An Echo Canceller with Frequency Dependent NLP Attenuation

Transcription

1 Examensarbete MEE 98-5 ER/B/D-98:32 n Echo Canceller with Frequency Dependent NLP ttenuation Niklas Nilsson June post filter attenuation 5 ttenuation (db) Master Thesis work at Ericsson Radio Systems B Supervisor Examiner Lars Karlström Jörgen Nordberg B/D - Speech Processing Products Department of Signal Processing Ericsson Radio Systems B, Kista, Sweden at the University of Karlskrona/Ronneby

2 Uppgjord tjst, namn - Prepared dept, name KI/ER/B/D Niklas Nilsson Godkänd tjst, namnteckning - pproved dept, signature KI/ER/B/DC Information copy Kontr - Checked INTERNL INFORMTION To KI/ER B (52) ER/B/D-98:32 Tillhör/Referens - File/Reference bstract In this thesis, three different post-filtering algorithms for acoustic residual echo attenuation have been studied and simulated to see if the algorithms work in case of network echoes. The simulations were carried out using real recorded speech signals. The results are presented by different plots. The post-filter was implemented in the frequency domain due to the lower computation complexity. The main drawback of this is that an additional time delay will occur, when using block wise calculation. Finally, results of the post-filter and those obtained with the Non Linear Processor (NLP) were compared to see if the NLP can be replaced by the post-filter or not. The second algorithm, the overweighted wiener filter yields a very high attenuation of the Near End (NE) signal if the overweight is too great. This means that some of the NE frequencies is represented in the Far End (FE) spectrum. The third algorithm gives the lowest attenuation in case of no NE noise and modulates the NE noise mostly. The first algorithm gives the best overall performance. The simulation results shows that the NE and FE signals must be fully separated in frequency to make the post-filter working at optimum. s the post-filter modulates the NE noise, comfort noise has been injected and tested with simulations using the third algorithm. The estimation and injection of comfort noise works best when the NE noise is of a fairly stationary nature. s the main requirement is to remove the entire residual echo, the NLP can not be replaced with the post filter as it leaves some residual echo behind.

3 INTERNL INFORMTION (52) ER/B/D-98:32 TBLE OF CONTENTS bstract bbreviations 3 INTRODUCTION 4 2 BCKGROUND 5 2. Echo origins Echo cancellers The modern echo canceller The adaptive linear filter The double talk detector The non linear processor The comfort noise generator 9 3 PURPOSE 4 POST-FILTER DESIGN 5 IMPLEMENTTION ND COMPUTTION OF FILTER- COEFFICIENTS 3 6 DESCRIPTION OF THE POST-FILTER LGORITHMS 5 6. First algorithm, the Wiener post-filter Second algorithm, the overweighted wiener post-filter Third algorithm, the post-filter using noise reduction techniques 6 7 SIMULTIONS 8 7. Simulations of the three algorithms The Wiener post-filter Results and discussions The overweighted Wiener post-filter Results and discussions Post-filter using noise reduction techniques Results and discussions Over-all discussions 36 8 COMFORT NOISE INJECTION Simulation examples Results and comments 4 9 COMPRISON OF THE NLP ND THE POST-FILTER Simulation examples Results and comments 42 CONCLUSIONS 45 SUGGESTIONS FOR FUTURE WORK 46 2 REFERENCES 47 ppendix 48

4 INTERNL INFORMTION (52) ER/B/D-98:32 bbreviations NLP GSM FE NE FIR LMS NLMS DT DTD CNG DFT FFT IFFT STFT FRF PSD CSD SNR SER PSTN NMT R VD PEF MSE MMSE Non Linear Processor Global System for Mobile communications Far end Near end Finite Impulse Response Least Mean Square Normalized Least Mean Square Double Talk Double Talk Detector Comfort Noise Generator Discrete Fourier Transform Fast Fourier Transform Inverse Fast Fourier Transform Short Term Fourier Transform Frequency Response function Power spectral Density Cross spectral Density Signal to Noise Ratio Signal to Echo Ratio Public Switched Telephony Network Nordic Mobile Telephony uto Regressive Voice ctivity detector Prediction Error Filter Mean Square Error Minimum Mean Square Error

5 INTERNL INFORMTION (52) ER/B/D-98:32 INTRODUCTION The adaptive linear filter in the echo canceller does not fully attenuate the echo, despite the fact, that it is set at optimal. The remaining residual echo needs further suppression. This task is handled by a Non Linear Processor (NLP). The NLP blocks the residual echo fully or partially. Unfortunately, the NLP cuts, especially weak parts of the near end signal, making the speech interrupted. n alternative way of suppressing residual echo would be to design an adaptive post-filter which attenuates the residual echo but leaves the near end signal unaffected. The object of this thesis is to study and make simulations of three different algorithms for acoustic residual echo reduction in accordance with []. Further, the results obtained are analysed to see if the algorithms work for network echoes and evaluate the possibilities of replacing the NLP with the post-filter.

6 INTERNL INFORMTION (52) ER/B/D-98:32 2 BCKGROUND 2. Echo origins The extent to which echo is a problem for the telephone user depends on the duration of the time delay involved. If the delay between the speech and the echo is short, the echo is not noticeable but perceived as a spectral distortion or reverberation. If the time delay exceeds, a few tens of milliseconds, the echo becomes noticeable and thereby annoying. Echoes are noticeable particularly in a telephone circuit that includes a geostationary satellite. Due to the high altitude of such a satellite, there is a one-way propagation time of about 25 ms between the two parties. This yields a round trip delay of 5 ms. In the Global System for Mobile communications (GSM) system, the echo is noticeable too. That is because there is a one way delay of about ms [2]. This delay is present due to speech coding and radio transmission, where channel coding and decoding are used. Echo has to be attenuated in accordance with the delay involved. Longer delays must be compensated with higher echo attenuation to prevent the echo to be heard. To see the cause of echoes, consider the telephone circuit depicted in Figure. 2-wire Subscriber Line 4-wire Transmission Network echo 2-wire Subscriber Line Subscriber echo Subscriber B Hybrid Hybrid Figure general telephone network with the two parties, the 4 wire connection, the 2-wire connection and the telephones. The telephone sets for subscribers and B are connected to their local station by a 2-wire subscriber line. That line serves the need for communication in both directions. The device which maintains the conversion between the 4-wire transmission network and the subscriber line is called a hybrid. hybrid is a bridge circuit as depicted in Figure 2. The 4-wire transmission network splits the transmission into two separate channels, one for each direction. The reason for having separate transmission paths for send and receive is among other things that signals transmitted over a long distance need amplification to maintain the level and compensate a signal loss. mplifiers are mainly operational

7 INTERNL INFORMTION (52) ER/B/D-98:32 in one direction only. nother reason of having a 4-wire connection is that digitally operated switches are using it. The main origin of echoes in telephone networks is an impedance mismatch at the hybrid. This means that there is coupling between the receive and send port of the hybrid. When a speech signal from the Far End (FE) encounters this mismatch, some energy of the speech signal will be returned to the speaker at the FE as an echo. Such echoes are usually called network echoes or hybrid echoes. Receive Balancing network Subscriber Send Figure 2 Hybrid circuit with receive and send ports, a speaker port and a balancing network. nother echo source is the acoustic coupling between the loudspeaker and the microphone. Some of the sound waves transmitted from the loudspeaker including room reflections are returned to the FE speaker as echo via the microphone. This type of echo is referred to as an acoustic echo.

8 INTERNL INFORMTION (52) ER/B/D-98:32 Figure 3 shows a typical acoustic echo situation. coustic echoes are present mainly in hands free equipments and tele conference systems. erberation Direct echo erberation Figure 3 coustic crosstalk between loudspeaker and microphone caused by direct echo and late reverberation. 2.2 Echo cancellers Two solutions of handling echo problems have been used so far. In the past, a device called echo suppressor [3] was widely used. That device basically operates as follows: When it detects a speech signal from the receive path it produces an attenuation towards the send path. The major drawback of using echo suppressors is the speech detection errors that occur. Speech detection errors will cause an attenuation of the Near End (NE) speech as well. In the late 6 s, echo cancellers [2] [3] [4] were introduced. basic echo canceller model is depicted in Figure 4. The echo canceller builds an estimate of the echo path through the hybrid. The estimated echo signal is then subtracted from the input signal and the echo is hopefully removed. Estimated echo path Echo source Subscriber line Figure 4 The basic echo canceller model.

9 INTERNL INFORMTION (52) ER/B/D-98: The modern echo canceller Figure 5 shows the parts in a modern echo canceller. R in Echo Canceller r(n) R out Far End daptive Linear Filter DTD Hybrid Near End s(n)+m(n) S out + NLP + CNG e(n) _ ^ d(n) S in s(n)+m(n)+d(n) DTD NLP CNG Double-talk Detector Non-linear Processor Comfort Noise Generator sn ( ) -The near end speech rn ( ) -The far end speech mn ( )-The near end noise dn ( ) -The echo signal dˆ ( n) -The echo signal as estimated by the echo canceller en ( ) -The residual echo signal Figure 5 The modern echo canceller model with its parts and signal definitions. Each part of the echo canceller is described in the following sections, The adaptive linear filter The central part of an echo canceller is the linear filter. The filter determines an estimate of the echo path. When the incoming speech signal from FE pass through this filter, a replica of the echo is obtained. The echo is cancelled by subtracting the estimated echo signal from the input signal. However, an echo canceller is a shared resource. t any moment in time, the signals passing through one channel in the echo canceller may come from a particular telephone call between two end stations. When a telephone call is done, the same channel is available for calls between two new end stations. This leads to

10 INTERNL INFORMTION (52) ER/B/D-98:32 different echo paths from call to call. When the echo path is changing in time, it is necessary to use an adaptive filter to estimate the echo path. The adaptive filter consists of a linear Finite Impulse Response (FIR) filter, usually controlled by a form of Least Mean Square (LMS) algorithm. The LMS algorithm updates the filter coefficients in time according to the changes in echo path impulse- or frequency response function. The most widely used adaptation algorithm is the Normalized Least Mean Square (NLMS) algorithm. NLMS is essential mainly where the input signal is subject to widely fluctuating power levels at the input of the filter. The convergence speed is another reason for using NLMS. Details about LMS and its variants can be found in references [3] [4] [5]. The length of a FIR filter is usually 52 taps, which corresponds to 64 ms at 8 khz sample rate. Therefore the impulse response of the echo path may not exceed 64 ms, otherwise the filter length for the estimate is too short and the echo attenuation becomes poor The double talk detector If near end speech is present, the adaptive algorithm will tend to diverge. This can cause a poor echo-path estimate which yields poor echo attenuation. The divergence problem still exists when speech signals from far end and near end are present simultaneously (Double Talk (DT)). Because of these problems, there is a need for a function that detects if there is NE speech present or not. This function is known as a Double Talk Detector (DTD). When the DTD is active, the adaptive filter will stop updating the filter coefficients The non linear processor In many cases, the echo suppression obtained by the adaptive FIR filter is not sufficient. These problems occur mainly when there are non linearities in the echo path according to bad balanced networks, long time delays and overload in the hybrid and so on. nother cause to bad echo suppression is when there is estimation errors. To overcome these problems, a device called a Non Linear Processor (NLP) is used. The NLP further reduces the echo by blocking the output signal, completely or partially, when it is dominated by residual echo The comfort noise generator If there is background noise at the NE, the NLP will remove both the residual echo and the background noise as well. This modulation of noise must be compensated to avoid poor speech quality due to unwanted silence periods in the speech. One way to do this is to add virtually generated noise to the output and thereby compensate for the modulated noise. The unit which handles this task is called a Comfort Noise Generator (CNG). The main task of the CNG is to create and inject noise that spectrally matches and is at the same energy level as the background noise at the NE.

11 INTERNL INFORMTION ER/B/D-98:32 (52) 3 PURPOSE The simplest NLP used today is based on a centre clipper [2] [4]. Figure 6 shows a basic centre clipper function. g(e(n)) T e(n) Figure 6 simple centre clipper function. The centre clipper is controlled by a preset threshold level T. When the residual echo signal is below that level, the centre clipper becomes active and the residual echo is cut off. Otherwise, it is let through. This method of cutting the residual echo will affect the near end signal in a negative manner. The listener at FE will experience the phone call very disturbing, as parts of the NE speech is removed as well. It is especially the weak parts of the NE signal, for example at the beginning and the end of words, which are accidentally cut off. n alternative method of suppressing residual echoes would be to replace the NLP with an adaptive post-filter. The post-filter will suppress the residual echo and leave the NE signal intact. This principle is described in [], where the performance of three different algorithms for residual acoustic echo reduction are compared. The purpose of this thesis is to make simulations and experiments in accordance with [] and see if the methods work for network echoes as well as for acoustic ones. Further, the possibilities of replacing the NLP with the postfilter are investigated in this work.

12 INTERNL INFORMTION ER/B/D-98:32 (52) 4 POST-FILTER DESIGN Figure 7 shows the post-filter attached to the echo canceller. ECHO CNCELLER R in R out FE daptive Linear Filter HYBRID d NE s+m s+m=out ^ ^ Post-filter _ S out e + d^ S in Sin = x = = s+m+d Computation of the spectral densities and filter coefficients e x ^ d ^ e = x-d = ^ = s+m+(d-d) Figure 7 The structure of the echo canceller with an additional post-filter and the filter coefficients computing block.

13 INTERNL INFORMTION ER/B/D-98:32 2(52) To describe the function and the design requirements of the post-filter, consider four sequences of a telephone call denoted in table. Table : Four sequences in a general telephone call Sequences Signals present Filter requirements NE signal FE signal Silence x(n)=m(n) No attenuation of residual echo needed 2 NE talk present 3 FE talk present 4 DT Double talk present x(n)=s(n)+m(n) x(n)=m(n)+d(n) x(n)=s(n)+m(n)+d(n) No attenuation of residual echo needed ttenuate the residual echo ttenuate the residual echo and leave the NE signal intact ambient background noise speech + ambient background noise ambient background noise speech + ambient background noise quiet quiet speech speech The algorithms of the post-filter are designed to follow these requirements without causing too much modulation of the NE-signal. Section 6 describes each algorithm according to the filter requirements.

14 INTERNL INFORMTION ER/B/D-98:32 3(52) 5 IMPLEMENTTION ND COMPUTTION OF FILTER COEFFICIENTS The post-filter was implemented in the frequency domain. Frequency domain implementation was chosen because of its lower computational complexity. The major drawback of frequency domain implementation is the additional time delay that occurs when using block wise calculation. This delay depends on the post-filter length of N taps, i e if N=52 the time delay will be 64 ms at a sample rate of 8kHz. Consider the filter coefficients computation block diagram depicted in figure 8. Hanning window w(p,n) x(n) e(n) ^ d(n) Sample/ Block x(p,n).. X x(p,n)w(p,n) FFT X(p,k).. X(p,k).. Spectrum estimation _ Γ xe Calculate coefficients E(p,k) H(p,k) Filter coefficients U(p,k) IFFT Filter coefficients u(p,n) u(p,n) Overlapadd out(p,n) block/ Sample ^ ^ out(n) = s(n)+m(n) Figure 8 The spectral densities and post-filter coefficients computation block diagram. First, the signals xn ( ), en ( ) and dˆ ( n) are divided into blocks with length of N taps. Each block p of the three signals is further windowed with a Hanning window. This is done to achieve smooth transitions in filter H when reconstructing the output signal in time domain. Then the Discrete Fourier Transform (DFT) as defined in equation () is performed to each windowed block to

15 INTERNL INFORMTION ER/B/D-98:32 4(52) get the Short Term Fourier Transforms (STFT:s) X( p, k), E( p, k) and Dˆ ( pk, ). Definition of the DFT with windowed signal blocks, N X( p, k) = x( p, n)w( p, n)e N, () n = where x( p, n) is the input signal block and w( p, n) is the windowing function(hanning). To further decrease the computation complexity, the Fast Fourier Transform (FFT) have been used. The FFT is a faster way of calculating the DFT. The filter coefficients are according to the Frequency Response Function (FRF) of the algorithms described in the next section, computed and copied into the post-filter. The output signal from the post-filter is inverse fourier transformed using the Inverse Fast Fourier Transform (IFFT) and synthesized using an overlap add technique [7] with an overlapping factor of 5%. Since all FRF:s can be expressed as functions of spectral densities, there is need of spectral estimation when using a limited set of data. It is assumed that the speech signals are short time stationary. rough estimate of the Power Spectral Density (PSD) Γ xx ( pk, ) is expressed as: Γˆ xx ( pk, ) = X( p, k) 2 (2) The expression X( p, k) 2 can be rewritten as X( p, k)x ( pk, ), where * denotes the complex conjugate. In a similar way, the expression of the Cross Spectral Density (CSD) estimate becomes: Γˆ xy ( pk, ) = X( p, k)y ( pk, ) (3) This estimate as well as the PSD estimate above yields too big variance. To reduce the variance, a general averaged CSD estimate is obtained using equation (4) Γ xy ( pk, ) = r -- X. (4) r i ( pk, )Y i ( pk, ) i = Here Γ xy ( pk, ) is the averaged CSD estimate and X i ( pk, ) is the STFT of x i ( pn, )w i ( pn, ). To reduce the complexity of computing the CSD and the memory requirements, the averaging in equation 2 can be done by low pass filtering as. Γ xy ( p +, k) = αγ xy ( pk, ) + ( α)x( p+, k)y ( p +, k), (5) where α is a forgetting factor. This factor is usually set between.65 and.9. By replacing index y with x, the averaged PSD estimate is obtained.

16 INTERNL INFORMTION ER/B/D-98:32 5(52) 6 DESCRIPTION OF THE POST-FILTER LGORITHMS 6. First algorithm, the Wiener post-filter In this case, the post-filter is designed according to an optimal Wiener filter model. This filter model is depicted in Figure 9. e(n) x(n) H u(n) - + ep(n) Figure 9 The Wiener filter model. The filter coefficients minimizes the Mean Square Error (MSE) [3] [5] between en ( ) and un ( ) using least squares filtering. In appendix, the filter is defined and derived. The estimated FRF of the post-filter is: Ĥ( pk, ) = Γ xe ( pk, ) Γ xx ( pk, ) (6) Where Γ xx ( pk, ) is the averaged PSD estimate of the signal X( p, k) and Γ xe ( pk, ) stands for the CSD between X( p, k) and the residual error signal E( p, k). Consider the sequences as in table. During ST where FE speech is present, the signal xn ( ) has more power than en ( ) and the filter is attenuating the residual echo signal because the term Γ xe ( pk, ) is smaller than Γ xx ( pk, ). In ST, where NE speech is present and during silence, the signals en ( ) and xn ( ) are identical. Hence the filter is reduced to an all pass filter and lets the residual echo signal pass through. During DT, frequencies which corresponds mainly to the echo are attenuated while other frequencies are less affected. This means that the echo canceller must give some attenuation in order to get the post-filter working effectively. This is because there must be a difference in energy level between xn ( ) and en ( ). Otherwise, the post-filter would always be an all-pass filter and thereby not be able to attenuate any residual echo.

17 INTERNL INFORMTION ER/B/D-98:32 6(52) 6.2 Second algorithm, the overweighted wiener post-filter This algorithm is a modified version of the one described above. This approach yields an additional degree of freedom to design the post-filter by adding an additional PSD term in the denominator of the FRF. This term is further amplified by a weighting factor, where >. The FRF estimate is expressed as: Ĥ( pk, ) = Γ xe ( pk, ) Γ xx ( pk, ) + Γ dˆ ( pk, ) dˆ (7) In this algorithm, Γ dˆ ( pk, ) stands for the averaged PSD estimate of the echo dˆ estimated by the echo canceller. The filter behaviour is similar to the Wiener post-filter described in the previous section. The difference is that with Γ dˆ ( pk, ) and in the denominator there is an additional attenuation of the dˆ residual echo. 6.3 Third algorithm, post-filter using noise reduction techniques This approach is based on noise reduction techniques [6]. These techniques are based on spectral subtraction. The post-filter is designed according to a non causal Wiener filter model [7] [9] as depicted in Figure. The echo estimated by the echo canceller dˆ ( n) is seen as the disturbance. x(n) + d(n) ^ u(n) H ^ x(n) Figure Block diagram of the non causal Wiener filter model with FRF as in equation 8. The filter coefficients obtained in this model minimizes the MSE between xn ( ) and xˆ ( n) so that xˆ ( n) xn ( ). This will result in an attenuation in the echo frequencies. In appendix B, this filter model is defined and described in detail. The FRF estimate of this algorithm is expressed as: Ĥ( pk, ) SER( p, k) = = + SER( p, k) Γ xx ( pk, ) , (8) Γ xx ( pk, ) + Γ dˆ ( pk, ) dˆ Γ where SER( p, k) xx ( pk, ) = is the Signal to Echo Ratio. ( pk, ) Γ dˆ dˆ

18 INTERNL INFORMTION ER/B/D-98:32 7(52) In ST, where NE speech is present and in silence, the signals en ( ) and xn ( ) are the same. This means that the term Γ dˆ ( pk, ) gives no contribution to dˆ Ĥ( pk, ) and thus, filter Ĥ( pk, ) will be an all-pass filter. During ST where the FE speaker is active, there is an echo situation. The term Γ dˆ ( pk, ) in the dˆ denominator which contains the echo frequencies is added to the term Γ xx ( pk, ). The denominator in Ĥ( pk, ) is greater than the numerator in the echo frequencies. Therefore, an attenuation of the residual echo signal is obtained. If in DT, the filter is attenuating the residual echo and letting the NE speech pass through as the NE has the same energy in xn ( ) and en ( ).

19 INTERNL INFORMTION ER/B/D-98:32 8(52) 7 SIMULTIONS In this section, simulation results of the three algorithms are presented and discussed. In all cases, real life recorded speech signals have been used. 7. Simulations of the three algorithms The simulation examples which were used are presented in Table 2. The simulation time in all examples apart from example 4 is set to 6 s to include all the sequences in accordance with table. In example 4 the simulation time is 56 s for the same reason. Examples to 3 are from a general Public Switched Telephony Network (PSTN) to PSTN conversation. In example 2, a stationary background NE car noise is present to see how the post-filter operates. Example 3 consists of a NE noise which is a non-stationary noise from a restaurant, here called babble noise. The fourth example is a live Nordic Mobile Telephony (NMT) mobile to PSTN telephone call and it is a very difficult case with lots of DT and non linearities. The non linearities give poor echo estimates and the post-filter is not able to attenuate the residual echo signal properly, as to be seen in the simulations. Simulations where NE is a male voice and FE is a female voice are not presented in the thesis, because they give similar results as the simulations according to Table 2. Table 2: Summary of the simulation examples No of simulation FE NE NE noise Name male voice female voice 2 male voice female voice none car noise General General 3 male voice female voice babble noise General 4 male voice male voice none NMT call ll simulations were carried out using a filter length of N=28 as default. This length seems to be the best one in view of the additional time delay and distortions as shown in the simulations, where different filter lengths are used. The forgetting factor α was set to.78. For each simulation, the signals are plotted in sample-domain. The momentary powers of the signals are plotted to see how the post-filter attenuation appears to be. The post-filter attenuation db ( ) is calculated using equation 9 and then plotted. The results obtained from each algorithm are discussed.

20 INTERNL INFORMTION ER/B/D-98:32 9(52) --- ( e N k ( n) ) 2 k = ( db) = log N db --- ( out N k ( n) ) 2 N k = (9) N Where --- e is the momentary power of the residual echo, N ( k ( n) ) 2 en ( ) N k = and --- out is the same of the output signal of the post-filter. N ( k ( n) ) 2 out( n) k = 7.. The Wiener post-filter Simulation example was carried out using different filter lengths to see how the length would affected the post-filter performance. The filter length was varied like this: N=32, 64, 28(default), 256, 52 and 24. The estimated PSD:s, CSD:s and the FRF:s were plotted according to figure 5 and Results and discussions In simulation examples to 3 (Figures to 3), the Wiener post-filter gives a fair attenuation of the residual echo without causing too much modulation of the NE-speech during DT. In the fourth example (Figure 4), where there are non linearities, the echo canceller attenuation will vary through time. This will result in varying post-filter performances. That is because the post-filter is dependent of the echo canceller attenuation. During DT, there is some audible modulation of the NE speech signal. That is probably because it is a male to male conversing situation, which yields a NE spectra represented at some of the frequencies of the FE spectra. More of this problem is stated in the next section, where the results from the overweighted Wiener filter is presented. The background noise is modulated both in simulation example 2 and 3. Simulation example 2 with car noise yields a very annoying speech quality after post- filtering. In simulation example 3, with NE babble noise, the quality is better than in the case of car noise. This is because the spectrum of that noise spans over a larger frequency range than the car noise. Then, the higher frequencies are not attenuated as much as the lower ones, since the residual echo lies mainly in the lower spectra.

21 INTERNL INFORMTION ER/B/D-98:32 2(52) In the cases, where the filter length N is set to N=32 and N=64, there is a nasty speech distortion of the NE-speech during DT. This is probably due to a low frequency resolution. Most of the real frequencies are not represented in the spectrum and this will result in spectral leakage. For lengths of N=28 and N=256, the post-filter is working properly without any audible distortions of the NE-speech during DT. In other words it is recommended to choose a filter length of N=28 according to the additional time delay caused by longer post-filter lengths. When the Filter length is N=52 and N=24 the filtered output signal sounds strange. The reason for that would be that with longer filter lengths, the signals loses the short-time stationary properties. If there are non stationary spectral variations of speech, then the performance of Wiener filters, which only work properly with stationary signals, become poor.

22 INTERNL INFORMTION ER/B/D-98:32 2(52) Simulation example 2 x x (a) (b) 5 post filter attenuation 2 power (db) ttenuation (db) 2 5 power (db) (c) (d) Figure (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

23 INTERNL INFORMTION ER/B/D-98:32 22(52) Simulation example 2: 2 x x (a) (b) 5 post filter attenuation power (db) ttenuation (db) 5 power (db) (c) (d) Figure 2 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

24 INTERNL INFORMTION ER/B/D-98:32 23(52) Simulation example 3: 2 x x (a) (b) 5 post filter attenuation power (db) ttenuation (db) power (db) (c) (d) Figure 3 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

25 INTERNL INFORMTION ER/B/D-98:32 24(52) Simulation example 4: 2 x x (a) (b) 5 post filter attenuation 2 power (db) ttenuation (db) 2 5 power (db) (c) (d) Figure 4 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (3 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

26 INTERNL INFORMTION ER/B/D-98:32 25(52) Simulation example run with different post-filter lengths. cross spectral densities gain db frequency (Hz) (a) power spectral densities gain db frequency (Hz) (b) Figure 5 (a) The cross spectral densities between xn ( ) and en ( ). (b) The power spectral densities of signal xn ( ) for filter lengths of N=32, 64, 28, 256, 52 and 24 taps.

27 INTERNL INFORMTION ER/B/D-98:32 26(52) Frequency Response functions gain db frequency (Hz) Figure 6 Frequency response functions of the Wiener post-filter of lengths N=32, 64, 28, 256, 52 and 24 taps.

28 INTERNL INFORMTION ER/B/D-98:32 27(52) 7..2 The overweighted Wiener post-filter The simulation example was carried out with =, 6 and 3 to see how the attenuation performances of the post-filter appears to be and how the NE signal is affected, especially during DT. Example is presented only, as it is enough to make the conclusions Results and discussions The overweighted version of the first algorithm seems to work well in all sequences except during DT, where the NE-speech is modulated in a non acceptable manner, especially when the parameter is set to 3, as shown in Figure 2. The NE signal is attenuated because some energy of the NE-speech signal lies on the same frequencies as the FE-speech signal. Then if the term Γ dˆ ( pk, ) is amplified by to a level that make the numerator greater then the dˆ denominator in Ĥ( pk, ), at common frequencies, the speech modulation becomes noticeable. Hence, the attenuation of the NE-speech becomes audible if is set too high. solution would perhaps be to find a way of controlling the parameter. During DT, parameter would be close to zero. During ST where FE is active, parameter should be high (>=3) to get maximum attenuation. Of course, this algorithm gives modulations of noise as the Wiener post-filter. In other words, the prerequisite for this algorithm to work perfectly is that the NE-speech signal should be totally separated from the FE-speech in frequency as sketched in figure 7. In this case, parameter can be set very high. Gain (DB) NE FE f Figure 7 n interpretation of how the spectrum of FE and NE should look like to make the post-filter working at optimum without any modulations of the NE signal.

29 INTERNL INFORMTION ER/B/D-98:32 28(52) Simulation example with =: 2 x x (a) (b) 5 post filter attenuation power (db) ttenuation (db) 5 power (db) (c) (d) Figure 8 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

30 INTERNL INFORMTION ER/B/D-98:32 29(52) Simulation example with =6: 2 x x (a) (b) 5 post filter attenuation 2 power (db) ttenuation (db) 5 2 power (db) (c) (d) Figure 9 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

31 INTERNL INFORMTION ER/B/D-98:32 3(52) Simulation example with =3: 2 x x (a) (b) 5 post filter attenuation 2 power (db) ttenuation (db) power (db) (c) (d) Figure 2 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

32 INTERNL INFORMTION ER/B/D-98:32 3(52) 7..3 Post-filter using noise reduction techniques Results and discussions This algorithm is not attenuating the residual echo as well as could be expected, neither for simulation example (Figure 2) nor 4 (Figure 24). The attenuation obtained is about 8 to 2 db compared with about 5 to 2 db in example 2 and 3 (Figure 22 and 23). This is probably due to the fact that there is correlation between xn ( ) and dˆ ( n). In example 2 and 3, where NE-noise is present, the noise is probably decorrelating the signal xn ( ) from dˆ ( n) and therefore, the post-filter performance is increasing in these cases. The results from example shows that the NE-speech modulation is not noticeable as the post-filter attenuation is low. In examples 2 and 3, the modulation of the NE-speech is tolerable because of the male-female conversation where the NE is fairly separated from FE in frequency. In example 4, the modulation of NE is noticeable due to the fact, that NE and FE sometimes lie in the same spectral range, but not as disturbing as in the first algorithm because of low post-filter attenuation. In order to get the non casual Wiener filter working properly, the signals xn ( ) and dˆ ( n) should be uncorrelated. s stated in appendix B, the signals xn ( ) and wn ( ) are assumed to be uncorrelated, and then the filter performance obtained in the simulations show that the theory corresponds to the simulation examples.

33 INTERNL INFORMTION ER/B/D-98:32 32(52) Simulation example : 2 x x (a) (b) 2 post filter attenuation 2 power (db) ttenuation (db) power (db) (c) (d) Figure 2 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

34 INTERNL INFORMTION ER/B/D-98:32 33(52) Simulation example 2: 2 x x (a) (b) 5 post filter attenuation power (db) ttenuation (db) power (db) (c) (d) Figure 22 Simulation results from example 2. (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

35 INTERNL INFORMTION ER/B/D-98:32 34(52) Simulation example 3: 2 x x (a) (b) 5 post filter attenuation power (db) ttenuation (db) power (db) (c) (d) Figure 23 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (8 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db.

36 INTERNL INFORMTION ER/B/D-98:32 35(52) Simulation example 4: 2 x x (a) (b) 5 post filter attenuation 2 power (db) ttenuation (db) 2 5 power (db) (c) (d) Figure 24 (a) The residual echo signal en ( ) and the output signal out( n) in sample domain. (b) The same signals at shorter duration (3 s) to see how the post-filter performance appears to be. (c) The momentary powers of en ( ) and out( n). (d) The post-filter attenuation in db

37 INTERNL INFORMTION ER/B/D-98:32 36(52) 7.2 Overall discussions ccording to the simulation results, the third algorithm gives the best attenuation in the examples where additive noise is present at the NE but with an increased modulation of the NE. The first algorithm gives the best overall performance. This algorithm is used to compare the post-filter performance with the NLP. In all cases when there is background noise present, the noise is modulated during the post-filtering. That is because the noise spectra is represented over the same frequencies as the residual echo. The noise as well as the residual echo is attenuated and this yields a poor speech quality, especially in example 2, where car noise is present. To solve this problem, comfort noise is injected as described in the next section. The third algorithm is used to see how the comfort noise injection appears to be. This is because this algorithm is modulating the noise very rapidly, in comparison with algorithm.

38 INTERNL INFORMTION ER/B/D-98:32 37(52) 8 COMFORT NOISE INJECTION s the background noise from NE is modulated when filtered by the post-filter, there is a need for injecting comfort noise to compensate for the modulation and fill out the spectral valleys in the residual echo signal. Figure 25 shows a variant of the CNG similar to []. x(n) VD True/False LPC estimate Noise level White noise generator W(n) F(z) h a m(n) ~ X m(n) ^ Hanning window m(n) ^ Sample/ Block m(p,n) ^ X w(p,n) m(p,n)w(p,n) ^ FFT M(p,k) ^ M(p,k) ^ G(p,k) IFFT Overlapadd Block/ Sample cn(n) Figure 25 The CNG block diagram and the filter G. The procedure of comfort noise estimation is based on linear prediction [5]. This is to predict a future value of a stationary random process from observation of past values of the process. Because the noise is to be predicted, it must be estimated during speech pauses. During the prediction, an estimate of the uto Regressive (R) filter coefficients is obtained. The R model is used because the parameters can be obtained as the solution of a set of linear equations.

39 INTERNL INFORMTION ER/B/D-98:32 38(52) The R filter used here is designed as: Fz ( ) = () z ( ) generated white noise process is filtered through Fz ( ) to obtain the NE noise estimate with a spectrum similar to the NE noise. This filtered noise is further amplified to the same level as the NE-noise and added to the output signal. To determine whether there is silence or not, there is need of a voice activity detector (VD). This VD detects if there is any speech in either directions. If the silence period is detected, the R coefficients can be estimated from the signal xn ( ) by a one-step predictor [5] using Yule-Walker equations [6] [7] as in equations as p R xx ( n) = f a R xx ( n a), () a = where R xx ( n) is the autocorrelation of the signal xn ( ). The R filter coefficients are f a and the filter length is p taps. Equation can be rewritten in matrix form of order p as: R xx ( ) R xx ( 2) R xx ( p) R xx ( ) R xx ( ) R xx ( 2) R xx ( p ) R xx ( ) R xx ( ) R xx ( ) R xx ( p 2) f 2 = = R xx ( ) R xx ( p ) R xx ( ) R xx ( ) f h p = g = Rf f = R g. (2) These equations are known as the normal equations. There are some methods of solving these equations. One solution is to use Levinson algorithm [5] [6] [7], which allows the recursive calculation of the predictor coefficients. To easier solve the normal equations, a Prediction Error Filter (PEF) [5] can be used. In the PEF of the one-step predictor, the order of the vectors and matrixes is p +. The matrix in equation 2 is rewritten as [5]:

40 INTERNL INFORMTION ER/B/D-98:32 39(52) h h p = R xx ( ) R xx ( ) R xx ( 2) R xx ( p) R xx ( ) R xx ( ) R xx ( ) R xx ( p ) R xx ( ) R xx ( p) R xx ( ) R xx ( ) p R xx ( ) h i R xx () i i = (3) s the vector in equation 3 has a much simpler form, a modified version of Levinson algorithm can be used. This algorithm is referred to as the Levinson- Durbin algorithm and is defined in [5] [7]. The R filter can either be designed in direct- or in Lattice form [7]. In Levinson-Durbin algorithm, the reflection coefficients [7] for a Lattice filter is obtained as well as the filter coefficients for a direct form realisation. The output noise m ( n) is further amplified to a level that corresponds to the power of the NE-noise itself. When the silence is over, the R coefficients obtained from this estimation procedure are frozen and used in the filter Fz ( ) until the next period of silence is detected by the VD. Then, the whole estimation procedure is restarted once again. Filter G( p, k) is designed as G( p, k) = H( p, k), (4) where H( p, k) is the FRF of the post-filter. This filter amplifies the noise at the frequencies that H( p, k) attenuates and vice versa. The output noise is therefore set to the same level as the input noise. fter that procedure the filtered noise is IFFT:ed, overlap-saved and finally added to the filtered residual echo signal. 8. Simulation examples The results of interest are those obtained from simulation example 2 and 3, as the NE-signal in example 2 consists of a fairly stationary car noise and example 3 has a non stationary background noise.

41 INTERNL INFORMTION ER/B/D-98:32 4(52) 8.2 Results and comments In example 2 (Figure 26), the output signal added with comfort noise sounds very good in comparison with the post-filter modulated one. Then the filter G( p, k) is working properly and the estimation procedure as well. The simulation results of example 3 (Figure 27) shows that it is difficult to build an estimate of the NE babble noise. This is because the noise is of a non stationary nature. The output signal does not sound annoying, anyway. To summarize the results, the estimation and injection of comfort noise is working well during the circumstances that the NE-noise is of a fairly stationary nature. I.e. the signal has a slow variation with time. When NE-noise is non stationary, the CNG tries to estimate the noise as far as possible. Simulation Example 2: output signal and comfort noise out(n) + cn(n) Figure 26 Results of example 2, where comfort noise is injected to the filtered output signal

42 INTERNL INFORMTION ER/B/D-98:32 4(52) Example output signal and comfort noise out(n) + cn(n) Figure 27 Results of example 3, where comfort noise is injected to the filtered output signal.

43 INTERNL INFORMTION ER/B/D-98:32 42(52) 9 COMPRISON OF THE NLP ND THE POST-FILTER In this section, the performance of the post-filter implemented as the third algorithm is compared with the NLP used in the ERICSSON network echo canceller. 9. Simulation examples The simulation examples to test are the NMT telephone call (example 4) and the male-female conversation with background car noise (example 2) to see the clipping effects and comfort noise injection. 9.2 Results and comments If considering the simulation results obtained from example 4 (Figure 28), the output from the NLP sounds very bad as the clipping threshold level of the NLP is set too high due to non linearities. The NE speech is modulated in such a way that parts of the conversation is more or less lost. With the post-filter, most of the conversation is understandable, but as stated before, it leaves some of the residual echo behind and the NE speech is fairly modulated when two men are talking to each other. In example 2 (Figure 29), the NLP does not affect the output as much as is the NMT conversation example. s the NLP is clipping the residual echo straight off, the comfort noise injection sounds more interrupted than the comfort noise obtained with filter G( p, k), which should inject as much comfort noise as the post-filter attenuates. Thereby, the comfort noise injection is smoother than the one obtained in NLP. The post-filter sounds best in the examples, where CN is injected. s the frequencies of NE and FE are better separated than in example 4, the NE speech is not modulated as much during DT. To summarize this, the NLP cuts the residual echo straight off. Then the injection of comfort noise sounds better with the post-filter. s the most important of residual echo suppression is to remove all the residual echo, the NLP can not be totally replaced with the post-filter. More research has to be done about how to control the parameter in the second algorithm. The solutions from [] would be of interest. Maybe the NLP and the CNG can be replaced with the residual echo suppressor as in [].

44 INTERNL INFORMTION ER/B/D-98:32 43(52) Example 4, the NMT telephone call example: output signal from post filter out(n) output signal from NLP NLP(n) Figure 28 Simulation results of example 4, where the post-filter is compared with the NLP.

45 INTERNL INFORMTION ER/B/D-98:32 44(52) The male-female conversation with background car noise output signal from post filter out(n) output signal from post filter with comfort noise out(n) + cn(n) (a) 3 output signal from NLP NLP(n) output signal from NLP with comfort noise NLP(n) + cn(n) (b) Figure 29 Simulation results of (a) the post-filter (b) the NLP in example 2, where the post-filter is compared with the NLP.

46 INTERNL INFORMTION ER/B/D-98:32 45(52) CONCLUSIONS Three different algorithms for acoustic residual echo suppression have been simulated for situations with network echoes. Simulations with different post-filter lengths shows that if the length is too short, then distortions of the NE-speech appears. That is because there is not enough frequency resolution in the spectrum to build a good estimate. If the post-filter is too long, then the windowed blocks loses the short time stationary properties. The Wiener filter methods does therefore not work properly. The post-filter lengths of N=28 and N=256 works best. The choice is thereby the filter length of N=28 in case of the additional time delay that occurs in the case of blockwise calculations. In order to get the algorithms to work perfect during DT, the NE-signal and the FE-signal have to be totally separated in frequency. This is not the fact in the reality. ccording to the simulation results, the third algorithm gives the best attenuation in the examples where additive noise is present at the NE but with an increased modulation of the NE. The first algorithm gives the best overall performance. The estimation and creation of comfort noise is working well during the circumstances that the NE-noise is of a fairly stationary nature, i.e. the periodic elements of the signal have slow variations with time. When NE-noise is non stationary, the CNG tries to estimate the noise as good as possible. In real echo situations, no residual echo is desired at all. The post-filter algorithms delivers an attenuation which is not good enough. One solution to this problem is to use a post-filter and a NLP together. The post-filter is used to suppress the residual and lowering the threshold level for the NLP which is clipping the remaining residual echo. Hence the NE speech would be less affected by the NLP and the speech quality may improve. But the price to pay is the additional time delay that occurs by the post-filter. nother solution would to use a method of decorrelation as in [] and maybe get rid of the NLP and the CNG.

47 INTERNL INFORMTION ER/B/D-98:32 46(52) SUGGESTIONS FOR FUTURE WORK Evaluate the possibilities of using a post-filter and a standard NLP together. Explore how to create and inject comfort noise when using this combination. Find accurate methods of controlling the parameter in algorithm 2 to get hard attenuation of residual echo without modulating the NE speech in a negative manner during DT. To evaluate possible methods of suppressing residual echoes based on sample per sample calculation and thus reduce the additional time delay that occurs when using blockwise calculation. Evaluate the method of using decorrelation as in [] and compare the results with those obtained in this thesis.

48 INTERNL INFORMTION ER/B/D-98:32 47(52) 2 REFERENCES [] Vale rie Turbin, ndre Gilloire and Pascal Scalart, "Comparison of three Post-filtering algorithms for acoustic echo reduction"., IEEE 997, pp [2] nders Eriksson, Gunnar Eriksson, Johnny Karlsen, nders Roxström and Teresa Vallon Hult, "Ericsson echo canceller - a key to improved speech quality". Ericsson iew No. 996 [3] Simon Haykin, "daptive filter theory., 3rd ed. Upper Saddle River, NJ: Prentice Hall 997 [4] K. Shenoi, "Digital Signal Processing in telecommunications"., Upper Saddle River, NJ: Prentice-Hall 997 [5] Peter M. Clarkson, "Optimal and daptive Signal Processing"., CRC Press, Inc. 993 [6] lberto Leon-Garcia, "Probability and Random Processes for Electrical Engineering"., Second Edition, ddison-wesley Publishing Company, Inc. 994 [7] John G.Proakis Dimitris G.Manolakis "Digital Signal Processing principles, algorithms and applications"., Second Edition, Macmillan Publishing Company 992 [8] yad Beghdad, coustic echo and noise reduction: a novel pproach., IWENC 97 pp [9] Jae S. Lim, Two-Dimensional Signal and Image Processing., Englewood cliffs, NJ: Prentice-Hall 99 [] Peter Eneroth and Thomas Gänsler, Frequency domain adaptive Echo Canceller with Post-Processing Residual Echo Suppression by Decorrelation., Signal Processing Report SPR-4, Department of pplied Electronics, Signal Processing Group, Lund University 997 [] Stefan Gustafsson, coustic Echo Compensation in the GSM Environment - Echo Cancellation Combined with the Echo Shaping Filter., Rheinisch-Westfälische Technische Hochschule achen 997

49 INTERNL INFORMTION ER/B/D-98:32 48(52) The optimal Wiener filter In figure 3, the optimal Wiener filter model block diagram is depicted. d(n) x(n) H y(n) - + e(n) Figure 3 The idea behind optimal filtering is to design a filter H with the impulse response hn ( ) in such a way that the output signal yn ( ) obtained will be as close as possible to the desired signal dn ( ) in the sense of minimum mean squared error(mmse). The performance of this filter is observed through the error signal en ( ) between dn ( ) and yn ( ). In accordance with figure 3, the signal en ( ) is expressed as en ( ) = dn ( ) yn ( ), (5) where yn ( ) for a filter length of m taps is yn ( ) = hn ( ) xn ( ) = m k = hk ( )xn ( k), (6) where stands for the convolution operator. The MSE is written as: J = E{ e 2 ( n) }, (7) where E{ e 2 ( n) } is the expectation value of e 2 ( n). The MMSE is obtained by differentiating J with respect to each filter coefficient and setting the partial derivatives J h() i to zero. The differentiation yields J hi () E e { ( n) }, (8) hi () 2E e ( n ) en ( ) = = hi () where en ( ) hi () will be xn ( i) which results in J hi () 2 E d ( n = ) hk ( )xn ( k) m k = xn ( i) (9)

50 INTERNL INFORMTION ER/B/D-98:32 49(52) Setting J h() i to zero gives m E d( n)xn ( i) + hk ( )xn ( k)xn ( i) k = = E{ x( n)dn ( + i) } = E hk ( )xn ( )xn ( k+ i) m k = The definition of the autocorrelation function of signal xn ( ) is R xx () i = E{ x( n)xn ( + i) } (2) (2) The cross correlation function between xn ( ) and dn ( ) is R xd () i = E{ x( n)dn ( + i) } (22) From equations 2 to 22, the expression becomes m R xd () i = hk ( )R xx ( i k) R xd () i = k = hi () R xx () i (23) This is known as the normal equations. Taking the fourier transform of equation gives Γ xx Γ xd ( f ) = H( f)γ xx ( f ), (24) where ( f ) is the PSD of xn ( ) and Γ xd ( f ) is the CSD between xn ( ) and dn ( ). Γ The result of this gives FRF of H( f) xd ( f ) = which is the one used in the first algorithm (equation 6). Γ xx ( f )

51 INTERNL INFORMTION ER/B/D-98:32 5(52) B The non causal Wiener filter and then, according to the definition of the cross correlation function in equation the term R ex () i =. The Fourier transform of this gives the CSD expres- Consider figure 3. Suppose there is a signal sn ( ) and a disturbance wn ( ). The signal xn ( ) is an observation of the noisy signal. xn ( ) and wn ( ) is assumed as stationary processes. The observation signal xn ( ) is expressed as xn ( ) = sn ( ) + wn ( ) (25) w(n) s(n) + x(n) H s(n) ^ Figure 3 The non causal Wiener filter block diagram, where sn ( ) is the signal, wn ( ) is the disturbance The goal is to design filter H with the impulse response of hn ( ) so that the signal sn ( ) is determined from xn ( ) as good as possible. The expression of the signal estimate ŝ( n) is given by ŝ( n) = xn ( ) hn ( ) (26) where is the convolution operator. n estimation error en ( ) is defined as en ( ) = sn ( ) ŝ( n) (27) The filter tries to minimize the error as stated above. To solve this estimation problem, the ortogonality principle can be used. In this problem, the error is to be minimized by saying that en ( ) is ortogonal with the input signal xn ( ). Then, the error en ( ) shall always be uncorrelated with any variable in the process xn ( ). The signals en ( ) and xn ( ) said to be ortogonal if n E{ e( n)xi ()} =, i (28) This means that E{ e( n)xn ( + i) } = (29)

52 INTERNL INFORMTION ER/B/D-98:32 5(52) sion Γ ex ( f ) = for every f. If the Fourier transform is taken of equation 27, the expression becomes E( f) = S( f) Ŝ( f ). (3) This expression can be rewritten as spectral densities E( f)x ( f ) = S( f)x ( f ) Ŝ( f )X ( f ) Γ ex ( f ) = Γ sx ( f ) Γ ŝx ( f ) ( f ) = ( f ) Γ ŝx Γ sx (3) From equation the expression is written as S( f) = H( f)x( f) S( f)x ( f ) = H( f)x( f)x ( f ) Γ ŝx ( f ) = H( f)γ xx ( f ) ( f ) = H( f)γ xx ( f ) Γ sx (32) Then the non causal Wiener filter H is expressed as H( f) = Γ sx ( f ) ( f ) Γ xx (33) If considering equation 23. ssume that sn ( ) and wn ( ) is uncorrelated which means that R sw () i = Γ sw ( f ) =. (34) Fourier transform of equation 25 gives X( f) = S( f) + W( f) Γ xx ( f ) = Γ ss ( f ) + Γ ww ( f ) (35) The cross correlation function between sn ( ) and xn ( ) can be expressed as R sx () i = E{ s( n)xn ( + i) } = E{ s( n) ( sn ( + i) + wn ( + i) )} = = E{ s( n)sn ( + i) } + E{ s( n)wn ( + i) } R sx () i = R ss () i + R sw () i (36) With the term R sw () i = set to zero, this yields R sx () i = R ss () i Γ sx ( f ) = Γ ss ( f ) (37)

53 INTERNL INFORMTION ER/B/D-98:32 52(52) Then the FRF is expressed as H( f) = Γ ss ( f ) ( f ) + Γ ww ( f ) Γ ss (38) The signal to noise ratio (SNR) is defined as SNR( f ) = Γ ss ( f ) ( f ) Γ ww (39) Then the final expression of the FRF becomes H( f) Γ ss ( f ) Γ ww ( f ) = H( f) = Γ ss ( f ) Γ ww ( f ) ( f ) ( f ) Γ ww Γ ww SNR( f ) SNR( f ) + (4) In the third algorithm (equation 8), the SNR is called SER