THE use of 3D sound technology is gaining ground on

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1 Estimation and Evaluation of Reduced Length Equalization Filters for Binaural Sound Reproduction Esben Theill Christiansen, Jakob Sandholt Klemmensen, Michael Mørkeberg Løngaa, Daniel Klokmose Nielsen, Christian Have Pedersen, Andreas Popp, and Søren Birk Sørensen Group 74, Aalborg University, 24 Abstract The objective of this study was to estimate two equalization filters, which flatten the frequency response of the reproduction chain used for binaural sound reproduction. These filters should be of a lower order than the reciprocal of the headphone Transfer Function (PTF) - the optimum filter. The PTFs were obtained from measurements on two headphones (Beyerdynamic DT99 Pro (DT99) and Monacor MD-3 (MD3)). These measurements were performed using the Maximum Length Sequence System Analyzer. Three different model structures and four different estimation methods were used to estimate the parameters of the equalization filters. The chosen models were: ARX, ARMAX and OE; the parameter estimation methods chosen were: PEM-LS, PEM-WLS, PEM-RLS and Steiglitz-McBride. The estimated filters for the DT99 were further evaluated by conducting a 3 Alternatives Forced Choice listening test. The four estimated equalization filters had orders that were significantly lower than the optimum equalization filter. For the DT99 the order was reduced to 24 and 4; for the MD3 the order was reduced to 35 and 48. Both filters for the DT99 were found using the ARMAX model and the PEM-RLS estimation method. The filters for the MD3 were found using the OE model and the Steiglitz-McBride estimation method. The listening test conducted showed no audible difference. Index Terms Headphone Transfer Function, Headphone Equalization, Binaural Sound Reproduction. I. INTRODUCTION THE use of 3D sound technology is gaining ground on both the consumer market and in the industry. Recently, the gaming industry has started to implement 3D sound via headphones in their newest computer games. This is used to provide an ability to locate events not only by the visual appearance but also by the sense of hearing. The main idea of 3D sound is to reproduce sound with respect to spatial features, which are information about the direction of sound created by the direct sound and its reflections from the surrounding environment. The spatial features are extracted by the human brain by using several clues e.g. coloration of sound and differences between the two ears. These differences can occur in time, phase, and amplitude []. In order to reproduce the spatial features, recordings must be performed with two microphones placed in the ears of a person or on an artificial head. Reproducing these sounds correctly at the eardrum of a listener is referred to as Binaural Technique. Optimum reproduction of spatial features is established with a flat frequency response, measured for the whole transmission chain. The transmission chain is depicted in figure and consists of the two recording microphones, an amplifier, an equalizer, and the headphone Transfer Function (PTF). The PTF is the transfer function measured from the terminals on the headphone to the entrance of the blocked ear canal. The transfer function (TF) is primarily influenced by the anatomical shape of the listeners ear and the electro-acoustical properties of the headphone. Amplifier Equalizer PTF Fig.. The transmission chain in a binaural reproduction system. The reproduction system consists of the amplifier, the equalizer, and the PTF and is also referred to as the reproduction chain. The equalizer is intended to compensate for the influences of the transmission chain, so that a flat frequency response is obtained. This is accomplished by determining the TF for the transmission chain. The TF is then inverted, and in this study it is referred to as the optimum equalization filter. In the transmission chain, it can be assumed that the only variable part is the PTF. The microphones are only used during the recording and will therefore be the same every time. The amplifier is not necessarily the same during recording and playback but is assumed to be flat in the audible frequency range. In addition, Møller et al. [2] states that individual equalization for each test person gives the best performance of binaural reproduction. However, it is noted that equalization based on an average of the PTF for each headphone may give acceptable results. Toft et al. [3] supports this approach by confirming, that it is possible to achieve good results with an average equalization of each headphone model. Toft et al. also concludes that it is not possible to create an applicable general equalization filter that covers different headphone models. The optimum equalizing filter in this study is therefore based on the average PTF of a headphone model. Equalization filters can be implemented between e.g. an amplifier and the headphone. This filter ensures a flat frequency response for the reproduction chain. This knowledge can be used in general synthesis of binaural signals without prior information about the audio playback equipment used. Implementation of equalization filters can be done digitally and it is desirable to construct this filter with a filter order that is as low as possible due to computational limitations. Reduction of filter order (i.e. length of digital filters) is investigated by Nielsen et al. [4], and it is concluded that it is possible to design lower order filters with satisfactory performance. However, it is not investigated whether there is

2 an audible difference between the optimum equalization filter and a lower order implementable filter. The purpose of this study is to investigate if there is an audible difference between the optimum equalizing filter of the PTF and two lower order filters. These two filters are chosen among several candidates found by parametric estimation methods with regard to minimizing the order of the filters. These filters will be experimentally evaluated by conducting listening tests on third party subjects. Monacor MD-3 Beyerdynamic DT99 Behringer HA44 Headphone Amplifier Sennheiser KE Microphone Power supply and Preamplifier II. METHODS This section will describe the steps followed throughout the study, starting with the measurements of the PTFs, followed by the preprocessing of the data. Subsequently, a model is inferred and the parameters are estimated. Finally the listening test is described. This test verifies if audible differences are present or not. A. Acquisition of data Preliminary measurements were carried out on 5 group members (aged 23-28). The PTFs were not directly measured but were derived by preprocessing the measured impulse responses. The impulse responses were measured on the blocked ear canal by using the Maximum-Length Sequence System Analyser (MLSSA), which for this purpose had the following settings: Sampling frequency: f s = 48 khz Antialiasing filter: 8 th Order Chebyshev, f c = 2 khz Sequence length: 495 samples 6 x concurrent preaveraging The selected preaveraging improved the signal-to-noise ratio by 2 db [5]. Measuring impulse responses with the MLSSA system required additional equipment; the setup is shown in figure 2. The measurement setup was placed so a minimum distance of m from the headphone to the floor, ceiling, table etc. was assured. The minimum distance implied that reflections from the surroundings did not occur before 6 ms had passed. The measurements took place in room B4-7 at the Department of Acoustics at Aalborg University. The microphone used for the measurements was a Sennheiser KE electret microphone (assumed flat in the interval Hz - khz [2]). This microphone was mounted in an earplug from EAR and placed in line with the entrance of the ear canal of the subject. Two different headphones were used for the measurements: ) Beyerdynamic DT99 Professional (DT99) with a frequency range from 5 Hz - 35 khz, reported by the manufacturer. 2) Monacor MD-3 (MD3) with a frequency range from 2 Hz - 8 khz, reported by the manufacturer. The headphones were set in place by the test person, measurements were repeated three times, and the headphones were taken off and put back on again between each measurement. Both channels on the headphone were measured, but with the DT99 the microphone was only placed in the left ear. The measurement on the right side was then carried out by MLSSA System Brüel & Kjaer Measuring amp. Type 2636 Clock generator 48 khz Fig. 2. Measurement setup for measurements of the impulse response from the input terminals of the headphone to the blocked ear canal. The microphone was mounted in an earplug and placed in the ear of the test subject. putting the right cup on the left ear. This simplification of the measurements was made, because it was assumed, that the ears were totally symmetrical. Putting the right cup on the left ear was not possible with the MD3; therefore, the microphone was placed in the right ear for measurements of the right canal of the MD3. After the measurements were performed, it was necessary to preprocess the data to find the averaged PTF. B. Preprocessing Preprocessing was accomplished in several steps in order to find the optimum equalization filter. Before transforming the impulse responses into the frequency domain the responses were truncated to 256 samples, corresponding to the first 5.33 ms of the impulse response. This truncation gave a frequency resolution of 87.5 Hz and could be performed, assuming that the remaining part of the impulse response was due to reverberations from the surroundings. The truncated response was transformed into the frequency domain by a 248-point Fast Fourier Transform and formed the PTF. This frequency spectrum contained components from the MLSSA system and the microphone amplifier. These components were therefore removed from the PTF by finding the TF for the MLSSA system and the amplifier, and following dividing the PTF with this TF. The microphone was assumed flat, hence the PTF was only divided by the sensitivity of the microphone. Averaging of the PTFs was attained over both channels on a sound level basis [2]. Averaging over both channels are valid due to the symmetrical properties of the ear and the headphone.

3 TABLE I MODEL STRUCTURES AND THEIR CORRESPONDING POLYNOMIALS. IN THE RIGHTMOST COLUMN THE EVALUATED PARAMETRIC ESTIMATION METHODS ARE PRESENTED FOR EACH MODEL. Structure Description A(q) B(q) C(q) F(q) Estimation method(s) ARX AutoRegressive with exogenous input PEM-LS, PEM-WLS ARMAX AutoRegressive Moving Average with exogenous Input PEM-RLS OE Output Error STMCB, PEM-RLS The optimum equalization filter was found by inverting the averaged PTF. Before inverting the PTF it was separated into a minimum phase part and an all-pass part. The allpass part was excluded as proposed by Minnaar [6]. The TF of the equalization filter was then found by inverting the minimum phase part of the averaged PTF, this assured that the equalization filter was stable [7]. The PTF is naturally damped at low and high frequencies, thus the equalization filter will have the opposite effect and amplify signals at these frequencies. The equalization filter was therefore bandwidth limited with a Butterworth bandpass filter in order to protect the headphone. C. Parametric models The obtained optimum equalization filter will be of high order. The primary aim of this study was to estimate reduced length versions of this filter. A model based approach was chosen for this purpose. Thus the optimum equalization filter was fitted to a set of parametric models using several parameter estimation methods. The general model structure is [8]: A(q)y(t) = B(q) u(t) + C(q)e(t), () F(q) where y(t) is the output, u(t) is the input, and e(t) is the noise in the system represented as a zero-mean Gaussian process. A(q), B(q), C(q), and F(q) are all polynomials of q, where q is a time shift operator. These polynomials are A(q) B(q) C(q) = + a q a na q na = + b q b nb q n b = + c q c nc q nc F(q) = + f q f nf q n f, where n a, n b, n c, and n f are the lengths of A(q), B(q), C(q), and F(q) respectively. From the general model structure three combinations were chosen; table I lists these combinations. The ARX was chosen due to its linear properties; hence, a simple linear regression could be obtained. The ARX model is described as, A(q)y(t) = B(q)u(t) + e(t), (2) The ARMAX model was selected for its additional modeling of the noise, A(q)y(t) = B(q)u(t) + C(q)e(t). (3) Finally the OE model was chosen. It separately models the dynamics of the system without using parameters on the noise model, y(t) = B(q) u(t) + e(t). (4) F(q) The ARMAX and OE model cause nonlinearity in the polynomial coefficients, which gives a more complicated nonlinear regression. However, the ability to handle fluctuations in the frequency response is improved. D. Parameter Estimation The polynomials of the selected models were estimated by the following set of estimation methods [8]: Prediction Error Method (PEM)-Least Squares (PEM-LS) PEM-Weighted Least Squares (PEM-WLS) PEM-Recursive Least Squares (PEM-RLS) Steiglitz-McBride Method (STMCB) The PEM was the general principle in the estimation of all models. PEM consists of an estimation by minimizing the sum of squared prediction errors, denoted as the performance function: V N (θ) = N (y(t) ŷ(t, θ)) 2, (5) 2N t= where θ is a parameter vector that comprises the polynomial coefficients, ŷ(t, θ) is the one-step predictor, and N is the sample size. Finding the parameters by minimization of (5), with respect to θ is the principle of the Prediction Error Method, θ o = arg min θ V (θ, Z N ), (6) where θ o is the optimum parameter vector for the data set, Z N. The PEM-LS was used for the ARX model. The minimizing problem in (6) could be solved analytically by linear regression. Hence the solution for PEM-LS was given as: [ N ] N ˆθ LS = ϕ(t)ϕ(t) T ϕ(t)y(t), (7) t= t= where ˆθ LS is the parameter vector, and ϕ(t) is the associated data vector. The format of the parameter vector, ˆθ LS, is, ˆθ LS = [â,..., â na,ˆb,...,ˆb nb ] T, and the format of the data vector, ϕ(t), is, ϕ(t) = [ y(t ),..., y(t n A ), u(t ),...,u(t n B )] T.

4 The PEM-WLS method was also used for the ARX model. The method was similar to PEM-LS, but introduced time weighting of the prediction errors. It was then investigated if time weighting performed better than the PEM-LS method. PEM-WLS has the solution, [ N ] N ˆθ WLS = β(n, t)ϕ(t)ϕ T (t) β(n, t)ϕ(t)y(t), (8) t= t= where β is the weighting function, implemented as β(n, t) = λ N t, with < λ <. It is seen that the PEM-LS solution is found when λ =. The PEM-RLS method was used for the ARMAX and the OE models. Since these models have nonlinear properties, a numerical approach to the minimization problem was chosen. The minimization method was implemented as a quasi-newton algorithm that was obtained by using the Levenberg-Marquardt search direction [9]. Hence, the parameter vector was estimated recursively as: ˆθ (k+) RLS = (k) ˆθ RLS R (k) V (ˆθ RLS θ ) (9) R = 2 (k) 2 V (ˆθ RLS θ ) + δi, where k denotes the k th iteration, R the search direction and δ a scalar adjusted iteratively. The initial values of ˆθ were found using a covariance method, the Prony estimate []. The STMCB method was evaluated as an alternative parameter estimation method for the OE model. This method approximates the OE model to an ARX model in order to find the polynomials of the OE model. The STMCB method was initialized by estimating F(q) with a Prony estimate []. The estimated polynomials were denoted ˆF (i) (q) and ˆB (i) (q) respectively. The STMCB method was implemented in the following steps: : Prefilter the data with ˆF (i) (q), y F (t) = ˆF (i) (q) y(t) u F(t) = 2: By linear regression (7) solve: ˆF (i) (q) u(t). ˆF (i+) (q)y F (t) = ˆB (i+) (q)u F (t) + e(t) where i denotes the i th iteration. Step and 2 were repeated until the parameters converged. If convergence was not obtained after a maximum number of iterations the parameters were rejected. As the PEM-RLS requires computations of the derivatives, the STMCB is a more computationally efficient method to estimate the parameters of the OE model. ) Selection of equalization filters: Two equalization filters were selected among the investigated models and methods. The investigated models were applied to the optimum equalization filter, and an iterative search included all combinations of polynomial orders up to 65 for each model and estimation method. The search resulted in several candidates for equalization filters. Among these candidates, the frequency deviation from the optimum frequency response, was the main selection criteria. The frequency deviation was calculated by: () Ĥ(f), where Ĥ(f) was the estimate of the optimum equalizing filter H(f). A frequency deviation of maximum ± db from a flat frequency response is not audible, according to Moore []. This limit was applied to the first of the two filters, and the lowest order filter that satisfied the stated criteria, within the frequency band from 5 Hz to 2 khz, was selected for the listening test. A lower filter order of the estimated filter was expected when increasing the maximum allowed deviation. Therefore, a different deviation limit of ±2 db was chosen for the second filter. However, this filter should not result in audible differences either, also stated by Moore. Hence, the filters were selected from the following criteria: DT99: MD3: Filter N(f) db, 5 Hz f 2 khz () Filter 2 N(f) 2 db, 5 Hz f 2 khz, (2) Filter 3 N(f) db, 5 Hz f 2 khz (3) Filter 4 N(f) 2 db, 5 Hz f 2 khz, (4) and were evaluated in the listening test. E. Listening test The listening test was conducted to investigate if there was an audible difference between the optimum equalizing filter and two lower order filters. The filters were only tested for the DT99. The test was carried out on 6 subjects, 5 males and female (aged 23-3). All the test subjects were chosen among students. To ensure that the test subjects had a normal hearing an audiometry test was made. Test subjects with a hearing loss greater than 2 db HL (Hearing Level) were rejected []. The room used for the test was a small cabin designed primarily for listening experiments. The test took place in room B5-4 at the Department of Acoustics at Aalborg University. The equipment used for the test was: Audiometer - Madsen Electronics Orbiter 922, Version 2 4-channel Headphone Amplifier - Behringer HA 44 Cd-player - Marantz Compact Disc player CD-32 Headphone - Beyerdynamic DT 99 Professional The used test method was a 3 Alternatives Forced Choice (3AFC) test. The test subject was presented with 3 stimuli - 2 were identical and was different. The test subject should then choose the stimulus which differed from the others. The used stimuli were generated from 2 sound sources played from 3 different directions giving a total of 6 sounds. The directions were artificially created by convolution of the sound sources with the Head Related Transfer Functions (HRTF) for the given directions. The HRTF s were provided by the Department of Acoustics at Aalborg University. The 6 sounds are listed below. Pink noise { 44,, 44 } azimuth - elevation Music { 44,, 44 } azimuth - elevation When one sound was presented a total of 24 stimuli were given resulting in 8 answers from the test subject. Six of these answers were from the comparison between the optimum

5 Fig. 3. Averaged PTF for MD-3 db re. Pa/V db re. Pa/V Averaged PTF for DT Average across the PTFs for the DT99 and MD3. The grey lines represent the individual PTFs. The black lines represent the average PTF. equalizing filter and the lower order filter; two were from the comparison between the optimum equalizing filter and a low pass filtered version of the sound. The low pass filtered sounds were added to keep the test subject motivated. Two seperate tests were conducted, one for each of the lower order filters. For each test a total of 288 answers were given. Of these, 72 were not included in the test statistics; this was done to ensure that the answers given for the lowpass filtered sounds had no influence on the test outcome. III. R ESULTS In this section the obtained results will be presented. First the obtained PTFs are presented, then the parameter models and the estimation methods are examined. Finally, the results of the listening test are presented. A. Headphone transfer functions (PTFs) The obtained PTFs and the averaged PTFs for DT99 and MD3 are shown in figure 3. The figure shows that the individual PTFs follow the average PTF for the lower frequencies up to approximately 7 khz for the DT99 and 6 khz for the MD3. For higher frequencies, the individual PTFs vary significantly. This corresponds to the measurements performed by Møller et al. [2]. B. Parameter estimation results The estimated parameter models were examined for both headphones. This was done in order to find filters that fulfilled the conditions previously listed in () to (4). ) DT99: The parameter estimation for DT99 with the ARX model did not produce any results with a frequency deviation less than ± db. Neither did any of the estimation methods for the OE model and therefore these models were rejected. Only the ARMAX model using the PEM-RLS method produced candidates with a frequency deviation less than ± db. For a frequency deviation less than ±2 db all the estimation methods produced valid results. The lowest order TABLE II L OWEST FILTER ORDER FOUND FOR DT99 Filter Model 2 ARMAX ARMAX Estimation method PEM-RLS PEM-RLS Order [na nb nc ] [6 24 4] [2 22 ] Error ± db ±2 db filter was found using the PEM-RLS method on the ARMAX model. The order of the selected filters are listed. in table II The first filter selected for DT99 is compared to the optimum equalization filter in figure 4. In figure 4(a) a frequency and phase deviation plot is depicted. Figure 4(b) shows the impulse response plot for both Filter and the optimum equalization filter for DT99. This plot shows no visible difference between the two responses. The comparison of Filter 2 and the optimum equalization filter is shown in figure 5. In figure 5(a) a frequency and phase deviation plot is depicted. It is seen that even though the frequency deviation limit was set to ±2 db, the lowest order filter only had a maximum deviation of ±.7 db. Figure 5(b) shows that deviations in the impulse response appeared between Filter 2 and the optimum equalization filter. The deviations were present after approximately 25 samples had passed and were caused by the frequency deviations in the lower frequencies. 2) MD3: The examination of the estimation methods gave considerably different results for the MD3 headphone. The results showed, that only the OE model produced valid TABLE III L OWEST FILTER ORDER FOUND FOR MD3 Filter Model 3 4 OE OE Estimation method STMCB STMCB Order [nb nf ] [5 33] [7 8] Error ± db ±2 db

6 Amplitude [db] 5 DT99 Error plot: ARMAX estimated with PEM-RLS, order [6 24 4] Ĥ(f) db Margin DT99: ARMAX estimated with PEM-RLS, order [6 24 4] h(n) ĥ(n) Angle [deg.] N(f) = H(f) Ĥ(f) Amplitude [.] Sample (a) Frequency deviation and phase error plots for Filter (b) Impulse response of Filter and the optimum filter response Fig. 4. Filter : Equalization filter for DT99 estimated with an ARMAX model with a PEM-RLS method. The filter order was [6 24 4]. Amplitude [db] DT99 Error plot: ARMAX estimated with PEM-RLS, order [2 22 ] Ĥ(f) 2 db Margin. N(f) = H(f) Ĥ(f) Amplitude [.] DT99: ARMAX estimated with PEM-RLS, order [2 22 ] h(n) ĥ(n) Angle [deg.] Sample (a) Frequency deviation and phase error plots for Filter 2 (b) Impulse response of Filter 2 and the optimum filter response Fig. 5. Filter 2: Equalization filter for DT99 estimated with an ARMAX model with a PEM-RLS method. The filter order was [2 22 ]. estimation results for Filter 3 and Filter 4. Therefore, the ARX and ARMAX models were rejected. In table III. the order of the selected filters for the MD3 are listed. The optimum equalization filter is compared to Filter 3 in figure 6. In figure 6(a) a frequency and phase deviation plot is depicted. As seen in figure 6(b) the estimated impulse response followed the inverse impulse response of the PTFs. The last filter selected for the MD3 is shown in figure 7. In figure 7(a) a frequency and phase deviation plot is depicted. Figure 7(a) shows the difference between the optimum equalization filter and Filter 4. It is seen that even though the frequency deviation limit was set to ±2 db, the lowest order filter only had a maximum deviation of about ±.8 db. The estimated impulse response of Filter 4 is shown in figure 7(b). C. Listening test The results of the listening test for both filters are shown in table IV. The results showed no audible differences from the optimum filters with a confidence level of 95%. TABLE IV RESULTS OF THE LISTENING TEST. FILTERS VS. CORRECT ANSWERS Filter Correct answers [%] Confidence interval [%] 3.6 [27.;39.6] [27.;39.6] IV. DISCUSSION We have investigated the possibility of reducing the lengths of equalization filters for binaural reproduction. The opti-

7 Amplitude [db] Angle [deg.] MD3 Error plot: OE estimated with STMCB, order [5 33] Ĥ(f) db Margin. N(f) = H(f) Ĥ(f) Amplitude [.] MD3: OE estimated with STMCB, order [5 33] h(n) ĥ(n) Sample (a) Frequency deviation and phase error plots for Filter 3 (b) Impulse response of Filter 3 and the optimum filter response Fig. 6. Filter 3: Equalization filter for MD3 estimated with an OE model with a STMCB method. The filter order was [5 33]. Amplitude [db] Angle [deg.] MD3 Error plot: OE estimated with STCMB, order [7 8] Ĥ(f) 2 db Margin. N(f) = H(f) Ĥ(f) Amplitude [.] MD3: OE estimated with STCMB, order [7 8] h(n) ĥ(n) Sample (a) Frequency deviation and phase error plots for Filter 4 (b) Impulse response of Filter 4 and the optimum filter response Fig. 7. Filter 4: Equalization filter for MD3 estimated with an OE model with a STMCB method. The filter order was [7 8]. mum equalization filters for DT99 and MD3 have been successfully reduced applying common parametric estimation techniques. The obtained filters were of order 24 to 48. For each headphone the two lowest order filters with an estimation error less than ± db and less than ±2 db respectively were chosen. The selected filters for the DT99 were compared with the optimum equalization filter in a 3AFC test. No audible differences were observed by the six test subjects. In general the human ear cannot detect relative frequency deviations within ± db. This research has shown that deviations less than ±.7 db in this case were not audible either as stated by Moore []. The impulse response of the optimum equalizing filter was used to estimate the parameters. This approach seems to favor higher frequency components, and therefore resulted in a higher rejection ratio than expected. A better pre-filtering in relation to the parameter estimation is expected to result in an even lower filter order. The investigation and research presented in this article reinforces indications made by Møller et al. [2], where it is noted that equalization based on average PTFs for each headphone may give acceptable results. Toft et al. [3] concludes that it is possible to achieve good results with an average equalization of each headphone, which is supported by this article as well. Nielsen et al. [4] also estimated lower order filters. Our research verifies the applicability of such a reduced order filter through listening tests. An investigation of an additional reduction in filter order

8 seems possible since verification of both filters indicated no audible difference. This will ultimately lead to the threshold value of audibility. Further verification of the obtained filters with respect to determining the directional deviations is suggested. REFERENCES [] H. Møller, Fundamentals of binaural technology, Applied Acoustics, vol. 36, pp. 7 28, 992. [2] H. Møller, D. Hammershøj, C. B. Jensen, and M. F. Sørensen, Transfer characteristics of headphones measured on human ears, J. Audio Eng. Soc., vol. 43, no. 4, pp , April 995. [3] J. C. Toft, S. Pedersen, A. Kristensen, and T. Sørensen, Equalization of headphones for use with 3d sound, 5th SEMCON 22, 22. [4] H. A. Nielsen, K. R. Pedersen, R. M. M. Petersen, K. H. Sørensen, and M. Sørensen, Evaluation and comparison of methods for real time equalization of stereo headphones for binaural sound reproduction, 6th SEMCON, 9th december 23, 23. [5] D. D. Rife and J. Vanderkooy, Transfer-function measurement with maximum-length sequences, J. Audio Eng. Soc., vol. 37, no. 6, pp , June 989. [6] P. Minnaar, Simulating an acoustical enviroment with binaural technology, Ph.D. thesis, 2. [7] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, st ed. Prentice Hall, 989. [8] L. Ljung, System Identification. Theory for the user, 2nd ed. Prentice Hall, 999. [9] W. Murray, Numerical Methods For Unconstrained Optimization, st ed. Academic Press, 972. [] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing, st ed. Prentice Hall, 992. [] B. C. J. Moore, An introduction to the psychology of hearing, 4th ed. Academic press, 997.