Chapter 7 LTI Discrete-Time Systems in the Transform Domain

Chapter 7

LTI Discrete-Time SysteLTI Discrete-Time Systems in the Transform Doms in the Transform Do

mainmain

Types of Transfer Functions

The time-domain classification of an LTI digital transfer function sequence is based on the length of its impulse response:

- Finite impulse response (FIR) transfer function

- Infinite impulse response (IIR) transfer function

Types of Transfer Functions

In the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications

(1) Classification based on the shape of the magnitude function |H (e jω) |

(2) Classification based on the the form of the phase function θ(ω)

§7.1 Classification Based onMagnitude Characteristics

One common classification is based on an ideal magnitude response

A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to one at these frequencies, and should have a frequency response equal to zero at all other frequencies

§7.1.1 Digital Filters with Ideal Magnitude Responses

The range of frequencies where the frequency response takes the value of one is called the passband

The range of frequencies where the frequency response takes the value of zero is called the stopband


Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:


Lowpass filter: Passband -0≤ω≤ωc

Stopband -ωc≤ω≤π Highpass filter: Passband -ωc≤ω≤π

Stopband -0≤ω≤ωc

Bandpass filter: Passband -ωc1≤ω≤ωc2

Stopband -0≤ω≤ωc1 and ωc2≤ω≤π Bandstop filter: Stopband -ωc1≤ω≤ωc2

Passband -0≤ω≤ωc1 and ωc2≤ω≤π


The frequenciesωc ,ωc1 ,andωc2 are called the cutoff frequencies

An ideal filter has a magnitude response equal to one in the passband and zero in the stopband, and has a zero phase everywhere


Earlier in the course we derived the inverse DTFT of the frequency response HLP(e jω) of the ideal lowpass filter:

nn

nnh c

LP ,sin

][

We have also shown that the above impulse response is not absolutely summable, and hence, the corresponding transfer function is not BIBO stable


Also, HLP[n] is not causal and is of doubly infinite length

The remaining three ideal filters are also characterized by doubly infinite, noncausal impulse responses and are not absolutely summable

Thus, the ideal filters with the ideal “brick wall” frequency responses cannot be realized with finite dimensional LTI filter


To develop stable and realizable transfer functions, the ideal frequency response specifications are relaxed by including a transition band between the passband and the stopband

This permits the magnitude response to decay slowly from its maximum value in the passband to the zero value in the stopband


Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband

Typical magnitude response specifications of a lowpass filter are shown the right

§7.1.2 Bounded Real TransferFunctions

A causal stable real-coefficient transfer function H(z) is defined as a bounded real

(BR) transfer function if

|HLP(e jω)|≤1 for all values of ω Let x[n] and y[n] denote, respectively, the in

put and output of a digital filter characterized by a BR transfer function H(z)

with X(e jω) and Y(e jω) denoting their DTFTs


Then the condition |H(e jω)|≤1 implies that22

)()( jj eXeY

nn

nxny22

][][

Integrating the above from –π to π, and applying Parseval’s relation we get


Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy implying that a digital filter characterized by a BR transfer function can be viewed as a passive structure

If |H(e jω)|=1, then the output energy is equal to the input energy, and such a digital filter is therefore a lossless system


A causal stable real-coefficient transfer function H(z) with |H(e jω)|=1 is thus called a lossless bounded real (LBR) transfer function

The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity


Example – Consider the causal stable IIR

transfer function

10,1

)(1

z

KzH

cos2)1()()()(

2

212

K

zHzHeH jez

j

where K is a real constant Its square-magnitude function is given by


The maximum value of |H(e jω)|2 is obtained when 2αcosω in the denominator is maximum and the minimum value is a obtained when 2αcosω is a minimum

For α > 0, maximum value of 2αcosωis equal to 2α at ω=0, and minimum value is -2α at ω=π


Thus, for α>0, the maximum value of |H(ej

ω)|2 is equal to K2/(1-α)2 at ω=0 and the minimum value is equal to K2/(1+α)2 at ω=π

On the other hand, for α<0, the maximum value of 2αcosω is equal to -2α at ω=πand the minimum value is equal to 2α at ω= 0


Here, the maximum value of |H(ejω)|2 is equal to K2/(1-α)2 at ω=π and the minimum value is equal to K2/(1-α)2 at ω=0

Hence, the maximum value can be made equal to 1 by choosing K=±(1-α), in which case the minimum value becomes (1-α)2/(1+α)2


is a BR function for K=±(1-α) Plots of the magnitude function for α=±0.5 w

ith values of K chosen to make H(z) a BR

function are shown on the next slide

10,1

)(1

z

KzH

Hence,


§7.1.3 Allpass Transfer Function

Definition An IIR transfer function A(z) with unity magni

tude response for all frequencies, i.e., |A(ejω)|2= 1, for all ω

is called an allpass transfer function An M-th order causal real-coefficient allpass

transfer function is of the form

MM

MM

MMMM

zdzdzd

zzdzddzAM

1

11

1

11

11

1)(

§7.1.3 Allpass Transfer Function If we denote the denominator polynomials of

AM(z) as DM(z):

DM(z)=1+d1z-1+···+dM-1z-M+1+dMz-M

then it follows that AM(z) can be written as:

Note from the above that if z=re jФ is a pole of a real coefficient allpass transfer function, then it has a zero at

)(

)()(

1

zD

zDzzA

M

MM

M

jer

z 1

§7.1.3 Allpass Transfer Function The numerator of a real-coefficient allpass t

ransfer function is said to be the mirror- image polynomial of the denominator, and vice versa

)()(~ 1 zDzzD M

MM

We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial DM

(z), i.e.,

)(~

zDM


implies that the poles and zeros of a real- coefficient allpass function exhibit mirror- image symmetry in the z-plane

)(

)()(

1

zD

zDzzA

M

MM

M

321

321

3 2.018.04.01

4.018.02.0)(

zzz

zzzzA

The expression


To show that we observe that

)(

)()(

11

zD

zDzzA

M

MM

M

)(

)(

)(

)()()(

1

11

zD

zDz

zD

zDzzAzA

M

MM

M

MM

MM

1)()()( 12

jezMMj

M zAzAeA

Therefore

Hence


Now, the poles of a causal stable transfer function must lie inside the unit circle in the z-plane

Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle

§7.1.3 Allpass Transfer Function Figure below shows the principal value of th

e phase of the 3rd-order allpass function

321

321

3 2.018.04.01

4.018.02.0)(

zzz

zzzzA

Note the discontinuity by the amount of 2π

in the phase θ(ω)


If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function θc(ω) indicated below

Note: The unwrapped phase function is a continuous function of ω

§7.1.3 Allpass Transfer Function The unwrapped phase function of any arbitr

ary causal stable allpass function is a continuous function of ω

Properties (1) A causal stable real-coefficient allpass tr

ansfer function is a lossless bounded real

(LBR) function or, equivalently, a causal stable allpass filter is a lossless structure

§7.1.3 Allpass Transfer Function (2) The magnitude function of a stable allpa

ss function A(z) satisfies:

1,1

1,1

1,1

)(

zfor

zfor

zfor

zA

)]([)(

cd

d

(3) Let τ(ω) denote the group delay function of an allpass filter A(z),i.e.,


The unwrapped phase function θc(ω) of a stable allpass function is a monotonically decreasing function of ω so that τ(ω) is everywhere positive in the range 0<ω<π

The group delay of an M-th order stable real-coefficient allpass transfer function satisfies:

Md 0

)(

§7.1.3 Allpass Transfer FunctionA Simple Application

A simple but often used application of an allpass filter is as a delay equalizer

Let G(z) be the transfer function of a digital filter designed to meet a prescribed magnitude response

The nonlinear phase response of G(z) can be corrected by cascading it with an allpass filter A(z) so that the overall cascade has a constant group delay in the band of interest


Since |A(ejω)|=1, we have

|G(ejω)A(ejω)|=|G(ejω)| Overall group delay is the given by the sum

of the group delays of G(z) and A(z)

G(z) A(z)


Example – Figure below shows the group delay of a 4th order elliptic filter with the following specifications: ωp=0.3π,δp=1dB, δs=35dB


Figure below shows the group delay of the original elliptic filter cascaded with an 8th order allpass section designed to equalize the group delay in the passband

§7.2 Classification Based on PhaseCharacteristics

A second classification of a transfer function is with respect to its phase characteristics

In many applications, it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passband

§7.2.1 Zero-Phase Transfer Function

One way to avoid any phase distortion is to make the frequency response of the filter real and nonnegative, i.e., to design the filter with a zero phase characteristic

However, it is not possible to design a causal digital filter with a zero phase


For non-real-time processing of real-valued input signals of finite length, zero-phase filtering can be very simply implemented by relaxing the causality requirement

One zero-phase filtering scheme is sketched below

H(z)x[n] v[n]

u[n]=v[-n]

H(z)u[n] w[n]

y[n]=w[-n]

§7.2.1 Zero-Phase Transfer Function It is easy to verify the above scheme in the f

requency domain Let X(ejω), V(ejω), U(ejω), W(ejω) and Y(ejω) den

ote the DTFTs of x[n], v[n], u[n], w[n],and y[n], respectively

From the figure shown earlier and making use of the symmetry relations we arrive at the relations between various DTFTs as given on the next slide


V(ejω)=H(ejω)X(ejω), W(ejω)=H(ejω)U(ejω)

U(ejω)=V*(ejω), Y(ejω)=W*(ejω) Combining the above equations we get

Y(ejω)=W*(ejω)=H*(ejω)U*(ejω)= H*(ejω)V(ejω)=H*(ejω)H(ejω)X(ejω)= |H(ejω)|2X(ejω)

H(z)x[n] v[n]

u[n]=v[-n]

H(z)u[n] w[n]

y[n]=w[-n]


The function filtfilt implements the above zero-phase filtering scheme

In the case of a causal transfer function with a nonzero phase response, the phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic in the frequency band of interest


The most general type of a filter with a linear phase has a frequency response given by

H(ejω)=e-jωD

which has a linear phase from ω=0 to ω=2π

Note also |H(ejω)|=1 τ(ω)=D

§7.2.2 Linear-Phase TransferFunction

The output y[n] of this filter to an input x[n]=Aejωn is then given by

y[n]=Ae-jωDejωn=Aejω(n-D)

If xa(t) and ya(t) represent the continuous-time signals whose sampled versions, sampled at t = nT ,are x[n] and y[n] given above, then the delay between xa(t) and ya(t) isprecisely the group delay of amount D


If D is an integer, then y[n] is identical to x[n], but delayed by D samples

If D is not an integer, y[n], being delayed by a fractional part, is not identical to x[n]

In the latter case, the waveform of the underlying continuous-time output is identical to the waveform of the underlying continuous-time input and delayed D units of time


If it is desired to pass input signal components in a certain frequency range undistorted in both magnitude and phase, then the transfer function should exhibit a unity magnitude response and a linear-phase response in the band of interest


Figure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband


Since the signal components in the stopband are blocked, the phase response in the stopband can be of any shape

Example – Determine the impulse response of an ideal lowpass filter with a linear phase response:

c

cnj

jLP

eeH

,0

0,)(

0


Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at

n

nn

nnnh c

LP ,)(

)(sin][

0

0

As before, the above filter is noncausal and of doubly infinite length, and hence, unrealizable


By truncating the impulse response to a finite number of terms, a realizable FIR approximation to the ideal lowpass filter can be developed

The truncated approximation may or may not exhibit linear phase, depending on the value of n0 chosen


If we choose n0=N/2 with N a positive integer, the truncated and shifted approximation

NnNn

Nnnh c

LP

0,

)2/(

)2/(sin][ˆ

will be a length N+1 causal linear-phase FIR filter


Figure below shows the filter coefficients obtained using the function sinc for two different values of N

§7.2.2 Linear-Phase Transfer Function

Zero-Phase Response Because of the symmetry of the impulse res

ponse coefficients as indicated in the two figures, the frequency response of the truncated approximation can be expresse as:

)(~

][ˆ)(ˆ 2/

0

LP

NjnjN

nLP

jLP HeenheH

where , called the zero-phase response or amplitude response, is a real function of ω

)(~ LPH

§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions

Consider the two 1st-order transfer function:

1,1,1

)(,)( 21

baaz

bzzH

az

bzzH

Both transfer functions have a pole inside the unit circle at the same location z=-a and are stable

But the zero of H1(z) is inside the unit circle at z=-b, whereas, the zero of H2(z) is at z=-1/b situated in a mirror-image symmetry


Figure below shows the pole-zero plots of the two transfer functions


However, both transfer functions have an identical magnitude function as

)()()()( 122

111

zHzHzHzH

cos

sintan

cos

sintan)](arg[ 11

1

abeH j

cossintan

cos1sintan)](arg[ 11

2

abbeH j

The corresponding phase functions are


Figure below shows the unwrapped phase responses of the two transfer functions for a=0.8 and b=-0.5


From this figure it follows that H2(z) has an excess phase lag with respect to H1(z)

The excess phase lag property of H2(z) with respect to H1(z) can also be explained by observing that we can write

)()(

2

11)(

1 zAzH

bz

bz

az

bz

az

bzzH


where A(z)=(bz+1)/(z+b) is a stable allpass function

The phase function of H1(z) and H2(z) are thus related through

)](arg[)](arg[)](arg[ 12 jjj eAeHeH

As the unwrapped phase function of a stable first-order allpass function is a negative function of ω, it follows from the above that H2(z) has indeed an excess phase lag with respect to H1(z)


Generalizing the above result, let Hm(z) be

a causal stable transfer function with all zeros inside the unit circle and let H(z) be another causal stable transfer function satifying |H(ejω)|=|Hm(ejω)|

These two transfer functions are then related through H(z) =Hm(z) A (z) where A (z) is a causal stable allpass function


The unwrapped phase functions of Hm(z) and H(z) are thus related through

)](arg[)](arg[)](arg[ jjm

j eAeHeH H(z) has an excess phase lag with respect t

o Hm(z) A causal stable transfer function with all zer

os inside the unit circle is called a minimum-phase transfer function


A causal stable transfer function with all zeros outside the unit circle is called a maximum-phase transfer function

A causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function


Example – Consider the mixed-phase transfer function

)5.01)(2.01(

)4.0)(3.01(2)(

11

11

zz

zzzH

functionAllpass

1

1

functionphaseMinimum

11

11

4.014.0

)5.01)(2.01()4.01)(3.01(2

)(

zz

zzzz

zH

We can rewrite H(z) as

§7.3 Types of Linear-Phase FIR Transfer Functions

It is impossible to design an IIR transfer function with an exact linear-phase

It is always possible to design an FIR transfer function with an exact linear-phase response

We now develop the forms of the linear- phase FIR transfer function H(z) with real impluse response h[n]

called the amplitude response, also called the zero-phase response, is a real function of ω


If H(z) is to have a linear-phase, its frequency response must be of the form

where c and β are constants, and ,)(H

nN

n

znhzH

0

][)(

)()( )( HeeH cjj

Let


For a real impulse response, the magnitude response |H(ejω)| is an even function of ω, i.e.,

)()( jj eHeH Since ,the amplitude)()( HeH j

)()( HH

response is then either an even function or an odd function of ω, i.e.


The frequency response satisfies the relation

)()( * jj eHeH

)()( )()( HeHe cjcj

If is an even function, then the above relation leads to

)(H

jj ee

or, equivalently, the relation

implying that either β=0 or β=π


Substituting the value of β in the above we get

)()( )( HeeH cjj

)()( )( jcj eHeH

)(

0

][)()( ncjN

n

jjc enheHeH

From

we have


Replacing ω with -ω in the previous equation we get

)(

0

][)( lcjN

l

elhH

)(

0

][)( nNcjN

n

enNhH

Making a change of variable l=N-n, we rewrite the above equation as


The above leads to the condition

As , we have )()( HH

)()( ][][ nNcjncj enNhenh

NnnNhnh 0,][][

Thus the FIR filter with an even amplitude ewsponse will have linear phase if it has a symmetric impulse response

with c=-N/2


The above is satisfied if β=π/2 or β=-π/2 Then

If is an odd function of ω, then from)(H

)()( )()( HeHe cjcj

we get as jj ee )()( HH

)()( )( HeeH cjj

)()( HjeeH jcj

reduces to


The last equation can be rewritten as

)(

0

][)()( ncjN

n

jjc enhjeHjeH

As , from the above we get)()( HH

)(

0

][)(

cj

N

ehjH


Making a change of variable l=N-n we rewrite the last equation as

)(

0

][)(

cj

N

ehjH

)(

0

][)( ncjN

n

enhjH

Equation the above with

we arrive at the condition for linear phase as


h[n]=h[n-N] , 0≤n≤N

with c=-N/2 Therefore, a FIR filter with an odd amplitude

response will have linear-phase response if it has an antisymmetric impulse response


Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functions

For an antisymmetric FIR filter of odd length, i.e., N even

h[N/2] = 0 We examine next the each of the 4 cases


Type 1: N=8 Type 2: N=7

Type 3: N=8 Type 4: N=7


Type 1: Symmetric Impulse Response with Odd Length

In this case, the degree N is even Assume N=8 for simplicity The transfer function H(z) is given by

87654

321

]8[]7[]6[]5[]4[]3[]2[]1[]0[)(

zhzhzhzhzhzhzhzhhzH


Because of symmetry, we have h[0]=h[8], h[1]=h[7], h[2]=h[6], h[3]=h[5]

Thus we can write

H[z]=h[0](1+z-8)+h[1](z-1+z-7)+ h[2](z-2+z-6)

+h[3](z-3+z-5)+ h[4]z-4

=z-4{h[0](z4+z-4)+h[1](z3+z-3)+h[2](z2+z-2)

+h[3](z+z-1)+h[4]}


The corresponding frequency response is then given by

]}4[)cos(]3[2)2cos(]2[2

)3cos(]1[2)4cos(]0[2{)( 4

hhh

hheeH jj

The quantity inside the braces is a real function of w, and can assume positive or negative values in the range 0||


The phase here is given by

θ(ω)=-4ω+β

where β is either 0 or π,and hence, it is a linear function of ω

The group delay is given by

4)(

)(

d

d

indicating a constant group delay of 4 samples


In the general case for Type 1 FIR filters, the frequency response is of the form

)()( 2/ HeeH jNj

)cos(]2

[2]2

[)(2/

1

nnN

hN

hHN

n

where the amplitude response , also called the zero-phase response, is of the form

)(H


which is seen to be a slightly modified version of a length-7 moving-average FIR filter

The above transfer function has a symmetric impulse response and therefore a linear phase response

]2

1

2

1[

6

1)( 654321

0 zzzzzzzH

Example – Consider


A plot of the magnitude response of H0(z) along with that of the 7-point moving-average filter is shown below

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

/

Mag

nitu

de

modified filtermoving-average


Note the improved magnitude response obtained by simply changing the first and the last impulse response coefficients of a moving-average (MA) filter

It can be shown that we can express

)1(6

1)1(

2

1)( 543211

0 zzzzzzzH

which is seen to be a cascade of a 2-point MA filter with a 6-point MA filter

Thus, H0(z) has a double zero at z=-1, i.e., (ω=π)


Type 2: Symmetric Impulse Response with Even Length

In this case, the degree N is odd Assume N=7 for simplicity The transfer function if of the form

H(z)=h[0]+h[1]z-1 +h[2]z-2 +h[3]z-3 +h[4]z-4

+h[5]z-5 +h[6]z-6 +h[7]z-7


Making use of the symmetry of the impulse response coefficients, the transfer function can be written as

H(z)=h[0](1+z-7)+h[1](z-1+z-6)

+h[2](z-2+z-5)+h[3](z-3+z-4)

=z-7/2{h[0](z7/2+z-7/2 )+h[1](z5/2+z-5/2 )

+h[2](z3/2+z-3/2 )+h[3](z1/2+z-1/2 )}


The corresponding frequency response is given by

)}2

cos(]3[2)2

3cos(]2[2

)2

5cos(]1[2)

2

7cos(]0[2{)( 2/7

hh

hheeH jj

As before, the quantity inside the braces is a real function of ω, and can assume positive or negative values in the range 0≤|ω|≤π


Here the phase function is given by θ(ω)=-7/2ω+βwhere again is either 0 or π

As a result, the phase is also a linear function of ω

The corresponding group delay is τ(ω)=7/2indicating a group delay of 7/2 samples


The expression for the frequency response in the general case for Type 2 FIR filters is of the form

)(~

)( 2/ HeeH jNj

))2

1(cos(]

2

1[2)(

~ 2/)1(

1

nnN

hHN

n

where the amplitude response is given by


Type 3:Antiymmetric Impulse Response with Odd Length

In this case, the degree N is even Assume N=8 for simplicity Applying the symmetry condition we get

H(z)=z-4{h[0](z4-z-4)+h[1](z3-z-3)

+h[2](z2-z-2)+h[3](z-z-1)}


It also exhibits a linear phase response given by

2

4)(

)}sin(]3[2)2sin(]2[2

)3sin(]1[2)4sin(]0[2{)( 2/4

hh

hheeeH jjj


where β is either 0 or π


The group delay here is τ(ω)=4

indicating a constant group delay of 4 samples In the general case

)(~

)( 2/ HjeeH jNj

)sin(]2

[2)(~ 2/

1

nnN

hHN

n

where the amplitude response is of the form


Type 4:Antiymmetric Impulse Response with Even Length

In this case, the degree N is even Assume N=7 for simplicity Applying the symmetry condition we get

H(z)=z-7/2{h[0](z7/2-z-7/2)+h[1](z5/2-z-5/2)

+h[2](z3/2-z-3/2)+h[3](z1/2-z-1/2)}



It again exhibits a linear phase response given by

θ(ω)=-7/2ω+π/2+β

where β is either 0 or π

)]}2

sin(]3[2)]2

3sin(]2[2

)]2

5sin(]1[2)]

2

7sin(]0[2{)( 2/2/7

hh

hheeeH jjj


The group delay is constant andis given byτ(ω)=7/2

)(~

)( 2/ HjeeH jNj

))2

1(sin(]

2

1[2)(

~ 2/)1(

1

nnN

hHN

n

In the general case we have

where now the amplitude response is of the form


General Form of Frequency Response In each of the four types of linear-phase FIR

filters, the frequency response is of the form

)(~

)( 2/ HejeeH jjNj The amplitude response for each of th

e four types of linear-phase FIR filters can become negative over certain frequency ranges, typically in the stopband

)(~ H


The magnitude and phase responses of the linear-phase FIR are given by

0)(~

,2

0)(~

,2)(

)(~

)(

HforN

HforN

HeH j

The group delay in each case isτ(ω)=N/2


Note that, even though the group delay is constant, since in general |H(ejω)| is not aconstant, the output waveform is not a replica of the input waveform

An FIR filter with a frequency response that is a real function of ω is often called a zero- phase filter

Such a filter must have a noncausal impulse response

§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions

Consider first an FIR filter with a symmetric impulse response: h[n]=h[N-n]

Its transfer function can be written as

nN

n

nN

n

znNhznhzH

00

][][)(

mN

m

NmNN

m

nN

n

zmhzzmhznNh

000

][][][

By making a change of variable m=N-n, we can write


Hence for an FIR filter with a symmetric impulse response of length N+1 we have

)(][ 1

0

zHzmh mN

m

)()( 1 zHzzH N

But,

A real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP)


Now consider first an FIR filter with an antisymmetric impulse response:

nN

n

nN

n

znNhznhzH

00

][][)(

)(][][ 1

00

zHzzmhznNh NmNN

m

nN

n

][][ nNhnh Its transfer function can be written as

By making a change of variable m=N-n, we get


Hence, the transfer function H(z) of an FIR filter with an antisymmetric impulse response satisfies the condition

)()( 1 zHzzH N

A real-coefficient polynomial H(z) satisfying the above condition is called a antimirror-image polynomial (AIP)

Hence, a zero at z=ξo is associated with a zero at

0z


It follows from the relation H(z)=±z-NH(z-1)

that if z=ξo is a zero of H(z), so is z=1/ξo Moreover, for an FIR filter with a real impuls

e response, the zeros of H(z) occur in complex conjugate pairs


Thus, a complex zero that is not on the unit circle is associated with a set of 4 zeros given by

jj er

zrez 1

,

jez

A zero on the unit circle appear as a pair

as its reciprocal is also its complex conjugate


Since a zero at z =±1 is its own reciprocal, it can appear only singly

Now a Type 2 FIR filter satisfies H(z)=z-NH(z-1)

with degree N odd Hence, H(-1)=(-1)-NH(-1)=-H(-1)

implying H(-1)=0 , i.e., H(z) must have a zero at z=-1


Likewise, a Type 3 or 4 FIR filter satisfies

H(z)=-z-NH(z-1) Thus H(-1)=-(1)-NH(1)=-H(1)

implying that H(z) must have a zero at z = 1 On the other hand, only the Type 3 FIR filter

is restricted to have a zero at since here the degree N is even and hence,

H(-1)=-(-1)-NH(-1)=-H(-1)


Typical zero locations shown below


Summarizing

(1) Type 1 FIR filter: Either an even number or no zeros at z =1 and z =-1

(2) Type 2 FIR filter: Either an even number or no zeros at z =1, and an odd number of zeros at z =-1

(3) Type 3 FIR filter: An odd number of zeros at z =1 and z =-1


(4) Type 4 FIR filter: An odd number of zeros at z =1, and either an even number of no zeros at z =-1

The presence of zeros at z =±1 leads to the following limitations on the use of these linear-phase transfer functions for designing frequency-selective filters


A Type 2 FIR filter cannot be used to design a highpass filter since it always has a zero z =-1

A Type 3 FIR filter has zeros at both z =1 and z =-1, and hence cannot be used to design either a lowpass or a highpass or a bandstop filter


A Type 4 FIR filter is not appropriate to design lowpass and bandstop filters due to the presence of a zero at z =1

Type 1 FIR filter has no such restrictions and can be used to design almost any type of filter

§7.4 Simple Digital Filters

Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications

We now describe several low-order FIR and IIR digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications

§7.4.1 Simple FIR Digital Filters

FIR digital filters considered here have integer-valued impulse response coefficients

These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive hardware implementations


Lowpass FIR Digital Filters The simplest lowpass FIR digital filter is the

2-point moving-average filter given by

z

zzzH

2

1)1(

2

1)( 1

0

The above transfer function has a zero at z=-1 and a pole at z=0

Note that here the pole vector has a unity magnitude for all values of ω


On the other hand, as ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2, the diameter of the unit circle, to 0

Hence, the magnitude response |H0(ejω)| is a monotonically decreasing function of ωfrom ω=0 to ω=π


The maximum value of the magnitude function is 1 at ω = 0, and the minimum value is 0 at ω = π, i.e.,

0)(,1)( 00

0 jj eHeH

)2/cos()( 2/0 jj eeH

The frequency response of the above filter is given by


can be seen to be a monotonically decreasing function of ω

)2/cos()(0 jeH

The magnitude response


The frequency ω=ωc at which

)(2

1)( 0

00jj eHeH c

since the dc gain G(0)=20log10|H(ej0)|=0

is of practical interest since here the gain G(ωc ) in dB is given by

dBe

eG

j

jc

c

32log20log20

log20)(

100

10

10

§7.4.1 Simple FIR Digital Filters Thus, the gain G(ω) at ω=ωc is approximately

3 dB less than the gain atω=0

As a result, ωc is called the 3-dB cutoff frequency

To determine the value of ωc we set

2

1)2/(cos)( 22

0 cj ceH

which yields ωc =π/2


The 3-dB cutoff frequency ωc can be consid

ered as the passband edge frequency As a result, for the filter H0(z) the passband

width is approximately π/2 Note: H0(z) has a zero at z=-1 or ω=π, which

is in the stopband of the filter


A cascade of the simple FIR filter

)1(2

1)( 1

0 zzH

results in an improved lowpass frequency response as illustrated below for a cascade of 3 sections


The 3-dB cutoff frequency of a cascade of M sections is given by

)2(cos2 2/11 Mc

For M = 3, the above yields ωc=0.302π

Thus, the cascade of first-order sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband

§7.4.1 Simple FIR Digital Filters A better approximation to the ideal lowpass f

ilter is given by a higher-order moving- average filter

Signals with rapid fluctuations in sample values are generally associated with high- frequency components

These high-frequency components are essentially removed by an moving-average filter resulting in a smoother output


Highpass FIR Digital Filters The simplest highpass FIR filter is obtained f

rom the simplest lowpass FIR filter by replacing z with -z

This results in

)1(2

1)( 1

1 zzH

§7.4.1 Simple FIR Digital Filters Corresponding frequency response is given

by)2/sin()( 2/

1 jj jeeH whose magnitude response is plotted below


The monotonically increasing behavior of the magnitude function can again be demonstrated by examining the pole-zero pattern of the transfer function H1(z)

The highpass transfer function H1(z) has a zero at z =1 or ω=0 which is in the


Improved highpass magnitude response can again be obtained by cascading several sections of the first-order highpass filter

Alternately, a higher-order highpass filter of the form

1

01 )1(1

)(M

n

nn zM

zH

is obtained by replacing z with -z in the transfer in function of a moving average filter


An application of the FIR highpass filters is in moving-target-indicator (MTI) radars

In these radars, interfering signals, called clutters, are generated from fixed objects in the path of the radar beam

The clutter, generated mainly from ground echoes and weather returns, has frequency components near zero frequency (dc)


The clutter can be removed by filtering the radar return signal through a two-pulse canceler, which is the first-order FIR highpass filter H1(z)=1/2(1-z-1)

For a more effective removal it may be necessary to use a three-pulse canceler obtained by cascading two two-pulse cancelers

§7.4.2 Simple IIR Digital Filters

Lowpass IIR Digital Filters We have shown earlier that the first-order ca

usal IIR transfer function

10,1

)(1

z

KH z

has a lowpass magnitude response for α>0


An improved lowpass magnitude response is obtained by adding a factor (1+z-1) to the numerator of transfer function

10,1

)1()(

1

1

z

zKH z

This forces the magnitude response to have a zero at ω=π in the stopband of the


On the other hand, the first-order causal IIR transfer function

01,1

)(1

z

KH z

has a highpass magnitude response for α< 0


However, the modified transfer function obtained with the addition of a factor (1+z-1) to the numerator

01,1

)1()(

1

1

z

zKH z

exhibits a lowpass magnitude response


The modified first-order lowpass transfer function for both positive and negative values of α is then given by

10,1

)1()(

1

1

z

zKH LP z

As ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2 to 0


The maximum values of the magnitude function is 2K/(1-α) at ω=0 and the minimum value is 0 at ω=π, i.e.,

0)(,1

2)( 0

j

LPj

LP eHK

eH

Therefore, |HLP(ejω)| is a monotonically decreasing function of ω from ω=0 to ω=π

§7.4.2 Simple IIR Digital Filters For most applications, it is usual to have a d

c gain of 0 dB, that is to have |HLP(ej0)| =1 To this end, we choose K=(1-α)/2 resulting i

n the first-order IIR lowpass transfer function

10,1

1

2

1)(

1

1

z

zzH LP

The above transfer function has a zero at i.e., at ω=π which is in the stopband


Lowpass IIR Digital Filters A first-order causal lowpass IIR digital filter h

as a transfer function given by

1

1

1

1

2

1)(

z

zzH LP

where |α|<1 for stability The above transfer function has a zero at

z =-1 i.e., at ω=π which is in the stopband


HLP(z) has a real pole at z =α As ω increases from 0 to π, the magnitude of

the zero vector decreases from a value of 2 to 0, whereas, for a positive value of α, the magnitude of the pole vector increases from a value of 1-α to 1+α

The maximum value of the magnitude function is 1 at ω= 0, and the minimum value is 0 at ω=π

§7.4.2 Simple IIR Digital Filters i.e., |HLP(ej0)| =1, |HLP(ejπ)| =0

Therefore, |HLP(ejω)| is a monotonically decreasing function of ω from ω=0 to ω=π as indicated below


The squared magnitude function is given by

)cos21(2

)cos1()1()(

2

22

jLP eH

22

222

)cos21(2

sin)21()1()(

d

eHd jLP

The derivative of |HLP(ejω)|2 with respect to ω is given by


d|HLP(ejω)|2/dω≤0 in the range 0≤ω≤π

verifying again the monotonically decreasing behavior of the magnitude function

To determine the 3-dB cutoff frequency we set

2

1)(

2cj

LP eH

in the expression for the square magnitude function resulting in


The above quadratic equation can be solved for α yielding two solutions

cc cos21)cos1()1( 22

2

1

)cos21(2

)cos1()1(2

2

c

c

21

2cos

c

or

which when solved yields

§7.4.2 Simple IIR Digital Filters The solution resulting in a stable transfer fun

ction HLP(z) is given by

c

c

cos

sin1

)cos21(2

)cos1()1()(

2

22

jLP eH

It follows from

that HLP(z) is a BR function for |α|<1


Highpass IIR Digital Filters A first-order causal highpass IIR digital filter

has a transfer function given by

1

1

1

1

2

1)(

z

zzH HP

where |α|<1 for stability The above transfer function has a zero at z=

1 i.e., at ω=0 which is in the stopband

§7.4.2 Simple IIR Digital Filters Its 3-dB cutoff frequency ωc is given by

cc cos/)sin1( which is the same as that of HLP(z)

Magnitude and gain responses of HHP(z) are shown below


HHP(z) is a BR function for |α|<1

Example – Design a first-order highpass digital filter with a 3-dB cutoff frequency of 0.8π

Now, sin(ωc)=sin(0.8π)=0.587785 and cos(0.8

π)=-0.80902 Therefore

α=(1-sinωc)/cosωc=-0.5095245


Therefore

1

1

1

1

5095245.01

124538.0

1

1

2

1)(

z

z

z

zzH LP


Bandpass IIR Digital Filters A 2nd-order bandpass digital transfer functio

n is given by

]2cos2cos)1(2)1(1[2

)2cos1()1(

)(

2222

2

2

jBP eH

21

2

)1(1

1

2

1)(

zz

zzH BP

Its squared magnitude function is


|HBP(ejω)|2 goes to zero at ω=0 and ω=π

It assumes a maximum value of 1 at ω=ω0, called the center frequency of the bandpass filter, where

)(cos 10

The frequencies ωc1 and ωc2 where |HBP(ejω)|2 becomes 1/2 are called the 3-dB cutoff frequencies


The difference between the two cutoff frequencies, assumingωc2 >ωc1 is called the 3-dB bandwidth and is given by

21

21 1

2cos

ccwB

The transfer function HBP(z) is a BR function if |α|<1 and |β|<1


Plots of |HBP(ejω)| are shown below


Example – Design a 2nd order bandpass digital filter with center frequency at 0.4π and a 3-dB bandwidth of 0.1π

Here β=cos(ω0)=cos(0.4π)=0.309017

and9510565.0)1.0cos()cos(

1

22

wB

The solution of the above equation yields:

α=1.376382 and α=0.72654253

§7.4.2 Simple IIR Digital Filters The corresponding transfer functions are

21

2

37638.17343424.01

118819.0)(

zz

zzH BP

21

2

72654253.0533531.01

113673.0)(

zz

zzH BP

and

The poles of H’BP(z) are at z=0.3671712±

j1.11425636 and have a magnitude>1


Thus, the poles of H’BP(z) are outside the unit circle making the transfer function unstable

On the other hand, the poles of H”BP(z) are at z= 0.2667655±j0.85095546 and have a magnitude of 0.8523746

Hence H”BP(z) is BIBOstable Later we outline a simpler stability test

§7.4.2 Simple IIR Digital Filters Figures below show the plots of the magnitu

de function and the group delay of H”BP(z)


Bandstop IIR Digital Filters A 2nd-order bandstop digital filter has a tran

sfer function given by

21

21

)1(1

21

2

1)(

zz

zzzH BS

The transfer function HBS(z) is a BR function if |α|<1 and |β|<1


Its magnitude response is plotted below


Here, the magnitude function takes the maximum value of 1 at ω=0 and ω=π

It goes to 0 at ω=ω0 , whereω0, called the notch frequency, is given by

ω0=cos-1(β)

The digital transfer function HBS(z) is more commonly called a notch filter


The frequenciesωc2 andωc1 where |HBS(ejω)|2 becomes 1/2 are called the 3-dB cutoff frequencies

The difference between the two cutoff frequencies, assuming ωc2 >ωc1 is called the 3-dB notch bandwidth and is given by

21

21 1

2cos

ccwB


Higher-Order IIR Digital Filters By cascading the simple digital filters discus

sed so far, we can implement digital filters with sharper magnitude responses

Consider a cascade of K first-order lowpass sections characterized by the transfer

1

1

1

1

2

1)(

z

zzH LP

§7.4.2 Simple IIR Digital Filters The overall structure has a transfer function

given by K

LP z

zzG

1

1

1

1

2

1)(

K

LP jG

)cos21(2

)cos1()1()(

2

22

The corresponding squared-magnitude function is given by

§7.4.2 Simple IIR Digital Filters To determine the relation between its 3-dB c

utoff frequencyωc and the parameter α, we set

2

1

)cos21(2

)cos1()1(2

2

K

c

c

c

cc

C

CCC

cos1

2sincos)1(1 2

which when solved for α, yields for a stable

GLP(z):


for K=1

KKC /)1(2

c

c

cos

sin1

where

It should be noted that the expression for αgiven earilier reduces to

§7.4.2 Simple IIR Digital Filters Example – Design a lowpass filter with a 3-d

B cutoff frequency at ωc=0.4π using a single first-order section and a cascade of 4 first-order sections, and compare their gain responses

For the single first-order lowpass filter we have

1584.0)4.0cos(

)4.0sin(1

cos

sin1

c

c

§7.4.2 Simple IIR Digital Filters For the cascade of 4 first-order sections, we

substitute K=4 and get

6818.122 4/)14(/)1( KKC

251.0)4.0cos(6818.11

)6818.1()6818.1(2)4.0sin()4.0cos(6818.11(1

cos1

2sincos)1(1

2

2

c

cc

C

CCC

Next we compute

§7.4.2 Simple IIR Digital Filters The gain responses of the two filters are sho

wn below As can be seen, cascading has resulted in a

sharper roll-off in the gain response

§7.4.3 Comb Filters

The simple filters discussed so far are characterized either by a single passband and/or a single stopband

There are applications where filters with multiple passbands and stopbands are required

The comb filter is an example of such filters

§7.4.3 Comb Filters In its most general form, a comb filter has a f

requency response that is a periodic function of ω with a period 2π/L, where L is a positive integer

If H(z) is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realization with L delays resulting in a structure with a transfer function given by G(z)=H(zL)

§7.4.3 Comb Filters

If |H(ejω)|exhibits a peak at ωp,then |G(ejω)| will exhibit L peaks at ωpk/L, 0≤k≤L-1 in the frequency range 0≤ω<2π

Likewise, if |H(ejω)| has a notch atω0, then |G(ejω)| will have L notches at ω0k/L, 0≤k≤L-1 in the frequency range 0≤ω<2π

A comb filter can be generated from either an FIR or an IIR prototype filter

§7.4.3 Comb Filters For example, the comb filter generated from th

e prototype lowpass FIR filter H0(z)=1/2(1+z-1) has a transfer function

)1(2

1)()( 00

LL zzHzG

|G0(ejω)| has L notchesat ω=(2k+1)π/L and L peaks at ω=2πk/L, 0≤k≤L-1, in the frequency range 0≤ω<2π

§7.4.3 Comb Filters For example, the comb filter generated from t

he prototype highpass FIR filter

H1(z)=1/2(1-z-1) has a transfer function

)1(

1)(

1

zM

zzH

M

|G1(ejω)| has L peaksatω=(2k+1)π/L and L notches at ω=2πk/L, 0≤k≤L-1, in the frequency range 0≤ω<2π

§7.4.3 Comb Filters Depending on applications, comb filters wit

h other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter

For example, the M-point moving average filter

)1(

1)(

1

zM

zzH

M

has been used as a prototype

§7.4.3 Comb Filters This filter has a peak magnitude at ω=0, an

d M-1 notches at ω=2πl/M, 1≤l≤M-1 The corresponding comb filter has a transfer

function

)1(

1)(

L

LM

zM

zzG

whose magnitude has L peaks at ω=2πk/L, 0≤k≤L-1 and L(M-1) notches at ω=2πk/LM, 1≤k≤L(M-1)

§7.5 Complementary TransferFunctions

A set of digital transfer functions with complementary characteristics often finds useful applications in practice

Four useful complementary relations are described next along with some applications

§7.5.1 Delay-Complementary Transfer Functions

A set of L transfer functions, {Hi(z)}, 0≤i≤L-1, is defined to be delay-complementary of each other if the sum of their transfer functions is equal to some integer multiple of unit delays, i.e.,

0,)(1

0

0

L

i

ni zzH

where n0 is a nonnegative integer


A delay-complementary pair {H0(z), {H1(z)}, can be readily designed if one of the pairs is a known Type 1 FIR transfer function of odd length

Let H0(z) be a Type 1 FIR transfer function of length M=2K+1

Then its delay-complementary transfer function is given by

)()( 01 zHzzH K


Let the magnitude response of H0(z) be equal to and 1±δp in the passband and less than or equal to δs in the stopband where δp and δs are very small numbers

Now the frequency response of H0(z) can be expressed as

)()( 00 HeeH jKj

where is the amplitude response)(0 H


Its delay-complementary transfer function H1

(z) has a frequency response given by)](1[)()( 011 HeHeeH jKjKj

Now, in the passband, ,1)(1 0 pp H

and in the stopband, ss H )(0

It follows from the above equation that in the stopband, and in the passband,

pp H )(1

ss H 1)(1 1


As a result, H1(z) has a complementary magnitude response characteristic to that of H0(z) with a stopband exactly identical to the passband of H0(z) , and a passband that is exactly identical to the stopband of H0(z)

Thus, if H0(z) is a lowpass filter, H1(z) will be a highpass filter, and vice versa


The frequency ω0 at which

5.0)()( 10 HH

the gain responses of both filters are 6 dB below their maximum values

The frequency ω0 is thus called the 6 dB crossover frequency


Example – Consider the Type 1 bandstop transfer function

)45541()1(64

1)( 121084242 zzzzzzzH BS

)45541()1(64

1)()(

121084242

10

zzzzzz

zHzzH BSBP

Its delay-complementary Type 1 bandpass transfer function is given by


Plots of the magnitude responses of HBS(z) and HBP(z) are shown below

§7.5.2 Allpass Complementary Transfer Functions

A set of M digital transfer functions, {Hi(z)}, 0≤i≤M-1, is defined to be allpass-complementary of each other, if the sum of their transfer functions is equal to an allpass function, i.e.,

1

0

)()(M

ii zAzH

§7.5.3 Power-Complementary Transfer Functions

A set of M digital transfer functions, {Hi(z)}, 0≤i≤M-1, is defined to be power-complementary of each other, if the sum of their square-magnitude responses is equal to a constant K for all values of ω, i.e.,

1

0

2allfor,)(

M

i

ji KeH


By analytic continuation, the above property is equal to

1

0

1 allfor,)()(M

iii KzHzH

for real coefficient Hi(z) Usually, by scaling the transfer functions, the

power-complementary property is defined for K=1


For a pair of power-complementary transfer functions, H0(z) and H1(z), the frequency ω0 where |H0(ejω0)|2= |H1(ejω0)|2 =0.5, is called the cross-over frequency

At this frequency the gain responses of both filters are 3-dB below their maximum values

As a result, ω0 is called the 3-dB cross-over frequency


Example – Consider the two transfer functions H0(z) and H1(z) given by

)]()([2

1)(

)]()([2

1)(

101

100

zAzAzH

zAzAzH

where A0(z) ana A1(z) are stable allpass transfer functions

Note that H0(z)+H1(z)=A0(z) Hence, H0(z) and H1(z) are allpass complement

ary


It can be shown that H0(z) and H1(z) are also power-complementary

Moreover, H0(z) and H1(z) are bounded-real transfer functions

§7.5.4 Double-Complementary Transfer Functions

A set of M transfer functions satisfying both the allpass complementary and the power- complementary properties is known as a doubly-complementary set

A pair of doubly-complementary IIR transfer functions, H0(z) and H1(z), with a sum of allpass decomposition can be simply realized as indicated below


)(

)()(

)(

)()( 1

10

0 zX

zYzH

zX

zYzH

A0(z)

A1(z)

●

●

●

X(z)Y0(z)

Y1(z)

1/2

-1


Example – The first-order lowpass transfer function

1

1

1

1

2

1)(

z

zzH LP

)]()([2

1

11

2

1)( 101

1

zAzAz

zzH LP

1

1

10 1)(,1)(

z

zzAzA

can be expressed as

where


Its power-complementary highpass transfer function is thus given by

1

1

1

1

10

11

21

11

21)]()([

21)(

zz

zzzAzAzHLP

The above expression is precisely the first- order highpass transfer function described earlier

Complementary TransferFunctions

Figure below demonstrates the allpass complementary property and the power complementary property of HLP(z) and HHP(z)

§7.5.5 Power-Symmetric Filtersand Conjugate Quadrature Filters

A real-coefficient causal digital filter with a transfer function H(z) is said to be a power- symmetric filter if it satisfies the condition

KzHzHzHzH )()()()( 11

where K>0 is aconstant


It can be shown that the gain function G(ω) of a power-symmetric transfer function at ω=π is given by

dBK 3log10 10

constanta)()()()( 11 zGzGzHzH

If we defined G(z)= H(-z), then it follows from the definition of the power-symmetric filter that H(z) and G(z) are power- complementary as


Conjugate Quadratic Filter If a power-symmetric filter has an FIR transf

er function H(z) of order N, then the FIR digital filter with a transfer function

)()( 1 zHzzG N

is called a conjugate quadratic filter of H(z) and vice-versa


It follows from the definition that G(z) is also a power-symmetric causal filter

It also can be seen that a pair of conjugate quadratic filters H(z) and G(z) are also power-complementary


Example – Let H(z)=1-2z-1+6z-2+3z-3 We form

100)345043()345043(

)3621)(3621()3621)(3621(

)()()()(

313

313

32321

32321

11

zzzzzzzz

zzzzzzzzzzzz

zHzHzHzH

H(z) is a power-symmetric transfer function

§7.8 Digital Two-Pairs

The LTI discrete-time systems considered so far are single-input, single-output structures characterized by a transfer function

Often, such a system can be efficiently realized by interconnecting two-input, two- output structures, more commonly called two-pairs

§7.8 Digital Two-Pairs

Figures below show two commonly used block diagram representations of a two-pair

Here Y1 and Y2 denote the two outputs X1 and X2 denote the two inputs,where the dependencies on the variable z has been omitted for simplicity

X1

Y1

Y2

X2

X1

X2

Y1

Y2

§7.8.1 Characterization The input-output relation of a digital two- pai

r is given by

2

1

2221

1211

2

1

X

X

tt

tt

Y

Y

2221

1211

tt

tt

In the above relation the matrix τ given by

is called the transfer matrix of the two-pair

§7.8.1 Characterization

It follows from the input-output relation that the transfer parameters can be found as follows:

02

2220

1

221

02

1120

1

111

12

12

,

,

XX

XX

XY

tXY

t

XY

tXY

t


An alternate characterization of the two-pair is in terms of its chain parameters

2

2

1

1

X

Y

DC

BA

Y

X

DC

BA

where the matrix Γ given by

is called the chain matrix of the two-pair


The relation between the transfer parameters and the chain parameters are given by

21

22112112

21

11

21

22

21

22211211

,,,1

,1,,

ttttt

Dtt

Ctt

Bt

A

ACt

At

ABCADt

ACt

§7.8.2 Two-Pair InterconnectionSchemes

Cascade Connection – Γ-cascade

2

2

1

1

2

2

1

1

X

Y

DC

BA

Y

X

X

Y

DC

BA

Y

X

DC

BA

DC

BA1X

1Y 2Y

2X

2Y 1X

1Y 2X Here


But from figure, X”1=Y’

2 and Y”1=X’

2 Substituting the above relations in the first e

quation on the previous slide and combining the two equations we get

2

2

1

1

X

Y

DC

BA

DC

BA

Y

X

DC

BA

DC

BA

DC

BA

Hence


Cascade Connection –τ-cascade

2

1

2221

1211

2

1

2

1

2221

1211

2

1

X

X

tt

tt

Y

Y

X

X

tt

tt

Y

Y Here

1X 1Y

2Y 2X

2Y

1X 1Y

2X

2221

1211

tt

tt

2221

1211

tt

tt


But from figure, X”1=Y’

1 and X’2 = Y”

2

Substituting the above relations in the first equation on the previous slide and combining the two equations we get

2

1

2221

1211

2221

1211

2

1

X

X

tt

tt

tt

tt

Y

Y

2221

1211

2221

1211

2221

1211

tt

tt

tt

tt

tt

tt

Hence,


Constrained Two-Pair

)(1)(

)()(

)(

22

211211

1

1

zGtzGtt

t

zGBAzGDC

XY

zH

It can be shown that

Y2

G(z)X1

X2Y1H(z)

§7.9 Algebraic Stability Test

We have shown that the BIBO stability of a causal rational transfer function requires that all its poles be inside the unit circle

For very high-order transfer functions, it is very difficult to determine the pole locations analytically

Root locations can of course be determined on a computer by some type of root finding algorithms

§7.9.1 The Stability Triangle

We now outline a simple algebraic test that does not require the determination of pole locations

The Stability Triangle For a 2nd-order transfer function the stability

can be easily checked by examining its denominator coefficients


denote the denominator of the transfer function

In terms of its poles, D(z) can be expressed as

22

111)( zdzdzD

221

121

12

11 )(1)1)(1()( zzzzzD

212211 ,)( dd

Let

Comparing the last two equations we get


Now the coefficient d2 is given by the product of the poles

Hence we must have

1,1 21

12 d

21 1 dd

The poles are inside the unit circle if

It can be shown that the second coefficient condition is given by


The region in the (d1, d2)-plane where the two coefficient condition are satisfied, called the stability triangle, is shown below


Example – Consider the two 2nd-order bandpass transfer functions designed earlier:

21

2

37638.17343424.01118819.0)(

zzzzHBP

21

2

72654253.0533531.01113673.0)(

zzzzHBP


In the case of H’BP(z), we observe that

d1=-0.7343424, d2=1.3763819 Since here |d2|>1, H’

BP(z) is unstable On the other hand, in the case of H”

BP(z), we observe that

d1=-0.53353098, d2=0.726542528 Here, |d2|<1 and |d1|<1+ d2, and hence H”

BP(z) is BIBO stable

§7.9.2 A Stability Test Procedure

Let DM(z) denote the denominator of an M-th order causal IIR transfer function H(z):

iM

i iM zdzD

0)(

)()(

)(1

zDzDz

zAM

MM

M

where we assume d2=1 for simplicity Defined an M-th order allpass transfer functi

on:


then it follows that

MM

MM

MMMMM

M zdzdzdzdzzdzdzdd

zA

1

12

21

1

12

21

1

1)(

M

i iM zzD1

1)1()(

M

i iM

Md1

)1(

Or, equivalently

If we express


Now for stability we must have, |λi|<1, which implies the condition |dM|<1

Define

kM= AM(∞)<1= dM

Then a necessary condition for stability of AM

(z), and hence,the transfer function H(z) is given by

12 Mk


Assume the above condition holds We now form a new function

)(1

)()(1

)()(1 zAd

dzAz

zAkkzA

zzAMM

MM

MM

MMM

)1(1

)2(2

11

)1()2(1

12

1 1)(

MM

MM

MMMM

M zdzdzdzzdzdd

zA

Substituting the rational form of AM(z) in the above equation we get


Hence, AM-1(z) is an allpass function of order M-1

Now the poles λo of AM-1(z) are given by the roots of the equation

11,1 2

Mid

dddid

M

iMMi

MoM k

A 1)(

where


If AM(z) is a stable allpass function, then

Hence 1)( oMA By assumption 12 Mk

Thus, if AM(z) is a stable allpass function, then the condition holds only if |λo | <1

1for,1

1for,1

1for,1

)(

z

z

z

zAM


Thus, if AM(z) is a stable allpass function and , then AM-1(z) is also a stable allpass function of one order lower

12 Mk

We now prove the converse, i.e., if AM-1(z) is a stable allpass function and , then AM

(z) is also a stable allpass function

12 Mk

Or, in other words AM-1(z) is a stable allpass function


To this end, we express AM(z) in terms of A

M-1(z) arriving at

)(1)(

)(1

11

1

zAzkzAzk

zAMM

MMM

MM k

A 1)( 011

0

By assumption holds12 Mk

If ζ0 is a pole of AM(z) , then


|AM-1(ζ0)|>|ζ0|

The above condition implies |AM-1(ζ0)|>1 if |ζ0|≥1

Assume AM-1(z) is a stable allpass function

Then AM-1(z) ≤1 for |z|≥1

Thus, for |ζ0|≥1, we should have

|AM-1(ζ0)| ≤1

Therefore, i.e., 1)( 011

0 MA


Thus there is a contradiction On the other hand, if |ζ0|<1 then from

AM-1(z) >1 for |z|<1

we have |AM-1(ζ0)|>1 The above condition does not violate the co

ndition |AM-1(ζ0)|>|ζ0|


Summarizing, a necessary and sufficient set of conditions for the causal allpass function AM-1(z) to be stable is therefore:

Thus, if and if AM-1(z) is a stable allpass function, then AM(z) is also a stable allpass function

12 Mk

(1) , and

(2) The allpass function AM-1(z) is stable

12 Mk

§7.9.2 A Stability Test Procedure Thus, once we have checked the condition

, we test next for the stability of the lower-order allpass function AM-1(z)

12 Mk

The process is then repeated, generating a set of coefficients:

and a set of allpass functions of decreasing order:

121 ,,...,, kkkk MM

1)(),(),(...,),(),( 0121 zAzAzAzAzA MM

§7.9.2 A Stability Test Procedure The allpass function AM(z) is atable if and onl

y if for i12 ik

12341)( 234

zzzz

zH

432

23

14

12

23

34

4

234

234

4

1

41

41

21

43

143

21

41

41

)(dzdzdzdz

zdzdzdzd

zzzz

zzzzzA

141)( 444 dAk Note:

Example – Test the stability of

From H(z) we generate a 4-th order allpass function


we determine the coefficients {d’i} of the thir

d-order allpass function A3(z) from the dcoefficients { d’

i} of A4(z):

31,1 2

4

44

iddddi

d ii

151

52

1511

11511

52

151

11

)(23

23

32

23

1

11

22

33

3

zzz

zzz

zdzdzdzdzdzd

zA

Using


Following the above procedure, we derive the next two lower-order allpass functions:

1151)( 333 dAk

10153

110153

)(

22479

224159

1224159

22479

)( 12

2

2

z

zzA

zz

zzzA

Note:


Since all of the stability conditions are satisfied, A4(z) and hence H(z) are stable

Note: It is not necessary to derive A1(z) since A2(z) can be tested for stability using the coefficient conditions

110153)(1

122479)(

12

22

Ak

Ak

Note:

Documents

Chapter 7 LTI Discrete-Time Systems in the Transform Domain