Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms

Generalized Linear Phase

Quote of the DayThe mathematical sciences particularly exhibit order, symmetry, and limitation; and these are

the greatest forms of the beautiful.

Aristotle

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall Inc.

Copyright (C) 2005 Güner Arslan

351M Digital Signal Processing 2

Linear Phase System• Ideal Delay System

• Magnitude, phase, and group delay

• Impulse response

• If =nd is integer

• For integer linear phase system delays the input

eeH jjid

jid

jid

jid

eHgrd

eH

1eH

nnsin

nhid

did nnnh

ddid nnxnnnxnhnxny



Linear Phase Systems• For non-integer the output is an interpolation of samples• Easiest way of representing is to think of it in continuous

• This representation can be used even if x[n] was not originally derived from a continuous-time signal

• The output of the system is

• Samples of a time-shifted, band-limited interpolation of the input sequence x[n]

• A linear phase system can be thought as

• A zero-phase system output is delayed by

Tjcc ejH and Ttth

TnTxny

jjj eeHeH



Symmetry of Linear Phase Impulse Responses• Linear-phase systems

• If 2 is integer – Impulse response symmetric

jjj eeHeH

nhn2h

=5

=4.5

=4.3



Generalized Linear Phase System• Generalized Linear Phase

• Additive constant in addition to linear term• Has constant group delay

• And linear phase of general form

jjjj eeAeH

constants and

of function Real :eA j

jj eHargdd

eHgrd

0 eHarg j



Condition for Generalized Linear Phase• We can write a generalized linear phase system response as

• The phase angle of this system is

• Cross multiply to get necessary condition for generalized linear phase

nsinnhjncosnhenheHnn

nj

n

j

sinejAcoseAeeAeH jjjjjj

ncosnh

nsinnh

cossin

n

n

0nsinnhnsinnh

0cosnsinsinncosnh

0cosnsinnhsinncosnh

nn

n

nn



Symmetry of Generalized Linear Phase• Necessary condition for generalized linear phase

• For =0 or

• For = /2 or 3/2

0nsinnhn

nhn2h0nsinnhn

nhn2h0ncosnhn



Causal Generalized Linear-Phase System• If the system is causal and generalized linear-phase

• Since h[n]=0 for n<0 we get

• An FIR impulse response of length M+1 is generalized linear phase if they are symmetric

• Here M is an even integer

Mn and 0n 0nh

nhnMh



Type I FIR Linear-Phase System• Type I system is defined with

symmetric impulse response

– M is an even integer

• The frequency response can be written as

• Where

Mn0for nMhnh

ncosnae

enheH

2/M

0n

2/Mj

njM

0n

j

M/21,2,...,kfor k2/Mh2ka

2/Mh0a



Type II FIR Linear-Phase System• Type I system is defined with


– M is an odd integer


• Where

Mn0for nMhnh

21

ncosnbe

enheH

2/1M

1n

2/Mj

njM

0n

j

/21M1,2,...,kfor

k2/1Mh2kb



Type III FIR Linear-Phase System• Type I system is defined with


– M is an even integer


• Where

Mn0for nMhnh

nsinncje

enheH

2/M

1n

2/Mj

njM

0n

j

M/21,2,...,kfor

k2/Mh2kc



Type IV FIR Linear-Phase System• Type I system is defined with


– M is an odd integer


• Where

Mn0for nMhnh

21

nsinndje

enheH

2/1M

1n

2/Mj

njM

0n

j

/21M1,2,...,kfor

k2/1Mh2kd



Location of Zeros for Symmetric Cases• For type I and II we have

• So if z0 is a zero 1/z0 is also a zero of the system

• If h[n] is real and z0 is a zero z0* is also a zero

• So for real and symmetric h[n] zeros come in sets of four• Special cases where zeros come in pairs

– If a zero is on the unit circle reciprocal is equal to conjugate– If a zero is real conjugate is equal to itself

• Special cases where a zero come by itself– If z=1 both the reciprocal and conjugate is itself

• Particular importance of z=-1

– If M is odd implies that

– Cannot design high-pass filter with symmetric FIR filter and M odd

1Mz zHzzHnMhnh

1H11H M

01H



Location of Zeros for Antisymmetric Cases• For type III and IV we have

• All properties of symmetric systems holds• Particular importance of both z=+1 and z=-1

– If z=1

• Independent from M: odd or even

– If z=-1

• If M+1 is odd implies that

1Mz zHzzHnMhnh

1H11H 1M

01H

01H1H1H



Typical Zero Locations



Relation of FIR Linear Phase to Minimum-Phase• In general a linear-phase FIR system is not minimum-phase• We can always write a linear-phase FIR system as

• Where

• And Mi is the number of zeros

• Hmin(z) covers all zeros inside the unit circle

• Huc(z) covers all zeros on the unit circle

• Hmax(z) covers all zeros outside the unit circle

zHzHzHzH maxucmin

iM1minmax zzHzH



Example• Problem 5.45

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Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms