design of a low-pass filter by multi-scale even order gaussian derivatives

Signal Processing 86 (2006) 3923–3933

Fast communication

Design of a low-pass filter by multi-scale even orderGaussian derivatives

Subhajit Karmakar�, Kuntal Ghosh, Sandip Sarkar, Swapan Sen

Microelctronics Division, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700064, India

Received 13 December 2005; received in revised form 21 June 2006; accepted 18 July 2006

Available online 17 August 2006

Abstract

In this paper, a new low-pass FIR filter design technique for achieving variable stopband attenuation without altering

the passband and stopband edges, is proposed. The filter function is a linear combination of multi-scale Gaussian

derivatives. In the frequency domain, the spectral modes corresponding to the continuous Gaussian derivatives have

monotonic tails. In discrete domain, this amounts to variable stopband attenuation which depends upon the truncation

length of the continuous kernel, the ‘scale’ and the order of Gaussian derivatives. The proposed algorithm consists of

derivation of some useful mathematical relations between the basic design parameters and the parameters of the Gaussian

derivatives and an optimization technique that finally produces an equiripple passband lowpass FIR filter with a higher

falloff rate at the beginning of the transition band. Such a filter with variable stopband attenuation may be effective at the

time of reconstruction of the desired passband signal in a S–D modulator output.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Gaussian derivatives; Equiripple filter; Low-pass filter

1. Introduction

It is well-known that the noise transfer functionof S–D modulator has a nature of increasingmagnitude with frequency and due to this natureof the quantization noise, S–D modulator is callednoise-shaper [1]. So, a filter with monotonicallydecreasing stopband may be effective at the time ofreconstruction of the desired passband signal.

There are several filters which can be used for thereconstruction of the signal from such kind ofnoises. These are Butterworth, Chebyshev, elliptic

and Bessel filters. But in many signal processingapplications it is desired that filter response will belinear throughout the band of interest. The Butter-worth and Chebyshev filters have a nearly linearphase response over about three-fourths of thepassband, and that of the elliptic filter is about one-half of the passband. On the other hand, Bessel filterhas the linearity of the phase response over a largerportion of the passband but at the expenses of apoorer gain response [2]. Usually FIR filters are oflinear phase and could be a better candidate for thisapplication. There are three conventional FIR filterdesign methodologies, namely ‘Optimal method’ byParks–McClellan algorithm, ‘Window method’ and‘Frequency Sampling method’. However, in thispaper, an alternative design methodology for

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0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.sigpro.2006.07.005

�Corresponding author. Tel.: +9133 23375345.

E-mail address: [email protected]

(S. Karmakar).

variable stopband attenuation FIR filter is proposedbased on a linear combination of multi-scaleGaussians derivatives. Gaussian derivatives havebeen well studied and applied in the domain ofimage processing [3–11]. The term multi-scale hasbeen used here to imply Gaussian derivatives havingdifferent standard deviations. By variable stopband,we mean that the stopband attenuation can bevaried without changing the passband and stopbandedge frequencies. This control over the stopband ispossible for the filters designed by our proposedmethodology because in the frequency domain, thespectral modes corresponding to the continuousGaussian derivatives have monotonic tails. Indiscrete domain this amounts to variable stopbandattenuation depending upon the truncation lengthof the continuous kernel, the ‘scale’ and the order ofGaussian derivatives. It is known that the ‘Optimalmethod’ usually designs a flat stopband floorwhereas ‘Window method’ produces a decayingstopband floor. But, it has been shown here thatnone of these conventional FIR design methodolo-gies is convenient for designing such a variablestopband FIR filter without altering passband andstopband edge frequencies. On the other hand,arbitrary response filter can generally be designed inprinciple by frequency sampling method, at theexpense of a very large number of taps compared toothers. Another advantage of using the Gaussianderivatives is that, these are symmetric in nature, sothat the discrete version of our proposed filter willalways have linear phase response.

Furthermore, we shall show that like optimalfilter, linear phase FIR filters of equiripple passbandcan also be designed with our proposed methodol-ogy and these filters have a higher falloff rate at thebeginning of the transition band.

The organization of the paper is as follows: InSection 2, the design of the filter has been presentedin detail. In Section 3, the performance of theproposed filter has been evaluated. Finally, there isa Conclusion section that also indicates possiblefuture directions of work.

2. Design of filter

The Gaussian derivatives in the spectral domainfollow a bandpass nature (Fig. 1), except the zero-thorder, which is lowpass in nature. So, their linearcombination may be used to design a low-pass filter.In our proposed methodology, only even orderderivatives have been used for the design to avoid

the mixing of symmetries. This filter design algo-rithm is divided into two parts. In the first part,simple mathematical relations between the basicfilter design parameters (passband ripple (dp),passband edge frequency (op), stopband edgefrequency (os) and stopband attenuation (ds)) andthe control parameters (‘scale’ and order ofderivatives) of Gaussian derivatives are established.The second part of the algorithm describes anoptimization procedure that makes the passband ofthe proposed filter equiripple.

2.1. The Gaussian derivatives and some

mathematical relations

The family of 1-D zero-mean Gaussian deriva-tives for a particular variance s2 is given by

g0ðxÞ ¼ expð�x2=2s2Þ, (1)

gnðxÞ ¼dn

dxng0ðxÞ. (2)

In spectral domain these are transformed as

G0ðo; sÞ ¼ s expð�s2o2=2Þ (3)

and

Gnðo; sÞ ¼ ð�joÞnG0ðoÞ. (4)

The Gaussian derivative function spectra arebimodal except the original Gaussian with modescentered at �on, where n is the derivative order. Thederivative order and the center of that spectral mode

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Fig. 1. A family of normalized Gaussian derivatives in the

frequency domain shows the bandpass nature except the original

Gaussian.

S. Karmakar et al. / Signal Processing 86 (2006) 3923–39333924

are related by the relation:

on ¼

ffiffiffinp

s. (5)

These spectral modes are distributed between 0and p.

At o ¼ on the spectral modes reach their max-imum values. Now putting Eq. (5) into Eq. (4) theamplitude of that spectral mode is obtained. Thenwe divide Eq. (4) with that amplitude, so that thenormalized spectral mode is expressed as

jGnðo;sÞj ¼ onsnn�n=2 exp �s2o2

2þ

n

2

� �. (6)

Now, to find a relation between the basic filterdesign parameters and the two control parameters,mentioned above, two approximations are herebymade.

Firstly, the spectral mode corresponding to thehighest order derivative of a family (i.e. all thederivatives having same scale) is approximated asthe passband edge frequency. As the family ofGaussian derivatives has monotonic stopband(Fig. 2):

jGqðosÞjojGpðosÞj8q; 0oqop,

where ‘p’ is the highest spectral mode correspondingto highest order derivative and the G– s are thespectral modes of a Gaussian derivative family.

Secondly, the transition band and the stopbandare also approximated by the same (i.e. highest)spectral mode. Under equiripple assumption, op is

always a minima. So ds is weighted by a factor of 1/(1�dp).

Let ‘p’ be the highest order derivative, then,

p ¼lnðds=1� dpÞ

ln a� a22þ 1

2

� � , (7)

where a ¼ os=op and hence the scale of the family is

s ¼ffiffiffipp

op. (8)

Once the highest order derivative (p) and the‘scale’ of the family are determined [vide Eqs. (7)and (8)], the two extreme spectral modes areassigned at the two end positions, i.e. the original

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8 x 106

Different order of derivative of a single family

Mag

nitu

de a

t π

Fig. 2. The contribution of the lower order derivatives become

smaller at o ¼ p for a particular family (s ¼ 4) of Gaussian

derivatives.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.04

0.05

0.06

0.07

0.08

0.09

0.1

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0.12

Passband edge frequency (ωp/2π)

Tan

sitio

n ba

nd(ω

s ωp)

/2π

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0.05

0.06

0.07

0.08

0.09

0.1

0.11


Tra

nsiti

on b

and

(ωs ω

p)/2

π

Fig. 3. (a) Change in transition band with respect to passband

edge frequency when no offset is incorporated (‘*’ denotes the

measured parameter after design and ‘o’ denotes the same before

design). (b) Change in transition band with respect to passband

edge frequency with an offset of 0.0064 (in normalized frequency)

(‘*’ denotes the measured parameter after design and ‘o’ denotes

the same before design).

S. Karmakar et al. / Signal Processing 86 (2006) 3923–3933 3925

Gaussian as lowest spectral mode around o ¼ 0 andthe pth order derivative as the highest spectral modearound op.

2.2. Optimization procedure

The second part of the algorithm deals with anoptimization procedure to select the intermediatespectral modes, so that their linear combinationyields the desired magnitude response given by thefollowing filter function:

HðjoÞ ¼P

n

knjGnðo;sÞj ¼ 1� dp for 0popop;

pds for oXos;

(9)

where dp is the passband ripple and ds is thestopband attenuation.

As n is even,

jGnðo;sÞj ¼ ð�1Þ�n=2Gnðo;sÞ

which means that H(jo) is always real.The passband of our proposed filter approximates

that of an ideal low-pass filter by an amount of errordp and the error function in the passband attainsmaximum at the extremal positions with alternatingsign. This would require computing the weightfactors at each iteration step and evaluation of theerror function at the selected extremas.

The weights are calculated on the basis ofequalization of the amplitudes of the extremalfrequencies to a specific value, say 17dp. If Gi be

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0.05

0.1

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0.25

0.3

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0.4

Pass

band

edg

e fr

eque

ncy

(ωp/2

π)

(bef

ore

& a

fter

des

ign)

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0.15

0.2

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0.3

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0.4

Passband edge frequency(ωp/2π)


Pass

band

edg

e fr

eque

ncy

(ωp/2

π)

(bef

ore

& a

fter

des

ign)

(a)

(b)

Fig. 4. (a) Passband edge frequency before and after the design

with no offset (‘*’ denotes the measured parameter after design

and ‘o’ denotes the same before design). (b) Passband edge

frequency before and after design with an offset of 0.0064 (in

normalized frequency) (‘*’ denotes the measured parameter after

design and ‘o’ denotes the same before design).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Stop

band

edg

e fr

eque

ncy

(ωs/

2π)

(be

fore

& a

fter

des

ign)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Stop

band

edg

e fr

eque

ncy

(ωs/

2π)

(b

efor

e &

aft

er d

esig

n)

(a)

(b)

Fig. 5. (a) Stopband edge frequency before and after the design

with no offset (‘*’ denotes the measured parameter after design

and ‘o’ denotes the same before design). (b) Stopband edge

frequency before and after design with an offset of 0.0064 (in


design and ‘o’ denotes the same before design).


the spectral modes and ki the weight factors, thenXi;j

aijki ¼ B, (10)

where BðiÞ ¼ 1� dp (amplitude of passband ripple)and

aij ¼ Giðoj ;sÞ ¼ oijs

ii�i=2 exp �s2o2

j

2þ

i

2

!,

8i; j ¼ 1 : n and oj are the extremal frequencies. Eq.(10) can be re-written as, AK ¼ B or K ¼ A�1B,provided det(A) is nonsingular. So finally, the ki’sare to be calculated from

a11 a12 . . . . . . a1n

a21 a22 . . . . . . a2n

..

. ...

. . . . . . ...

..

. ...

. . . . . . ...

an1 an2 . . . . . . ann

0BBBBBBBB@

1CCCCCCCCA

k1

k2

..

.

..

.

kn

0BBBBBBBB@

1CCCCCCCCA¼

1� dp1� dp1� dp

..

.

1� dp

0BBBBBBB@

1CCCCCCCA.

(11)

Our filter is a linear combination of even orderGaussian derivatives. So, if there are ‘N’ numbers ofderivatives in the combination then there should beat least ‘N’ number of extremal positions in theerror function. But from equiripple low-pass filterdesign criterion we know that the op is always aminima whereas, o ¼ 0 can be either a maxima or aminima. So there exist two possibilities, i.e.

(1) If o ¼ 0 is a maxima and op is minima: Then foreven ‘N’ there must be at least N�2 number ofextrema in the interval 0oooop and for odd‘N’ there will be at least N�1 number of extremain the same interval.

(2) If o ¼ 0 and op both are minima: Then for odd‘N’ there must be at least N�2 number ofextrema in the interval 0oooop and for even‘N’ there will be at least N�1 number of extremain the same interval.

Furthermore, the error in the passband is definedas

EðoÞ ¼ HðjoÞ �W ðoÞDðoÞ, (12)

where DðoÞ ¼ 10popop, is the passband of anideal low-pass filter. The values of W(o) are 1� dp;where E(o) attains maximum with alternating sign.Minimization of the maximum error ‘e’ defined as

� ¼ jEðoÞjmax (13)

yields the desired passband. The optimization isdone by searching new extremal points of the errorfunction in a dense grid (1/1000th of an integraldivision) from 0popop and minimizing the max-imum error ‘e’ iteratively.

2.3. The detailed algorithm of filter design

The algorithm is implemented through thefollowing steps:

(i) Input: op, dp, os and �20 log10(ds) ¼ stopbandattenuation (dB), are supplied as input parametersto the algorithm.

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(H(ω

q)H

(ωp)

)/(ω

q ω

p)/2

π

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.45.5

5

4.5

4

3.5

3

2.5

2


(H(ω

q)H

(ωp)

)/(ω

q ω

p)/2

π

(a)

(b)

Fig. 6. (a) Relative deviation of the passband edge magnitude

ðHðoqÞ �HðopÞÞ with respect to the deviation in passband edge

ðoq �opÞ when no offset is incorporated (‘*’ denotes the

measured parameter after design). (b) Relative deviation of the

passband edge magnitude ðHðoqÞ �HðopÞÞ with respect to the

deviation in passband edge ðoq �opÞ with an offset of 0.0064 (in


design).


(ii) Calculation of s: Scale of the derivative familyis calculated with Eqs. (7) and (8).

(iii) Initialization of N: The algorithm starts withthe initial condition N ¼ 2.

(iv) Calculation of the orders of derivative: Theinterval 0popop is uniformly divided into N

number of frequencies. The peaks of the spectralmodes of the Gaussian derivatives are assigned atthese frequencies and corresponding order ofderivatives are calculated from Eq. (5). The ordersare quantized to the nearest even integers.

(v) Recalculation of s: s for the lowest and highestspectral modes, centered on o ¼ 0 (dc) and thepassband edge frequency, are kept unaltered. Inorder that the intermediate spectral modes mayretain their previous positions even after quantiza-tion to the nearest even integer, the scale s for eacheven order derivative is now to be changedaccording to Eq. (5). Thus instead of a singleGaussian derivative family, the use of multi-scaleGaussians are necessitated.

(vi) Weighted addition: Initial weights for sum-ming up the spectral modes of the correspondingorder of Gaussian derivatives are calculated by Eq.(11), where the choices of initial extremal frequen-cies are the spectral modes corresponding toGaussian derivatives.

(vii) Evaluation of new extremal frequencies: It isexpected that the weighted addition may change the

extremas in the magnitude response. So, the newextremal frequencies are searched for in the range0popop in a dense grid.

(viii) Selection of the extremal frequencies: Thepseudocode of the selection algorithm is givenbelow. next denotes number of extremal frequency.

If next4ndev in the interval 0oooop

(1) Perform step (x)Elseif nextoN in the interval 0oooop

If next ¼ N�2 and N is ‘Even’(1) Include o ¼ 0 (dc) and op to the extremal

frequency(2) Calculate E(o) by Eq. (12) on a dense grid(3) Include error at dc & op

(4) Perform step (ix)Elseif next ¼ N�1 and N is ‘Even’

(1) Include o ¼ 0 (dc) to the extremal frequency(2) Calculate E(o) by Eq. (12) on a dense grid and

include error at dc(3) Change sign of the error at dc(4) Perform step (ix)

Elseif next ¼ N�1 and N is ‘Odd’(1) Include o ¼ 0 (dc) to the extremal frequency(2) Calculate E(o) by Eq. (12) on a dense grid and

include error at dc(3) Perform step (ix)

EndElse

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200

150

100

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0

50

ω (0 to π)

Mag

nitu

de (

dB)

Gaussdev (300dB)Gaussdev (200dB)Gaussdev at100dB

Fig. 7. The magnitude response of stopband floor of the proposed filter for three different truncation lengths. Design specifications are

op ¼ p/5, peak to peak passband ripple ¼ 0.007dB, os ¼ p/3.4, ds ¼ 100 dB, respectively.


Perform step (x)End(ix) Minimization of maximum error:If eo10�6 and H (jo)40 for all o

The program is terminated to the best approx-imationElseife410�6 and H (jo)40 for all o

Keep the extremal frequencies from the previousstep and perform steps (vi)–(viii)Else

Perform step (x)End(x) Increment of N: N is increased by a unitand the order of derivatives is recalculated fromstep (iv).

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Gdev (100dB)Gdev (200dB)firpm (200dB)Kaiser (200dB)design stopband edge at 0.3927

Frequency Sampling technique

0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48

(a)

(b)

Fig. 8. (a) Performance comparison of variable stopband filters designed by ‘Kaiser Window method’, ‘Optimal method (‘firpm’)’ and our

proposed methodology for the design specifications op ¼ p/16, peak to peak passband ripple ¼ 0.007 dB, os ¼ p/8 (0.3927), minimum

stopband attenuation ds ¼ 100 dB and variable stopband attenuation is 100–200dB. (b) Expanded transition band of variable stopband

filters designed by ‘Frequency Sampling method’, for the design specifications op ¼ p/16, peak to peak passband ripple ¼ 0.007dB,

os ¼ p/8 (0.3927), minimum stopband attenuation ds ¼ 100 dB and variable stopband attenuation is 100–200dB (at p/6.844).


2.4. Filter kernel

The kernel of the proposed filter can be obtainedby the linear combination of the multi-scaleGaussian derivatives obtained from the aboveprocedure. The continuous kernel is then discretizedin unit interval. The different orders of Gaussianderivatives are computed at the sampling pointswith the following recursion relation:

gnðxÞ ¼ �x

vgn�1ðxÞ �

n� 1

vgn�2ðxÞ, (14)

where v ¼ s2 and gnðxÞ is the sampled ‘nth order’Gaussian derivative.

3. Results

To check the performance of the proposedalgorithm, we have compared the design parametersbefore and after the design. For the sake of thiscomparison three transition bands have beenconsidered in three different regions of passbandfrom 0 to p with dp ¼ 0:0002 and stopbandattenuation of about 100 dB. We have calculatedthe weight factors at each iteration, using Eq. (11).When the transition band is too small it creates aproblem in calculating the inverse numerically(using MATLAB version 7.0.1) as the contributionof the lower order derivative at the farthest extremabecomes very small. That is why towards half the

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2

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nnitu

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-6

-4

-2

0

2

4

6x10-3

x10-3

ω (0 to π)

Mag

nnitu

de (

dB)

GaussdevfirpmKaiser

GaussdevfirpmKaiser

GaussdevfirpmKaiser

GaussdevfirpmKaiser

(a) (b)

(d)(c)

Fig. 9. (a) Comparison of the magnitude response of three design methodology for the design specifications op ¼ p/16, peak to peak

passband ripple ¼ 0.007dB, os ¼ p/8, ds ¼ 100dB. NGdev ¼ 363, Nfirpm ¼ 156, NKaiser ¼ 207. (b) Comparison of the magnitude response

in the passband of (a). (c) Comparison of the magnitude response of three design methodology for the design specifications op ¼ p/16,peak to peak passband ripple ¼ 0.007dB, os ¼ p/8, ds ¼ 200 dB. NGdev ¼ 929, Nfirpm ¼ 261, NKaiser ¼ 430. (d) Comparison of the

magnitude response in the passband of (c).


normalized frequency, the transition band has beenincreased for the sake of convergence of thealgorithm. In this algorithm, op is allowed to moveaway (downwards) in magnitude rather than remainstrictly at (1�dp). As a consequence, the transitionband after design also undergoes a change. Fig. 3ashows the effect of this change in transition bandwith respect to the original passband edge fre-quency. The transition band changes mainly due tothe change in passband edge frequency but there is aslight change in stopband edge also. It is clear fromFig. 4a that passband edge frequency has driftedtowards a lower value (oq) after the design while thedrift in stopband edge frequency (Fig. 5a) isnegligible. To prevent the drift in passband edgefrequency, the basic assumption that the highestspectral mode will be around op, is to be changed byan offset 0.0064 (in normalized frequency). As aresult, the change in transition band becomes lowerthan the previous cases which can be observed fromFigs. 3b, 4b and 5b for the constant offsetmentioned above. Moreover Figs. 3b and 4b,suggest that the offset should be dynamicallychanged to minimize the change of transition bandwhen the passband edge frequency goes towardshalf the normalized frequency. Fig. 6a and b show arelative deviation of the passband edge magnitudeðHðoqÞ �HðopÞÞ with respect to the deviation inpassband edge ðoq � opÞ for both the cases, i.e.,

without an offset and with an offset. Fig. 6bindicates the role of the offset in providing a smallerrelative deviation.

Performance of the discrete kernel of our pro-posed filter is also compared with two standard FIR

filters namely ‘Optimal FIR filter’ by Parks–McClel-lan algorithm and the ‘Kaiser Window’ method.Here we have used two functions (‘firpm’ and ‘fir1’)of ‘MATLAB’ for the comparison.

It has been already mentioned that our motiva-tion is to design a FIR filter with a variablestopband attenuation while keeping the passbandand stopband edges almost fixed; Fig. 7 shows thatwith the increase of truncation length of the discretekernel, the stopband of the filter attains thisproperty. The truncation lengths are taken as somemultiples of the ‘highest scale’. In Fig. 8a, compar-ison of filter responses for variable stopbandattenuation designed with three methodologies (by‘Optimal method’, ‘Window method’ and ‘Proposedmethod’) is presented. Two plots are given for ourproposed methodology. One plot (dotted red line) isfor the filter designed to the specification and theother plot (solid red line) is for filter designed to thesame specifications but the stopband floor issuppressed further to 200 dB. It is evident fromthese two plots that the design stopband edge didnot change for the two different stopband floors.This is achieved by simply changing the truncation

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350

300

250

200

150

100

50

0

50

ω (0 to π)

Mag

nitu

de (

dB)

KaiserfirpmGaussdev

Fig. 10. Comparison of the magnitude response of three design methodology for the design specifications op ¼ p/16, peak to peak

passband ripple ¼ 0.007 dB, os ¼ p/8, ds ¼ 300 dB.


length of the continuous kernel. In the other twomethodologies, there is no control in their design toassign a specific frequency with a specific amplitude,in the transition band, so that the stopbandattenuation can changed from 100 to 200 dB with-out changing the stopband edge (0.3927) at 100 dB.It is evident from Fig. 8 that the contour of thetransition band cannot pass through the designated

stopband edge (0.3927) at 100 dB. Also, by the‘Frequency sampling method’ the mentioned designcriteria could not be met (Fig. 8b) even at theexpenses of a large number of taps (16K). ‘MA-TLAB’ function ‘fir2’ has been used for the designby ‘Frequency Sampling method’.

Moreover, with the proposed methodology FIR

low-pass filter with equiripple passband can be

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ω (0 to π)

ω (0 to π)

Mag

nitu

de (

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GaussdevfirpmKaiser

0 0.1 0.2 0.3 0.4 0.5 0.6

-6

-4

-2

0

2

4

6

x 10-3

Mag

nitu

de (

dB)

Gaussdev

firpm

Kaiser

(a)

(b)

Fig. 11. (a) Comparison of the magnitude response of three design methodology for the design specifications op ¼ p/5, peak to peak

passband ripple ¼ 0.007 dB, os ¼ p/2, ds ¼ 150dB. (b) Comparison of the magnitude response in the passband of (a).


designed. A short comparison with the conventionalFIR filters has been also performed. Fig. 9a and cshow the 100 and 200 dB attenuation for the threedesigns. In Fig. 10 it is observed that ‘firpm’ doesnot converge for 300 dB attenuation whereas ourproposed filter and the ‘Kaiser Window’-baseddesign converge to the desired specification. It isalso clear from both the figures that our proposedfilter has a faster falloff at the beginning of thetransition band and till about the middle of it. Theperformance of the three designs for a widertransition band is also shown in Fig. 11a whereour proposed filter has much faster falloff at thetransition band than other two. The peak to peakpassband ripple for all the design is kept at 0.007 dBand it is apparent from Figs. 9b, d and 11b that thepassband ripple is much higher in ‘firpm’ design andmuch lower for ‘Kaiser Window’-based designwhereas the proposed filter is much closer to thatvalue after discretization. In the examples (Fig. 9aand c) it is observed that the kernel length of ourproposed filter (NGdev) is much higher than theother two (Nfirpm and NKaiser) because in our designalgorithm we have to restrict ourselves only to evenorder derivatives at the expense of a higher ‘scale’and the truncation length is assigned as somemultiple of the ‘highest scale’.

4. Conclusion

In this paper, we have proposed an algorithm todesign a low-pass filter with linear combination ofmulti-scale even order Gaussian derivatives. Themagnitude response in the passband of the contin-uous filter kernel is almost equiripple and thestopband is monotonically decreasing. Comparativeperformance of the discretized kernel of ourproposed filter shows a faster falloff at the begin-ning of the transition band compared to well-knownfilter design techniques. Also, depending upon thelength of the filter kernel, the stopband attenuationcan be varied without changing the passband andstopband edges. This cannot be achieved by ‘Kaiserwindow’-based design and ‘Optimal design’ techni-ques, though the kernel length (number of tap)

required in our case is greater. A further improve-ment of the optimization algorithm is possible bycarrying out future work to find a suitable rejectionscheme for eliminating superfluous extrema(next4N). Moreover, the same analysis can beextended for the design of bandpass filter, becausein spectral domain Gaussian derivatives are ofbandpass nature (except the original Gaussian).

Acknowledgement

The authors would like to acknowledge thecontributions from Mr. Sabyasachi Siddhanta,Mr. Rivu Sarkar, Prof. Kamales Bhaumik and allthe members of Microelectronics Division, SINP.

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design of a low-pass filter by multi-scale even order gaussian derivatives

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