1310 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO.

4, APRIL 2005

A Piloted Adaptive Notch Filter

Yong Ching Lim, Fellow, IEEE, Yue Xian Zou, Member, IEEE, and N. Zheng

Abstract—In the implementation of an adaptive notch filter will outperform a fixed step-size algorithm if appropriate step
using the least mean squares (LMS) algorithm, the zero of the sizes are obtainable. Several techniques for obtaining the step
filter is steered toward the input sinusoid based on the gradient sizes have been reported in the literature to support the variable
information. The convergent may be speeded up if a larger step
size is used when the zero of the notch filter is far away from the step-size LMS (VSLMS) algorithm [6]–[12]. All of these al-
frequency of the input sinusoid. The gradient provides informa- gorithms derive the step-size based on the estimation errors at
tion on the direction where the zero should be steered but does several time instants (i.e., time domain averaging).
not provide information on the distance between the zero and Harris et al. [6] suggested a method that makes use of the sign
the frequency of the sinusoid. Conventional variable step-size
change between consecutive gradient estimates to determine the
algorithms determine the step size based on a (linear/nonlinear)
weighted average of the gradient estimate at several sampling step size. If the signs of the gradient estimates for several con-
instances (time domain averaging). In this paper, we propose a secutive input samples are unchanged, it is assumed that the dis-
new method for extracting information on the distance between tance between the zero of the notch and the frequency of the
the frequency of the input sinusoid and the zero of the notch. We sinusoid is large; the step size is then increased by a predeter-
use three (or more) notches, namely, a main notch and two (or
more) pilot notches implemented with minimal additional cost.
mined factor. If the signs of the gradient estimates alternate for
The pilot notches are used to analyze the gradient estimates at several consecutive samples, it is assumed that the zero of the
the same sampling instance but at several frequency points as notch filter has converged; the step size is thus decreased by a
the main notch. Simulation results show that our new piloted predetermined factor.
notch technique is significantly superior to step-size determination The step size may also be made a function of the gradient es-
based on a time-averaging technique. Novel theoretical analysis
is presented. Our method can be used in conjunction with most timate. After an adaptive algorithm has converged, the expected
existing algorithms to determine the step size. value of the gradient estimate will tend to zero. Hence, the mag-
Index Terms—Adaptive notch filter, fast convergence, least mean nitude of the expected value of the gradient estimate is a measure
squares algorithm, low misadjustment, pilot notches, steering di- of the state of convergence. The expected value of the gradient
rection, variable step-size algorithm. estimate can be estimated by lowpass filtering the gradient esti-
mate. Let denote the lowpass filtered value of the gradient
estimate at time . The step size can be made proportional to the
I. INTRODUCTION
magnitude of or to a function of the magnitude of .

I N the least mean squares (LMS) based adaptive notch filter

[1]–[5], the notch frequency is updated every sampling in-
stance and eventually converges to the frequency of the input
This approach has been adopted by Karni et al. [7] and Shan et
al. [8].
Farhang [19] and Mathews and Xie [13] proposed using the
sinusoid. There are many reports on the analysis, improvement product of consecutive gradient estimates to update the step size.
of convergence speed, and complexity reduction for adaptive This is developed based on the fact that for an adaptive algo-
filters [14]–[16]. The convergence rate and the misadjustment rithm, after convergence has been achieved, the solution will
after convergence are influenced by the adaptation step size [2]. wander around the optimum point; this causes the gradient es-
A larger adaptation step size will result in a faster convergence timate to change sign rapidly. If the product of consecutive gra-
rate but will yield a larger misadjustment. Ideally, a large step dient estimates is negative, then there is a change in the sign of
size should be used when the distance between the zero of the these two consecutive gradient estimates.
notch and the frequency of the sinusoid is large in order to im- There is another class of very important notch filters—the in-
prove the convergence rate. A small step size should be em- finite impulse response (IIR) adaptive line enhancer with con-
ployed if the zero of the notch has converged to the frequency of trolled bandwidth. Farhang [20] proposed a novel technique that
the sinusoid. As a consequence, a variable step-size algorithm is eminently suitable for controlling the bandwidth of the line
enhancer such that its bandwidth is increased when its frequency
Manuscript received July 3, 2003; revised April 16, 2004. This work was response peak is far from the input sinusoid’s frequency to as-
supported in part by Temasek Laboratories at NTU, Nanyang Technological sure fast convergence. The bandwidth is decreased when the line
University, Singapore Polytechnic, and National University of Singapore. The
associate editor coordinating the review of this manuscript and approving it for enhancer has locked on to the input sinusoid. Excellent results
publication was Dr. Behrouz Farhang-Boroujeny. had been obtained. The study of the adaptive notch filter with
Y. C. Lim is with the School of Electrical and Electronic Engineering, time-varying bandwidth has also been reported in [21] and [22].
Nanyang Technological University, Singapore 639798 (e-mail: ele-
limyc@pmail.ntu.edu.sg). In [23], an implementation using Constantinides [24] transfor-
Y. X. Zou is with the School of Electrical and Electronic Engineering, Singa- mation was proposed. A technique for implementing the adap-
pore Polytechnic, Singapore 139651. tive notch filter using an allpass filter section was presented in
N. Zheng is with the Department of Electrical and Computer Engineering,
National University of Singapore, Singapore 119260. [25]. The adaptive notch filter is a special class of IIR adaptive
Digital Object Identifier 10.1109/TSP.2005.843742 filters, an excellent treatment of which is available in [26].
1053-587X/$20.00 © 2005 IEEE
LIM et al.: PILOTED ADAPTIVE NOTCH FILTER 1311

Fig. 1. Structure of an IIR adaptive notch filter.

In this paper, we propose a new concept for estimating the dis-

tance between the notch frequency and the frequency of the sinu-
soid, leading to a new method in step-size determination. It should
be noted that our technique produces additional information on
the distance between the notch frequency and the frequency of
the sinusoid; it does not replace existing techniques; it can be ap-
plied concurrently with existing techniques. Our technique uses
three (or more) notches, namely, a main notch and two (or more)
Fig. 2. Pole-zero positions of the IIR notch filter.
pilot notches implemented with minimal additional cost. The zero
of the main notch is sandwiched between the zeros of the pilot
notches. The pilot notches are used to provide the information on The time-varying transfer function of the notch filter at
whether the distance between the frequency of the sinusoid and time is given by
the zero of the main notch is large. If the frequency of the sinusoid
is sandwiched between the zeros of the pilot notches, the signs of (1)
the gradient estimates for the pilot notches will point in opposite
directions. This situation can be detected by examining the signs where is the adjustable weight at time , and is the
of the gradient estimates of the three notches. If they are not all distance of the pole from the origin. The poles and zeros of
equal, it is an indication that the frequency of the sinusoid is sand- the notch filter are shown in Fig. 2. The notch frequency at
wiched between the zeros of the two pilot notches; a smaller step time , which is denoted by , is given by the zeros of
size is preferred. If all the signs of the gradient estimates are the and, thus, is given by
same, a larger step size is selected.
This paper is organized as follows. Section II presents a very (2)
brief description of the adaptive notch filter for the purpose of
defining notations. The concept of the pilot notches is presented A large number of excellent algorithms [28]–[30] for updating
in Sections III and IV. Simulation results demonstrating the supe- has been developed. A direct application of Widrow’s
riority of our new piloted notch technique are shown in Section V. technique will lead to
New theoretical analysis on the steering directions obtained by
the notches when the input signal is a sinusoid contaminated with (3)
white noise is discussed in Section VI. Novel theoretical analysis
where is the adaptation step size at time . In the con-
on the probability of producing a correct estimate for the steering
ventional fixed step size LMS algorithm, is fixed and is
direction is presented in Section VII. The effectsof notch position,
not a function of . We will use (3) as a reference for discus-
signal-to-noise ratio (SNR), and pole radius on steering direction
sion, but all currently existing methods for updating , in-
estimation are presented in Sections VIII– X, respectively. Novel
cluding those variable bandwidth algorithms, may be used in-
theoretical analysis on the improvement in steering direction es-
stead. The implementation complexity, theoretical analysis, and
timation that is obtainable jointly from the main and pilot notches
performance figures will certainly depend on the actual algo-
is discussed in Section XI. The applications of the piloted notch
rithm used for updating . Examples of applying the pi-
technique in conjunction with various weight update equation are
loted notch concept to other algorithms for updating are
presented in Section XII.
illustrated in Section XII.
A disadvantage of such a formulation is that there is a lack of
II. ADAPTIVE NOTCH FILTER information as to how far away the notch frequency is from the
Fig. 1 shows the structure of a second-order IIR adaptive frequency of the sinusoid. Algorithms that makes use of infor-
notch filter. It consists of a pole followed by a zero [27]. An ad- mation on sign changes in [6], [13], [19] are in fact
vantage of having the pole precede the zero over that of the zero attempting to estimate the distance of the notch frequency from
preceding the pole is that after convergence, the pole enhances the frequency of the sinusoid through time domain averaging.
the SNR, which in turn reduces the bias of the zero estimate. In this paper, we introduce pilot notches to perform analysis at
1312 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

In (7) and (8), and have positive values and are functions
of . They correspond to the effect of and on
and . The values of and need not be determined
accurately since the pilots only serve as rough indicators of the
frequency of the sinusoid. Substituting (7) and (8) into (5) and
(6), respectively, (z) and (z) become

(9)

(10)

Let and be the time domain outputs of and

, respectively. Thus

(11)
(12)

Fig. 3. Pole-zero positions of the piloted notch filter.

The implementation structure for obtaining and is
shown in Fig. 4. Since the values of and have no effect
on the accuracy of the estimate of the frequency of the sinu-
several frequency points near the main notch. This provides ad- soid, they can be selected as a convenient power-of-two so that
ditional information for determining the distance between the and can be obtained from that of by simple
notch frequency and the frequency of the sinusoid. shift-add operations [31]–[33]; in this case, the additional cost
for obtaining and is minimal.
III. PILOT NOTCHES
We will refer to the notch presented in the above section as IV. ADAPTATION STEP SIZE
the main notch. We will now introduce two pilot notches into The function of the pilot notches is to provide information
the system. The notch frequency of one of them is higher on whether the frequency of the sinusoid is far from that of
than and the other one is lower than that of the main notch. the main notch. The signs of ,
See Fig. 3. Furthermore, we arbitrarily choose and provide information on the relative posi-
tion of the frequency of the sinusoid and that of the notches.
(4) This issue will be discussed further in Sections VI and VII. If
is a pure sinusoid without additive noise, a positive value
We also assume that the frequency of the input sinusoid is larger of implies that the frequency of the sinusoid
than and less than the difference between half of the sam- is higher than the notch frequency of . Obviously, in
pling frequency and , i.e., the notch frequency of the lower this case, the frequency of the sinusoid must also be higher
frequency pilot is larger than zero, and the notch frequency of than the notch frequencies of and , and hence,
the higher frequency pilot is less than half of the sampling fre- and must also be positive. Simi-
quency. Let the -transform transfer function of the two pilot larly, if , and are all
notches at time be and , respectively, where negative, the frequency of the sinusoid is lower than the notch
frequency of . As a result, if sign
sign sign , it is implied that
(5) the frequency of the sinusoid is either higher than the notch fre-
quency of or lower than that of . This means
and that the frequency of the sinusoid is far from the location of the
(6) zero of the main notch. Thus, in order to reduce the time re-
quired to steer the main notch frequency to the frequency of the
sinusoid, a larger value of is used.
Note that and share the same denominator as If the frequency of the sinusoid is very close to the main
in order to reduce the computational complexity. notch, it must be sandwiched between the notch frequencies of
Although it is possible to compute and based and , respectively. In this case,
on predetermined and , such computations will require will be negative, and will be positive. The sign
operations involving the computation of and , of will depend on whether the frequency of the
which are computationally complex. Hence, we define sinusoid is higher than the main notch frequency. In either case,
the signs of , and
(7) will not be equal. Under these circumstances, a smaller
(8) is used in order to minimize misadjustment.
LIM et al.: PILOTED ADAPTIVE NOTCH FILTER 1313

Fig. 4. Piloted notch filter. If  and  are integer power-of-two, the additional cost for obtaining e (n) and e (n) is minimal.

Fig. 5. Structure of the lattice notch filter.

Our new technique makes use of the pilots to obtain additional

information on the steering direction by analyzing the steering
directions at the same time instant at several distinct frequency
points.

V. SIMULATION RESULTS
In this section, we show a comparison for the convergence
speeds of four algorithms:
a) conventional fixed step-size LMS algorithm; Fig. 6. Comparison between the convergent speeds for various adaptive
b) our piloted notch variable step-size LMS algorithm; algorithms.
c) time domain averaging variable step-size algorithm;
d) lattice LMS algorithm. where was 0.0065. This value of was selected so that the
The input signal SNR was 10 dB. The frequency of the input si- conventional lattice and the conventional (direct form) LMS al-
nusoid was one eighth sampling frequency. The notch frequency gorithms produced the same mean square weight error. Note
was initialized at three eighths sampling frequency. The pole ra- that the convergence property for the lattice algorithm is dif-
dius was 0.9. ferent from that of the conventional (direct-form) LMS algo-
For the conventional fixed step-size LMS algorithm, the step rithm. In order for the comparison of convergence speed to be
size was . For our piloted notch variable step-size algo- meaningful, the mean square error should be made the same.
rithm, the step sizes were and . For the pilots, Fig. 6 shows the convergence of for the above four al-
. For time domain averaging variable step-size gorithms. It can be seen from Fig. 6 that our new piloted notch
algorithm, the step sizes were selected based on the product of algorithm converges significantly faster than the other three al-
consecutive gradient estimates . If gorithms. Another figure of merit of an algorithm is the mean
the product was positive, a larger step size was used; square error after convergence has been attained. It can be seen
otherwise, a smaller step size was used. The struc- from Fig. 6 that all the algorithms had converged after .
ture for the lattice LMS algorithm was the two-multiplier lattice (There is no eigenvalue problem in this case because there is
shown in Fig. 5. In Fig. 5, is the first reflection coeffi- only one parameter.) The mean square error ,
cient of the lattice; this notation is used in order to facilitate the where is the desired value of the weight, for all four al-
comparison of results with other algorithms. The transfer func- gorithms were computed using values of for ranging
tion for the lattice notch filter was [18] from 1000 to 1200 and tabulated in Table I. As can be seen from
Table I, our new piloted notch variable step-size algorithm out-
(13) performs the other three algorithms significantly. The success
of our piloted notch algorithm lies in the ability of the pilots to
where is the z-transform of . was updated detect the separation between the notch frequency and the fre-
using quency of the input sinusoid. A larger step size was correctly
selected when the separation between the notch frequency and
(14) the frequency of the input sinusoid was large, and a smaller step
1314 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

TABLE I

Wn n
MEAN SQUARE WEIGHT ERROR WAS COMPUTED USING VALUES OF
( ) FOR RANGING FROM 1000 TO 1200

Wn
Fig. 8. Average step size. The average value is computed as a weighted average
of ( ) weighted by a 401-point Hamming window function.

lattice algorithm and the result will be reported when it is avail-

able.) Note that our piloted notch concept does not replace the
existing LMS algorithm; it merely adds pilots to the existing
LMS algorithm, extracting further information through the pi-
lots. It can be seen from Fig. 6 that using the pilots alone will
outperform using time averaging alone, but of course, pilots can
be incorporated into the time averaging algorithm to outperform
using piloted algorithm alone. There is no limit to the number
of ways the piloted algorithm can be mixed with other existing
Wn
Fig. 7. Average step size. The average value is computed as a weighted average
of ( ) weighted by a 41-point Hamming window function. algorithms.

size was correctly selected if it was small. This can be seen from VI. STEERING DIRECTION ANALYSIS
the average value of the step size shown in Fig. 7. In Fig. 7, The main function of the pilots is to provide information on
the average step size was a weighted average of weighted whether the frequency of the sinusoid is far away from the notch
by a 41-point Hamming window function. As can be seen from frequency. Such information is obtained from the steering direc-
Fig. 7, the average step size for our piloted notch algorithm was tion provided by the pilot notches. If the steering directions of all
large before it converged and small after it converged. The time the pilot notches as well as the main notch are all in the same di-
domain averaging step-size selection technique based on con- rection, the frequency of the sinusoid is said to be far away from
secutive sign change does not provide a very accurate decision the notch frequency; otherwise, it is near the notch frequency.
for step-size selection. It is not immediately obvious from Fig. 7 If more than two pilots are used, grey-level information on the
that for the time domain averaging technique, a larger step size distance is available. The number of grey levels, of course, de-
was selected before convergence, and a smaller step size was pends on the number of pilots. The steering direction is central
selected after convergence. The effect of step-size selection for to the development. The effect of noise on the accuracy of the
the time domain averaging technique becomes visible if a 401 information provided by the steering direction estimate is thus
(instead of a 41) point Hamming window function was used to important to know.
compute the weighted average value of , as shown in Fig. 8. In this section, we consider the relationship between the
It can be seen from Fig. 8 that the time domain averaging tech- product term and the relative position of the
nique did select a larger average step size before convergence frequency of the input sinusoid with respect to the frequency
and a smaller step-size after convergence, although the differ- of the notch for the case where the input is a sinusoid with
ence is not very significant. additive white Gaussian noise. Refer to Fig. 1. Let
In principle, our piloted notch concept can be incorporated
into any existing algorithm to obtain a piloted version of that al- (15)
gorithm if an efficient way of incorporating the pilots into such
existing algorithm can be found. In certain cases, it is straight- where is a zero mean white Gaussian noise with variance
forward to incorporate the pilots efficiently, but in certain cases, , and is a random phase constant. In order to simplify theo-
it is not. For example, it is straightforward to incorporate pi- retical analysis, we will assume that can be approximated
lots efficiently into the “time averaging” algorithm, but incor- by a time invariant constant . In this case, let
porating the pilots efficiently into the lattice algorithm is not so
straightforward. (We are investigating the possibility of piloted (16)
LIM et al.: PILOTED ADAPTIVE NOTCH FILTER 1315

where The noise is thus a filtered version of . We assume that

is also Gaussian distributed with zero mean. Its variance
is thus given by

(17a)
(17b)
(26)
and is a filtered version of . We wll assume that
is Gaussian distributed with zero mean. Thus, the autocorrela- The product term is given by
tion function of can be obtained from the inverse Fourier
transformation of its power spectrum density and is thus given
(27)
by
where
(18)

where is given by (28)

The first term of the right-hand side of (27) is the value

(19)
of for a pure sinusoidal input without noise.
The term represents the effect of noise on the value of
is the transfer function of the allpole section of the notch . Since the steering direction of the notch is
filter, and is the power spectrum density of the input determined by the sign of , it is important to
white noise, which is a constant equal to . The autocorrelation understand the effect of on the sign of . If
of can be derived by applying the Residue Theorem to (18). will have no
, and are useful in subsequent analysis; effect on the sign of . If the sign of and that of
therefore, we obtain are the same, will also have
no effect on the sign of . The sign of
will be affected if , i.e.,
, and at the same time,
and have different signs.
Substituting for in (28) and using (25), becomes

(20)

(21a) (29)

Since the mean values of , and

(21b)
are zero, the mean value of , which is denoted by , can
and be obtained from (29) as
(22a)
(30)

(22b) Substituting (20) and (21a) into (30) yields

The output of the notch filter is given by

(31)

(23)
Fig. 9 shows the versus plots for and for
where and .
From (31), it is straightforward to show that
(24)
when
and when (32)
(25) when
1316 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

Fig. 9. Mean of d(n) versus W plot. Fig. 10. Variance of d(n ) versus W plot when r = 0:5 and
input noise power = 1.

It can be seen from Fig. 9 that when is close to unity, is

approximately equal to zero, except when the notch frequency is
near d.c. or half of the sampling frequency, i.e., when .
For small , we will let .
Using the result of (28), the variance of is thus given by

(33)

Multiplying both sides of (25) by , taking expectation,

and considering the results of (20) and (21), we have

(34)

(35)

Fig. 11. Variance of d(n ) versus W plot when r = 0:7 and

From the above equation, it is clear that the correlation of
input noise power = 1.
and approaches zero when the pole radius approaches
unity. Thus, from (33) and (34), the variance of is given by
derivation, we have assumed that is small. The condition
for to be small is that either or is not close to 2.

VII. PROBABILITY OF PRODUCING A CORRECT ESTIMATE FOR

THE STEERING DIRECTION

We have shown that the steering directions of the pilot notches

can be obtained from the signs of and
, respectively. We have also shown how these steering
(36) directions provide information on whether the frequency of the
input sinusoid is sandwiched between the two pilot notches.
Figs. 10–12 show the comparisons of the computer simulated When the input sinusoid of the adaptive notch filter is corrupted
results and the theoretical results obtained from (36) when the by a white Gaussian noise, the signs of and
pole radius , and , respectively. In Figs. 10–12, may not indicate correctly whether the frequency
for both the simulation and theoretical results, we assume that of the input sinusoid is higher or lower than that of the pilot
the input noise has unit power. The results show that (36) notch. In this section, we present an analysis on the probability
provides an accurate estimate of when the pole radius is close that the sign of provides the correct steering di-
to unity or when the notch frequency of the filter is not near d.c. rection for the main notch; the results for and
or half of the sampling frequency. This is expected since, in the can be obtained in a similar way.
LIM et al.: PILOTED ADAPTIVE NOTCH FILTER 1317

The factor determining the steering direction is the sign of

. Assuming that the sign of is independent of that of
, the probability that the sign of is positive
is denoted by and is given by

(43)

where and are the probabilities that and

are positive, respectively. From (39), the probability that is
positive is given by

(44)

Fig. 12. Variance of d(n) versus W plot when r = 0:9 and and from (42), the probability that is positive is given
input noise power = 1. by

We assume that the input noise is Gaussian with zero mean

and variance . Since is a linearly filtered version of a
Gaussian distributed signal (by assumption), is Gaussian
distributed with zero mean. Thus, from (25), , which is the
sum of , is also Gaussian distributed.
The variance of is given by (26). From (23), at any time (45)
instant is Gaussian with expected value , where
Substituting (37), (38), (26), and (20) into (44) and substituting
(37) (40) and (20) into (45), it can be seen that both and are
functions of a) the frequency of the input sinusoid , b) the
We would like to highlight the distinction between the expected notch position , c) the input SNR, and d) the pole radius .
value of at any time and the expected value of over Substituting (44) and (45) into (43), it is clear that , which is
all . While the expected value of at time is given by the probability of the sign of being positive, is also
(37), the expected value of over all is zero. The variance a function of these four quantities, viz. , input SNR, and .
of is the variance of , i.e., A discussion on the effects of these items on the probability of
the steering direction will be presented in the following sections.
(38)

The probability density function (pdf) of is given by VIII. EFFECT OF NOTCH POSITION ON THE
STEERING DIRECTION
Figs. 13 and 14 show the probability that sign
(39)
gives the correct steering direction as a function of notch posi-
tion when the frequencies of the input sinusoid are
Let and be the expected value and variance of and , respectively. The input SNR is 0 dB, and the pole ra-
at time , respectively. From (16), it is straightforward to show dius . The steering direction is correct if
that is positive when and is negative when
. A solid line and an array of symbols are plotted on both
(40) Figs. 13 and 14. The solid lines correspond to results predicted
and by (43). The arrays of symbols correspond to results obtained
from computer simulation runs. It can be seen from Figs. 13
(41)
and 14 that the predictions provided by (43) agree very closely
where is given in (20). The pdf of a sinusoid is not Gaussian. with computer simulation results.
Although is not Gaussian, in order to simplify theoretical The curves shown in Figs. 13 and 14 can be explained intu-
analysis, we will treat as Gaussian. Thus, the pdf of itively as follows. When the initial notch frequency is far from
can be expressed as the frequency of the sinusoid, the signal is swamped by noise.
The probability of obtaining a correct steering direction is low.
As the notch moves toward the sinusoid, the sinusoid is en-
(42) hanced by the pole, and the probability of producing a correct
steering direction increases. However, when the notch position
1318 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

0
Fig. 13. Probability of sign[x(n 1)e(n)] producing the correct estimate of 0
Fig. 15. Probability of sign[x(n 1)e(n)] producing the correct estimate of
the steer direction versus notch position plot. The input SNR = 0 dB, the pole the steer direction versus SNR plot. The notch position is 0:5 , the pole radius
radius r = 0:9, and the input sinusoid’s frequency is 0:25 . r = 0:9, and the input sinusoid’s frequency is 0:25 .

steering direction can also be obtained from (43)–(45). By fixing

the input sinusoid’s frequency , the notch position and the
pole radius , the probability of producing a correct steering di-
rection as a function of the input SNR can be predicted from
(43) using the results of (20), (26), (27), (38), (40), (44), and
(45). The results predicted using (43) are shown in Fig. 15 to-
gether with computer simulation results. In Fig. 15, the notch
position of the filter is fixed at , and the frequency of the
input sinusoid is , whereas the pole radius is selected to
be . The steering direction is correct if is
negative. It is obvious that when the input SNR is low, i.e., the
input signal is noisy, the steering direction becomes less accu-
rate as an indicator of the position of the frequency of the input
sinusoid. In Fig. 15, the result represented by each symbol is
obtained by averaging 100 independent simulation runs.

0
Fig. 14. Probability of sign[x(n 1)e(n)] producing the correct estimate of
X. EFFECT OF THE POLE RADIUS ON STEERING DIRECTION
the steer direction versus notch position plot. The input SNR = 0 dB, the pole The effect of the pole radius on the probability of the
radius r = 0:9, and the input sinusoid’s frequency is 0:8 .
steering direction can also be obtained from (43)–(45) for any
given input frequency , notch position , and input SNR.
is very near the frequency of the sinusoid, because of the par-
The results corresponding to notch position at , input
abolic nature of the performance surface of the objective func-
frequency , and input SNR of 0 dB are shown in
tion, the probability of producing a correct steering direction
Fig. 16.
becomes low again. This is because the gradient of the perfor-
It can be seen from Fig. 16 that the closer the poles of the
mance surface due to the signal approaches zero as the notch ap-
notch filter to the unit circle, the lower the probability that the
proaches its optimum position and, thus, is easily swamped by
steering direction points to the correct direction when .
noise. It is interesting to note from Figs. 13 and 14 that the prob-
This is because, when , the noise component near the
ability of producing a correct steering direction is always larger
pole is greatly enhanced if the pole is very close to the unit
than 0.5, i.e., the steering direction obtained from
circle. This implies that a smaller pole radius will result in a
is always better than a random guess. In Figs. 13 and 14, the re-
better estimate for the steering direction, which in turn will re-
sult represented by each symbol is obtained from averaging 100
sult in a faster convergence speed of the notch filter. However, it
independent simulation runs.
is known that a smaller pole radius will cause bad steady-state
performance [17]. In order to have a fast convergence and a
IX. EFFECT OF INPUT SNR ON STEERING DIRECTION good steady-state performance, a variable pole radius method
The influence of pole radius on the performance of notch fil- may be used, i.e., choose a smaller when the notch is far
ters has been discussed in [20]–[22]. The effect of the input SNR from the frequency of the input sinusoid to achieve a faster tran-
on the probability that the sign of yields the correct sient response and a larger after convergence to obtain a better
LIM et al.: PILOTED ADAPTIVE NOTCH FILTER 1319

Fig. 17. P 0P versus P plot for various values of t.

0
Fig. 16. Probability of sign[x(n 1)e(n)] producing the correct estimate of
the steer direction versus pole radius r plot. The notch position is 0:5 , the input
SNR = 0 dB, and the input sinusoid’s frequency is 0:25 . positions of the filter are far from the frequency of the input
sinusoid, . In order to simplify notation, we will
steady-state performance. The consistency between the results substitute into (46). Hence, we have
predicted by (43) and those obtained from simulation runs for
various are shown in Fig. 16. In Fig. 16, the result represented (47)
by each symbol is obtained by averaging 100 independent sim-
ulation runs. Thus

(48)
XI. JOINT STEERING DIRECTION ESTIMATION
From (48), it can be seen that for
In the previous sections, we have presented a discussion on is positive, implying that is always larger than , i.e., the
the probability that the sign of correctly predicting three notches together provide a more accurate estimate for the
the steering direction of an IIR notch filter when the input signal steering direction than does the main notch alone.
is a sinusoid contaminated with white Gaussian noise. In this
section, we show that the main notch and the pilot notches col-
B. Notch Positions Are in the Vicinities of the Maxima of the
lectively produce a better steering direction estimate than the
Curves in Figs. 13 and 14
main notch does alone.
Consider the piloted notch filter structure shown in Fig. 1. Let From Figs. 13 and 14, it can be seen that when the notch
the probability that the steering direction of the main notch, i.e., positions of the filter are in the vicinities near the two maxima
the sign of , giving a correct steering direction be of the curve, may be larger than both and , i.e.,
. Let the probabilities that the steering directions of the pilot , where is positive. To simplify notation, we will
notches, i.e., the signs of and , substitute into (46). This leads to
are correct be and , respectively. Since the notch filter is
steered in the direction to which at least two of the three notches (49)
are pointed, the probability of producing a correct steering di-
rection for a piloted notch filter is given by Thus

(50)

Since the positions of the pilot notches are close to the main
(46) notch, is small. Fig. 17 shows the curves of for
, and , respectively. It can be seen
from Fig. 17 that for , if .
It can also be seen from Figs. 13 and 14 that in the vicinities
A. Notch Positions Are Far From the Frequency of the Input
near the maxima of the curves, the values of are significantly
Sinusoid
larger than 0.6, and therefore, the three notches together provide
Since the positions of the pilot notches are close to the main a more accurate estimate for the steering direction than does the
notch, it can be seen from Figs. 13 and 14 that when the notch main notch alone.
1320 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

C. Notch Positions Are Very Close to the Frequency of the

Sinusoid
When the notch positions of the filter are very close to the
frequency of the input sinusoid, we have one of the following:
, , or
is less than both and but is larger than 0.5. From (46), we
have

(51)

If , then

(52)

If , then Fig. 18. Comparison between the convergent speeds for piloted and nonpiloted
least mean p-power adaptation algorithms. The SNR was 10 dB, and p = 2. The
weight mean square error of the piloted result was over an order of magnitude
smaller than that of the nonpiloted result.
(53)

If is less than both and , then both (50) and (53) are larger step size would be selected only if the pilots indicated
true. Since both and are less than unity and larger than 0.5, (52) that the frequency of the sinusoid was outside the notches of
and (55) are positive, and hence, . The above analysis the two pilots in three consecutive samples; this very greatly
shows that the pilot notches together with the main notch pro- reduced the chance of selecting the larger step size after
vide more accurate information for the steering direction than convergence had been established. The mean square weight
does the main notch alone. error for the nonpiloted notch was 0.001 26, and that for
the piloted notch was 0.000 09; this was about an order of
XII. IN CONJUNCTION WITH OTHER magnitude improvement in the mean square weight error! The
WEIGHT UPDATE EQUATIONS piloted notch converged (ignoring the overshoot) at a speed
that was about an order of magnitude faster than that of the
The piloted notch concept can be applied in conjunction with
nonpiloted notch, as can be seen from Fig. 18.
other weight update equations. Consider, for example, the least
Comparing the nonpiloted results of Figs. 6 and 18, it can be
mean -power method [29], where
seen that (3) yields a faster convergence compared with the least
mean -power method because (3) adjusts the weight
using data filtered by the pole; the pole enhances the sinusoid
for even (54a) and causes an improvement in the SNR, especially after conver-
sign gence. This translates into a faster convergence for a given mean
for odd (54b) square error.
When the SNR is very low, the pilots will be fooled by noise
The signs of , and more easily, particularly if the separation between frequencies
are used to determine the steer directions for the main and pilot of the pilots is very small. This problem may be reduced by
notches for selecting the step size . A simulation for placing the notches further apart. A simulation was done for the
was done to demonstrate the piloted notch technique when case with 0 dB SNR, and the results were shown in Fig. 19.
used in conjunction with (54). The step size for the nonpiloted The step size for the nonpiloted notch was selected to be 0.28
least mean -power filter was 0.085; this step size was selected to give a comparable mean square weight error as that for the
experimentally to produce a mean square weight error similar previous 10-dB SNR nonpiloted notch. The step sizes for the
to that of the conventional nonpiloted LMS algorithm shown in piloted notch were 0.28 10, 0.28, and . Let be the
Table I. The step sizes for the piloted least mean -power filter number of consecutive samples where the pilots indicated that
were and 0.085 10, respectively. The frequency of the frequency of the sinusoid was outside the frequencies of the
the input sinusoid, SNR value, and the values for and are pilots. The largest step size was selected only if . The
the same as those in Section V. middle step size was selected if . The smallest step
We will make use of this example to illustrate a further size was selected if . The values of and were
improvement to the piloted notch technique. Ideally, after the increased to in order to increase the separation between
notch has converged, the smaller step size should be selected. the notches. The mean square weight error for the nonpiloted
Fig. 7 shows that the larger step size was occasionally selected notch was 0.001 26, and that for the piloted notch was 0.000 25;
after convergence had been achieved. This was due to the this is about a factor-of-five improvement. The piloted notch
presence of noise that fooled the pilots. In the simulation converged at a speed about three times that of the nonpiloted
results shown in Fig. 18, we imposed the condition that the notch, as can be seen from Fig. 19.
LIM et al.: PILOTED ADAPTIVE NOTCH FILTER 1321

Fig. 19. Comparison between the convergent speeds for piloted and nonpiloted Fig. 20. Comparison between the convergent speeds for piloted and nonpiloted
least mean p-power adaptation algorithms. The SNR is 0 dB, and p = 2. The notch with gradient filtering. The SNR is 10 dB. The piloted notch converges at
weight mean square error of the piloted result is one fifth that of the nonpiloted a speed that is 50 times that of the nonpiloted notch. The weight mean square
result. error of the piloted notch result is slightly smaller than that of the nonpiloted
notch.
The origin, pole, and zero of some adaptive notch do not lie
on the same straight line. An example of this type of notch filter values for and are the same as those in Section V. The
is the gradient algorithm reported in [30]. The transfer function piloted notch was expected to converge 50 times as fast as
of the notch is given by the nonpiloted notch. The mean square weight error for the
piloted notch was 0.000 08, and that for the nonpiloted notch
(55) was 0.000 10. The results are shown in Fig. 20.
Fig. 21 shows the transient behavior of a piloted notch
when tracking a sinusoid whose frequency changed between
Note that (55) becomes (13) if is replaced by and one eighth sampling frequency and three eighth sampling fre-
is replaced by . The variable quency in a square wave manner. The input signal SNR was 30
at time is updated using the recursion dB. The weight update algorithm was the least mean -power
algorithm with . The piloted notch uses four pilots. The
(56)
outermost pilots have . The innermost pilots
The estimate is obtained from have . The value of was 0.9. The step size
for the nonpiloted algorithm was 0.01. Step-size selection for
the piloted notch was done as follows. If the outermost pilots
(57)
indicated that the frequency of the sinusoid was outside them
for three consecutive samples, the step size was 100 0.01. If
where is the -transform of , and is the -trans- the innermost pilots indicated that the frequency of the sinusoid
form transfer function of the partial gradient filter. can was outside them for three consecutive samples (but the outer
be any suitable fancy function. As our intention is only to il- most indicated otherwise), the step size was 10 0.01. If the
lustrate that the piloted technique is superior to the nonpiloted innermost pilots indicated that the frequency of the sinusoid
technique, the specific form for is immaterial. As an il- was within them for three consecutive samples, the step size
lustration, we chose was 0.8 0.01. The step size was 0.01 otherwise. The mean
square weight errors were 0.000 000 09 for both the piloted and
(58) nonpiloted algorithms. Fig. 21(a) shows a comparison between
the speed of convergence for both the piloted and nonpiloted
where notches. The range for is expanded in
Fig. 21(b) to show the transient behavior. Note that the transient
(59) behavior is determined by the weight update algorithm.
The input signal SNR was 10 dB. The frequency of the sinusoid
was the same as that used in the previous examples, and XIII. CONCLUSION
was 0.9. The step size for the nonpiloted algorithm was 0.001.
The step sizes for the piloted algorithm were 50 0.001 and In this paper, we proposed a piloted notch filter structure con-
0.9 0.001. The larger step size was selected only if the pilots sisting of a main notch and two pilot notches. We have shown
indicated that the frequency of the sinusoid was outside the that the piloted notch filter gives a more accurate estimate on the
notches of the two pilots in three consecutive samples. The steering direction of the notch filter. The pilot notches are very
1322 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005

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[29] S.-C. Pei and C.-C. Tseng, “Adaptive IIR notch filter based on least mean Yue-Xian Zou (S’96–M’01) received the M.Sc. de-
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N. Zheng, photograph and biography not available at time of publication.

Yong Ching Lim (S’80–M’80–SM’92–F’00) re-

ceived the A.C.G.I. and B.Sc. degrees in 1977 and
the D.I.C. and Ph.D. degrees in 1980, all in electrical
engineering, from Imperial College, University of
London, London, U.K.
From 1980 to 1982, he was a National Research
Council Research Associate with the Naval Postgrad-
uate School, Monterey, CA. From 1982 to 2003, he
was with the Department of Electrical Engineering,
National University of Singapore. Since 2003, he has
been with the School of Electrical and Electronic En-
ginerering, Nanyang Technological University, Singapore, where he is currently
a Professor. His research interests include digital signal processing and VLSI
circuits and systems design.
Dr. Lim received the 1996 IEEE Circuits and Systems Society’s
Guillemin-Cauer Award, the 1990 IREE (Australia) Norman Hayes Award,
1977 IEE (U.K.) Prize, and the 1974–1977 Siemens Memorial (Imperial
College) Award. He served as a lecturer for the IEEE Circuits and Systems
Society under the distinguished lecturer program from 2001 to 2002 and as an
associate editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from
1991 to 1993 and from 1999 to 2001. He also served as an Associate Editor for
Circuits, Systems, and Signal Processing from 1993 to 2000. He served as the
Chairman of the DSP Technical Committee of the IEEE Circuits and Systems
Society from 1998 to 2000. He served in the Technical Program Committee’s
DSP Track as the Chairman of ISCAS’97 and ISCAS’00 and as a Co-chairman
of ISCAS’99. He is a member of Eta Kappa Nu.