New Optimized IIR Low-Pass Differentiators - DriveHQ

2010 International Conference on Signal Acquisition and Processing

New Optimized IIR Low-Pass Differentiators Amir Tahmasbi

Shahriar B. Shokouhi

Department of Electrical Engineering, Iran Univ. of Science and Technology, Narmak, Tehran, Iran Email: [email protected]

Department of Electrical Engineering, Iran Univ. of Science and Technology, Narmak, Tehran, Iran Email: [email protected] an order-4 LPD is developed whose frequency response is very close to order-30 Parks-McClellan LPDs. In the second phase, the nominator polynomial coefficients of resulting transfer function keep constant while the denominator polynomial coefficients is being changed in such a manner that yields better filter parameters; in fact, this phase is an optimizing procedure using Genetic Algorithm. However, an appropriate fitness function is defined in the frequency domain to optimize both magnitude and phase responses. The total weighted least square error is used as a fitness function for Genetic algorithm [3]. Optimizing procedure has yielded interesting results; indeed, the order-4 proposed LPDs yield a frequency response which is almost equal to order-30 Parks-McClellan LPDs, and also yield almost linear phase in the pass-band. It is shown that the resulting LPDs yield steeper roll-off properties and smaller magnitude error than Al-Alaoui’s one.

Abstract—In this paper a novel approach is proposed for approximating Parks-McClellan low-pass differentiators using optimized low-order IIR filters. Indeed, a suitable IIR filter is designed for approximating Parks- McClellan Low pass differentiator using modified Al-Alaoui’s method, and then denominator polynomial coefficients of resulting transfer function optimized by Genetic algorithm. A suitable fitness function is defined to optimize both magnitude and phase responses; moreover, appropriate weighting coefficients and GA parameters are reported for several cut-off frequencies. It is shown that the order-4 proposed low-pass differentiators yield a frequency response which is almost equal to order-30 Parks-McClellan low-pass differentiators. Furthermore, they yield steep roll-off properties, small magnitude error and almost linear phase in the pass-band; the percentage error of magnitude response is less than 0.5%. Keywords- Al-Alaoui method; Low Pass Differentiator; IIR Filter; Genetic Algorithm.

I.

II.

INTRODUCTION

This paper describes a new method for approximating Parks-McClellan low pass differentiators using optimized IIR filters. In many applications, differentiation is followed by low-pass filtering [1], [2]; Differentiation is used to extract information about sharp transients in the signal while low-pass filtering is used to reject high frequency noises [1]. The most important advantage of FIR low-pass differentiators is the perfect linear phase; even though, needing to high filter orders makes them unsuitable for realtime applications. Therefore, low-order IIR low-pass differentiators with a linear phase in the pass-band should be used instead of corresponding FIR filters to solve the realtime problem. In the rest of paper we will use LPD abbreviation instead of Low Pass Differentiator. The basic concept to design the proposed IIR LPDs contains two important phases. In the first phase, an IIR LPD is designed using modified Al-Alaoui’s approach; indeed, an IIR LPD is developed whose numerator polynomial, up to a gain constant, is a linear phase FIR filter. So this filter not only has the linear phase characteristics of an FIR filter, but also has the steep roll-off properties of an IIR filter [1]. For instance, in this paper an order-3 Elliptic low-pass filter with 0.1dB ripple in the pass-band and 40dB stop-band attenuation is cascaded with Al-Alaoui operator; therefore, 978-0-7695-3960-7/10 $26.00 © 2010 IEEE DOI 10.1109/ICSAP.2010.22

AN OVERVIEW ON IIR LPDS AND OPTIMIZING METHODS

Although IIR filters have been used to meet the desired magnitude specifications with smaller orders than corresponding FIR filters [1], [4], the disadvantage of them is the nonlinear phase; therefore, low-order IIR LPDs with a linear phase in the pass-band should be used instead of corresponding FIR filters. Several methods have been introduced by different authors for approximating FIR filters with IIR ones; for instance, a well known method is Model Reduction approach. The order reduction of this approach is no lower than 12/60 [4]. Another one is Al-Alaoui’s approach; in this method the order reduction for approximating Selesnick’s LPD is 4/29 and 5/30 [1]. On the other hand, several methods have been introduced by different authors for optimizing IIR filters. Some of these methods are listed below: 1- Appling least-squares (LS) methods to obtain the IIR LPDs by optimizing both the numerator and denominator coefficients of transfer function [1]. 2- The time-domain methods of Prony and Shank [5]. 3- The frequency-domain approach which utilizes the iterative Fletcher and Powell method [5]. 4- The constrained least-squares method [6]. 205

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TABLE I. LPDI ORDER-4 COEFFICIENTS

ωc (rad / s)

0.5

0.6

0.7

b1 b2 b4 b5 a1 a2 a3 a4

0.0907 0.1604 -0.1604 -0.0907 0.5241 0.5499 0.0732 -0.0003

0.1278 0.2396 -0.2396 -0.1278 1.1072 0.8079 0.1838 0.0126

0.1716 0.3326 -0.3326 -0.1716 1.6529 1.2188 0.3456 0.0289

Al-Alaoui employed the time-domain methods of Prony and Shank for approximating Selesnick’s low-pass differentiator, but these methods yielded unsatisfactory results for both the amplitude and phase responses [1]; moreover, he employed the frequency-domain method of Fletcher and Powell; this method yielded unsatisfactory results for the magnitude response while yielded a good approximation for the phase response [1]. Eventually, he used the constraint least-squares method and got good results [1]. In this paper genetic algorithm is proposed for optimizing IIR LPDs. III.

Figure 1. Magnitude responses of order-4 LPD I vs. order-30 ParksMcClellan LPD and order-4 Al-Alaoui LPD.

B. Second Step In this section LPDI has been optimized. With some simplifications we can write from (1):

H (z) =

BASIC CONCEPT OF OPTIMIZED IIR LPDS

A. First Step In this step Al-Alaoui’s approach has been used to develop an IIR LPD whose frequency response is close to Parks-McClellan LPD. Needless to say, this approach is a cascading approach and employs a two-step procedure; obtain a low-order IIR differentiator whose transfer function has a numerator that represents a linear phase FIR filter, and then cascade the differentiator with an IIR low-pass filter whose numerator also represents a linear phase IIR filter [1]. According to this method, an order-3 Elliptic low-pass filter with 0.1dB ripple in the pass-band and 40dB stop-band attenuation cascaded with Al-Alaoui operator [1]; therefore, an order-4 low pass differentiator is developed whose frequency response is very close to order-30 ParksMcClellan LPDs. We have named it LPDI. Its transfer function is given as follow: H(z) =

b1 + b 2 z − 1 + b 4 z − 3 + b5 z − 4 . 1 + a1 z − 1 + a 2 z − 2 + a 3 z − 3 + a 4 z − 4

1 + c1 z -1 + c 2 z -2 + c 3 z -3 + c 4 z -4 B( z) = x5 . A( z ) 1 + x 1 z -1 + x 2 z - 2 + x 3 z - 3 + x 4 z - 4

(2)

Where cn = bn +1 / b1 , n = 1,..., 4 and bn is the nth numerator coefficient in the (1) and x1 ,..., x5 are variables. The numerator coefficients keep fixed and denominator coefficients and gain factor are being changed in such a manner that the frequency response is going to be improved; moreover, optimized filter should yield steeper roll off properties while group delay stays fair. The proposed optimization approach is a frequency domain approach; indeed, the objective is minimizing least squares error of magnitude and group delay responses. The transfer function of a low-pass differentiator can be expressed as a combination of the second order parts [7]. The equation is given here. K

H (z) = G∏

(1)

1 + β k 1 z -1 + β k 2 z -2

k = 1 1 + α k1 z

-1

+ α k2 z - 2

(3)

.

H (ω ) = GA (ω ) e jθ (ω ) .

Table I shows its coefficients for several cut-off frequencies. Fig. 1 shows the magnitude response in compare with order-4 Al-Alaoui and order-30 ParksMcClellan LPDs. However, the group delay and the percentage error of magnitude response are like Al-Alaoui’s one and omitted for brevity.

A (ω ) =

K

(4)

z -1 + β

1+ β

z -2

∏ 1 + α k 1 z -1 + α k 2 z - 2 k =1

k1

k2

. z = e jω

The group delay is expressed as [7],

206


(5)

τ g (ω ) = −

d θ (ω ) = τ g (z) dω

z = e jω

dz ⎡ d θ (z ) ⎤ = −⎢ ⎥ ⎣ dz ⎦ z = e jω d ω

For ω c = 0.7π λ = 0.005 , 0 ≤ ω n ≤ ωc + 0.02π ⎧1 ⎪ Wn = ⎨0.6 , ωc + 0.02π ≤ ω n ≤ ωc + 0.17π ⎪0.52 , ωc + 0.17π ≤ ω n ≤ π ⎩

(6)

Also it can be expressed as follow [1], [7]. ⎧⎪ K ⎡

β k1 z + 2 β k 2

τ g ( z ) = Re ⎨ ∑ ⎢

2

⎪⎩ k = 1 ⎢⎣ z + β k1 z + β k2

⎤ ⎫⎪ ⎥⎬ z + α k1 z + α k2 ⎥⎦ ⎪⎭

α k 1 z + 2α k 2

−

2

As discussed above, the objective is minimizing least squares error of magnitude and group delay responses; therefore, the optimally criterion is to minimize the following error [1]:

n =1

L

[

(8)

For ωc = 0.5π λ = 0.001 ⎧0.9 ⎪ Wn = ⎨ 0.7 ⎪ 0.19 ⎩

Where ε is the total weighted least square error over all frequencies: ω1 , ω2 ,..., ω L For 0 ≤ ωn ≤ π

ω n = nπ / L G p is the 4-K dimensional vector of the coefficients

⎧0.8 ⎪ Vn = ⎨ 0.7 ⎪ 0.5 ⎩

And λ , {Wn } and {Vn } are weighting factors selected by the designer. L is the number of frequency components. The error in magnitude at a frequency like ω n is GA(ωn ) − Ad (ωn ) , where Ad (ω n ) is the desired magnitude response (in this case ParksMcClellan) and the delay error is τ g (ωn ) − τ g (ω0 ) − τ d (ωn ) ,

{α k1} , {α k 2 } , {β k1} and {β k 2 } .

, 0 ≤ ω n ≤ ωc , ωc ≤ ω n ≤ ωc + 0.17π , ωc + 0.17π ≤ ω n ≤ π , 0 ≤ ω n ≤ ωc , ωc ≤ ω n ≤ ω c + 0.17π , ωc + 0.17π ≤ ω n ≤ π

TABLE II. GA PARAMETERS FOR OPTIMIZED LPD I

where τ g (ω 0 ) is the filter delay at some nominal center

frequency in the pass-band, and τ d (ω n ) is the desired delay response of the filter relative to τ g (ω 0 ) [1]. Genetic Algorithm is used for minimizing the total G weighted least square error. Indeed, ε ( p, G ) is the fitness function and it should be minimized by finding the suitable x1,…, x5 [4]. IV.

, 0 ≤ ωn ≤ ωc , ω c ≤ ω n ≤ ω c + 0.17π , ω c + 0.17π ≤ ω n ≤ π , 0 ≤ ω n ≤ ωc , ωc ≤ ω n ≤ ω c + 0.17π , ωc + 0.17π ≤ ω n ≤ π

⎧1 ⎪ Vn = ⎨0.83 ⎪ 0 .7 ⎩

2

]

λ ∑ V n τ g (ω n ) − τ g (ω 0 ) − τ d (ω n ) n =1

For ω c = 0.6π λ = 0.003 ⎧ 1 ⎪ Wn = ⎨0.6 ⎪ 0.5 ⎩

2

L

G

ε ( p , G ) = (1 − λ )∑ W n [GA (ω n ) − Ad (ω n )] +

, 0 ≤ ω n ≤ ωc + 0.02π , ωc + 0.02π ≤ ω n ≤ ωc + 0.17π , ωc + 0.17π ≤ ω n ≤ π

⎧1 ⎪ Vn = ⎨0.8 ⎪0.7 ⎩

(7)

ω c (rad / s)

0.7

0.6

0.5

Population Size Generation StallGenLimit StallTimeLimit TolFun

34 2000 500 800

30 2000 500 200

30 2000 500 Inf

10 − 80

10 − 80

10 − 90

TolCon

10 − 80

10 − 80

10 − 80

ε min

0.04

0.2

0.3

Note: The default values of MATLAB Genetic Algorithm Toolbox are selected for other parameters like CrossoverFraction, MutationFcn and so on, and they have not been showed in the table.

SIMULATION RESULTS

In this paper L=1000, so the optimization procedure is applied on 1000 frequency components; greater L increases the accuracy. Genetic algorithm parameters are selected in such a manner which yield the best magnitude and phase responses. These parameters are tabulated in table II for three different cut off frequencies. Also the weighting coefficients has been calculated and reported as follows for three different cut off frequencies.

Fig. 2 shows the magnitude response of order-4 optimized LPDI vs. order-30 Parks-McClellan LPD and order-4 Al-Alaoui LPD, for normalized cut-off frequency 0.7. It is obvious that proposed low-pass differentiator has steeper roll-off properties than Al-Alaoui’s one, in addition, its magnitude response is almost equal to Parks-McClellan LPD. Fig.3 shows the percentage error of magnitude

207


response; it proves that optimized LPDI has smaller percentage error in the pass band than Al-Alaoui’s one; indeed, the percentage of magnitude error is less than 0.5%. Fig.4 shows the group delay; the deflection of group delay near the cut-off frequency from its initial value for optimized LPDI is 3.471 − 0.7792 ≈ 2.9 and for Al-Alaoui’s LPD is 2.895 − 0.409 ≈ 2.5 . Therefore, the group delay has not been changed significantly with respect to Al-Alaoui’s LPD. Fig.5 shows the magnitude response of optimized LPDI for normalized cut-off frequency 0.6. It is clear that its frequency response is almost equal to order-30 ParksMcClellan LPD. The magnitude error and group delay is like those for ω c = 0.7 and omitted for brevity. Results for optimized LPDI for different values of ωc are tabulated in table III.

Figure 2. Magnitude response of order-4 Optimized LPD I vs. order30 Parks-McClellan LPD and order-4 Al-Alaoui LPD. for wc = 0.7.

TABLE III. ORDER-4 OPTIMIZED LPD I COEFFICIENTS

ω c (rad / s)

0.7

0.6

0.5

x1 x2 x3 x4 x5 C1 C2 C3 C4

1.5598 1.1511 0.30918 0.042473 0.16235 1.93823 0 -1.93823 -1 0.5 %

0.73813 0.73488 0.071074 0.07258 0.099494 1.8748 0 -1.8748 -1 1.2%

0.1342 1.7176 -0.4142 0.1750 0.1088 1.7694 0 -1.7694 -1 1%

Max. Magnitude Error

Figure 3. Magnitude error of order-4 Optimized LPDI vs. order-30 Parks-McClellan LPD and order-4 Al-Alaoui LPD.

Figure 4. Group delay of order-4 Optimized LPD I vs. order-30 Parks-McClellan LPD and order-4 Al-Alaoui LPD.

Figure 5. Magnitude response of order-4 Optimized LPDI vs. order-30 Parks-McClellan LPD and order-4 Al-Alaoui LPD. for wc = 0.6.

208


V.

REFERENCES

CONCLUSION

A novel approach is introduced for approximating ParksMcClellan low-pass differentiators using optimized loworder IIR filters. Indeed, a suitable IIR filter is designed for approximating Parks- McClellan LPD using modified AlAlaoui’s method, and then denominator polynomial coefficients of resulting transfer function optimized by Genetic algorithm. However, an appropriate fitness function is defined in the frequency domain to optimize both magnitude and phase responses. The total weighted least square error is used as a fitness function for Genetic algorithm. For instance, in this paper an order-3 Elliptic low-pass filter with 0.1dB ripple in the pass-band and 40dB stop-band attenuation is cascaded with Al-Alaoui operator, so an order4 LPD is developed. The resulting transfer function is optimized using GA. It is shown that the final optimized IIR LPDs yield a frequency response which is almost equal to order-30 Parks-McClellan LPDs, and also yield almost linear phase in the pass-band; furthermore, these LPDs yield steeper roll-off properties and smaller magnitude error than Al-Alaoui’s one; the percentage error of magnitude response in the pass band is less than 0.5%.

[1]

[2] [3]

[4]

[5]

[6]

[7]

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