Chapter 4 Digital Processing of Continuous-Time Signals

Chapter 4

Digital Processing of ConDigital Processing of Continuous-Time Signalstinuous-Time Signals

§4.1 Introduction

Digital processing of a continuous-time signal involves the following basic steps:

(1) Conversion of the continuous-time signal into a discrete-time signal,

(2) Processing of the discrete-time signal,

(3) Conversion of the processed discrete- time signal back into a continuous-time signal

§4.1 Introduction

Conversion of a continuous-time signal into digital form is carried out by an analog-to- digital (A/D) converter

The reverse operation of converting a digital signal into a continuous-time signal is performed by a digital-to-analog (D/A) converter

§4.1 Introduction

Since the A/D conversion takes a finite amount of time, a sample-and-hold (S/H) circuit is used to ensure that the analog signal at the input of the A/D converter remains constant in amplitude until the conversion is complete to minimize the error in its representation

§4.1 Introduction

To prevent aliasing, an analog anti-aliasing filter is employed before the S/H circuit

To smooth the output signal of the D/A converter, which has a staircase-like waveform, an analog reconstruction filter is used

§4.1 Introduction

Since both the anti-aliasing filter and the reconstruction filter are analog lowpass filters, we review first the theory behind the design of such filters

Also, the most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype

Anti-Aliasing

filter

Digitalprocessor D/A

ReconstructionfilterA/DS/H

Complete block-diagram

§4.2 Sampling of Continuous-Time Signals

As indicated earlier, discrete-time signals in many applications are generated by sampling continuous-time signals

We have seen earlier that identical discrete- time signals may result from the sampling of more than one distinct continuous-time function


In fact, there exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal

However, under certain conditions, it is possible to relate a unique continuous-time signal to a given discrete-time signal


If these conditions hold, then it is possible to recover the original continuous-time signal from its sampled values

We next develop this correspondence and the associated conditions

§4.2.1 Effect of Sampling in the Frequency Domain

Let ga(t) be a continuous-time signal that is sampled uniformly at t = nT, generating the sequence g[n] where

g[n]=ga(nT), -∞ <n<∞

with T being the sampling period The reciprocal of T is called the sampling fre

quency FT , i.e.,

FT=1/T


Now, the frequency-domain representation of ga(t) is given by its continuos-time Fourier transform(CTFT):

dtetgjG tjaa

)()(

nnjj engeG ][)(

The frequency-domain representation of g[n]

is given by its discrete-time Fourier transform (DTFT):


To establish the relation between Ga(jΩ) and G(ejω) ,we treat the sampling operation mathematically as a multiplication of ga (t) by a periodic impulse train p(t):

n

nTttp )()(

)(

)()(

tp

tgtg p


p(t) consists of a train of ideal impulses with a period T as shown below

n

aap nTtnTgtptgtg )()()()()(

The multiplication operation yields an impulse train:


g p (t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value g a (nT) of g a (t) at the instant


There are two different forms of Gp(jΩ): One form is given by the weighted sum of th

e CTFTs of δ(t-nT) :

n

nTjap enTgjG )()(

ktj

k TnTejk

TnTt

)(1

)(

where ΩT =2π/T and Φ(jΩ) is the CTFT of φ(t)

To derive the second form, we make use of the Poisson’s formula:


For t=0ktj

k TnTejk

TnTt

)(1

)(

k Tnjk

TnT )(

1)(

Now, from the frequency shifting property of the CTFT, the CTFT of g a (t) e-jΨt is given by Ga

(j(Ω+Ψ))

reduces to


Substituting φ(t)=ga(t)e-jΨt in

k Tnjk

TnT )(

1)(

k Tan

nTja kjG

TenTg ))((1)(

we arrive at

By replacing Ψ with Ω in the above equation we arrive at the alternative form of the CTFT G p (jΩ ) of g p (t)


The alternative form of the CTFT of g p(t) is given by

kTaTp kjGjG )()( 1

Therefore, G p (jΩ ) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of G a (jΩ ) , shifted by integer multiples of ΩT

and scaled by 1/T


The term on the RHS of the previous equation for k=0 is the baseband portion of G p (jΩ ), and each of the remaining terms are the frequency translated portions of

G p (jΩ ) The frequency range

22TT

is called the baseband or Nyquist band


Assume ga(t) is a band-limited signal with a CTFT Ga(jΩ) as shown below

Ga(jΩ)

Ωm-Ωm

Ω0

ΩT-ΩT 2Ω

T

3ΩT 0

P(jΩ)

Ω

1 … …

The spectrum P(jΩ) of p(t) having a sampling period T=2π/ ΩT is indicated below


Two possible spectra of G p (jΩ ) are shown below


It is evident from the top figure on the previous slide that if ΩT>2Ωm,there is no overlap between the shifted replicas of

Ga(jΩ) generating Gp(jΩ) On the other hand, as indicated by the figure

on the bottom, if ΩT<2Ωm ,there is an overlap of the spectra of the shifted replicas of Ga(jΩ) generating Gp(jΩ)


Hr(jΩ)ga(t)gp(t)

p(t)

)(ˆ tga

If ΩT>2Ωm , ga(t) can be recovered exactly from gp(t) by passing it through an ideal lowpass filter Hr(jΩ) with a gain T and a cutoff frequency Ωc greater than Ωm and less than ΩT -Ωm

as shown below


The spectra of the filter and pertinent signals are shown below


On the other hand, if ΩT<2Ωm ,due to the overlap of the shifted replicas of Ga(jΩ) , the spectrum Ga(jΩ) cannot be separated by filtering to recover Ga(jΩ) because of the distortion caused by a part of the replicas immediately outside the baseband folded back or aliased into the baseband


Sampling theorem – Let ga(t) be a band- limite

d signal with CTFT Ga(jΩ) =0 for |Ω|>Ωm

Then ga(t) is uniquely determined by its samples

ga(nT) ,-∞≤n≤∞ if

ΩT≥2Ωm

where ΩT=2π/T


The condition ΩT≥2Ωm is often referred to a

s the Nyquist condition

The frequency ΩT/2 is usually referred to as t

he folding frequency


Given ga(nT) , we can recover exactly ga(t) by generating an impulse train

n

ap nTtnTgtg )()()(

and then passing it through an ideal lowpass filter Hr(jΩ) with a gain T and a cutoff frequency Ωc satisfying

m < c < (T - m )


The highest frequency Ωm contained in ga(t) is usually called the Nyquist frequency

since it determines the minimum sampling

frequency ΩT=2Ωm that must be used to fully recover ga(t) from its sampled version

The frequency 2Ωm is called the Nyquist rate


Oversampling – The sampling frequency is higher than the Nyquist rate

Undersampling – The sampling frequency is lower than the Nyquist rate

Critical sampling – The sampling frequency is equal to the Nyquist rate

Note: A pure sinusoid may not be recoverable from its critically sampled version


In digital telephony, a 3.4 kHz signal bandwidth is acceptable for telephone conversation

Here, a sampling rate of 8 kHz, which is greater than twice the signal bandwidth, is used


In high-quality analog music signal processing, a bandwidth of 20 kHz has been determined to preserve the fidelity

Hence, in compact disc (CD) music

systems, a sampling rate of 44.1 kHz, which is slightly higher than twice the signal bandwidth, is used


Example - Consider the three continuous- time sinusoidal signals:

ga(t)=cos(6πt)

ga(t)=cos(14πt)

ga(t)=cos(26πt) Their corresponding CTFTs are:

G1(jΩ)=π[δ(Ω-6π)+δ(Ω+6π)]

G2(jΩ)=π[δ(Ω-14π)+δ(Ω+14π)]

G3(jΩ)=π[δ(Ω-26π)+δ(Ω+26π)]


These three transforms are plotted below


These continuous-time signals sampled at a rate of T = 0.1 sec, i.e., with a sampling frequency ΩT=20πrad/sec

The sampling process generates the continuous-time impulse trains, g1p(t),

g2p(t), and g3p(t) Their corresponding CTFTs are given by

31,)(10)( k Tp kjGjG


Plots of the 3 CTFTs are shown below


These figures also indicate by dotted lines the frequency response of an ideal lowpass filter with a cutoff at Ωc=ΩT/2=10π and a gain T=0.1

The CTFTs of the lowpass filter output are also shown in these three figures

In the case of g1 (t), the sampling rate satisfies the Nyquist condition, hence no aliasing


Moreover, the reconstructed output is precis

ely the original continuous-time signal

In the other two cases, the sampling rate do

es not satisfy the Nyquist condition, resulting

in aliasing and the filter outputs are all equal

to cos(6πt)


Note: In the figure below, the impulse appearing at Ω = 6π in the positive frequency passband of the filter results from the aliasing of the impulse in G2(jΩ) at Ω = −14π

Likewise, the impulse appearing at Ω = 6π in the positive frequency passband of the filter results from the aliasing of the impulse in G3(jΩ) at Ω = 26π


We now derive the relation between the

DTFT of g[n] and the CTFT of gp[t] To this end we compare

n

njj engeG ][)(

n

nTjap enTgjG ][)(

and make use of g[n]=ga[nT] ,-∞<n< ∞

with


Observation: We have

Tpj jGeG /)()(

Tj

p eGjG )()(

k

Tap kjGT

jG ))((1

)(

From the above observation and

or, equivalently,


we arrive at the desired result given by

ka

kTa

Tk

Taj

T

kj

TjG

T

jkT

jGT

jkjGT

eG

)2

(1

)(1

)(1

)( /


The relation derived on the previous slide can be alternately expressed as

)(1

)(

k TaTj jkjG

TeG

Tj

p eGjG )()(

Tpj jGeG /)()(

it follows that G(ejω) is obtained from Gp(jΩ) by applying the mapping Ω=ω/T

or from

From


Now, the CTFT Gp(jΩ) is a periodic function

of Ω with a period ΩT=2π/T

Because of the mapping, the DTFT G(ejω)

is a periodic function of ω with a period 2π

§4.2.2 Recovery of the Analog Signal

We now derive the expression for the output

ĝa(t) of the ideal lowpass reconstruction filter Hr(jΩ) as a function of the samples g[n]

The impulse response hr (t) of the lowpass reconstruction filter is obtained by taking the inverse DTFT of Hr(jΩ) :

)( jHr

c

cT

,0,


Thus, the impulse response is given by

np nTtngtg )(][)(

tt

t

deTdejHth

T

c

tjtjrr

c

c

,2/

)sin(2

)(21)(

The input to the lowpass filter is the impulse train gp (t) :


Therefore, the output ĝa(t) of the ideal lowpass filter is given by:

TnTt

TnTtngtg

na /)(

]/)(sin[][)(ˆ

nrpra nTthngtgthtg )(][)()()(^

*

Substituting hr (t) =sin(Ωct)/(ΩTt/2) in the above and assuming for simplicity

c= T/2= /T, we get


The ideal bandlimited interpolation process is illustrated below


It can be shown that when Ωc= ΩT/ 2 in

2/

)sin()(

t

tth

T

cr

TnTt

TnTtngtg

na /)(

]/)(sin[][)(ˆ

)(][)( rTgrgrTg aa for all integer values of r in the range-∞<r<∞

we observe

hr(0)=1 and hr(nT)=0 for n≠0 As a result, from


The relation

)(][)(ˆ rTgrgrTg aa

holds whether or not the condition of the sampling theorem is satisfied

However, ĝa(rT)=ga(rT) for all values of t only if the sampling frequency ΩT satisfies the condition of the sampling theorem

§4.2.3 Implication of the Sampling Process

Consider again the three continuous-time signals: g1(t) =cos(6πt) ,g2(t) =cos(14πt),

and g3(t) =cos(26πt)

The plot of the CTFT G1p (jΩ) of the sampled version g1p(t) of g1(t) is shown below


From the plot, it is apparent that we can recover any of its frequency-translated versions cos[(20k±6)πt] outside the baseband by passing g1p(t) through an ideal analog bandpass f

ilter with a passband centered at Ω= (20k±6)π


For example, to recover the signal cos(34πt), it will be necessary to employ a bandpass filter with a frequency response

otherwise,0)34()34(,1.0

)(

jHr

where ∆ is a small number


Likewise, we can recover the aliased baseband component cos(6πt) from the

sampled version of either g2p(t) or g3p(t) by passing it through an ideal lowpass filter with a frequency response:

otherwise,0)6()6(,1.0

)(

jHr


There is no aliasing distortion unless the original continuous-time signal also contains the component cos(6πt)

Similarly, from either g2p(t) or g3p(t) we can recover any one of the frequency-translated versions, including the parent continuous-time signal g2 (t) or g3(t) as the case may be, by employing suitable filters

§4.3 Sampling of Bandpass Signals

The conditions developed earlier for the unique representation of a continuous-time signal by the discrete-time signal obtained by uniform sampling assumed that the continuous-time signal is bandlimited in the frequency range from dc to some frequency Ωm

Such a continuous-time signal is commonly referred to as a lowpass signal

§4.3 Sampling of Bandpass Signals There are applications where the continuou

s- time signal is bandlimited to a higher frequency range ΩL≤|Ω|≤ΩH with ΩL> 0

Such a signal is usually referred to as thebandpass signal

To prevent aliasing a bandpass signal can of course be sampled at a rate greater than twice the highest frequency, i.e. by ensuring

ΩT ≥2ΩH

§4.3 Sampling of Bandpass Signals However, due to the bandpass spectrum of

the continuous-time signal, the spectrum of the discrete-time signal obtained by sampling will have spectral gaps with no signal components present in these gaps

Moreover, if ΩH is very large, the sampling

rate also has to be very large which may not be practical in some situations


A more practical approach is to use undersampling

Let ΔΩ = ΩH - ΩL define the bandwidth of the bandpass signal

Assume first that the highest frequency ΩH

contained in the signal is an integer multiple of the bandwidth, i.e.,

ΩH =M(ΔΩ )


We choose the sampling frequency ΩT to satisfy the condition

T = 2() = 2H/M

which is smaller than 2ΩH , the Nyquist rate

Substitute the above expression for ΩT in

kTaTp kjGjG )()( 1

§4.3 Sampling of Bandpass Signals This leads to

k kjjGjG aTp )()( 21

As before,Gp (jΩ) consists of a sum of Ga (jΩ) and replicas of Ga (jΩ) shifted by integer multiples of twice the bandwidth ∆Ω and scaled by 1/T

The amount of shift for each value of k ensures that there will be no overlap between all shifted replicas →no aliasing


Figure below illustrate the idea behind

-H -L L H0

Gp(j)

-H -L L H0

Gp(j)

§4.3 Sampling of Bandpass Signals As can be seen, ga(t) can be recovered from

gp(t) by passing it through an ideal bandpass filter with a passband given by ΩL≤|Ω|≤ΩH and a gain of T

Note: Any of the replicas in the lower frequency bands can be retained by passing

gp(t) through bandpass filters with passbands ΩL≤ - k(ΔΩ) ≤|Ω|≤ΩH - k(ΔΩ) , 1≤k≤ M-1 providing a translation to lower frequency ranges

§4.4.1 Analog Lowpass Filter Specifications

Typical magnitude response |Ha(jΩ)| of an

analog lowpass filter may be given as indicated below


In the passband, defined by 0≤Ω≤Ωp, we require 1-p |Ha(j)| 1+ p , || p

i.e., |Ha(jΩ)| approximates unity within an error of ±δp

In the stopband, defined by Ωs≤Ω<∞, we require |Ha(j)| s , s

i.e., |Ha(jΩ)| approximates zero within an error of δs


Ωp – passband edge frequency Ωs – stopband edge frequency δp – peak ripple value in the passband δs – peak ripple value in the stopband Peak passband ripple

dBpp )1(log20 10

dBss )(log20 10 Minimum stopband attenuation


Magnitude specifications may alternately be given in a normalized form as indicated below


Here, the maximum value of the magnitude in the passband assumed to be untiy

21/1 – Maximum passband deviation,

given by the minimum value of the magnitude in the passband

1/A – Maximum stopband magnitude


Two additional parameters are defined –

(1) Transition ratio k = p/s

For a lowpass filter k<1

121

A

k(2) Discrimination parameter

Usually k<<1

§4.4.2 Butterworth Approximation

The magnitude-square response of an N-th order analog lowpass Butterworth filter is given by

Nc

a jH 22

)/(1

1)(

First 2N-1 derivatives of |Ha(jΩ)|2 at Ω=0 are equal to zero

The Butterworth lowpass filter thus is said to have a maximally-flat magnitude at Ω=0


Gain in dB is G (Ω)=10 log10|Ha(jΩ)|2 As G(0)=0 and

G (Ωc )= 10 log10(0.5)=−3.0103 -≅ 3dB

Ωc is called the 3-dB cutoff frequency


Typical magnitude responses with Ωc=1

0 1 2 30

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Butterworth Filter

N = 2N = 4N = 10


Two parameters completely characterizing a Butterworth lowpass filter are Ωc and N

These are determined from the specified bandedges Ωp and Ωs, and minimum

21/1 passband magnitude , and maximum stopband ripple 1/A

§4.4.2 Butterworth Approximation Ωc and N are thus determined from

22

2

11

)/(11)(

N

cppa jH

22

2 1)/(1

1)(A

jH Ncs

sa

)/1(log)/1(log

)/(log]/)1[(log

21

10

110

10

2210

kkA

Nps

Solving the above we get

§4.4.2 Butterworth Approximation Since order N must be an integer, value obt

ained is rounded up to the next highest integer

This value of N is used next to determine Ωc by satisfying either the stopband edge or the passband edge specification exactly

If the stopband edge specification is satisfied, then the passband edge specification is exceeded providing a safety margin


Transfer function of an analog Butterworth lowpass filter is given by

N

Nc

NN

Nc

Na

pssdssDCsH

1

1

0)()(

)(

Nep NNjc

1,]2/)12([

Denominator DN(s) is known as the Butterworth polynomial of order N

where

§4.4.2 Butterworth Approximation Example – Determine the lowest order of a

Butterworth lowpass filter with a 1-dB cutoff frequency at 1kHz and a minimum attenuation of 40 dB at 5kHz

Now1

1

1log10

210

401

log10210

Awhich yields A2=10,000

which yields ε2=0.25895and

§4.4.2 Butterworth Approximation Therefore

51334.19611 2

1

A

k

51

p

s

k

2811.3)/1(log

)/1(log

10

110 ＝k

kN

We choose N=4

Hence

and

§4.4.3 Chebyshev Approximation The magnitude-square response of an N-th

order analog lowpass Type 1 Chebyshev filter is given by

1),coshcosh(

1),coscos()(

1

1

N

NTN

)/(1

1)( 22

2

pNa

TsH

where TN(Ω) is the Chebyshev polynomial of order N:

§4.4.3 Chebyshev Approximation Typical magnitude response plots of the ana

log lowpass Type 1 Chebyshev filter are shown below

0 1 2 30

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Type 1 Chebyshev Filter

N = 2N = 3N = 8

§4.4.3 Chebyshev Approximation

If at Ω=Ωs the magnitude is equal to 1/A, then

2222 1

)/(11)(

ATjH

psNsa

)/1(cosh)/1(cosh

)/(cosh)/1(cosh

11

1

1

21

kkA

Nps

Order N is chosen as the nearest integer greater than or equal to the above value

Solving the above we get

§4.4.3 Chebyshev Approximation The magnitude-square response of an N-th

order analog lowpass Type 2 Chebyshev (also called inverse Chebyshev) filter is given by

22

2

)/(

)/(1

1)(

sN

psNa

T

TjH

where TN(Ω) is the Chebyshev polynomial of order N

§4.4.3 Chebyshev Approximation Typical magnitude response plots of the ana

log lowpass Type 2 Chebyshev filter are shown below

0 1 2 30

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Type 2 Chebyshev Filter

N = 3N = 5N = 7

§4.4.3 Chebyshev Approximation The order N of the Type 2 Chebyshev is det

ermined from given ε, Ωs, and A using

)/1(cosh)/1(cosh

)/(cosh)/1(cosh

11

1

1

21

kkA

Nps

6059.2)/1(cosh

)/1(cosh1

11

k

kN

Example – Determine the lowest order of a Chebyshev lowpass filter with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz -

§4.4.4 Elliptic Approximation The square-magnitude response of an ellipti

c lowpass filter is given by

)/(1

1)( 22

2

pNa

RjH

where RN(Ω) is a rational function of order N satisfying RN(1/Ω)=1/RN(Ω), with the roots of its numerator lying in the interval 0< Ω<1 and the roots of its denominator lying in the interval 1< Ω<∞

§4.4.4 Elliptic Approximation

For given Ωp , Ωs ,ε, and A, the filter order can be estimated using

)/1(log)/4(log2

10

110

k

N

21' kk

)'1(2'1

0 kk

130

90

500 )(150)(15)(2

where

§4.4.4 Elliptic Approximation

Example - Determine the lowest order of a elliptic lowpass filter with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHzNote: k = 0.2 and 1/k=196.5134

Substituting these values we get

k’ = 0.979796, ρ0 =0 00255135,ρ= 0 0025513525

and hence N = 2.23308 Choose N = 3

§4.4.4 Elliptic Approximation Typical magnitude response plots with Ωp=1 a

re shown below

0 1 2 30

0.2

0.4

0.6

0.8

1

Mag

nitu

de

Elliptic Filter

N = 3N = 4

§4.4.6 Analog Lowpass Filter Design Example – Design an elliptic lowpass filter of lo

west order with a 1-dB cutoff frequency at 1 kHz and a minimum attenuation of 40 dB at 5 kHz

Code fragments used[N, Wn] = ellipord(Wp, Ws, Rp, Rs, ‘s’);[b, a] = ellip(N, Rp, Rs, Wn, ‘s’);with Wp = 2*pi*1000;

Ws = 2*pi*5000; Rp = 1; Rs = 40;

§4.4.6 Analog Lowpass Filter Design Gain plot

§4.5 Design of Analog Highpass Bandpass and Bandstop Filters

Steps involved in the design process: Step 1 – Develop of specifications of a prototype analog lowpass filter HLP(s) from specifications of desired analog filter HD(s)using a frequency transformation

Step 2 – Design the prototype analog lowpass filter

Step 3 – Determine the transfer function HD(s) of desired analog filter by applying the inverse frequency transformation to HLP (s)

§4.5 Design of Analog Highpass Bandpass and Bandstop Filters

Let s denote the Laplace transform variable of prototype analog lowpass filter HLP(s) and ŝ denote the Laplace transform variable of desired analog filter HD(ŝ)

The mapping from s-domain to ŝ-domain is given by the invertible transform s=F(ŝ)

Then

)(ˆ

)ˆ(

1)ˆ()(

)()ˆ(

sFsDLP

sFsLPD

sHsH

sHsH

HLP (s) and is the passband edgep

§4.5.1 Analog Highpass Filter Design

Spectral Transformation:

ss pp

ˆ

ˆ

ˆpp

frequency of HHP (ŝ) On the imaginary axis the transformation is

where Ωp is the passband edge frequency of


ˆ

ˆpp


Example - Design an analog Butterworth highpass filter with the specifications:

Fp=4 kHz, Fs=1 kHz,αp=0.1 dB,αs=40 dB Choose Ωp=1 Then

41000

4000ˆ

ˆ

ˆ2

ˆ2

s

p

s

ps

F

F

F

F

Analog lowpass filter specifications: Ωp=1, Ωs=4, αp=0.1 dB αs=40 dB


Code fragments used[N, Wn] = buttord(1, 4, 0.1, 40, ‘s’);[B, A] = butter(N, Wn, ‘s’);[num, den] = lp2hp(B, A, 2*pi*4000);

Gain plots

§4.5.2 Analog Bandpass Filter Design

Spectral Transformation

w

pB

ˆ

ˆˆ 220

upper passband edge frequencies of desired bandpass filter HBP (ŝ)

HLP (s), and and are the lower and2ˆ

p1ˆ

p

where Ωp is the passband edge frequency of


On the imaginary axis the transformation is

w

pB

ˆ

ˆˆ 220

where is the width of passband and is the passband center frequency of the bandpass filter

passband edge frequency ±Ωp is mapped into and , lower and upper passband edge frequencies

12ˆˆ

ppwB

0

1ˆ

p 2ˆ

p


w

pB

ˆ

ˆˆ 220


If bandedge frequencies do not satisfy the above condition, then one of the frequencies needs to be changed to a new value so that the condition is satisfied

2121ˆˆˆˆ

sspp

Stopband edge frequency ±Ωs is mapped into and , lower and upper stopband edge frequencies

Also,2

ˆs1

ˆs


increase any one of the stopband edges or decrease any one of the passband edges as shown below

Case 1:

to make we can either2121ˆˆˆˆ

sspp 2121

ˆˆˆˆsspp


(1) Decrease to

larger passband and shorter leftmost transition band

1ˆ

p 221ˆ/ˆˆ

pss

(2) Increase to

No change in passband and shorter leftmost transition band

1ˆ

s 221ˆ/ˆˆ

spp


Note: The condition

can also be satisfied by decreasing

which is not acceptable as the passband is reduced from the desired value

212120

ˆˆˆˆsspp

2ˆ

p

Alternately, the condition can be satisfied by increasing which is not acceptable as the upper stopband is reduced from the desired value

2ˆ

s


decrease any one of the stopband edges or increase any one of the passband edges as shown below

Case 2:to make we can either

2121ˆˆˆˆ

sspp

2121ˆˆˆˆ

sspp


(1) Increase to

larger passband and shorter rightmost transition band

2ˆ

p 121ˆ/ˆˆ

pss

(2) Decrease to

No change in passband and shorter rightmost transition band

121ˆ/ˆˆ

spp 2ˆ

s


Note: The condition

can also be satisfied by increasing

which is not acceptable as the passband is reduced from the desired value

1ˆ

p2121

20

ˆˆˆˆsspp

Alternately, the condition can be satisfied by decreasing which is not acceptable as the lower stopband is reduced from the desired value

1ˆ

s


Example – Design an analog elliptic bandpa

ss filter with the specifications:

Now and621 1028ˆˆ pp FF 6

21 1024ˆˆ ss FF

Since we choose2121ˆˆˆˆ

sspp FFFF

571428.3ˆ/ˆˆˆ2211 pssp FFFF

dB22,dB1,kHz8ˆ,kHz3ˆ,kHz7ˆ,kHz4ˆ

2

121

sps

spp

F

FFF


We choose Ωp=1 Hence

4.137/25

9-24s

）（

Analog lowpass filter specifications: Ωp=1, Ω

s=1.4, αp=1dB,αs=22dB


Code fragments used

[N, Wn] = ellipord(1, 1.4, 1, 22, ‘s’);

[B, A] = ellip(N, 1, 22, Wn, ‘s’);

[num, den]= lp2bp(B, A, 2*pi*4.8989795, 2*pi*25/7); Gain plot

§4.5.3 Analog Bandstop Filter Design

Spectral Transformation

where Ωs is the stopband edge frequency of HLP(s), and and are the lower and upper stopband edge frequencies of the desired bandstop filter HBS(ŝ)

s1 s2

20

2s1s2

s ˆˆ

)ˆˆ(ˆ

s

ss


On the imaginary axis the transformation

220

s ˆˆ

ˆ

wB

where is the widrth of stopband and is the stopband center frequency of the bandstop filter

Stopband edge frequency is mapped into and , lower and upper stopband edge frequencies

s1s2ˆˆ wB

s1 s2s

0


Passband edge frequency ±Ωp is mapped into and , lower and upper passband edge frequencies

p1 p2


If bandedge frequencies do not satisfy the above condition, then one of the frequencies needs to be changed to a new value so that the condition is satisfied

212120

ˆˆˆˆˆsspp

Also,


Case 1: To make we can either incr

ease any one of the stopband edges or decrease any one of the passband edges as shown below

2121ˆˆˆˆ

sspp

2121ˆˆˆˆ

sspp


(1) Decrease to

larger high-frequency passband and shorter rightmost transition band

221ˆ/ˆˆ

pss 2ˆ

p

(2) Increase to

No change in passband and shorter rightmost transition band

221ˆ/ˆˆ

spp 2ˆ

s


Note: The condition

can also be satisfied by decreasing

which is not acceptable as the low – frequency passband is reduced from the desired value

Alternately, the condition can be satisfied by increasing which is not acceptable as the stopband is reduced from the desired value

212120

ˆˆˆˆˆsspp

1ˆ

p

1ˆ

s


Case 1: To make we can either decre

ase any one of the stopband edges or increase any one of the passband edges as shown below

2121ˆˆˆˆ

sspp

2121ˆˆˆˆ

sspp


(1) Increase to larger passband and shorter leftmost transition band

121ˆ/ˆˆ

pss 1ˆ

p

(2) Decrease to

No change in passband and shorter leftmost transition band

121ˆ/ˆˆ

spp 1ˆ

s


Note: The condition

can also be satisfied by increasing

which is not acceptable as the high – frequency passband is decreasd from the desired value

Alternately, the condition can be satisfied by decreasing which is not acceptable as the stopband is decreased

212120

ˆˆˆˆˆsspp

2ˆ

p

2ˆ

s

Documents

Chapter 4 Digital Processing of Continuous-Time Signals