Minimum Phase FIR Filter Design From Linear ... - Semantic Scholar

MINIMUM PHASE FIR FILTER DESIGN FROM LINEAR PHASE SYSTEMS USING ROOT MOMENTS Tania Stathaki, Anthony Constantinides,

Georgios Stathakis

Signal Processing and Digital Systems Section, Imperial College, UK

.

ABSTRACT In this contribution Finite Impulse

we propose a method for a minimum

Response (FIR) Iilter design from a given linear

phase FIR function concentrating factorisation approach applied

on

with the same amplitude very

high

procedures

taken

degree

polynomials

involves

Theorem

of the transfer

function.

polynomial coefficients problem. The concept solving

which facilitate is developed in

to the celebrated

Newton

the above prohlem,

scheme

arc

very

computational

directly

In addition

to

as

far

as

robustness

H,r,in(‘)

Hmax(:) H,(z)

.

of the methods

is the maximum

group

phase FIR

An influential

function

filters has

representative

is based on the Remcz exchange

of

Direct

phase FIR

design

with

digital

filter

dctcrminc an FIR digital filter which response but is of minimum phase”

prespecified

transfer

phase

function

he to factorise

problem.

the given

and replace each of the zeros outside

with its reciprocal.

This, in principle

function

has identical

At tirst glance this may appear to be a trivial would

The factor

digital

liltcr

transfer

timctions

have

that a real given

has zeros at location

z,-, , I / zn , :i

~(0) =nl2.

applications

communication Often

to

A

typical

FIR

digital

filter

may be of length 200 or more for a ran&c with

stringent

specifications

as

in

is not viable.

in many applications

unimportant

amplitude

or irrelevant.

the phase response For example

is either

in some speech

processing areas it is not significant. This form of freedom in the design of the filters is not normally taken into

Indeed a

FIR transfer

consideration

.

by existing

FIR filter design methods.

For linear phase FIR tiltcrs.

the unit circle

at Icast. would

is

telecommunications. For such filters the group delay may be undesirable particularly when it approaches about when bidirectional human-to-human 200ms.

.

naive approach

H(z).

and

for Iz,~+ I

transfer function

algorithm.

problem: a linear

H(z)

The group delay of an n th order linear phase FIR transfer

response is possible 131. In this paper we address the following “Given

phase part of

phase. From this follows

FIR transfer function

.

Response (FIR) digital

attention.

delay.

phase part of

contains all roots that arc on the unit circle,

Linear

However. most procedures assume a linear phase response with the consequence that the resulting filters do not have the lowest

or

H,,,i,(Z)H,,,(~)H,(Z)

non-minimum

and

INTRODUCTION

considerable

b. . ) ]

d

are concerned.

The design of Finite Impulse attracted

) n(--j r~~~~lt][




- 2cos~rz

+(S,,,

+ I) = J@‘%(8)

The

= e j(n!+oll~2)s[A(@]2

H(P~*)

group

delay

~(0)

as a fraction

~(0) = (n; +$)

= 4.

Thus, in principle,

of

given transfer function

the sampling

a minimum

we can follow

period

is

phase version

Hmin(:)

or dcterminc

multiplicity

its zeros into the unit

and make each of its zeros of

fundamental

relationships

Step 3 the transfer function

[H,,(z)]*

as T(Z)=

+ nh, = (n - 2)h,

S,

+ h,S,-,

known

or S2 + h,S,

+ h2Sm-2+...+nhm

as

H,(z)

procedures

h&vron

+ 2h2 = 0

= 0

(4) long, the successive

values of the root moments.

II’ the signal is of finite

then for m > n

+ 112Sm-2+...+/1,,S,_,,

S,

+ h,S,-,

duration = 0

as above can bc used to calculate

S,,,

H(r”)I.

Both Step I and Step 2 imply at first glance that a root linding procedure may be required. However, as already pointed out, finding

the

= (n - m)h,,, or

+ h2Smm2+...+mh,,,

The same relationship Then we shall have IT(ej’)l=l

root

WC obtain

When the signal treated by this means is infinitely above equation is repeatedly used to calculate

H,,(z)

Construct

expressions

and generally S,, + h,S,-,

2.

Step 2 Find

(3) the last two

S, + nh, = (n - I)h, or S, + h, = 0

the steps below.

and rctlect

+

identities S2 + h,S,

H”,,,(z)

Either determine

+(n-2)h2i”-’

+(n-l)h,~“-~

by equating

following

of the

Step 1 circle.

=n$-

of equation (I) we have

...+(n-m)h,Zn-m-l+... Hence

Hmax(e.iH)I

I Hmin(r”)I=I

Moreover,

to obtain

H’(z)

n-m-’ +. .

+ h2Sm-2+...+nh,)z

By direct differentiation

r=, Hence

+ h,S,-,

are known

to

be inaccurate

and

for

m < 0 by inserting

n-l,

n-2,

either

positive

successively

n-3:..ctc. or

values

It should

negative

for

be noted

values

that

m

of

m equal

arc

S,,

to for

evaluated

unreliable for large order polynomials. Factorisation without root finding forms also the basis of the procedure developed in

recursively from the coefficients of equation (I) alone. The above relationships also follow from the delinition

[1 1.[31.[4]. In [ I],[41 USC is made of the real cepstral parameters as in [S]. where the ccpstral aliasing problem is rccopnised and careful procedures are recommended to reduce

differential

its effects. In [3] they approach

to hc finite and, hence. they can bc applied to infinite duration signals. It is suflicicnt at this juncture to observe that both

the

Lagrange

interpolation

the factorisation point

of

problem

view.

In

the

from above

ccpstrum

However

and

are csscntially

in [6] n is assumed to be finite

is only a minor point as the iteration

finite

duration

signals

and

procedures it is assumed that the zeros of the transfer function on the unit circle arc a priori known. We make no such

exponential

assumption

the same way 171. To facilitate

in our present paper.

An alternative without

and direct

having

to go through

possible through

root

the Root Moments

81. or the differential

construction estimation

procedure

procedures

of a given polynomial

is [7-

cepstrum 161.

4. In relation

polynomial

polynomials

typically

given

in (2) are referred

Newton

in

161.

known.

This

in equ.(4) do not require n

infinite

duration

type interpretation

signals

01

can hc treated in

the exposition,

the parameters This terminology

from the differential

ccpstrum.

4.2 Implications and Interpretation one can interpret

transformation

as in equ.(l)

a priori

to as the root moments.

emphasises the deviation

Essentially

ROOT MOMENTS

entire function

of the

included

{S,,,}

the set of equations {h,}

of the coefficients

of the same cardinality.

(4) as a

to the parameter

The transformations

set

arc one-to-

defined a set of parameters given by one and hence we can have the following S,

explicitly

to compute

determined

directly

are known

They

arc

dominant a wider

S,,

related

to

perspective signal

many

signal

can be

The parameters

processing

H(z)

operations.

ccpstrum.

However

to think of them purely in this sense since enables

processing

us to provide

problems

answers to many

that have been, hitherto,

the polynomial

H’(:)=x-

’

H(z)

(I)

of Root Moments as a product

of factors we can

and given that H(ri ) = 0 we have

,=, i - r; H’(z)

Corollary 1 degree {ri} {Sm}

Given

exists

polynomial

equation

there

Corollary 2 there

in

m = I:..,n,

{h,}

of the

which

has

a set of coefficients

polynomial

i= I;...n.

existence corollaries.

exists

(I) a

set

of

n th roots

parameters

S,, = n , given by cqu.(2).

Convcrscly a set of

given a set of root moments

coefficients

as in equ.(l)

{h,.)

determinable

r = I:...n.

{-In!} for

recursively

a

through

cqu.(4). The proofs are self evident from the above analysis.

4.3 Root Moments of Products of Signals

171.

4.1 Iterative Estimation By writing

hi

as the root moments of the polynomial

amongst which is the differential

unattainable

write

(2)

in that thcsc parameters

from the coefficients

it would be limiting digital

= ir;” i=l

rj is the ith root of (I). The roots of (I) are not needed

where

S,

= r,m +qm+...+r,“’

= n,1”-’ +(S,

+nh,)~~-~

In our main problem WC need the following

result. Assume that

the root moments of the polynomial

are Sit’)

root moments of the polynomial root sl”’ m

+(A’,

+h,S,

+r~h~)i”-~+...

moments

of

= Sf,(Z) +sp, ,n

the

product

f,(z) .f;(z)

are S,J;‘(‘) ,f’(:)

and the Then the

= ,/;(Z),/?(Z)

are

4.4 Non Iterative Estimation The Newton

Identities

yield

signal, encompassing

of Root Moments

the root

not only

moments

tf(p(ek )& )=,jns, Dm(hi,“-‘(ek )}

of the entirc

those roots that lit

within

the

unit circle but also those that are outside the unit circle. However, it is often the case that a specific factor of a given polynomial

H(z)

is required.

such as the minimum

DFT (n - i)hip”-‘-’ i

) = ,-j’

Dm

t I-

roots of the required

defined

as z = p(B)eJe

factor of H(z)

residue theorem

contain

Then it follows

that the root moments

the

from the

h;prtmi(ek

)

of this factor

(5)

The algorithm

Transform

THE ALGORITHM

5.

relies on the direct extraction

of the appropriate

factors from the FIR linear phase transfer implement

from the fact

+ jpce) p”(e) 1

p(e) @f

With N a power of 2 WC can use the Fast Fourier (FFT) algorithm.

are given by

This is evident

(f3, )

in a

manner.

Let a closed contour

Cauchy

&ek

phase

factor and in this case its root moments can be determined different

(11)

and hence

T(z)

function

needed to

above.

Step 1: “dz and the contribution

to the integration

rj’s that lit within

are those due to those

(It is assumed that we have no zeros on

The integration

the contour

from

integration

the coefficients

quite conveniently

through

of

will H(z)

have to be effected and this can hc done

the use of the DFI

as it is shown

a careful

(5) becomes for

z = p(19)P

around

ccntred

Again

selected

but a good

carefully,

Discretisation

of equation

(6) suitable

for DFT

(7)

k=Ol. ’

Therefore.

N-l

for

where

an

N-point

we have the inverse DFT

(8)

the resulting

is the unit circle

C: [iI=

root moments from the above. correspond

of the minimum

phase component

of

H(z)

I then to those

In this case we

The

required

s j ,,,,,, (d rcL”-’ ff cek1J(m+l)H, m c N k=Ofwk) either

cqu.(7)

or

for

the special

form

of

parameters

are the root moments

FIR digital

transfer

cqu.(8)

(9) the

of s(Ok ) can be done through the USCof the DFI

function

So(m),

filter transfer function

has

polynomial

the

root

obtained as S(m)=

Step 4: From Step 3 and from the Newton FIR

ldcntitics

transfer

moments

SI(m)+S?(m)

function

we form of

the

degree

s, (0) + S2(0).

5.1 Estimation of the Radii of the Integration The radii of integration in the above algorithm must he chosen so as to enclose the appropriate zeros of the given FIR digital filter

have the special form of (8)

For

the

So

a

of the radius.

Step 3: use requires

‘*“’

of integration

as the reciprocal

must hc I yields

which has its zeros on the unit circle.

required

If the contour

function

and of radius

in Step

produces

= Si, (m) + St)(m)

S(m)=2Si,(m)+

@.=zk ’ N

radius

to the root

transfer

at the origin selection

good sclcction

of that factor of the original

g(e) = H'(p(e)ajs) p(@)dpo+ jp(e) p"(e) de H(p(@P) i 1

computation

of the contour

the radius of the contour

intcgralion

Sz(m)

where

a circle

greater than unity.

ThC

transform.

importance

Step 2:

correspondingly

values

choice

moments of that part of the original FIR which has its zeros inside the unit circle. Integrate

below. Equation

with

and of radius less

is of crucial

gives S, (m) = Si” (m) which parameters correspond

r- .) In practice directly

r

Integrate around a circle ccntred at the origin than unity. The radius of the contour and it is examined separately below.

transfer

integration

function.

contour

while for S2(fn)

r

Thus

S,(m)

for

the radius

must be such that

of the

I > r 1 max(l q, I ) ,

the radius of the integration

contour

r

must

be chosen such that I < r < min(l rOU,I ) For equiripple

piecewise constant tiltcrs

the required

radii can

be estimated as follows.

also. It is observed that on z = p(B)e@

Let us remove

we can write

the linear

phase factor

response to yield only a real function.

Ht(fle)J@) = J(n-l)~“-’ Ccn - i)hipnmi-l(c),-NJ i=O

which for 6 = ok can be computed

shift vertically

half way between

values.

the initial

Since

result of these operations

as

modulus

H’(p(e, je.i4 ) = $-I)4 Similarly

we have

DFT tn - i)hip”-‘-‘(ek ) I

(10)

unchanged. I+ 6

and

almost

cvcrywhere.

namely 1- 6

The

the frequency

ripple except

now WC

and minimum

is cquiripple

the

of equiripplc

variation

response will

cverywhcrc

band. In Fig. 2 WC indicate function.

function

he a real function

a normaliscd

almost

its maximum

transfer will

from

This function

remains

vary hetwecn

in the transition

the zeros of the shifted

transfer

A

reasonahlc

evcrywherc.

representation

of

this

zero

pattcm

almost

The above transfer

function

is cquiripple.

characteristic

C m;u =2+(n”

+L) lln

between the values

and Clllin =2-(a”

The ripple width of C(z)

FIR

by the parameter n.

is equiripple

values which yield the ripple variation

filter

6=

The

quantity

6

of the shifted

approximated

to

phase design

result

“Optimal Truns.

Design on ASSP.

“A Fast Procedure

to Design

IEEE

Trclns. on

Phase FIR Flltcrs”,

A. and A.T. Fam. “The Differential

Cepstrum

Dcfmition and Prop&es”, Proc. 1.EE.E Int. Symposium Circuits and Systems. pages 77-80, April I98 I. [7]

Stathaki T.. “Cumulant Based and Algchraic for Signal Modelling”, PhlI Thesis. University

[8]

Todhuntcr

is an a priori

to be located

For small ripple

0

width

-

Techniques of London.

I..

Theory

of

f2pution.s.

McMillan

Figure 1: Shifted frequency response

o.5rlyp

1

WC show

in Fig.

3 along

a specitic

with

._..j.

i

minimum

further

results

as

that the reduction

in

indicated below.

5.2 Variations in the Procedures It may he the case in a system application the group delay obtained

by the above algorithms

is more than

the required amount. Then we can improve the non linearity the phase response as follows. . The root moments corresponding to the stop band transmission .

in

Figure 2: Zeros of shifted zero-phase

zeros remain the same as above.

response

From the rest of the zeros WC can select an appropriate number

in conjugate

in an arbitrary

form, for real transfer

functions,

fashion.

Fig. 3 shows such an example,

from which

it is seen that the

group delay here is more than the minimum maximum

value. It is almost equiripple

but less than its

to a constant.

Further

work is ncccssary to explore the options open here.

EXPERIMENTAL

6.

RESULTS

Figure I shows the amplitude of the shifted frequency response while Figure 2 shows its zeros. It is seen that these zeros are similar

of those of

C(z).

Figure

responses for the minimum solution

for

selected

on alternate

method

opcratcs

questions

which

the

maximum

as expected.

Moreover,

there

implications

of equ.( 12).

contour is

7.

for

phase

However,

the

the need

Boite R. and H. Leich,

Herman Recursive Electronics

0.

-1.5

delay

-1

-0.5

zeros

have are

1.5

Figure 3: Group delay responses

many

in relation

implementation

of

for

further

exploring

1

been

that the proposed there

0 0.5 Real oart

to

cqu.(9). the

REFERENCES “A New Procedure

of High Order Minimum Filters”, Signul Processing, [2]

the group

that need to hc addressed particularly of

[I ]

3 shows

phase and for another intermediate

hasis. It is evident

the choice

and Digital Letters.

B

1882.

I-J.4 .................!

n

H.W.

for the Design

Phase FIR Digital or CCD vol. 3, pages 101-108. 1981, Schussler,

“Design

Filters With Minimum vol. 6, pages 329-330. 1970.

of

on

1994.

as

the above can

ol vol.

Prentice Hall, 1993.

Polydoros

0.6

2 s

u =

IEEE

Hence we can estimate the radius of

the zeros arc expected

+$$I]~.

C.J.,

Circuits and Systems, vol. 29. no. 5. pages 327-33 I. 19X2. Nikias CL. and A. Petropulu, Higher Order Spectra

London

design parameter. on which

[S]

and

lln known

Wellckens

Minimum

Analysis.

(un +‘,

the circle

and

Equiripplc

[6]

+L). cl”

can be found from its maximum

2

Y.

141 Mian G.A. and A.P. Naider,

linear phase and its

zeros are located on two circles controlled The amplitude

Kamp

Minimum Phase FIR Filters”, 3 I, pages 922-926. 1983.

C(z)=(:“-u”)(Z”-1) un

minimum

[3]

is given by

Non

Phase”, 0

0.5

1

1.5

2

2.5

3

Co.