digital filter specifications

1 Professor A G Constantinides

Digital Filter SpecificationsDigital Filter Specifications• We discuss in this course only the magnitude

approximation problem• There are four basic types of ideal filters with

magnitude responses as shown below

1

0 c –c

HLP(e j)

0 c –c

1

HHP (e j)

11–

–c1 c1 –c2 c2

HBP (e j)

1

–c1 c1 –c2 c2

HBS(e j)


Digital Filter SpecificationsDigital Filter Specifications• These filters are unealizable because their

impulse responses infinitely long non-causal

• In practice the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances

• In addition, a transition band is specified between the passband and stopband


Digital Filter SpecificationsDigital Filter Specifications• For example the magnitude response

of a digital lowpass filter may be given as indicated below

)( jeG


Digital Filter SpecificationsDigital Filter Specifications• In the passband we require

that with a deviation

• In the stopband we require that with a deviation

1)( jeG

0)( jeG s

pp 0

s

ppj

p eG ,1)(1

ssjeG ,)(


Digital Filter SpecificationsDigital Filter SpecificationsFilter specification parameters

• - passband edge frequency

• - stopband edge frequency

• - peak ripple value in the passband

• - peak ripple value in the stopband

p

s

sp


Digital Filter SpecificationsDigital Filter Specifications

• Practical specifications are often given in terms of loss function (in dB)

•

• Peak passband ripple

dB

• Minimum stopband attenuation

dB

)(log20)( 10 jeGG

)1(log20 10 pp

)(log20 10 ss


Digital Filter SpecificationsDigital Filter Specifications• In practice, passband edge frequency

and stopband edge frequency are specified in Hz

• For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using

TFF

F

F pT

p

T

pp

2

2

TFFF

F sT

s

T

ss 2

2

sFpF


Digital Filter SpecificationsDigital Filter Specifications

• Example - Let kHz, kHz, and kHz

• Then

7pF 3sF25TF

56.01025

)107(23

3

p

24.01025

)103(23

3

s


• The transfer function H(z) meeting the specifications must be a causal transfer function

• For IIR real digital filter the transfer function is a real rational function of

• H(z) must be stable and of lowest order N for reduced computational complexity

Selection of Filter TypeSelection of Filter Type

1z

NN

MM

zdzdzddzpzpzpp

zH

22

110

22

110)(


Selection of Filter TypeSelection of Filter Type• For FIR real digital filter the transfer

function is a polynomial in with real coefficients

• For reduced computational complexity, degree N of H(z) must be as small as possible

• If a linear phase is desired, the filter coefficients must satisfy the constraint:

N

n

nznhzH0

][)(

][][ nNhnh

1z


Selection of Filter TypeSelection of Filter Type• Advantages in using an FIR filter -

(1) Can be designed with exact linear phase,

(2) Filter structure always stable with quantised coefficients

• Disadvantages in using an FIR filter - Order of an FIR filter, in most cases, is considerably higher than the order of an equivalent IIR filter meeting the same specifications, and FIR filter has thus higher computational complexity


FIR DesignFIR Design

FIR Digital Filter DesignFIR Digital Filter Design

Three commonly used approaches to FIR filter design -

(1) Windowed Fourier series approach

(2) Frequency sampling approach

(3) Computer-based optimization methods


Finite Impulse Response Finite Impulse Response FiltersFilters

• The transfer function is given by

• The length of Impulse Response is N

• All poles are at .

• Zeros can be placed anywhere on the z-plane

1

0).()(

N

n

nznhzH

0z


FIR: Linear phaseFIR: Linear phase

• Linear Phase: The impulse response is required to be

• so that for N even:

)1()( nNhnh

1

2

12

0).().()(

N

Nn

nN

n

n znhznhzH

12

0

)1(12

0).1().(

N

n

nNN

n

n znNhznh

12

0)(

N

n

mn zznh nNm 1



• for N odd:

• I) On we have for N even, and +ve sign

1

21

0

21

21

).()(

N

n

Nmn z

NhzznhzH

1: zC

12

0

21

21

cos).(2.)(N

n

NTj

Tj NnTnheeH



• II) While for –ve sign

• [Note: antisymmetric case adds rads to phase, with discontinuity at ]

• III) For N odd with +ve sign

12

0

21

21

sin).(2.)(N

n

NTj

Tj NnTnhjeeH

2/0

21

)( 21

NheeH

NTj

Tj

23

0 21

cos).(2

N

n

NnTnh



• IV) While with a –ve sign

• [Notice that for the antisymmetric case to have linear phase we require

The phase discontinuity is as for N even]

2

3

0

21

21

sin).(.2)(

N

n

NTj

Tj NnTnhjeeH

.02

1

Nh



• The cases most commonly used in filter design are (I) and (III), for which the amplitude characteristic can be written as a polynomial in

2cos

T



For phase linearity the FIR transfer function must have zeros outside the

unit circle



• To develop expression for phase response set transfer function

• In factored form

• Where , is real & zeros occur in conjugates

nnzhzhzhhzH ...)( 2

21

10

)1().1()( 12

1

11

1

zzKzH i

n

ii

n

i

1,1 ii K



• Let

where

• Thus

)()()( 21 zNzKNzH

)1ln()1ln()ln())(ln(2

1

11

1

1

n

ii

n

ii zzKzH

)1()( 11

11

zzN i

n

i )1()( 12

12

zzN i

n

i



• Expand in a Laurent Series convergent within the unit circle

• To do so modify the second sum as

)1

1ln()ln()1ln(2

1

12

1

12

1zzzi

n

ii

n

ii

n

i



• So that

• Thus

• where

)1

1ln()1ln()ln()ln())(ln(2

1

1

1

12

n

i i

n

ii zzznKzH

mNm

m

mNm z

ms

zms

znKzH2

1

1

2 )ln()ln())(ln(

1

1

1n

i

mi

Nms

1

1

2n

i

mi

Nms



• are the root moments of the minimum phase component

• are the inverse root moments of the maximum phase component

• Now on the unit circle we have

and

jez

)()()( jj eAeH

1Nms

2Nms


Fundamental RelationshipsFundamental Relationships

• hence (note Fourier form)

jmNm

m

jmNmj e

ms

ems

jnKeH2

1

1

2)ln())(ln(

)())(ln())(ln())(ln( )( jAeAeH jj

mms

ms

KANm

m

Nm cos)()ln())(ln(

2

1

1

mms

ms

nNm

m

Nm sin)()(

2

1

1

2



• Thus for linear phase the second term in the fundamental phase relationship must be identically zero for all index values.

• Hence • 1) the maximum phase factor has zeros which are

the inverses of the those of the minimum phase factor

• 2) the phase response is linear with group delay equal to the number of zeros outside the unit circle



• It follows that zeros of linear phase FIR trasfer functions not on the circumference of the unit circle occur in the form

1 ijie


Design of FIR filters: WindowsDesign of FIR filters: Windows

(i) Start with ideal infinite duration

(ii) Truncate to finite length. (This produces unwanted ripples increasing in height near discontinuity.)

(iii) Modify to

Weight w(n) is the window

)(nh

)().()(~

nwnhnh


WindowsWindows

Commonly used windows • Rectangular 1

• Bartlett• Hann• Hamming• • Blackman•• Kaiser

21 NnN

n21

Nn2

cos1

Nn2

cos46.054.0

Nn

Nn 4

cos08.02

cos5.042.0

)(1

21 0

2

0 JNn

J


Kaiser windowKaiser window

• Kaiser window

β Transition width (Hz)

Min. stop attn dB

2.12 1.5/N 30

4.54 2.9/N 50

6.76 4.3/N 70

8.96 5.7/N 90


ExampleExample• Lowpass filter of length 51 and 2/ c

0 0.2 0.4 0.6 0.8 1

-100

-50

0

/

Gai

n, d

B

Lowpass Filter Designed Using Hann window

0 0.2 0.4 0.6 0.8 1

-100

-50

0

/G

ain,

dB

Lowpass Filter Designed Using Hamming window

0 0.2 0.4 0.6 0.8 1

-100

-50

0

/

Gai

n, d

B

Lowpass Filter Designed Using Blackman window


Frequency Sampling MethodFrequency Sampling Method

• In this approach we are given and need to find

• This is an interpolation problem and the solution is given in the DFT part of the course

• It has similar problems to the windowing approach

2/ c

1

0 12

.1

1).(

1)(

N

k kNj

N

ze

zkH

NzH

)(kH)(zH


Linear-Phase FIR Filter Linear-Phase FIR Filter Design by OptimisationDesign by Optimisation

• Amplitude response for all 4 types of linear-phase FIR filters can be expressed as

where

)()()( AQH

4Typefor),2/sin(

3Typefor),sin(

2Typefor/2),cos(

1Typefor,1

)(

Q



• Modified form of weighted error function

where

)]()()()[()( DAQW E

])()[()()()(

QDAQW

)](~)()[(~ DAW

)()()(~ QWW

)(/)()(~ QDD



• Optimisation Problem - Determine which minimise the peak absolute value of

over the specified frequency bands

• After has been determined, construct the original and hence h[n]

)](~)cos(][~)[(~)(0

DkkaWL

k

E

][~ ka

R

)( jeA][~ ka



Solution is obtained via the Alternation Theorem

The optimal solution has equiripple behaviour consistent with the total number of available parameters.

Parks and McClellan used the Remez algorithm to develop a procedure for designing linear FIR digital filters.


FIR Digital Filter Order FIR Digital Filter Order EstimationEstimation

Kaiser’s Formula:

• ie N is inversely proportional to transition band width and not on transition band location

2/)(6.14

)(log20 10

ps

spN



• Hermann-Rabiner-Chan’s Formula:

where

with

2/)(

]2/))[(,(),( 2

ps

psspsp FDN

sppsp aaaD 1031022

101 log])(log)(log[),(

])(log)(log[ 61052

104 aaa pp ]log[log),( 101021 spsp bbF

4761.0,07114.0,005309.0 321 aaa

4278.0,5941.0,00266.0 654 aaa

51244.0,01217.11 21 bb



• Fred Harris’ guide:

where A is the attenuation in dB

• Then add about 10% to it

2/)(20 ps

AN



• Formula valid for

• For , formula to be used is obtained by interchanging and

• Both formulae provide only an estimate of the required filter order N

• If specifications are not met, increase filter order until they are met

sp

sp p s

digital filter specifications

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