, respectively, the output and

the input sequences Design Objective: Choose suitable values of

0 and 1 so that the output is a sinusoidal sequence with a frequency 0.4 rad/sample


Now, the frequency response of the FIR filter is given by

j

jjjj

jj

jjj

e

eeee

eeehehheH

)cos2(

22

)1(]2[]1[]0[)(

10

10

12

0

2

§3.8.7 The Concept of Filtering The magnitude and phase functions are

|H(ej)|= |20cos+1| () = - In order to block the low-frequency compone

nt, the magnitude function atω=0.1 should be equal to zero

Likewise, to pass the high-frequency component, the magnitude function at ω=0.4 should be equal to one


Thus, the two conditions that must be satisfied are

|H(ej0.1)|=20cos(0.1)+1=0

|H(ej0.4)|= 20cos(0.4)+1=1 Solving the above two equations we get

0=-6.76195 1=13.456335


Thus the output-input relation of the FIR filter is given by

y[n]=-6.76195(x[n]+x[n-2])+13.456335x[n-1]

where the input is

x[n]=cos(0.1n)+cos(0.4n)µ[n] Program 3_3.m can be used to verify the filt

ering action of the above system


Figure below shows the plots generated by running this program


The first seven samples of the output are shown below

§3.8.7 The Concept of Filtering From this table, it can be seen that, neglecti

ng the least significant digit,

y[n]=cos(0.4(n-1)) for n≥2 Computation of the present value of the outp

ut requires the knowledge of the present and two previous input samples

Hence, the first two output samples, y[0]

and y[1], are the result of assumed zero input sample values at n=-1 and n=-2


Therefore, first two output samples constitute the transient part of the output

Since the impulse response is of length 3, the steady-state is reached at n =N=2

Note also that the output is delayed version of the high-frequency component cos(0.4n) of the input, and the delay is one sample period

§3.9 Phase and Group Delays If the input x[n] to an LTI system H (e jω) is a

sinusoidal signal of frequency ω0: nnAnx o ),cos(][

n

neHAny ooj o ，))(cos()(][

Then, the output y[n] is also a sinusoidal

signal of the same frequency ω0 but lagging in phase by θ(ω0 ) radians:

§3.9 Phase and Group Delays

We can rewrite the output expression as

)(cos)(][ opoj neHAny o

o

oop

)()(

where

is called the phase delay The minus sign in front indicates phase lag


Thus, the output y[n] is a time-delayed version of the input x[n]

In general, y[n] will not be delayed replica

of x[n] unless the phase delay τp (ω0) is an integer Phase delay has a physical meaning only with respect to the underlying continuous- time functions associated with y[n] and x[n]

§3.9 Phase and Group Delays When the input is composed of many sinuso

idal components with different frequencies that are not harmonically

related, each component will go through different phase delays

In this case, the signal delay is determined using the group delay defined by

dd

g

)()(


In defning the group delay, it is assumed that the phase function is unwrapped so that its derivatives exist

Group delay also has a physical meaning only with respect to the underlying continuous-time functions associated with y[n] and x[n]

§3.9 Phase and Group Delays A graphical comparison of the two types of d

elays are indicated below


Example – The phase function of the FIR filter y[n]= =0x[n]+1x[n-1]+0x[n-2] is θ

(ω)=-ω Hence its group delay is given by τg (ω)=1 ve

rifying the result obtained earlier by simulation

§3.9 Phase and Group Delays Example – For the M-point moving-average

filter

otherwise,010,/1

][MnM

nh

2/

0

22

)1()(

M

kM

kM

21)( M

g

the phase function is

Hence its group delay is


Physical significance of the two delays are better understood by examining the continuous-time case

Consider an LTI continuous-time system with a frequency response

)()()( ajaa ejHjH

)cos()()( ttatx ca

and excited by a narrow-band amplitude modulated continuous-time signal


a(t) is a lowpass modulating signal with a band-limited continuous-time Fourier transform given by

0,0)( jA

and cos(Ωct) is the high-frequency carrier


We assume that in the frequency range

Ωc-Ωo<|Ω|<Ωc+Ωo the frequency response of the continuous-time system has a constant magnitude and a linear phase:

)()()(

)()()()(

cgccpc

accaa cd

d

)()( caa jHjH

Also, because of the band-limiting

constraint Xa(jΩ)=0 outside the frequency

range Ωc-Ωo<|Ω|<Ωc+Ωo


]))[(])[((21)( cca jAjAjX

Now, the CTFT of xa(t) is given by

§3.9 Phase and Group Delays As a result, the output response ya(t) of the

LTI continuous-time system is given by ))((cos())(()( cgccga ttaty

Assuming |Ha(jΩc)|=1 As can be seen from the above equation, the gro

up delay τg (Ωc) is precisely the delay of the envelope a(t) of the input signal xa (t), whereas,the phase delay τp(Ωc) is the delay of the carrier

§3.9 Phase and Group Delays The figure below illustrates the effects of the

two delays on an amplitude modulated sinusoidal signal

§3.9 Phase and Group Delays The waveform of the underlying continuous-t

ime output shows distortion when the group delay is not constant over the bandwidth of the modulated signal

If the distortion is unacceptable, an allpass delay equalizer is usually cascaded with the LTI system so that the overall group delay is approximately linear over the frequency range of interest while keeping the magnitude response of the original LTI system unchanged


Phase Delay Computation Using MATLAB Phase delay can be computed using the fun

ction phasedelay Figure below shows the phase delay of the

DTFT

j2-j-

-j2j

e7265.00.5335e-1

)e-0.1367(1)H(e


Phase Delay Computation Using MATLAB Group delay can be computed using the fun

ction grpdelay Figure below shows the group delay of the

DTFT

j2-j-

-j2j

e7265.00.5335e-1

)e-0.1367(1)H(e

Chapter 3 Discrete-Time Fourier Transform. §3.1 The Continuous-Time Fourier Transform Definition...

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Transcript of Chapter 3 Discrete-Time Fourier Transform. §3.1 The Continuous-Time Fourier Transform Definition...