lecture 7 lti discrete-time systems in the transform domaintania/teaching/dsp/lecture 7 lti...

LTI DiscreteLTI Discrete--Time SystemsTime Systems

in Transform Domainin Transform DomainSimple FiltersSimple Filters

Comb Filters Comb Filters (Optional reading)(Optional reading)

AllpassAllpass Transfer FunctionsTransfer Functions

Minimum/Maximum Phase Transfer FunctionsMinimum/Maximum Phase Transfer Functions

Complementary Filters Complementary Filters (Optional reading)(Optional reading)

Digital TwoDigital Two--Pairs Pairs (Optional reading)(Optional reading)

Tania Stathaki

811b

[email protected]

Simple Digital FiltersSimple 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

Simple FIR Digital FiltersSimple 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

• 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

z

zzzH

2

11 1

21

0+

=+= − )()(


• 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 is

a monotonically decreasing function of ωfrom ω = 0 to ω = π

|)(| 0ωjeH


• The maximum value of the magnitude

function is 1 at ω = 0, and the minimum

value is 0 at ω = π, i.e.,

• The frequency response of the above filter

is given by

01 00

0 == |)(|,|)(| πjj eHeH

)2/cos()( 2/0 ω= ω−ω jj eeH


• The magnitude response

is a monotonically decreasing function of ω)2/cos(|)(| 0 ω=ωjeH

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

First-order FIR lowpass filter


• The frequency at which

is of practical interest since here the gain in dB is

since the DC gain is

cω=ω

)(2

1)( 0

00jjeHeH c =ω

dB32log20)(log20 100

10 −≅−= jeH

)(log20)ω(Gω

10cj

c eH=

0)(log20 0

10 =jeH


• Thus, the gain G(ω) at is approximately 3 dB less than the gain at ω=0

• As a result, is called the 3-dB cutoff

frequency

• To determine the value of we set

which yields

cω=ω

cω

cω

2/π=ωc

2

1220 )2/(cos|)(| =ω=ω

cj ceH


• The 3-dB cutoff frequency can be

considered as the passband edge frequency

• As a result, for the filter the passband

width is approximately π/2

• The stopband is from π/2 to π

• Note: has a zero at or ω = π,which is in the stopband of the filter

cω

)(zH0

)(zH0 1−=z


• A cascade of the simple FIR filter

results in an improved lowpass frequency

response as illustrated below for a cascade

of 3 sections

)()( 1

21

0 1 −+= zzH

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

First-order FIR lowpass filter cascade


• The 3-dB cutoff frequency of a cascade of

M sections is given by

• For M = 3, the above yields

• 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

)2(cos2 2/11 Mc

−−=ω

π=ω 302.0c


• A better approximation to the ideal lowpass

filter is given by a higher-order Moving

Average (MA) filter

• Signals with rapid fluctuations in sample

values are generally associated with high-

frequency components

• These high-frequency components are

essentially removed by an MA filter

resulting in a smoother output waveform


Highpass FIR Digital Filters

• The simplest highpass FIR filter is obtained

from the simplest lowpass FIR filter by

replacing z with

• This results in

)()( 1

2

11 1 −−= zzH

z−


• Corresponding frequency response is given

by

whose magnitude response is plotted below

)2/sin()( 2/1 ω= ω−ω jj ejeH

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

First-order FIR highpass filter


• The monotonically increasing behavior of

the magnitude function can again be

demonstrated by examining the pole-zero

pattern of the transfer function

• The highpass transfer function has a

zero at z = 1 or ω = 0 which is in the

stopband of the filter

)(zH1

)(zH1


• 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

is obtained by replacing z with in the

transfer function of an MA filter

z−

nMn

n

MzzH −−

=∑ −= 10

11 1)()(

Simple IIR Digital FiltersSimple IIR Digital Filters

Lowpass IIR Digital Filters

• A first-order causal lowpass IIR digital

filter has a transfer function given by

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

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

α−

+α−= −

−

1

1

1

1

2

1)(

z

zzHLP

1−=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 to

• The maximum value of the magnitude

function is 1 at ω = 0, and the minimum

value is 0 at ω = π

α−1 α+1

)(zHLP


• i.e.,

• Therefore, is a monotonically

decreasing function of ω from ω = 0 to ω = πas indicated below

0|)(|,1|)(| 0 == πjLP

jLP eHeH

|)(| ωjLP eH

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

α = 0.8α = 0.7α = 0.5

10-2

10-1

100

-20

-15

-10

-5

0

ω/π

Gain, dB α = 0.8

α = 0.7α = 0.5


• The squared magnitude function is given by

• The derivative of with respect

to ω is given by

)cos21(2

)cos1()1(|)(|

2

22

ωα−α+

ω+α−=ωj

LP eH

2|)(| ωjLP eH

22

222

)cos21(2

sin)21()1(|)(|

α+ωα−

ωα+α+α−−=

ω

ω

d

eHd jLP


in the range

verifying again the monotonically decreasing

behavior of the magnitude function

• To determine the 3-dB cutoff frequency we set

in the expression for the squared magnitude

function resulting in

0ω/)(2

ω ≤deHd j

LP π≤ω≤0

2

1)(2

ω =cj

LP eH


or

which when solved yields

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

2

1

)cos21(2

)cos1()1(2

2

=ωα−α+

ω+α−

c

c

cc ωα−α+=ω+α− cos21)cos1()1( 22

21

2cos

α+

α=ωc


• The solution resulting in a stable transfer

function is given by

• It follows from

that is a BR function for |α| < 1

)cos21(2

)cos1()1(|)(|

2

22

ωα−α+

ω+α−=ωj

LP eH

)(zHLP

)(zHLP

c

cωω−

=αcossin1


Highpass IIR Digital Filters

• A first-order causal highpass IIR digital filter

has a transfer function given by

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

• It is a BR function for |α| < 1

−−+

= −

−

1

1

1

1

2

1)(

z

zzHHP α

α


• Its 3-dB cutoff frequency is given by

which is the same as that of

• Magnitude and gain responses of

are shown below

cc ωω−=α cos/)sin1(

cω

)(zHLP

)(zHHP

10-2

10-1

100

-20

-15

-10

-5

0

ω/π

Gain, dB

α = 0.8α = 0.7α = 0.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

α = 0.8α = 0.7α = 0.5

Example 1-First Order HP Filter

• Design a first-order highpass filter with a 3-

dB cutoff frequency of 0.8π

• Now,

and

• Therefore

587785.0)8.0sin()sin( =π=ωc80902.0)8.0cos( −=π

5095245.0cos/)sin1( −=ωω−=α cc

Example 1-First Order HP Filter

• Therefore,

+

−=

−

−

1

1

509524501

12452380

z

z

..

−

−+=

−

−

1

1

1

1

2

1

z

zzHHP

α

α)(


Bandpass IIR Digital Filters

• A 2nd-order bandpass digital transfer


• Its squared magnitude function is2)( ωj

BP eH

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

)2cos1()1(2222

2

ωα+ωα+β−α+α+β+

ω−α−=

α+α+β−

−α−= −−

−

21

2

)1(1

1

2

1)(

zz

zzHBP


• goes to zero at ω = 0 and ω = π• It assumes a maximum value of 1 at ,

called the center frequency of the bandpass

filter, where

• The frequencies and where

becomes 1/2 are called the 3-dB cutoff

frequencies

2|)(| ωjBP eH

2|)(| ωjBP eH

oω=ω

)(cos 1 β=ω −o

1cω 2cω


• The difference between the two cutoff

frequencies, assuming is called

the 3-dB bandwidth and is given by

• The transfer function is a BR

function if |α| < 1 and |β| < 1

12 cc ω>ω

)(zHBP

112 cos−=ω−ω= ccwB

α+

α21

2


• Plots of are shown below|)(| ωjBP eH

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

β = 0.34

α = 0.8α = 0.5α = 0.2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

α = 0.6

β = 0.8β = 0.5β = 0.2

Example 2-Second Order BP Filter

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

• Here

and

• The solution of the above equation yields:α = 1.376382 and α = 0.72654253

9510565.0)1.0cos()cos(1

22

=π==α+

αwB

309017.0)4.0cos()cos( =π=ω=β o


• The corresponding transfer functions are

and

• The poles of are at z = 0.3671712 and have a magnitude > 1

±114256361.j

21

2

376381734342401

1188190

−−

−

+−

−−=

zz

zzHBP

...)('

21

2

72654253053353101

1136730

−−

−

+−

−=

zz

zzHBP

...)("

)(' zHBP


• Thus, the poles of are outside the

unit circle making the transfer function

unstable

• On the other hand, the poles of are

at z = and have a

magnitude of 0.8523746

• Hence, is BIBO stable

)(' zHBP

)(" zHBP8095546026676550 .. j±

)(" zHBP


• Figures below show the plots of the

magnitude function and the group delay of

)(" zHBP

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

0 0.2 0.4 0.6 0.8 1-1

0

1

2

3

4

5

6

7

ω/π

Group delay, samples


Bandstop IIR Digital Filters

• A 2nd-order bandstop digital filter has a

transfer function given by

• The transfer function is a BR

function if |α| < 1 and |β| < 1

α+α+β−

+β−α+= −−

−−

21

21

)1(1

21

2

1)(

zz

zzzHBS

)(zHBS


• Its magnitude response is plotted below

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

α = 0.8α = 0.5α = 0.2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

β = 0.8β = 0.5β = 0.2


• Here, the magnitude function takes the

maximum value of 1 at ω = 0 and ω = π• It goes to 0 at , where , called the

notch frequency, is given by

• The digital transfer function is more

commonly called a notch filter

oω=ω oω

)(cos 1 β=ω −o

)(zHBS


• The frequencies and where

becomes 1/2 are called the 3-dB cutoff

frequencies

• The difference between the two cutoff

frequencies, assuming is called

the 3-dB notch bandwidth and is given by

2|)(| ωjBS eH1cω 2cω

12 cc ω>ω

112 cos−=ω−ω= ccwB

α+

α21

2


Higher-Order IIR Digital Filters

• By cascading the simple digital filters

discussed so far, we can implement digital

filters with sharper magnitude responses

• Consider a cascade of K first-order lowpass

sections characterized by the transfer

function

α−

+α−= −

−

1

1

1

1

2

1)(

z

zzHLP


• The overall structure has a transfer function

given by

• The corresponding squared-magnitude


K

LPz

zzG

−+−

= −

−

1

1

α1

1

2

α1)(

K

jLP eG

ωα−α+

ω+α−=ω

)cos21(2

)cos1()1(|)(|

2

22


• To determine the relation between its 3-dB

cutoff frequency and the parameter α, we set

which when solved for α, yields for a stable :

cω

2

1

)cos21(2

)cos1()1(2

2

=

ωα−α+

ω+α−K

c

c

)(zGLP

c

cc

C

CCC

ω+−−ω−ω−+

=αcos1

2sincos)1(1 2


where

• It should be noted that the expression given

above reduces to

for K = 1

KKC /)( 12 −=

c

cωω−

=αcossin1

• Design a lowpass filter with a 3-dB cutoff

frequency at 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

π=ω 4.0c

1584.0)4.0cos(

)4.0sin(1

cos

sin1=

ππ+

=ωω+

=αc

c

Example 3-Design of an LP Filter


• For the cascade of 4 first-order sections, we

substitute K = 4 and get

• Next we compute

6818122 4141 ./)(/)( === −− KKC

c

cc

C

CCC

ω+−−ω−ω−+

=αcos1

2sincos)1(1 2

)4.0cos(6818.11

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

π+−−π−π−+

=

2510.−=


• The gain responses of the two filters are

shown below

• As can be seen, cascading has resulted in a

sharper roll-off in the gain response

10-2

10-1

100

-20

-15

-10

-5

0

ω/π

Gain, dB

K=4

K=1Passband details

10-2

10-1

100

-4

-2

0

ω/π

Gain, dB K=1

K=4

Comb FiltersComb 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


• In its most general form, a comb filter has a

frequency 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 )()( LzHzG =


• If exhibits a peak at , then

will exhibit L peaks at ,

in the frequency range

• Likewise, if has a notch at ,

then will have L notches at ,

in the frequency range

• A comb filter can be generated from either

an FIR or an IIR prototype filter

|)(| ωjeH

|)(| ωjeH

|)(| ωjeG

|)(| ωjeGpω

oω

Lkp /ω

Lko /ω

10 −≤≤ Lk

10 −≤≤ Lk

π<ω≤ 20

π<ω≤ 20


• For example, the comb filter generated from

the prototype lowpass FIR filter

has a transfer function

• has L notches

at ω = (2k+1)π/L and Lpeaks at ω = 2π k/L,

)( 1

21 1 −+ z=)(zH0

)()()( LL zzHzG −+== 121

00

10 −≤≤ Lk , in thefrequency range

π<ω≤ 20

|)(| 0ωjeG

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

Comb filter from lowpass prototype


• Furthermore, the comb filter generated from

the prototype highpass FIR filter

has a transfer function

• has L peaks

at ω = (2k+1)π/L and Lnotches at ω = 2π k/L,

|)(| 1ωjeG

)( 1

21 1 −− z=)(zH1

)()()( LL zzHzG −−== 121

11

10 −≤≤ Lk , in thefrequency range

π<ω≤ 200 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

Comb filter from highpass prototype


• Depending on applications, comb filters with other

types of periodic magnitude responses can be

easily generated by appropriately choosing the

prototype filter

• For example, theM-point moving average filter

has been used as a prototype

)1(

1)(

1−

−

−−

=zM

zzH

M


• This filter has a peak magnitude at ω = 0, and

notches at ,

• The corresponding comb filter has a transfer

function

whose magnitude has L peaks at ,

and notches at

,

1−M M/2 lπ=ω 11 −≤≤ Ml

)1(

1)(

L

LM

zM

zzG

−

−

−−

=

Lk/2π=ω10 −≤≤ Lk )( 1−ML

LMk/2π=ω )( 11 −≤≤ MLk


Definition

• An IIR transfer function A(z) with unity

magnitude response for all frequencies, i.e.,

is called an allpass transfer function

• An M-th order causal real-coefficient

allpass transfer function is of the form

ω=ω allfor,1|)(| 2jeA

M

M

M

M

MM

MMM

zdzdzd

zzdzddzA

−+−−

−

−+−−−

++++++++

±=1

1

1

1

1

1

1

1

1)(

K

K


• If we denote the denominator polynomials of

as :

then it follows that can be written as:

• Note from the above that if is a pole of a

real coefficient allpass transfer function, then it

has a zero at

)(zDM)(zAMM

M

M

MM zdzdzdzD −+−−

− ++++= 1

1

1

11)( K

)(zAM

)(

)()(

zD

zDzM

M

MM

zA1−−

±=

φ= jrez

φ−= jr ez 1


• The numerator of a real-coefficient allpass

transfer function is said to be themirror-

image polynomial of the denominator, and

vice versa

• We shall use the notation to denote

the mirror-image polynomial of a degree-M

polynomial , i.e.,

)(zDM~

)(zDM

)()( zDzzD MM

M−=

~


• The expression

implies that the poles and zeros of a real-

coefficient allpass function exhibitmirror-

image symmetry in the z-plane

)(

)()(

zD

zDzM

M

MM

zA1−−

±=

321

321

32.018.04.01

4.018.02.0)( −−−

−−−

−+++++−

=zzz

zzzzA

-1 0 1 2 3

-1.5

-1

-0.5

0

0.5

1

1.5

Real Part

Imaginary Part


• To show that we observe that

• Therefore

• Hence,

)(

)()(

1

1

−− ±=

zD

zDzzA

M

M

M

M

)(

)(

)(

)()()(

1

11

−

−−− =

zD

zDz

zD

zDzzAzA

M

M

M

M

M

M

MM

1|)(| =ωjM eA

1)()(|)(| 12 == ω=−ω

jezMMj

M zAzAeA


• 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

• A causal stable real-coefficient allpass transfer

function is a lossless bounded real (LBR)

function or, equivalently, a causal stable allpass

filter is a lossless structure


• The magnitude function of a stable allpass

function A(z) satisfies:

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

<>==><

1for1

1for1

1for1

z

z

z

zA

,

,

,

)(

)]([)( ωθ−=ωτω cd

d


• The unwrapped phase function of a

stable allpass function is a monotonically

decreasing function of ω so that τ(ω) iseverywhere positive in the range 0 < ω < π

• The group delay of an M-th order stable

real-coefficient allpass transfer function

satisfies:

)(ωθc

π=ω∫ ωτπ

Md0

)(

AllpassAllpass Transfer FunctionTransfer Function

A 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 allpassfilter A(z) so that the overall cascade has a constant group delay in the band of interest

AllpassAllpass Transfer FunctionTransfer Function

• Since , we have

• Overall group delay is the given by the sum

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

1|)(| =ωjeA

|)(||)()(| ωωω = jjj eGeAeG

G(z) A(z)

MinimumMinimum--Phase and MaximumPhase and Maximum--

Phase Transfer FunctionsPhase Transfer Functions

• Consider the two 1st-order transfer functions:

• Both transfer functions have a pole inside the

unit circle at the same location and are

stable

• But the zero of is inside the unit circle

at , whereas, the zero of is at

situated in a mirror-image symmetry

11121 <<== +

+++ bazHzH

azbz

azbz ,,)(,)(

az −=

bz −=

bz 1−=

)(zH1

)(zH2



• Figure below shows the pole-zero plots of

the two transfer functions

)(1 zH )(2 zH



• However, both transfer functions have an

identical magnitude as

• The corresponding phase functions are

)()()()( 1

22

1

11

−− = zHzHzHzH

ωcos

ωsintan

ωcos1

ωsintan)](arg[ 11ω

2 +−

+= −−

ab

beH j

ωcos

ωsintan

ωcos

ωsintan)](arg[ 11ω

1 +−

+= −−

abeH j



• Figure below shows the unwrapped phase

responses of the two transfer functions for

a=0.8 and b=-0.5

0 0.2 0.4 0.6 0.8 1-4

-3

-2

-1

0

1

2

ω/π

Phase, degrees

H1(z)

H2(z)



• From this figure it follows that has

an excess phase lag with respect to

• Generalizing the above result, we can show

that a causal stable transfer function with all

zeros outside the unit circle has an excess

phase compared to a causal transfer

function with identical magnitude but

having all zeros inside the unit circle

)(2 zH

)(1 zH



• A causal stable transfer function with all zeros

inside the unit circle is called aminimum-phase

transfer function

• A causal stable transfer function with all zeros

outside the unit circle is called amaximum-

phase transfer function

• Any nonminimum-phase transfer function can be

expressed as the product of a minimum-phase

transfer function and a stable allpass transfer

function

Complementary Transfer FunctionsComplementary Transfer Functions

• 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


Delay-Complementary Transfer Functions

• A set of L transfer functions, ,

, 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.,

where is a nonnegative integer

)}({ zHi10 −≤≤ Li

0,)(1

0

≠ββ= −−

=∑ onL

ii zzH

on


• A delay-complementary pair

can be readily designed if one of the pairs is

a known Type 1 FIR transfer function of

odd length

• Let be a Type 1 FIR transfer function

of lengthM = 2K+1

• Then its delay-complementary transfer


)}(),({ 10 zHzH

)()( 01 zHzzH K −= −

)(0 zH


• Let the magnitude response of be

equal to in the passband and less than

or equal to in the stopband where and

are very small numbers

• Now the frequency response of can be

expressed as

where is the amplitude response

)(0 zH

pδ±1

sδ pδ

sδ)(0 zH

)()( 00 ω= ω−ω HeeH jKj ~

)(0 ωH~


• Its delay-complementary transfer function

has a frequency response given by

• Now, in the passband,

and in the stopband,

• It follows from the above equation that in

the passband, and in the

stopband,

)(1 zH

)](1[)()( 011 ω−=ω= ω−ω−ω HeHeeH jKjKj ~ ~

,1)(1 0 pp H δ+≤ω≤δ−~

ss H δ≤ω≤δ− )(0~

pp H δ≤ω≤δ− )(1~

ss H δ+≤ω≤δ− 1)(1 1~


• As a result, has a complementary

magnitude response characteristic, with a

stopband exactly identical to the passband

of , and a passband that is exactly

identical to the stopband of

• Thus, if is a lowpass filter, will

be a highpass filter, and vice versa

)(1 zH

)(0 zH

)(0 zH

)(1 zH)(0 zH


• At frequency at which

the gain responses of both filters are 6 dB

below their maximum values

• The frequency is thus called the 6-dB

crossover frequencyoω

oω

5.0)()( 10 =ω=ω oo HH~~

Example 4

• Consider the Type 1 bandstop transfer function

• Its delay-complementary Type 1 bandpass transfer


)45541()1(64

1)( 121084242 −−−−−− +−++−+= zzzzzzzHBS

)45541()1(64

1 121084242 −−−−−− +++++−−= zzzzzz

)()( 10 zHzzH BSBP −= −

Example 4

• Plots of the magnitude responses of

and are shown below

)(zHBS)(zHBP

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Magnitude

HBS(z) H

BP(z)


Allpass Complementary Filters

• A set of M digital transfer functions, ,

, is defined to be allpass-

complementary of each other, if the sum of

their transfer functions is equal to an allpass

function, i.e.,

)}({ zHi10 −≤≤ Mi

)()(1

0

zAzHM

ii =∑

−

=


Power-Complementary Transfer Functions

• A set of M digital transfer functions, ,

, 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.,

)}({ zHi10 −≤≤ Mi

ω=∑−

=

ω allfor,)(1

0

2KeH

M

i

ji


• By analytic continuation, the above

property is equal to

for real coefficient

• Usually, by scaling the transfer functions,

the power-complementary property is

defined for K = 1

)(zHi

ω=∑−

=

− allfor,)()(1

0

1 KzHzHM

iii


• For a pair of power-complementary transfer

functions, and , the frequency

where , 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, is called the 3-dB cross-

over frequency

oω

oω

)(0 zH )(1 zH

5.0|)(||)(| 21

20 == ωω oo jj eHeH


• Consider the two transfer functions

and given by

where and are stable allpass

transfer functions

• Note that

• Hence, and are allpass

complementary

)(0 zH

)(1 zH

)]()([)( 1021

0 zAzAzH +=

)(0 zA )(1 zA

)]()([)( 1021

1 zAzAzH −=

)()()( 010 zAzHzH =+

)(0 zH )(1 zH


• It can be shown that and are

also power-complementary

• Moreover, and are bounded-

real transfer functions

)(0 zH )(1 zH

)(0 zH )(1 zH


Doubly-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, and , with a

sum of allpass decomposition can be simply

realized as indicated below

)(0 zH )(1 zH

+

+)(1 zA

)(0 zA

)(zX

)(0 zY

)(1 zY1−

2/1

)(

)(00 )(

zX

zYzH =

)(

)(11 )(

zX

zYzH =

Example 5

• The first-order lowpass transfer function

can be expressed as

where

= −

−

α−+α−

1

1

1

1

2

1)(z

zLP zH

)]()([)( 102

1

11

2

11

1

zAzAzHz

zLP +

= =

α−+α−+ −

−

1

1

11

)( −

−

α−+α−=z

zzA,)( 10 =zA

Example 5

• Its power-complementary highpass transfer

function is thus given by

• The above expression is precisely the first-

order highpass transfer function described

earlier

=− −

−

α−+α−−=

1

1

11

2

1102

1 )]()([)(z

zHP zAzAzH

= −

−

α−−α+

1

1

1

1

2

1

z

z


Power-Symmetric 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

where K > 0 is a constant

KzHzHzHzH =−−+ −− )()()()( 11


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

• If we define , then it follows

from the definition of the power-symmetric

filter that H(z) and G(z) are power-

complementary as

constanta)()()()( 11 =+ −− zGzGzHzH

)()( zHzG −=

dBK 3log10 10 −


Conjugate Quadratic Filter

• If a power-symmetric filter has an FIR

transfer function H(z) of order N, then the

FIR digital filter with a transfer function

is called a conjugate quadratic filter of

H(z) and vice-versa

)()( 11 −−= zHzzG


• 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 6

• Let

• We form

• H(z) is a power-symmetric transfer

function

)3621)(3621( 32321 zzzzzz ++−++−= −−−

)()()()( 11 −− −−+ zHzHzHzH

321 3621)( −−− ++−= zzzzH

)3621)(3621( 32321 zzzzzz −++−+++ −−−

)345043( 313 −− ++++= zzzz

100)345043( 313 =−−+−−+ −− zzzz

Digital TwoDigital Two--PairsPairs

• 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


• Figures below show two commonly used

block diagram representations of a two-pair

• Here and denote the two outputs, and

and denote the two inputs, where the

dependencies on the variable z have been

omitted for simplicity

1Y 2Y

1X 2X

1Y

2Y

1X

2X1Y

1X 2Y

2X


• The input-output relation of a digital two-

pair is given by

• In the above relation the matrix ττττ given by

is called the transfer matrix of the two-pair

=

2

1

2221

1211

2

1

X

X

tt

tt

Y

Y

=

2221

1211

tt

ttττττ


• It follows from the input-output relation that

the transfer parameters can be found as

follows:

02

112

01

111

12 ==

==XX

X

Yt

X

Yt ,

02

222

01

221

12 ==

==XX

X

Yt

X

Yt ,


• An alternative characterization of the two-

pair is in terms of its chain parameters as

where the matrix ΓΓΓΓ given by

is called the chain matrix of the two-pair

=

2

2

1

1

X

YDCBA

Y

X

=

DCBAΓΓΓΓ

- -


• The relation between the transfer parameters and

the chain parameters are given by

A

Bt

At

A

BCADt

A

Ct −==

−== 22211211 ,

1,,

21

22112112

21

11

21

22

21

1t

ttttD

t

tC

t

tB

tA

−==−== ,,,

TwoTwo--Pair Interconnection SchemesPair Interconnection Schemes

Cascade Connection - ΓΓΓΓ-cascade

• Here

'1Y

'1X

"1Y

'2Y "

2Y"1X

"2X'

2X

''

''

DCBA

- -

""

""

DCBA

--

=

'2

'2

''

''

'1

'1

X

Y

DCBA

Y

X

=

"2

"2

""

""

"1

"1

X

Y

DCBA

Y

X


• But from figure, and

• Substituting the above relations in the first

equation on the previous slide and

combining the two equations we get

• Hence,

'2

"1 YX = '

2"1 XY =

=

"2

"2

""

""

''

''

'1

'1

X

Y

DCBA

DCBA

Y

X

=

""

""

''

''

DCBA

DCBA

DCBA


Cascade Connection - ττττ-cascade

• Here

'1Y'

1X"1Y

'2Y

"2Y

"1X

"2X

'2X

'22

'21

'12

'11

tt

tt

- - --

"22

"21

"12

"11

tt

tt

=

"2

"1

"22

"21

"12

"11

"2

"1

X

X

tt

tt

Y

Y


• But from figure, and

• Substituting the above relations in the first

equation on the previous slide and

combining the two equations we get

• Hence,

'1

"1 YX = '

2"2 YX =

=

'2

'1

'22

'21

'12

'11

"22

"21

"12

"11

"2

"1

X

X

tt

tt

tt

tt

Y

Y

=

'22

'21

'12

'11

"22

"21

"12

"11

2221

1211

tt

tt

tt

tt

tt

tt


Constrained Two-Pair

• It can be shown that

1Y

1X 2Y

2XG(z)

H(z)

)(

)()(

1

1

zGBA

zGDC

X

YzH

⋅+⋅+

==

)(1

)(

22

211211

zGt

zGttt

−+=

lecture 7 lti discrete-time systems in the transform domaintania/teaching/dsp/lecture 7 lti...

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