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A new design of low pass filter by Gaussian derivative family #Subhajit Karmakar, Kuntal Ghosh, Ratan K. Saha, Sandip Sarkar, Swapan Sen Microeletronics Division, Saha Institute of Nuclear Physics 1/AF, Bidhanangar, Kolkata-64, India. s ubhaiit. karmak-Uar asaha. ac. in

Abstract

Exploiting the models of Human Visual System based on Gaussian derivatives and their non-localization property in spectral domain, a new design of low pass filter is proposed in this work. The filter is designed by a weighted combination of the Gaussian derivative family which makes the passband of this filter almost equiripple. This design principle can be extendedfor the bandpass filter also because of the bandpass nature of the Gaussian derivatives in spectral domain. This work will be applied in the design of adaptive digital filters for sigma-delta based ultrasound beamformer. Key words: Gaussian derivatives, optimal filter, FIR 1.

INTRODUCTION

2. DESIGN OF FILTER

We know that the frequency response of an ideal lowpass filter is a box function having an impulse response that extends infinitely in the inverse domain. But most of the digital filters designed, are of finite length for reducing computational complexity. As a result, the ideal frequency response is relaxed by including a transition band between the pass band and the stop band to permit the magnitude response to decay more gradually from its maximum value in the pass band to the zero value in the stop band. Moreover the magnitude response is allowed to vary by a small amount both in the pass band and stop band to obtain what is known as an equiripple filter [8]. The two types of stable, realizable digital filters are the Finite Impulse Response (FIR) and the Infinite Impulse Response (IIR) filters. Though, there exist various methods of designing stable and realizable digital filters the most widely used technique for the design of optimized equiripple finite impulse response filter is ParksMcClellan algorithm [9, 10]. The attempt in the present work has been to adopt an altemative method to design an optimized lowpass filter with a family of Gaussian derivatives. In computer aided filter design, Parks-McClellan algorithm calculates the minimum order filter coefficients when the pass-band edge frequencies, stopband edge frequency, amount of pass-band and stopband ripples are supplied as input, whereas in our proposed algorithm, the pass-band edge frequency is approximated as the highest spectral mode of a particular family of Gaussian derivatives, the lowest spectral mode being the original Gaussian. The intermediate spectral modes have been determined in such a way that their weighted combination of all the modes taken together makes the pass-band equiripple. The control over the transition band edge is determined by the slope of the highest order spectral mode for that particular design. A. Gaussian Derivatives

The idea of Gaussian derivative based modeling of the Human Visual System (HVS) [1] has been applied to design various image processing filters, in image coding and image transform [2]. Appledom has applied combination of various orders of Gaussian derivatives in designing 'near-ideal' interpolation filters [3]. Though the maximum order of Gaussian derivative found in HVS, is ten [1], still higher orders have been applied previously[2], as also in the present work, where we have proposed a new design of a low pass filter by the combination of Gaussian derivatives for the purpose of I-D signal processing. Properties of Gaussian filters are well studied in signal processing [4]. It is easy to see that the various orders of Gaussian derivatives are nothing but the product of the Hermite polynomial and the original Gaussian function [5, 6]. Modifications of Hermite function based orthogonal pulses have found applications in ultra functions are wi!deband communicationscan[7]. These setasofbasis functions. also be used linearly independent and Moreover, any family of 1-D Gaussian derivatives is localized in temporal domain but in spectral domain the centers of this family are separated from one another. This non-localization property of the Gaussian derivatives in spectral domain has been explored in this work to design a The family of 1-D Gaussian derivatives centered at the origin lowpass filter that is almost equiripple in pass band. This design incorporates all the design parameters required to for a particular variance &2 iS (1) design a specific filter. go (x) = exp(-x2

/2&2)

0-7803-9588-3/05/$20.00 C)2005 IEEE 177

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(3)

Gn (w,o) = (-_jo)nGo(0C)

(4)

(5)

These spectral modes are distributed between 0 and t.

Gn (wt), a)

A

=

wnacn

C2 2 2

-n12 exp(- 5t

+

n

-)

(6)

2

n= a-(M -m

Where n > m B. Filter function =

Z

2nnl2)n-m exp(-1/2)

a, a2,

a,2 a22

..

.

a,n

k,

an2n

k2

(7)

fl

p p

~a,

.

.

.3

2

(10)

The pass-band edge frequency is approximated as the highest spectral mode of a particular family of Gaussian derivatives, the lowest spectral mode being the original Gaussian. The control over the transition band edge and the stopband is determined by the slope of the highest order spectral mode for that particular design. We are now going to take two approaches towards the designing of our filter. We have adopted the second approach after a brief analysis of the problems of the initial approach.

Approach-I

kn G, (w a)

2

1±3

D. Choice ofthe spectral modes

A

is expressed as Gn (w, a) and

Intersection of two consecutive Gn (w, a) is denoted by

|H(jc)|

.4z

Go (w, a) = a exp(-&wC2/22)

go (x)

Vnn

|G, (w a)

t

V i, j = 1: n and O)j are the maximal frequencies. Equation (8) can be re-written as, AK = B or, K = A 'B, provided det(A) is non singular. So finally, the ki -s are to be calculated from:

d

The Gaussian derivative function spectra are bimodal ex cept the original Gaussian with modes centered at + (0n where n is the derivative order. The derivative order and the center of that spectral mode are related by the relation:

Normalized

2 2

and ai = G(

=

In spectral domain these are transformed as, and

= aexp(i)ii-i2

(2)

gn (x)

Design parameters for Set- I (Fig. I b) a =3 (8) A Order of derivative =0, 5, 17, 37, 65 = kn Gn(Ca)) Spectral modes evaluated for the corresponding derivative order= 0.7454, 1.3744, 2.0276, 2.6874 When n is odd k is purely imaginary and for n even it is real. Passband0, edge frequency = 2.6874 k is always real. two consecutive spectral modes = 2/cs between Separation The magnitude response of the designed filter is almost = 1.2892 dB Passband ripple equiripple in the pass-band and monotonically decreasing in the stopband. 14 .I C. Weight factor calculation I12 n

Since the positions of the maximal frequencies are known from (5) for a particular design, the weights have been calculated on the basis of equalization of the amplitudes of these maximal frequencies to a specific value, say 8p. If there are Gi numbers of spectral modes and ki are the weight factors, then

Eaik1 i,j

Iz IF,

i,

11

t

i

"",

.4.1 .. ji,

-,I

'I,

It, .. As

I14bI.

E~06. 0.4

=

B

(9)

02

where, B(i) = 1±p

0

0

1

2

3

4

5

Frequency (to (rad/sec)

6

7

Fig. la: Magnitude response of the designed lowpass filter along with different spectral modes for the order 0, 4, 17 37 & 65.

178


domain. Though in the beginning of the design process the spectral modes were assigned equispaced, the quantization of the order of derivatives makes spacing between them unequal. According to this separation evaluated orders of derivatives were 0, 4, 17, 37 and 65 (for set-1). But the weighted combination of these orders of derivatives does not yield the desired equiripple in the passband (Fig. la, 1c). A slight modification in the choice of these orders (for set- 1) has been made. The new orders of derivatives are 0, 5, 17, 37, and 65. If the weighted combination of the above order of derivatives were to yield almost equiripple in the passband, the resultant ripple amplitude is too high 1.289 dB. There is as such no control over this parameter. The weight factors are calculated on the basis of equalizing the amplitude of the maximal frequencies in the passband to get the desired ripple value 1 ± S5 Also it is assumed that the maximal frequencies occur exactly at the peak frequencies of the spectral modes. This is not always true.

Design parameters for Set-2 (Fig. I c) ay =3 Order of derivative = 0, 7, 22 & 45; Spectral modes evaluated for the corresponding derivative order = 0, 0.881917, 1.56347, 2.0276 and 2.236 Passband edge frequency = 3.08 Separation between two consecutive spectral modes - 2/cs Passband ripple 0.1 dB 1.4

(iv)

212

-

\\\ ,

o

0.6

(v)

0,2 0

0

1

2

3

4

Frequency (f) (rad/sec)

5

6

7

Approach-LI

Fig. lb: Magnitude response of the designed lowpass filter along with different spectral modes for the order 0, 5, 17 37 & 65.

'6' is not chosen arbitrarily. It is calculated on the basis of passband edge frequency, stopband edge frequency and the stopband attenuation.

(i)

1.4

in(S5)

1.2-

as

1

(Inab--+-) 2 2

t: ';\, \e

(11)

where 'p' is the highest order of derivative for the design and -20loglo(65) = stopband attenuation (dB),

~0,6

a=

04

02.j

'

=

0

1

2

3

4

Frequency (o (rad/sec)

5

6

7

,= stopband edge frequency,

passband edge frequency and

IF

(ii)

Disadvantages of the above design

(iii)

5,

(12)

p

Fig. lc: Magnitude response of the designed lowpass filter along with different spectral modes for the order 0, 7, 22 & 45.

(i) (ii)

a)

'cy' is chosen arbitrarily. Stopband edge frequency is not included as a design parameter, rather it is to be estimated graphically (Fig. lb, lc) from the resultant spectrum of the filter. It depends on the slope of the highest order spectral mode (Fig. Ib, Ic). Separation between two consecutive spectral modes is chosen intuitively considering the inverse relationship between time and frequency

Maximum number of extremal frequencies in the pass-band is decided by the number of Gaussian derivative assigned in between the dc and the pass-band edge frequency. If there is a total of L number of derivatives from dc to passband edge frequency, then the number of extremal frequencies < 2L-1. As the two ends (the dc and the pass-band edge frequency) have already been decided there could be (L2) number of derivatives assigned in between them. These order of derivatives are calculated at (2L-1) number of equidistant frequencies from dc to ap. For a particular design specification number of extremal

179


frequencies depends on the pass-band ripple and the 'scale' (a) of the Gaussian derivatives. In the present methodology, we have restricted our design to a sub-class of optimum number of extremal frequencies. 'L' is chosen in such a way so that the design algorithm converges towards the desired ripple in the pass-band. In the optimum FIR filter design technique, the lo,ation of the extremal frequencies are optimized iteratively with the help of Remez exchange algorithm, for our case, the extremal frequencies are obtained iteratively using (9, 10) by minimizing the maximum absolute error (e) between the desired filter response and the weighted ideal one at those frequencies. -W()D(Comax = min E=

minIE(W)l

H(jw)j

for 0.< Where, W(tv) = B and D(w)=I in the pass-band.

final

A 'A

.8)

"! 08 Y QZ

2z 0.41"0,5r

*1

0.3 02 '

11

II

0-1 .e oLa

0

r

O'S

1

o

XA-

-

i'5

--

,-

2

@>

Frequency (o (rad/sec)

25

3

Fig. 2a: Magnitude response of the designed lowpass filter along with different spectral modes for the order 0, 1, 3, 7, 11, 18, 25, 35 & 45.

The magnitude response (Fig. 3a) of the lowpass filter designed with even order of Gaussian derivatives on the other hand shows that passband is not equripple. The frequency sampled FIR kernels (Fig. 2c, 3b) show the desired filter responses (Fig. 2d, 3c).

180


TABLE 1

Pass-band

Os

2.2361 2.14

±0.1 ±0.1

3.08 3.05

2.29

0.28

2.24

-0.103

Design Parameters

(op

Initial

I. Final (n odd & even) II. Final (n even) & Highest order 46 III. Final (n even) & Highest order 44

ripple (dB)

Stopband Ao)= Attn. lcpllni - O)p;finall (dB) Deviation

Maximum Pass-band

ripple (dB)

50.21 50.66

0 0.0961

3.1

50.62

0.0289

0.279

3.06

50.79

0.0361

0.389

0.1 0.1

102 V-

50

1s.015

-so, 1 00)

-

-100 0

0,99k

'

/

Fi.d Mantud

I

0

5'S

1

2

(rad/sc)

2.5

3

an

0s5

phs

resos

th keelin of->>v ;F-ige-.2c+9..

2 25 3 t.5 Freouency ( (red/sec Fig. 2d: Magnitude and phase response of the kernel in Fig. 2c. 0

2

15

Frequency (.) (radlsec)

-

15 Frequency

V1000 rf

1-

0.985 r

1

0,5

Fig. 2b: Magnitude response of the passband of the filter in Fig. 2a.

1

0.8

07'-

056

0i8-

fApinat

,-

0)

10 3 0.6

03

)04.

E

< 0.2

0.4

N

50 1

-02

0

5

.i

10

1s

02

20

25

30

0

35

Sample index Fig. 2c: FIR filter kernel obtained by frequency sampling the lowpass filter in Fig. 2a

0

'-1-11

0t5

1

--:.

15s

2

Frequency (.1 (rad/sec)

2v5

-"A

3

Fig. 3a:: Magnitude response of the designed lowpass filter along with different spectral modes for the order 0, 2, 4, 6, 12, 18, 28, 36 & 46.

181


bandpass nature (except the original Gaussian). The properties of the designed filter kemel in discrete domain need further comparison with that of a standard optimized filter.

0.8 r^ 0s7 0.6 0'5'

ACKNOWLEDGEMENT

The authors would like to acknowledge the help and support from all members of Microelectronics Division, SINP.

4) 0.4-

E

0.3r

I2Z

0 X

REFERENCES

~

'

+

>

-02 0

5

10

0

\Tr

15

20

35

30

25

Sample index

Fig. 3b: FIR filter kernel obtained by frequency sampling the lowpass filter in Fig. 3a 50

.50.

1000

0>5

1

2

25 2.;

3

2S Frequencyo.t (rad/see)

25

3

15

Frequency (rad/see)

0-'z

~1000, 4. .3000 0

1 QO

i

Fig. 3c: Magnitude and phase response of the kernel in Fig. 3b.

4. CONCLUSION

Nature uses Gaussian derivative based receptive fields in visual signal processing [1, 6]. Consequently, the Gaussian derivative family has found application in computer vision too [1]. Bloom has applied Gaussian derivatives of much higher order than found in nature for the purpose of image coding [2], while Appledom has used combination of comparatively lower order Gaussian derivatives in designing 'near-ideal' interpolation filters [3]. The present work attempts to intregrate these approaches in the field of optimal filter designing while trying to incorporate some ideas of Parks-McClellan algorithm in digital signal processing. Our future attempt would be to apply this work in the design of adaptive digital filters for sigma-delta based ultrasound beamformer because of its simple design principle. The same analysis can be done for the design of bandpass filter also because in spectral domain Gaussian derivatives are of

[1] R. A. Young, "The Gaussian derivative model for spatial vision: I. Retinal mechanism", Spatial Vision, vol. 2, pp. 273-293, 1987. [2] J. A. Bloom and T. R. Reed, "A Gaussian DerivativeBased Transform", IEEE Trans. On Image Processing, vol. 5, No. 3, March 1996. [3] C. R. Appledom, "A new approach to the interpolation of sampled data", IEEE Trans. Med. Imag., vol. 15, pp. 369-3 76, 1996. [4] A. I. Zverev, Handbook of Filter Synthesis, John Wiley and Sons Inc., New York, 1967, pp. 67-71. [5] J. Martens, "The Hermite transform-theory", IEEE Trans. Acoust., Speech, Signal Processing, vol 38, pp. 1595-1606, 1990. [6] J. J. Koenderink and A. J. van Doom, "Receptive field families", Biological Cybernetics, vol 63, pp. 291-297, 1990. [7] Ghavami M., L. B. Michael and R. Kohno, "Hermite function based orthogonal pulses for ultra wideband communication", Proc. International Symposium on Wireless Personal Multimedia (WPMC), Alborg, Denmark, pp. 437-440, Sept. 2001. [8] S. K. Mitra, Digital Signal Processing; A ComputerBased Approach, 2nd Edition, Tata McGraw-Hill Publishing Company Limited, New Delhi, 2001, pp. 222-223. [9] T. W. Parks and J. H. McClellan, "Chebyshev approximation for nonrecursive digital filters with linear phase", IEEE Trans. On Circuit Theory, CTl9,pp. 189-194, 1972. [1O] L. R. Rabiner, J. H. McClellan and T. W. Parks, "FIR digital filter design techniques using weighted Chebyshev approximation", Proc. of The IEEE, vol. 63, No. 4, pp.595-609, April 1972. [ll]L. J. van Vilet, I. T. Young and P. W. Verbeek, "Recursive Gaussian Derivative Filters", Proc. of the 14t/h International Conf on Pattern Recognition, ICPR'98, Brisbane, Australia, 16-20 Aug. 1998, IEEE Computer Society Press, vol. I, No. 4, pp.509514.

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