Biol. Cybernetics 29, 97 104 (1978) Biological Cybernetics 9 by Springer-Verlag 1978

On the Choice of Noise for the Analysis of the Peripheral Auditory System

C. Swerup Department of Environmental Medicine, Karolinska Institute, Stockholm, Sweden

Abstract. The cross-correlation between output and input of a system containing nonlinearities, when that system is stimulated with Gaussian white noise, is a good estimate of the linear properties of the system. In practice, however, when sequences of pseudonoise are used, great errors may be introduced in the estimate of the linear part depending on the properties of the noise. This consideration assumes special importance in the analysis of the linear properties of the peripheral auditory system, where the rectifying properties of the haircells constitute a second order nonlinearity. To explore this problem, a simple model has been de- signed, consisting of a second order nonlinearity with- out memory and sandwiched between two bandpass filters. Different types of pseudonoise are used as input whereupon it is shown that noise based on binary m- sequences, which is commonly used in noise gene- rators, will yield totally incorrect information about this system. Somewhat better results are achieved with other types of noise. By using inverse-repeat sequences the results are greatly improved. Furthermore, certain anomalies obtained in the analysis of responses from single fibers in the auditory nerve are viewed in the light of the present results. The theoretical analysis of these anomalies reveals some information about the organization of the peripheral auditory system. For example, the possibility of the existence of a second bandpass filter in the auditory periphery seems to be excluded.

Introduction

Today, much effort is being devoted to analyzing nonlinear physiological systems. Noise has become a very popular stimulus for the study of these systems. Methods for analyzing nonlinear systems have been developed to an unsatisfactory degree until recent

years when great advances have been made. The methods of Lee and Schetzen (1961) designed to obtain the Wiener-kernels by cross-correlation are one exam- ple of methods which have successfully been applied to a number of physiological systems (for a review, see McCann and Marmarelis, 1975).

Many of the relations in noise theory are based on the assumption that the input is Gaussian white noise (GWN). However, when analyzing physiological sys- tems in practice, GWN has never been used. One of the reasons is that the long duration of the GWN signal required to obtain a good statistical accuracy would call for enormous computational resources. One way to use GWN and at the same time maintain com- putation needs at a realistic level might be to use the reverse correlation method (de Boer and Kuyper, 1968; de Boer, 1973). However, one mostly uses se- quences of pseudonoise, which more or less resemble GWN. Instead of stimulating the system with one long period of noise, a short sequence of pseudonoise is repeated again and again, and the response is averaged (see e.g. Moller, 1974; O'Leary and Honrubia, 1975).

The theoretically derived formulas that are valid for GWN can not of course be applied to pseudonoise without further notice. Here is given one simple exam- ple of the consequences. It has been shown that the cross-correlation between output and input of a system stimulated with Gaussian signals does not change in shape when an amplitude distortion is introduced. This finding is the basis for the theorem of Bussgang (Bussgang, 1952), which has also been borne out in another way by Price (1958). See also de Boer (1976).

A simple system containing a delay circuit of 1 ms and an amplitude distorting nonlinearity of the form y= u + 2u 2, see Figure 1, was stimulated with a noise sequence based on binary m-sequences, which will be

Abbreviations : GWN = Gaussian white noise ; FMN = Filtered noise based on binary m-sequences; GPN=Gaussian pseudonoise; BEN = Binary equirandom noise; BPNL = Bandpass nonlinear

0340-1200/78/0029/0097/$01.60

98

x(t)) ~ y(t)>

Fig. 1. A simple linear system followed by an amplitude distortion

b

I I I I I I 1 2 3 4 5 6 MSEC

Fig. 2. a Shows the expected cross-correlation for the system in Figure 1. b Shows the cross-correlation, when the system was tested with noise based on binary m-sequences (FMN)

x(t)> ~ y(t)>

Fig. 3. The BPNL-model. A zero-memory nonlinearity (ZNL) is sandwiched between two bandpass filters (BP1 and BP2)

described below (FMN). The cross-correlation was computed and is shown in Figure 2b. The expected cross-correlation for GWN with the same bandwidth as input would of course be a peak at 1 ms delay, see Figure 2a, as the nonlinearity should not change the shape of the cross-correlation according to the theo- rem of Bussgang. However, the curve obtained shows quite other features, and were it to be interpreted without care, a totally erroneous idea about the nature of the system under test would arise.

The extra peaks seen in Figure 2b depend on the noise itself and have no correspondence in the system. It is evident that in a more complex system these peaks might be filtered to smooth curves, which might lead to serious misinterpretations of the properties of the sys- tem. Thus even the linear analysis of a system contain- ing nonlinearities must be made with great care.

The BPNL-Model

A simple model of the peripheral auditory system consists of a bandpass filter followed by a rectifier, a

lowpass filter and a triggering device. The bandpass filter corresponds to the frequency selectivity of the basilar membrane, the rectifier to the haircells and the lowpass filter to the smoothing of the neural excitatory process (Weiss, 1966). In this model the second filter is thus a lowpass filter instead of another bandpass filter as proposed by others. Pfeiffer (1970) suggested that a bandpass filter as the second filter would explain two- tone inhibition of cochlear nerve fibers. The theory of the second bandpass filter has been further developed by Evans (1975). Nonlinearities may be assumed to occur at different levels of the peripheral auditory system. By proposing that the system consists of a chain of alternating linear and no-memory nonlinear networks, analytical expressions for the identification of the different elements of this chain can be derived for stimulation with GWN (Korenberg, 1973). For this reason, it seems attractive to apply the BPNL-model to the peripheral auditory system.

This model consists of a zero-memory nonlinearity sandwiched between two bandpass filters, see Figure 3. With a zero-memory nonlinearity is meant a network for which the output can be expressed as a power series of the input

v(t) = a o + a 1 u(t) + azU(t) 2 + a3u(t) 3 + a4u(t) 4 4- . . . . (1)

Zero memory means that there is no delay in the system, i.e. the amplitude distortion is instantaneous.

When applying the BPNL-model to the peripheral auditory system, the even order terms of (1) cannot be negligible, because of the rectifying properties of the haircells. Though these even order terms are not interesting in the linear analysis of this system, they will nevertheless complicate it, as will be seen in the subsequent discussion.

The theoretical formula for the cross-correlation between output and input for a BPNL-system contain- ing nonlinearities of the second order is shown in (2). The only assumption made for the input x(t) in the derivation of this formula is that it has zero mean value. For proof, see the appendix.

r:,y(z) = a 1 ~ hl(v)h2(~)rxx(a + v - z)dvd~

-t- a 2 f f f hl(v)hl(#)h2(~)

rxxx(~ + v - ~, ~r + # - z)dvdl~&r. (2)

Equation (2) consists of two terms. The first term contains r~x(Z), which is the first order autocorrelation of the input signal. The second term contains r~x(a, "c), which is the second order autocorrelation of the input. The autocorrelation computed from the product of two x-factors is called the first order. The next one, computed from three x-factors, is called the second order etc. The nomenclature in this field is very varying.

Up to now, no assumptions have been made about the properties of the input signal except zero mean value. If x(t) is GWN with unity power spectral density, the first order autocorrelation is equal to a delta function and all high, even order autocorre- lations, become identical to zero, i.e.

rxx('c) = 6(~) and rx~x(~, ~) = 0. (3)

Equation (2) is then reduced to

r xy('c ) = a 1 ~ hz(o)ha (z - a)d~ (4)

which is simply the convolution of the two filters. De Boer has recently pointed out that in the ana-

lysis of a BPNL-system the cross-correlation between output and input will reflect the linear properties, if GWN is used as input (de Boer, 1976). This means that the cross-correlation should depend only on the first term of (2). If the system is tested with pseudonoise, however, the terms following the first term in (2) may cause disturbances. As shown by Gardiner (1966) it is possible to determine the linear impulse response of a compound system containing a no-memory nonli- nearity by performing different tests on the system whereby binary noise with different amplitude factors is used. The linear term is then arrived at via a system of equations. Even so, it seems quite unsatisfactory to test a physiological system with different inputs, since problems with biological stability etc. will always arise. The ideal must be to have a perfect stimulus signal and to be able to average the output as long as needed for statistical reasons and as long as the biological system permits.

As the stimulus will always be some sort of pseudonoise, the theoretical expressions should be derived without assuming any properties of the signal. Once the theoretical expressions have been derived, an assessment must be made of how the different proper- ties of the signal, e.g. aberrations from GWN, will influence the analySis.

Description of the Model Studied

To elucidate the difficulties incurred in analyzing the linear part of a BPNL-system containing second order nonlinearities, a simple system was constructed and tested with different types of pseudonoise. The results were then used in part to explain certain anomalies obtained in the analysis of responses from single auditory nerve fibers.

To make the system more similar to a physiological system, the signal was first delayed i ms, which does not affect the general results. For the sake of simplicity the two bandpass filters were chosen to be identical, with a center frequency of 6.25kHz and frequency slopes of 6 dB/octave (Fig. 4). The nonlinearity chosen

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xCt)> ~ yCt)>

Fig. 4. The system under test

was v=uq-4u 2, i.e. ao=0, a l= l and a2=4 in (1). Setting a o different from 0 will simply shift the DC- level of the output and is of no special interest in this case. a 2 was set at 4 in order to provide a visualization of the disturbances resulting from the second term in (2). The second term will be greatly affected by the amplitude of the noise. To render the tests for the different types of noise comparable, all noise sequences generated had zero mean value and were normalized so that they had a unity power spectral density. This procedure also implies that the computed cross- correlations are estimates of the first order Wiener- kernels for this system according to Lee and Schetzen (1961). Furthermore the noise sequences formed were set to have about the same frequency content, well above the limits of the system analyzed. The time interval for sampling was always the same, 20 ~ts, and the stimulus was 680 or 681 steps long, which cor- responds to a duration of 13.6 ms (except the ternary signal, which, because of its type of generation, had 728 steps and thus a duration of 14.6 ms). The parameters of the noise were thus chosen in such a way as to ensure good statistical estimates of the cross- correlation according to the ordinary rules for analyz- ing linear systems (see e.g. Bendat and Piersol, 1971).

The simulations were run on a NORD-10 com- puter. The circular cross-correlations were computed without performing any smoothing, which might ob- scure the results. Figure 5a shows the theoretical im- pulse response of the system, i.e. the computed con- volution of the two identical filters, which was then delayed 1 ms. The cross-correlation of the system for the different types of noise is shown in Figures 5b~t and 6 and comments on the results are presented in the following sections.

Results for Different Types of Noise

1. Filtered Noise Based on Binary m-Sequences (FMN)

A sequence of FMN, the type of noise commonly used in noise generators, was generated in a Hewlett Packard noise generator (Type 3722 A). FMN consists of a lowpass-filtered binary signal of the m-sequence type which is generated by feedback and modulo-two addition of certain stages in a shift register to form the next bit generated. Thus the binary sequence will consist of O's and l's and, after a certain number of steps, the signal will repeat itself. The binary sequence consisted of 4095 steps, each 3.33gs long, giving a duration of 13.6ms for each period. After lowpass

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t 1

,.ii '/vl'v'lvtlr ,l ,lv ,tl,Fvlv

Fig. 5. a Shows the theoretical impulse response for the system in Figure 4. b--d Shows the cross-correlation, when the system was tested with the following types of noise : b noise based on binary m- sequences (FMN), e Gaussian pseudonoise (GPN), d binary equiran- dora noise (BEN)

0

I

;Vvvv,llVvv'vvvvFvvwvv'vv'vvv'vvv~vU'vv"v'v',V'v'

01 ~'-Alr"'~vv'v,,vv,'v~ Vvv'V~Yvv~{,, ''~ vvv",

I

F i L ;MsEc

Fig. 6a--d. The cross-correlation for the system in Figure 4 tested with inverse-repeat sequences. The inverse-repeat sequences were formed from the following types of noise : a noise based on binary m- sequences (FMN), b Gaussian pseudonoise (GPN), c binary equiran- dom noise (BEN). In d a ternary m-sequence was used

filtering the signal had an upper cutoff frequency of 15 kHz. This signal was then sampled in 681 steps with an interval of 20 gs.

From the cross-correlation in Figure 5b it is seen that there are additional peaks that result from the noise itself. These peaks depend on the generation of the binary m-sequence and will not decrease if the stimulus length is increased. They arise because of the anomalies of the binary- m-sequences in their second order autocorrelation. The first theoretical work on these sequences was by Zierler (1959), who investigated their basic properties. The high order autocorrelation functions of the binary m-sequences, especially the properties of the third order autocorrelation, which has special importance in the evaluation of the second

order Wiener-kernels of a system, have been the subject of further investigation (Barker and Pradisthayon, 1970). Even the second order autocor- relation shows serious anomalies, however. The second order autocorrelation for this signal was computed, and as expected, showed sharp peaks in the two- dimensional plane of analysis.

For GWN the second order autocorrelation is identical to zero. For a pseudonoise sequence the second order autocorrelation should vary randomly because of the increased statistical variance caused by the finite sequence length. The sharp peaks that are obtained for the m-sequence are caused by the type of generation of the signal. The anomalies will resist filtering and no simple relation for the high order

properties of a filtered signal can be derived. Thus the properties of each signal have to be analyzed by computing the high order autocorrelation functions.

2. Gaussian Pseudonoise (GPN)

In order to get a better signal and to keep the different elements totally independent statistically, each step in the signal was generated by a random number gene- rator and transformed into a Gaussian process. The cross-correlation function in Figure5c, however, shows that no information about the system is won. This is of course the result of the effort to generate GWN with such a small number of degrees of freedom. Granted, signals of this type will approximate true GWN if the length of the signal is increased, but some errors will always arise due to the amplitude trun- cation that must be carried out for practical reasons (Marmarelis, 1972). Thus when short signals are being dealt with, it is futile to generate them in this manner.

3. Binary Equirandom Noise (BEN)

Binary equirandom noise is the simplest type of the so- called constant-switching-pace symmetric random sig- nals introduced by Marmarelis (1975). Such signals remain constant during a short time interval. With every step, they attain statistically independent values according to a symmetric probability density function. Marmarelis has shown that the symmetry of this function renders all even order autocorrelations identi- cal with zero. It is obvious that this will not be exactly true for signals of finite length since they are governed not by a random but by a pseudorandom process.

BEN is a signal that has the value + 1 or - 1, each having a probability of 1/2. The binary type of sti- mulus has several advantages for computational pur- poses, as the two levels can represent "on" and "off' in a logical circuit, and simple equipment can be used to construct an automatic cross-correlator. While com- mercially available cross-correlators are able to set the input signal at two levels, they can only do so through amplitude distortion of the signal, a disadvantage avoided through automatic cross-correlation with BEN. This must be a considerable advantage, as amplitude distortion of non-Gaussian signals may lead to errors as shown above.

In the present study the BEN was generated with the same random number generator as was used for the GPN above. For positive values of the random number, the signal was set at + 1 and for negative values at - 1. The signal, which is shown in Figure 7a, can be viewed as an amplitude distorted version of the GPN. The resolution in the cross-correlation estimate of the system when tested with BEN shows a great increase compared to what was the case with GPN (Fig. 5d). This depends on the diminished power of the second order autocorrelation.

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j HUIEINI[Ulil NIUIIIIi]HImil Ilil lnlU i _,IilItlIIHIIUHIIUIUNIUIIIIHIIIIlIIIlIIIIIll IHI IIUIIIINIIUI

b

J N I IIII IIJ I III 1 II IIll IIII ill| NIl 1 II -'t HH llll INIIIJHI Hill] IH[I Nil 11 !1 Hi HUIIIt

/ I I I I q I 0 1 2 3 4 5 6 MSEC

Fig. 7a and b. The first part of two of the noise signals used. a Shows the binary equirandom noise (BEN). b Shows the ternary m- sequence. Note the different amplitudes necessary to obtain unity power spectral density

4. Introducin 9 the Inverse-Repeat Sequences

As seen above, errors are obtained when FMN, GPN or BEN are used for the analysis of this system because the second order autocorrelation is not equal to zero as is the case for GWN. It is evident that noise based on binary m-sequences is useless even for the linear analysis of a nonlinear system containing non- negligible second order nonlinearities. Likewise, it seems fruitless to try to generate GWN for such short sequences as those used here. BEN did not seem entirely useless in this context, however.

In order to obtain signals with good statistical properties up to the second order, another class of signals was formed, namely the inverse-repeat se- quences. For this purpose, each of the signals described above was changed in the following manner. The second half of the signal was set to be equal to the first half but with the opposite sign. Because of this asym- metry, all even order autocorrelations will be identical to zero, regardless of the stimulus length, cf. the above discussion concerning BEN. This principle has earlier been applied to the binary m-sequences in order to improve their high order characteristics (Ream, 1970). By inversion of alternate bits in the generation, these sequences were given the same property, namely, all even order autocorrelation functions were rendered identical with zero. The deterministic nature of these signals, however, will still result in anomalies of the high order odd autocorrelations (Simpson, 1966). These anomalies cannot be eliminated by filtering or any other method.

By forming the inverse-repeat sequences of the signals used here the following advantages are gained :

1. There will be no problems with the even order autocorrelations, as these are identical to zero. This is a great advantage in the linear analysis of a system like this, containing as it does even order nonlinearities.

2. All efforts can be devoted to obtaining good properties of the first, third, fifth order etc. The de- mands made upon the odd order autocorrelations

i

102

b

0 6 10 MSEC

Fig. 8a and b. The cross-correlation between the response and stimulus for a single auditory nerve fiber. The stimulus used was FMN. In a the analysis was made for one polarity of the stimulus. In b the analysis was performed on the difference between two measure- ments in which the input had different polarities (published with the permission of Aage Moller)

depend on the degree of nonlinearity of the system being analyzed and on the order to which the kernel- functions are to be computed.

The disadvantage will be an increased variance in the linear analysis caused by the decreased number of degrees of freedom, as the 680 points will no longer be statistically independent.

Each signal has to be tested in detail for its properties. The first order autocorrelation function of an inverse-repeat signal will contain a positive peak for zero delay and the same peak with the reverse sign for timeshift T/2, where T is the length of the signal. The settling time of the system analyzed in this case must obviously be less than T/2. This is also true in general, since the length of the stimulus should be chosen to be a factor 10 or so above the settling time of the system in order to obtain a good statistical estimate of the cross-correlation (see e.g. Bendat and Piersol, 1971).

The inverse-repeat sequences were formed from all sequences above, meaning that the first 340 values were repeated with the opposite signs, thus forming new sequences 680 steps in length. The cross- correlation of the system when it was tested with these signals is seen in Figure 6a-c. The improvement in resolution is evident. What is won is that the second term of (2) is identical to zero, and the price paid is that the first term will undergo an increased variance. To what degree this can be tolerated must be examined in each case, e.g. by tests on simple models like the present one.

5. Ternary m-Sequences The perfect noise up to the second order is obtained by using ternary m-sequences. These are generated in a way similar to the binary m-sequences. In this case the feedback signals are added modulo-three to form the next bit in a shift register. The signal will thus contain three levels, e.g. + 1, 0 and -1 . Due to the type of generation, the signal will be of the inverse-repeat type, and thus all even order autocorrelations are zero. The first order autocorrelation is perfect with a positive delta function at zero delay and a negative at T/2 and zero elsewhere, meaning that the first and second order properties will be identical to those of GWN. However, the third order autocorrelation suffers anomalies as shown by Gyftopoulos and Hooper (Gyftopoulos and Hooper, 1964; Hooper and Gyftopoulos, 1967).

To test the present system a ternary signal was generated with 728 steps, i.e. slightly longer than the 680 or 681 steps above. The signal is shown in Figure 7b. In the generation of the signal feedback was taken from stages 5 and 6 according to Godfrey (1966). Figure 6d shows the cross-correlation obtained when the ternary signal was used as input. As expected from the properties of the signal, the curve obtained is identical to the theoretical curve in Figure 5a.

Application of Present Results to Observations from Auditory Measurements

The difficulties in analyzing the linear part of a BPNL- system containing second order nonlinearities are clear from the examples presented in the foregoing sections. The anomalies induced by using noise based on binary m-sequences are exemplified in Figure 8a. The figure illustrates the cross-correlation between the period histogram and the stimulus, when measuring from a single auditory nerve fiber in the anesthetized rat. The stimulus here was the same sequence as the FMN described above. For details on that experiment, see Moller (1977). The rectification performed by the haircells causes a second order term to be introduced. When the BPNL-model is used, this means that the factor a 2 is great enough, or alternatively, that the noise power is strong enough, to allow the second term of (2) to affect the analysis.

As geen from Figure 8a, besides the typical oscil- lations in the cross-correlation due to bandpass filter- ing, a slowly oscillating component is also present, which is not negligible. This component is caused by the noise itself and corresponds to the second term of (2). The following observations may serve to prove that this component thus has nothing to do with the linear properties of the system analyzed:

1. The oscillations in the cross-correlation re- gularly appear in these recordings and always have a similar appearance.

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2. The polarity of the oscillations is reversed when the polarity of the stimulus is reversed. This is due to the fact that the second term in (2) contains three x- factors, while the first term contains only two. Thus the oscillations change sign but the linear part of the cross- correlation will not change.

3. The same type of oscillations can be obtained using a very simple model consisting of a bandpass filter followed by a second order nonlinearity of the type described above and a lowpass filter. This model was tested with the FMN-sequence whereupon similar oscillations appeared (Fig. 9a).

To eliminate these anomalies, Moller (1977) came upon the ingenious idea of stimulating the system with reverse polarity, whereafter he analyzed the difference between the two responses (Fig. 8b). This shows the cross-correlation using half of the difference of the two responses, which should correspond to the first term of (2). Working with the difference in this manner will cause all anomalies depending on the even order terms of the nonlinearity to vanish from the result. What will remain will be a cross-correlation depending entirely on the odd order terms of the nonlinearity. Gardiner's method, described above, would yield a similar result.

However, that all information about the even order terms should be taken away is not entirely satisfactory for the further investigation of the different com- ponents of the peripheral auditory system. Moreover, it would be more desirable from the physiological point of view to obtain the same information in one single measurement. Figure 9b shows what happened, when the model was stimulated with an inverse-repeat sequence formed from FMN. Figure 9c shows the result of using the ternary m-sequence as a stimulus, which is the same as the theoretical impulse response of this second order model.

It is worth noting that these slow oscillations could not be synthesized when the second filter was a bandpass filter. To obtain the oscillations the filter had to transmit low frequencies. Linear analysis with GWN as input does not give any information at all about the distribution of the filtering between the two filters, but instead will always measure the convolution of the two filters, see (4). Thanks to the second order anomalies of the FMN, information about the second filter can be obtained, as the analysis will be sensitive to the squaring action taking place between the two filters. It seems clear from this simple analysis that the second filter could not be a bandpass filter but has to be a lowpass filter. This type of analysis of the second filter is not as advanced as a second order Wiener- kernel analysis of the peripheral auditory system, which could reveal the distribution of the filtering between the two filters (Korenberg, 1973). For an analysis of the latter type, recordings must be made with a different type of noise.

1-

I I 5 Ill MSEC

Fig. 9a--e. Analysis of a simple model to demonstrate the same anomalies in the cross-correlation as in Figure 8 caused by noise based on binary m-sequences (FMN). In a FMN was used, in b inverse-repeat FMN, and in e a ternary m-sequence

Discussion

As has been shown in this paper, even the linear analysis of a BPNL-system might lead to considerably different results depending on what type of stimulus is used. The linear analysis of a system containing simple nonlinearities might contain large errors caused by the noise itself. Great care has to be taken in interpreting the results of such an analysis. The theoretical relations valid for GWN cannot be applied to pseudonoise without further notice.

Noise based on binary m-sequences, commonly used to test various systems, is not suited even for the linear analysis of systems containing second order nonlinearities. As the anomalies induced by this type of noise will depend on the power of the noise, different transfer functions for different levels of the input

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signal are obtained. This might lead to erroneous conclusions about nonlinearities in the system.

The use of ternary m-sequences as stimulus seems attractive from the mathematical point of view. However, it might seem wrong from a physiological standpoint to stimulate a system with a signal contain- ing only three levels. To render the signal more physiological, these sequences can be filtered. The properties of the filtered versions must be examined in each individual case, as no general formulas can be derived. One way to obtain the high order properties of a signal is through Laguerre-filtering as outlined by Simpson (1966).

For the linear analysis of the peripheral auditory system, the inverse-repeat sequences seem to offer the best solution for eliminating the distorting action of the haircell rectification. Further investigations are needed in order to obtain information as to how the high order statistical properties of the different types of noise affect the second order kernel analysis of systems of this type. Such information will be of great impor- tance in the efforts to define the characteristics of the different components of the peripheral auditory sys- tem, e.g. to solve the controversial problem of the second filter.

Appendix

Derivation of the theoretical formulas for the cross-correlation between output and input of a second order BPNL-system. For definitions see Figure 3. The bandpass filters BP 1 and BP2 have the impulse responses hl(z ) and h2(z), respectively. It is assumed that the input x(t) has zero mean value.

Input-output relations for the different steps in this system:

u(t) = ~ h 1 (v)x(t - v)dv

v(t) = ~ alu(t) i = a o + al u(t ) + a2u(t) 2

y(t) = ~ h2(a)v ( t - a)da.

The relation between output y(t) and input x(t) is easily derived.

y(t) = aoSh2(a)d~ + a 1 ~Sh~(v)h2(a)x(t - a - v)dvda

+ a2~hl (v )h l (#)h2(a)x ( t - a - v )x ( t - a - #)dvd#da.

After some simple computations the expression for the cross- correlation rxy(z ) between output and input becomes :

r~y(z) = y(t )x(t - - z) = a 1 ~ h 1 (v)h2(a)r~(a + v - z)dvda

+ a 2 ~h l (v )h l (p )hE(c r ) r~(a + v - z, a + # - z )dvd#d~.

Acknowledgements . I am greatly indebted to Docent Aage Moller for having introduced me to the field of noise analysis of physiological systems and for having put his registrations at my disposal. The valuable assistance of Pamela Boston and Irene Unander-Scharin is greatly acknowledged. I want to thank Dr. Hans Nilsson for valuable criticism during the preparation of the manuscript. Computer time was made available partly by a grant from the Swedish Natural Science Research Council and partly by the Karolinska Institute.

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Received: February 2, 1978

Dr. C. Swerup Dept. of Environmental Medicine, Karolinska Inst. S-10401 Stockholm, Sweden