A NONLINEAR ADAPTIVE NOISE CANCELLER

Michael J. Cokerand Donald N. Simkins

ESL Incorporated

Sunnyvale. California S[t2]

ABSTRACT where

The purpose of this paper is to describe a nonlinear system structure that can be used to perform adaptive interference cancellation. It is shown that a nonlinear extension of the conventional tapped delay line filter is amenable to adaptation with the LBS algorithm, and can be used to represent a class of nonlinear systems. Experimental results are presented illustrating cases where nonlinear interference cancelling is superior to conventional linear cancelling.

The LBS algorithm (1) is a gradient d9scent method that iteratively seeks the minimum value of Eft 1 by adjusting the at each sample time k. It is elements of the weight vector clear from equation (I) that the mean output power is a quadratic function of N, and therefore has a global mnimum value and no local minima. The quadratic nature of SEt which is a result of the filter structure, is the key to the success of the LBS algorithm. filter structure (including nonThus,2for any linear) for which E[t ] quadratic in the adaptive parameters, LBS may be used to adapt the filter.

]

It can be seen from Figure 1 that the effectiveness with which the linear adaptive filter can estimate the additive interference n0 from n is determined by how "close' to linear is the relationship between the two. If n0 can not be approximated closely by any linear transformation of n1, then cancellation will be poor. This may be the case for a number of reasons. For example, be a noise-corrupted transformation of n0. 01 may Or the relationship between n0 and n1 may be linear but not invertible; that is, n may have been obtained from n0 by a transformation hat is not reversible, such as a lowpams linear

INTRODUCTION

Sn conventional interference cancelling, the estimation process is linear. This paper describes a nonlinear interference canceller that offers the potential for improved performance over the linear canceller. The nonlinear canceller is related in structure to the linear one, and includes it as a

filter. Another possibility is that n1 may be a nonliner transforIf this were the case, then it appears that the n0. adaptive canceller potentially could work better if it were possible for the adapative filter to be nonlinear as well.

mation of

special case. Review of Adaptive Interference Cancelljqg

In its simplest form, the adaptive interference canceller has two inputs (Figure 1). One, called the primary, contains the signal of interest s(k) plus interference n (k). The other input, called the reference, contains some (genrally unkooum) transformation of n (k); call it n1(k). It is assumed that a is uncorrelated wits both n and n1, and that all signals are wide-sense stationary. The reference is used as the input to a linear filter, typically a tapped delay line with variable weights. The output of the canceller, t(k), is the difference between the filter output y(k) and the primary input d(k). This difference signal is used with an adaptive algorithm, such as the LBS algorithm, to adjust the weights of the adaptive filter so as to minimize the amount of n(k) present in t(k), thereby maximizing the ratio of signal power to noise power at

NONLINEAR INTERFERENCE CANCELLATION

As an illustration of a suitable nonlinear filter structure, let us retain the finite memory restriction of the tapped delay line filter, but remove the constraint of linearity. Thus, the output of the optimum finite—memory filter is y(k) = where is as defined previously, and f is a (generally non11k linear) scaler function of The Taylor by (2) f(X)

the output. Because the only correlation existing among the inputs to the canceller is between n1 and n/ the mean power of the

t

is

Bk2] = E[s2]+

E[(n-y)2]

(4)

Ito m.{I Xr

=

(0) + +

The output t(k) may be written as (2)

where W(k) is the vector of tap weights at time k, and 11(k) is the vector of delayed input values in the delay line at time k. Thus the mean output power also is given by

470

CHI559—4!80!0000—0470$00.75© 1980 IEEE

=

(1)

Thus, to maximize the outpt signal-to-noise ratio, the canceller must minimize is equivalent to minimizing E[(n0-y)2], which the total output power E[t ].

= d(k) - WT(k)x(k)

series expansion for f(X) about the vector 0 is given

using differential operator notation. may be written f(X)

t(k)

(5)

denotes expected value.

is

The generalization of the conventional linear structure to the nonlinear one described appears to provide a significant performance advantage in certain cases. At the same time, the complicated analyses usually associated with optimal nonlinear systems and the requirement to measure higher—order statistical parameters are avoided by virtue of the LBS algorithm.

output

e)d2] - 2pT ÷ WTRW P = E[dX],R = E[WWT], and E[] =

n 'c-' r=1

r

s'0

n 0 's-' 4._i i=r 3=1

rl

O

xr +

s'

By

expending (4), this

0

E E w(2?1 xri r=i x.

i=r

13 r,i,3. xrxx.

(5) +

where

Ii =

f(0),

2

W1 r =

This problem was simulated by computer using both a conventional linear canceller and a nonlinear canceller. The "unknown" memoryless nonlinearity was chosen to be a sine function. The interference consisted of uncorrelated samples with a uniform density from -e/2 to o/2, so that the invertible range of the sine function was not exceeded. Both the linear and nonlinear filters were provided with six coefficients; the linear filter consisted of a five—tap delay line filter plus bias weight, and the nonlinear filter was described by the memoryless relationship

Psr X=0 r=i

Ox2

r

—

(2)

Wri

22f(X)

,

Ox.

ri

etc.

5

y(k)

W1 W1 1 2

2)

wj If we

Wl2

E W'(k)

x'(k)

i=0

Both cancellers were adapted over 135,000 samples. Additionally, the optimal solutions and residual errors for both cancellers were determined analytically. Table 1 shows the resulting weight vectors and residual interference powers. The interference reduction of the nonlinear canceller is superior to that of the linear canceller by 13 dB.

If we now assume that the desired function f(X) may be approximated sufficiently closely by truncatingequation (5) to p terms(w may rearrange the elements of the weighting structures [W ], m0,l,...p-l to form a "super" weight vector = wT —

=

W(l) ... 3

(2) Wl,2

n

I

Table 1. w(2) n,n

Weight Vectors and Residual Noise Fowers for the Nemoryless NonlinearityBxperiment

likewise form a "super" argument vector

aC4

xT =

x2 203 T

[l:x1

...

X1X2 x1x3

Lfl0

Linear

..1

3 .

L55

Xl containingall possible cross products of to order p-l, .n up = wTx = y f(X)

i=1,2,.. f(X)

Neulunear

E[r2J

=

BEd2)

—30.5

-.2ss2

0

0

0

.s;ss .00022

—35.0

The electrical signal generated by the heart, the eleclrocardiograph signal (BcG), frequently contaminates other biological signals of interest. For example, in fetal electrocardiography, the BCG of the mother is unavoidableand much stronger than that of the fetus. Adaptive filters have been used successfully both to cancel the maternal BCG from the fetal signal, and to enhance the fetal BCG after cancellation of the maternal signal (3).

E[(d-y)2] (7)

P = EjdX]and I = 6ixxTi. This shows (assuming constancy for all moments of the elements of N up to order p-l) that the expected value of the squared error is a quadretic function of

The primary channel is obtained from a lead placed on the body directly over the source of the signal of interest. This lead generally receives a strong interfering FOG signal. The reference channel may be obtainedby placing a lead at any point on the body that has a strong BcG and is also far enough away from the area of the desired signal to prevent its reception.

For this reason, the LNS algorithm can be

= 51(k) ÷ 2p

-55.4

.0757 .6996 .ssso -.2864 .0039 .0194 .00029 0

0

-07.2

.00402

.0006

0

essue,,

cancellation of BCG From Biological Signals

(6)

where

W(ktl)

.187 .0030 .0030 .0010 .0093 0064

then we may write

-

the elements of used to adapt !'

Theereejial

0

sesiduas

.ffj_0

and

=

Theeretical

.0456

wa

w

t(k) x(k)

Thus under the above assumptions and for proper choice of the adaptive constant p, the nonlinear canceller will converge to a solution providing the maximum output signal-to-noise ratio possible with the given filter structure. Note that the conventional linear canceller with a bias weight amounts to implementing only the first two terms of equation (5); (0) is the bias weight and are the weights of the tapped delay line. Clearly, the implementation of (6) and (B) requires many more weights than the conventional adaptive filter, for any terms of equation (5) beyond the first two. In order to reduce the amount of computationnecessary, the experimental results described in the sequel were obtained using a modified version of equation (6). This canceller implements

only linear terms and quadratic terms of the form .n (i.e., no cross terms).

,

il,2,

AFFLICATIONEXANFLES The Nemoryless Nonlinearity As a simple but illustrative example of nonlinear interference cancelling, consider the following problem. Let the primary channel contain the signal of interest with additive white noise. Assume that the only reference available consists of the interference waveform distorted by an unknown memoryless For simplicity, assume that both the primary and nonlinearity. the reference channels contain no other noise signals. Thus, the optimum solution for the adaptive filter is the exact inverse of the unknown nonlinearity.

471

The EOn signal consists of isolated pulses that appear simultaneously in the primary and reference channels. In each channel, the pulses are very similar over time. Nowever, at different sensor locations, the pulses have different shapes, due to varyingprojections of the cardiac electric vector (4). In order to cancel the BOG in the primary, the adaptive filter must transform the pulse shape of the reference BOG to that of the primary. Analysis of recorded BOG waveforms has shown that often the reference BOG pulses are shaped such that very little energy is present in some spectral region where the pulses in the primary have significant energy. Because of the presence of "muscle noise", the linear canceller has very little information at these frequencies with which to reconstruct the pulse shape of the primary BOG. Therefore, cancellation will be poorer at these frequencies than at frequencies where significant reference BOG energy exists. The inability of the conventional canceller to completelyremove the BOG may be considered to be an example of the reference being a linearly non-invertible (near-singularity followed by additive noise) transformation of the pulses in the primary. In general, a nonlinear filter can not undo such a transformation; but pulsed interference represents a special case, for which it is possible for the nonlinear canceller to perform better than the linear canceller. The reason for this can be explained by the models of Figure 2. As in Figure 2a, each of the primary and reference pulse trains may be considered to be the response of a linear filter to a train of impulses (noise has been ignored). The modified nonlinear canceller described above also squares the reference signal. This squared signal is also a pulse train, with the pulses arriving at the

same times as for the other two signals, and may be regarded as being generated in the same way (Figure 2b). It is unlikely that both the reference signal and its square will have spectral nulls at the same frequency. From this point of view, the modified nonlinear canceller looks much like a multiple reference linear canceller (1), and in general will show better performance than

a

It is felt that there are important unanswered questions regarding the convergence characteristics of the nonlinear adaptive filter. Although the filters used for this study were very well-behaved, it is conceivable that problems could arise when using a high—ordernonlinear filter with signals whose statistics change appreciably with time.

single reference canceller.

An obvious disadvantage of the nonlinear filter in its most general Taylor series form is the increased computation requirement. However, it seems likely that specific applications may be addressed using variationsof the general structure, or basis functions other than powers of the input, that require less computation.

Figure 3 shows an example of an BCG signal before and after cancellation using both a conventional linear adaptive canceller and the modified nonlinear adaptive canceller. The linear adaptive filter consisted of 200 weights plus a bias weight. The nonlinear filter had an additional 200 'quadratic" weights. Signals were lowpass filtered at 500 Hz and sampled at 1200 Hz. The reference signal is shown in Figure 4. Note that the P and T waves of the ECG in Figure 3 are nearly absent from the reference, thus preventingthem from being cancelled significantly. Both channels also contain 60 Hz interference (and harmonics thereof) and low-level muscle noise. Both filters were adapted over 200,000 samples (about 167 sec real time). The nonlinear canceller shows improved reduction of the BCG over that of the linear canceller.

NBFBRBNCBS (1)

12, Decemher 1976. (2)

B. Deutsch,

Bnglewood Cliffa, New Jersey

Prentice-Hall, 1965.

If we plot the magnitude OFT's of the primary, the linear residual, and the non—linear residual together (Figure 5) we see that the three spectra closely coincide at frequencies where little ECG energy is present; thus, neither canceller has a significant effect on the primary signal at these frequencies. However, the spectrum of the nonlinear residual shows considerably less energy than the other two spectra in the region of 50 Hz. Furthermore, cancellation of the 50 Hz interference is better for the nonlinear canceller.

(3)

B. Ferrara, "The Time-Sequenced Adaptive Filter," Information Systems Laboratory, Stanford University, 1978 (Ph.D. dissertation).

(4)

The reason that the pulse was not entirely cancelled is revealed by means of a spectral analysis of the primary and reference signals. Figure 6 shows the magnitude OFT's of the reference and primary pulses shown in Figures 3 and 4. The spectra are typical of those observed in the recordings made for this study. Note that the spectrum of the reference pulse rolls off rapidly from about 40 Hz to about 55 Hz, forming a deep null in the spectrum around 50-55 Hz. The spectrum of the primary pulse rolls off less rapidly, and does not have a deep null in this region. Thus, that portion of the primary pulse in the vicinity of 50 Hz can not be cancelled by the linear canceller. Indeed, the magnitude OFT of the linear residual (Figure 5) shows that most of the umcancelled energy is in the region of 50 Hz. Figure 6 also shows the magnitude spectrum of the square of the reference pulse of Figure 4. Nnle that the squared reference has relatively more energy in the vicinity of 50 Hz than the reference itself. Simulated BCG Cancellation

A more quantitative measure of the nonlinear canceller performance relative to the linear canceller was obtained by using simulated BCG signals for both the primary and reference inputs. Figure 7 is a block diagram of the simulation. The primary channel is the sum of a periodic sequence of triangle pulses, y(k), and a uniformly distributed random sequence x(k). y(k) is 10 dB weaker in power than x(k). The reference channel is formed by passing y(k) through a nolch filter and then adding white noise, n(k). n(k) is uncorrelated with y(k) and is 25 dB below y(k) in power. In order to cancel y(k) from the primary channel, the adaptive filter must recreate y(k) from w(k) , the output of the notch filter. Since some spectral components of y(k) are not present in w(k), this inversion cannot be done via linear filtering. Figure B shows y(k) and the outputs of the linear and nonlinear filters. The performance for each canceller was computed by measuring the excess poer above the minimum possible power tevel, B (x(k) , where n'(k) is n(k) after passing through the adaptive filter. The linear canceller was capable of suppressing the interference by only 6.4 dB, whereas the nonlinear canceller provided 22.3 dB of interference suppression.

n'(k)(

5UNI55fly

B. Nidrow, et al, "Adaptive Noise Cancelling Principles and Applications," Proceedings of the IBEB, vol. 63, No.

AND CONCLUSIONS

A nonlinear filter structure that is amenable to adaptation using LNS has been described. The nonlinear filter is a natural extension of the linear tapped delay line structure, and is based on a Taylor series representation of a nonlinear system. Bxamples are given where the nonlinear adaptive canceller shows performance superior to that of the conventional linear canceller. It is apparent that there are many potential applications of nonlinear adaptive filters in areas other than noise cancelling, such as nonlinear system modelling.

472

N.B. Nomble, et al, "Data Compression for Storing and Transmitting BCO's/VCD's," Proceedings of the IBBB, vol. 65, No. 5, Nay 1977.

OigOal Model

0)0)

+

Abortive IteFnFOerO marcher —I

1 0(k) a03(k(

:

yrimxry

I

3(0)

0 N

C

20

60

40

Q

100

120

FREQUENCF

lOAd

Figure

5.

340

lAd

Energy Spectraof the Signals Of Figure3. The Too Residuals Otffor Significantly Only Between40 end 40 Hr

Figure 2. a) Signal todel for the Modified NonlinearTaneehler. hI Equivaleot eodel

1.5 0.5 0

O.5 a

p

e

1 1.5

as

- ae

so PfloaEwrF em 100

sOs

two

isa

Spectraof the Pris000. ccferenoe(Figures 3 and 4), aod the Squored Figure 6. Energy Referenoe (not nheon). Note the Null in the Rnfarenoe Spectrum at About 50 Ho. The Squared Referenoe Has Relatively Moreanergyin tie engionOf 40—60 Ha Than pons the Reference Itself

2 2.5 3

di.dJ_

1' F 0.5' 0'

uato'e-

0

LAL 200

400

600

600

1000

SAFIPLEH

Figure

of Primary, Linear Residual, and NonlinearResidual Signals. The Figure 3. Example 000linear Cunrellar ShowsSuperiorROE Cancellation

Diagrom of the adoptiveoco Caaoehlang Eopericent trainS Ninrnlated Wavefarron

7.

2

1.5

0.5 p

0

10

ifD 1.5 2.5 200

400

600

9#IIPLES

600

1000

Figure 4. Reference Corresponding to Figure3. This Signal Cnetaion the ECu and 60 Ba Signal (and Harmonica) Interferenoo. Note the obnenoe of Signif5000t P

Figure

8

0) SinulatedECG Ieterferenoe. A) Oatpat of linear Adaptive Falter After l,ORE.000 Adopts. o) Ontpot of Nenhinear adaptive Falter After 750.000 Adapts

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