A Comb Filter with Adaptive Notch Bandwidth for

Periodic Noise Reduction

Yosuke SUGIURA Tokyo University of Science

Tokyo, Japan

Arata KAWAMURA Osaka University

Osaka, Japan

Naoyuki AIKAWA Tokyo University of Science

Tokyo, Japan Email: sugiura@te.noda.tus.ac.jp Email: kawamura@sys.es.osaka-u.ac.jp Email: ain@te.noda.tus.ac.jp

Abstract-This paper proposes a comb filter with adaptive notch bandwidth. The comb filter is used to reduce a periodic noise from an observed signal. To extract the desired signal completely, we should appropriately design the notch bandwidth of the comb filter. Specifically, we should design the notch bandwidth to be equal to the fluctuation bandwidth. In this paper, to automatically reduce only the periodic noise with the frequency fluctuation, we propose the comb filter which achieves the adaptive notch bandwidth. Simulation results show the effectiveness of the proposed comb filter.

I. INTRODUCTION

In speech processing, image processing, biomedical signal processing, and many other signal processing fields, one of the most important issues is to extract a desired signal from an observed signal which contains noise. In some practical situations, the noise has often periodicity, i.e. , the noise consists of a fundamental frequency and its harmonic frequencies. Examples of such noise are a car engine noise, a hum noise in an electrocardiogram, an acoustic feedback in hearing aids, and so on. A comb filter is used to reduce the periodic noise, since its frequency response becomes zero at equally spaced frequencies which are called as notch frequen­cies [1]-[4]. Specifically, we can remove the periodic noise completely by aligning the first (lowest) notch frequency with the fundamental frequency of the noise.

In a practical situation, the periodic noise often has a fre­quency fluctuation which is a short-term frequency variation with respect to time. The frequency fluctuation appears as the frequency broadening centered at the harmonic frequency, and the fluctuation bandwidth tends to be large in high frequency range. To remove such practical noise, we adjust the notch bandwidth for each of notch frequencies, where the notch bandwidth denotes the elimination bandwidth of the comb filter where the filter gain is less than 1/ J2 around notch frequency. An appropriate notch bandwidth should be equal to the fluctuation bandwidth. Unfortunately, in many cases, the fluctuation bandwidth are unknown, and they often change with time. Hence, it needs to develop a method which automatically adjusts the notch bandwidth so that it provides the desired signal from the observed signal.

In this paper, we propose a comb filter which employs an adaptive notch bandwidth for appropriately reducing the periodic noise. The adaptive notch bandwidth is achieved by

978-1-4799-0434-1/13/$31.00 ©20 13 IEEE

adjusting the comb filter's feedback gain which determines the notch bandwidth. When the desired signal is a white signal, adjusting the feedback gain to minimize the correlation of the comb filter output signal gives an appropriate bandwidth. To evaluate the correlation of the comb filter output signal, we introduce an adaptive line enhancer (ALE) [5] which estimates the current comb filter output signal using past several input samples. The ALE output power becomes small when the comb filter output is identical to the desired signal. On the other hand, the ALE output power becomes large when the comb filter output signal includes the periodic noise. Hence, we utilize the ALE output power to update the adaptive notch bandwidth. Simulation results show that the proposed comb filter can adjust the notch bandwidth according to the fluctuation bandwidth of the noise.

II. A CONVENTIONAL COMB FILTER

In this section, we explain a conventional comb filter [1]. The conventional comb filter is shown in Figure 1, where x ( n) is the input signal at the time n, and y( n) is the output signal of the comb filter. The transfer function of this comb filter is given by

Cconv(Z) = 1 - Qconv(z)(1 - g), 1 - b 1 + z-N

QconAz) = -2- . 1 - bz-N '

(1)

(2)

where N is a natural number, b (- 1 < b < 1) is the notch bandwidth parameter, and 9 (0 ::; 9 ::; 1) is the notch gain parameter. The filter Qconv(z) is an inverse comb filter which enhances the harmonic frequency components. The amplitude freq uency response of Q con v (z) is represented by

I ( jW) I _ 1 1 - b 1 + z-N I Qconv e - -2- ' 1 _ brN

1 + cos(Nw) (3) 1 + b2 - 2bcos(Nw)'

From this equation, we see that IQconv(ejW)I takes extreme values at w = 7Jj(l = 0,1", . ,N). The maximum values and minimum values of IQconv(ejW)I are represented by

WMax (m) = 2m1f/N, (4)

rcrcs 2013

xen)

and

Fig. 1. Structure of the conventional comb filter.

Wmin(m) = (2m + l)7f/N, (5)

respectively. Here, m = 0, I, . . . , l N /2 J. The comb filter has three important factors, i.e., the notch

frequency, the notch bandwidth, and the notch gain, which determine the comb filter's frequency response completely. The notch frequency gives the minimum comb filter gain. As see from (1), the minimum gain of the comb filter depends on the maximum of 1 Qconv ( ejW) I, e.g. , the frequency at which the comb filter gain ICconv(ejW)1 takes the minimum is equal to the frequency at which 1 Q con v ( ejW) 1 takes the maximum. Hence, from (4), we obtain the m-th notch frequency as

27fm wm =

N ' m = 0, I, ". , l N /2 J. (6)

We see that the first notch frequency WI is uniquely deter­mined by N. Next, we reveal the notch gain which is the filter gain at the notch frequency. By substituting (4) into (1), the notch gain is obtained by

(7)

We see from (7) that all the notch gains are the same value g,

independently of Wm. Lastly, we show the notch bandwidth

which is the frequency range satisfying ICconv(ejW) 1 < 1/v'2. Solving ICconv(ejW)1 = 1/v'2 for w, we have the notch bandwidth at Wm as

2 -1 ( 2b ) W m =

N cos 1 + b2

. (8)

We see from (8) that the notch bandwidths are the same at all notch frequencies.

Figure 2 shows 1 Cconv (ejW) 1 for the bandwidth parameters b = 0 and 0.5, where we put N = 8 and g = 0.3. Here, we see that the notch bandwidth becomes narrow as b goes toward to 1. We also see that the notch bandwidths and gains are the respective same value for all notch frequencies.

To reduce only the periodic noise from the observed signal, we should appropriately design the notch frequency, the notch gain, and the notch bandwidth of the comb filter. Specifically, we design the first notch frequency to be equal to the lowest

o � ________ -L __________ � ________ ��

o 2 Normalized Frequency

3

Fig. 2. Amplitude frequency responses of Cconv(z) with b = 0 and 0.5.

frequency of noise, design the notch gain according to the frequency amplitude of the noise, and design the notch bandwidth to be equal to the fluctuation bandwidth of the noise.

III. A COMB FILTER WITH ADAPTIVE NOTCH

BANDWIDTH

For reducing only the periodic noise, we estimate the fluctuation bandwidth of the noise, and then design the notch bandwidth to be equal to the fluctuation bandwidth. Unfor­tunately, in many practical cases, the fluctuation bandwidth are unknown, and they often change with time. Thus, in this section, we derive an adaptive method for the notch bandwidth. Note that we assume that the desired signal is a wideband random signal. For simplicity, we also assume that the notch frequency and notch gain are appropriately designed. Now, we represent the observed signal by the sum of the desired signal and the periodic noise, i.e. ,

x(n) = d(n) + s(n) , (9)

where d( n ) denotes the desired wideband random signal and s (n ) denotes the periodic noise. When the notch bandwidth becomes equal to the fluctuation bandwidth of the noise, the comb filter removes only the periodic noise, i.e. , the output signal of the comb filter y(n ) becomes y(n ) = d(n ) . On the other hand, when the notch bandwidth and the fluctuation bandwidth are different, the comb filter contains the residual periodic noise, i.e. , y(n ) = d(n ) + s(n ) . Since the notch bandwidth only depends on the parameter b as shown in (8), we adjust b so that s(n) is reduced. To evaluate s(n), we set an adaptive line enhancer (ALE) [5] at the comb filter output.

The additional ALE predicts the current output signal of the comb filter y( n ) by using the several past samples of y( n ) . When y(n ) = d(n), the ALE cannot predict y(n ) . Then, the output power of the ALE becomes approximately O. On the other hand, when y(n ) = d(n ) + s(n ), the ALE can predict

y( n ) and thus its output power becomes large. Hence, we can use the output power of the ALE as a criterion for the power of s(n ) . The block diagram of the additional ALE is shown in Figure 3, where H(z) denotes the ALE. The ALE H(z)

yen) = den) + sen) e(n)

x(n)

is given by

, , , , , , ,

I , ' L ________________________ .J

Fig. 3. Block diagram of the ALE.

y(n) }-........ ---.---+

Fig. 4. Structure of the proposed comb filter.

U-l

H(z) = L aiz-i, i=O

(10)

where U is the filter order and ai is the filter coefficient of H(z). The signal s(n) is the ALE output signal given by

U-l

s(n) = L aiy(n -D -i), (11) i=O

where the integer D (> 0) denotes the input delay. The signal e( n) denotes the prediction error signal given by

e(n) = y(n) - s(n) . (12)

When H(z) has the optimal filter coefficients, the ALE extracts the non-periodic components included in y(n), i.e. , e( n) � d. On the other hand, in this case, the output signal of H(z) extracts the periodic components included in y(n), i.e., s(n) � s. We update ai by using the LMS algorithm [6], i.e.,

e(n)y(n - i-D) hi(n + 1) = hi(n) + fJh U-l 2' (l3)

Ll=O y(n -I - D)

where fJh is a step size parameter such that 0 < fJh < 2 is satisfied.

To obtain the appropriate notch bandwidth, we update b to minimize the the output error e2 (n) . The structure of the comb filter with the adaptive notch bandwidth is shown in Figure 4, where u( n) is the signal represented as u( n) =

x(n)-bu(n-N) . Based on the simplified recursive prediction error (SPRE) algorithm [7], we update b along the gradient

of e2(n) . The SPRE algorithm is one of the LMS algorithm for the IIR filter. The gradient of e2 (n) is defined as

n 2 ( ) = 8e2(n)

= ()8e(n)

v e n -8b

2e n 8b

= 2e(n)· (1 - g){ u(n) + u(n - N)

_ 1 - b 8U(n) }

. (14) 2 8b

Taking the derivative of u(n), we obtain

8��n) = :

b {x(n) + bu(n - N)

} = ( _ N) b

8u(n - N) un +

8b .

According to [7], we can approximate (16) by

8u(n) � u(n - N)

8b .

Substituting (17) in (14), we obtain \7e2(n) as

(15)

(16)

(l7)

\7e2 (n) = 2e(n) . (1-g){ u(n) + 1 � b

u(n - N)}

. (18)

Then, the normalized SPRE algorithm for the adaptation of b is obtained as

( 1-9) { l+b } b(n + 1) = b(n) - fJb

T(n) u(n) + -2-u(n - N) ,

(19)

T(n) = aT(n - 1) + ( 1 - a) { u2 (n) + u2 (n - N) } , (20)

where fJb is a step size parameter such that 0 < fJb < 2 is satisfied. When we consider the stability of the comb filter, i.e., -1 < b < 1, we can obtain the normalized stable SPRE algorithm for the adaptation of b as { (3 b(n) :s; -(3

b(n + 1) = �n ; , �(3 < b(n) < (3 , (3, b(n) :s; (3

(21)

- (1-9) { l+b } b(n) = b(n) - fJb

T(n) u(n) + -2-u(n - N) ,

(22)

T(n) = aT(n - 1) + ( 1 - a){ u2 (n) + u2 (n - N)} . (23)

When we set fJb to a larger value, the convergence speed becomes faster and the fluctuation of the value of b around the convergence value becomes larger. Conversely, when we set fJb to a smaller value, the convergence speed becomes slower and the fluctuation of the value of b around the convergence value becomes smaller. The parameter a is a forgetting factor such that 0 :s; a < 1 is satisfied. For the stability of the comb filter, we introduce (3( < 1) which is a positive constant close to 1.

-30 iii' �-40 E :::l

t5 -50 Q) c..

(f) Q; -60 � o

a... -70

TABLE I AN EXAMPLE OF A TABLE

j I I I I I

o Normalized Frequency

Fig. 5. Power spectrum of the observed signal.

IV. SIMUL ATION

To confirm the effectiveness of the proposed comb filter, we carried out an evaluation experiment for reducing the periodic noise signal from the observed signal which consists of the desired wideband signal and the single periodic noise signal. We used a white Gaussian signal as a desired signal and used the periodic noise whose fundamental frequency is 27r /20. We produced the observed signal by adding the periodic noise to the desired signal with SNR=OdB. The power spectrum of the observed signal is shown in Figure 5, where the frequency amplitudes of the periodic noise gradually decrease with increasing frequency.

The parameters required in the proposed comb filter are sUlmnarized in Table I. Here, Since we focus on the capability of the adaptive notch bandwidth, we assume that the notch frequency and the notch gain are appropriately designed.

We evaluated the convergence performance of the notch bandwidth parameter b. Figure 6 illustrates the convergence property of b for two input signals, where one input signal is equal to Figure 5, and another is the signal which is introduced the frequency fluctuation into Figure 5. We set the fluctuation bandwidth to 27r /20 X 10-3. In Figure 6, the horizontal axis denotes the sample number and the vertical axis denotes the value of b. We see that the parameter b converges to a small value (b = 0.82) when existing the frequency fluctuation in the periodic noise. On the other hand, when the periodic noise has no frequency fluctuation, b converges toward a upper value (b = f3 = 0.99). It means that the notch bandwidth automatically becomes wide when existing the frequency fluctuation in the periodic noise. From this result, we see that the proposed comb filter can achieve the adaptive notch bandwidth.

V. CONCLUSION

In this paper, we proposed a new comb filter which has an adaptive notch bandwidth for reducing the periodic noise. The

.... OJ

...... OJ

E ro .... ro

a..

Fig. 6.

o

For periodic noise with no fluctuation

For periodic noise with fluctuation

2 4 6 8 Samples ( x 104)

10

Convergence performance of the parameter b for two input signals.

adaptive notch bandwidth has been established by introducing an additional ALE which predicts the comb filter output signal. The adaptive notch bandwidth is adaptively updated to minimize the ALE output power. The proposed comb filter can adjust the adaptive notch bandwidth according to the fluctuation bandwidth of the noise.

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