-1- THEEXCESSPHASEINLOUDSPEAKERROOM

A BSTRACTThe impulse response c[n] characterizing the system of a loudspeaker and a microphone placed in a closed room contains information on the electroacoustical properties of loudspeaker and microphone as well as the acoustical properties of the room.The corresponding transfer function C(z) offers information on the magnitude and phase of the system when regarded as a digital filter. A part of this phase, the excess phase, gives rise to problems when reaching for mathematically perfect equalization of the system. It is a fact however that phenomena which appear through analysis of C(z) might not be perceivable to the human ear.We have shown that, unfortunately, the excess phase contribution is audible in con-trolled listening tests. That is, it can not be neglected in the design of equalization systems.1.I NTRODUCTIONIn high-fidelity reproduction of sound a crucial point is to obtain as little coloration as possi ble due to the reproducti on system. We wi ll defi ne the term colorati on as ampli tude as well as phase distortion. Whenever signals are recorded, steps are taken to ensure that the recordi ng system i tself does not contri bute si gni fi cantly to the recordi ng - at least not in the frequency range where human beings perceive sound. Microphones and analog/di gi tal recordi ng equi pment can i n general be desi gned to affect the signals only very little.(2.1) But when the recordi ng i s played back colorati on effects are certai nly i mposed. These effects come from the necessary voltage-to-pressure transducer, a loudspeaker or a pair of headphones. Additionally, the sound waves generated by the loudspeaker cone must travel trough a room to reach the ear. Hence the acoustical charachteristics of the room plays a role too, and the effect due to these is normally more significant than that of the loudspeaker itself.The system consisting of loudspeaker and room can be modelled as a digital filter, by accepting some limitations, henceforth referred to as C(z). The goal is then to design an electrical equalization filter H(z) that cancels theimpacts from the loudspeaker/room system, see fig. 1.Varoi us attempts have been made to i ncorporate equali zati on i n reproducti on systems, see [1] - [5]. Some subtle listening tests e.g. [6] - [8] have shown that also phase changes influence the perception. Few of these investigations however concern excess phase in the transfer function of loudspeaker and listening room. A recent work in this field [9] indicates through preliminary listening tests that phase changes in such transfer functions are in fact audible even with common music recordings as test material. A remarkable result si nce nonli near phase changes mi ght gi ve ri se to difficulties in equalization tasks.2.S YSTEM IMPULSE RESPONSESDesign of equali zati on fi lters requi res knowledge of the loudspeaker/room system either as temporal or spectral information. When an electrical impulse is applied to the loudspeaker it produces a sound pressure impulse, and when this travels through the room and i s recorded by a mi crophone the output, called c[n], can be regarded as unambi guous information on the colorations introduced in that specific playback path By Z-transforming this impulse response thus obtaining the transfer function C(z) the deviations from the ideal situation, i.e. unity gain for all frequencies and linear phase, are revealed. The Z-transformed response, allthough non-parameter zed, can be regarded as one generated by a di gi tal fi lter, and i t i s qui te obvi ous that the i deal equalization filter must then comply with eq. 2.1.The spectral i nformati on i n C(z) i s remarkably changed i f the loudspeaker and mi crophone positions are varied just slightly [10]. Furthermore, the human hearing works binaurally. Hence two transfer functions are evoked and for higher frequencies(2.2)(2.3) these are indeed not alike because of aural shadowing of the head. Even stationarity can not be assumed. When the head moves, the two transfer functions become time dependent.What we consider here is only the theoretical /mathematical part of the equalization task and even then di ffi culti es wi ll most certai nly ri se attempti ng to carry out the di vi si on i n eq. 2.1. Consi der C(z) as the rati o of two rati onal polynomi als wi th M respecti vely N distinct roots as in eq. 2.2. Since C(z) is in fact not a genuine parametric filter we must presume the numbers M and N to be large (perhaps infinite) and non-determinable. Nevertheless this model is resonable.When some of the zeros z in C(z), i.e. the roots of B(z) fall outside the unit circle injthe Z-plane, the inversion is simply not possible. The poles become zeros and vice versa through inversion, it is therefore a basic requirement that the roots of B(z) must be located entirely inside the unit circle. Unfortunately, that is most often not the case.Systems with zeros inside the unit circle only are called minimum phase systems, and we shall use the term excess phase systems for those with zeros outside the unit circle only. Hence, any transfer function C(z) can be decomposed into a product of aminimum phase part and an excess phase part according to eq. 2.3 with C(z) possiblyepalso containing a pure delay.Now, an i mportant feature of the decomposi ti on i n eq. 2.3 i s that all magni tudei nformati on of C(z) is held in C(z) whereas the magnitude of C(z) always will bemp epunity and thus C(z) consti tutes an allpass fi lter. Accordi ng to the arrangement of epzeros of minimum phase functions we can invert C(z) but not C(z), see [2] and [12],mp epmeaning we can compensate for the magnitude and the minimum phase of the system transfer function but not the excess phase.If excess phase however do not contribute with any significant audible phenomena we do of course not need to worry about i t not bei ng equali zed. Si nce excess phase usually do undertake an extensi ve part of the enti re phase of a loudspeaker/room transfer function it is worth while to investigate its audibility in order to get a measure of how severe this commonly accepted omission in fact is.(3.1)3.H OMOMORPHIC DECONVOLUTIONThe experi ments thus requi re a method to separate the mi ni mum phase and the excess phase parts of an impulse response. Such separation can be accomplished by employing the techniques of homomorphic deconvolution [11]. Homomorphic deconvolut on takes advantage of a si gnal analysi s domai n i n whi ch transformed si gnals are called cepstra. It can be shown that if a signal contains minimum phase only then its cepstrum will turn out to be causal. Similarly, given a causal cepstra it is ensured that it represents a time domain signal containing minimum phase only. Consequently the minimum phase part of a signal can be extracted by first forming the cepstrum, then delete any non-causal information, and finally turning back to the time domain. Transformati on from the ti me domai n can be executed by employi ng the Discrete Fourier Transform in the following three steps:•Perform Fourier Transform of the sequence s[n].•Take the natural logarithm of the absolute value.^•Perform inverse Fourier Transformation yielding the cepstrum s[n].Because of the finite resolution in computable L-points Discrete Fourier Transforms this algorithm can only be an approximation [12]. In mathematical terms it becomes:The most important property of this transformation is perhaps the introduction of a nonlinear operator - the logari thm. The cepstrum exhi bi ts as a consequence the property of mappi ng the convoluti on operati on i n the ti me domai n i nto addi ti on through the three relationships of the filtering operation:time domain-convolution / deconvolution frequency domain-multiplication / division cepstral domain-addition / subtractionThe extracti on of C(z) from C(z) i nvolves deconvoluti on, and now i t becomesmppossi ble to perform deconvoluti on si mply by subtracti on, operati ng i n the cepstral domai n. Hence, if the non-causal part of the cepstrum is subtracted (i.e. removed), the resulting causal cepstrum represents a minimum phase system. The scheme of phase splitting linear systems is depicted in fig. 2. The minimum phase part is found in the cepstral domain by multiplying s[n] with an appropriate window l, leaving only the^mincausal part of the cepstrum, see eq. 3.2, where u[n] i s the uni t step functi on. The(3.2)(3.3)(3.4)(3.5)(3.6) excess phase part is then determined in the frequency domain by dividing the spectrum of the minimum phase part into the original spectrum according to eq. 2.3. Allthough the above mentioned method is employed through this work another way of obtaining the minimum phase part of an impulse response is to take advantage of the properti es of the Hi lbert Transform (HT), see e.g. [13]. Thi s transform li nks together magni tude and mi ni mum phase, so that mi ni mum phase can be deri ved unambi guously from the magnitude, see eqs. 3.3 - 3.6. If one is restricted to operate with relatively small lengths of DFT the Hilbert Transform method is more accurate.4.L ISTENING TEST CONDITIONSIf we take an isolated look at the magnitude of the frequency spectrum of an excess phase part we will expect an almost unmeasurable impact. It is known however that the human hearing system simultanously employs both time- and frequency domain information. Methods for simultanous time- and frequency analysis do exist [9], [14] -[17] but no unambigous connection between these methods and the hearing seems to have been proven. So i t i s necessary to turn to a subjecti ve analysi s method, - the li steni ng test, see e.g. [18]. There are numerous ways to plan and i mplement such tests, and standards have been made to ensure some kind of objectivity, see e.g. [19]. The test in this work is very simple. It will just be a matter of judging if there is a difference between two signals or not.4.1 Two related experimentsThe excess phase part of an impulse response can now be extracted and regarded as a filter, and it is obvious to compare a signal modified by this filter with an unmodified version. In real li fe however the excess phase exi sts uni solated i n conjuncti on wi th mi ni mum phase and magnitude which leads to the question: Will the isolated effects of excess phase be masked sufficiently by minimum phase and magnitude?Hence two experimental scenaria are of interest given the excitation signal s[n] and a loudspeaker/room i mpulse response c[n]. One i n whi ch an unmodi fi ed si gnal i s compared with the exces phase filtered signal, and another in which a minimum phase filtered signal is compared with a signal filtered by the entire response:EXPERIMENT 1 A s[n]B s[n]c[n]ephEXPERIMENT 2 A s[n]c[n]mphB s[n]c[n]c[n]mph eph4.2 Excitation signalsIn order to comply with statistics and in order to reveal possible signal dependencies we chose three different excitation signals - everyday well known types of signals:a) A female voiceb) A male voicec) A guitar soloAll three are recorded anechoi cally, i.e no si gni fi cant transfer functi on colourati on i mposed, and are sampled from the Compact Disc, "Music for Archimedes", CD B&O 101 1992. It is common to use artificial signals as well, e.g. white noise but since this experiment is characterized by a somewhat random phase, it is considered not well suited. The signals are of length app. 6 sec. each which is a reasonable compromise between the short term memory of the ear and time to get a good impression of the signal properties.4.3 Loudspeaker & roomsIn fi gs. 3-5 are shown the characteri sti cs of the loudspeaker, a KEF 107. Its magni tude deviation from 0dB is less than ±2dB in the range 60Hz to 12kHz. It is assumed that the phase contri buti on from the mi crophone, bei ng a B&K 4133, i s negligible in the range 60Hz to 12kHz.We chose for the experiment not only one room impulse response. In fact we tested for three di fferent si tuati ons: Two i nherently di fferent i mpulse responses from two di fferent rooms called R1 and R2, these being of length 300msec. and for room R1 also a shorter version of 150msec. In figs. 6-11 are shown the three characteristics in both time and frequency domains. Room R1 is a standard listening room with reverberation time 0.45sec. Room R2 is larger and less damped having a reverberation time of 0.6sec.After 300msec. the response of room R1 is down 40dB (20dB for 150msec.), and for room R2 i t i s down 30dB. The mi ni mum- and excess phase parts are found by the method depicted in sec. 3 and the time domain response, the magnitude spectrum, and the phase spectrum of these are shown in figs. 12-29.4.4 Obtaining impulse responsesThe impulse responses c[n] are obtained by use of a PC based MLSSA system. That is a Max i mum Length Sequence no i se burst, denoted a[n], is generated and transm i tted through a loudspeaker positioned in the room. By a microphone positioned elsewhere in the room the resulting sound pressure, b[n], is recorded, and because of the special cross correlation properties between a[n] and b[n], the spectral properties of the impulse response c[n] equal those of b[n]. The delay in c[n] due to the physical distance between loudspeaker and microphone is removed since it does not represent any colourat i on. Each response i s obta i ned as the average of 20 independent measurements in order to cancel out additive random noise. The noise burst a[n] needs not to be measured since it is artificially generated and therefore well known.We are not reachi ng for thorough determi nati on of the room acousti cs but only searchi ng for an arbitrary but representative loudspeaker/room transfer function. Thus the only constraint on the arrangement of loudspeaker and microphone is to place the latter in a diffuse sound field.All signals and responses are sampled at 48kHz and bandlimited upwards to 12kHz and downwards to 60Hz, si nce no si gni fi cant si gnal energy appears outsi de these boundari es.4.5 Test sequencesEach test signal applied to each test object in the two tests yield 18 AB pairs. To enhance statistics and to reveal the amount of consistency each pair is presented both forwards AB and backwards BA. Thus 36 test sequences emerge. The test sequencesare recorded on DAT in random order, and the task for the test persons is after each sequence to judge whether or not they beli eve i n a di fference between A and B. Of course, there is for all 36 sequences a difference - the question is whether or not it is audible.4.6 Implementation of the test10 persons were submitted to the experiment. They were not tested for their hearing capability. It must be assumed that when the phase modifications are imposed in the wide frequency range 60Hz to 12kHz, it will not be necessary to evaluate for frequency dependent hearing losses. All test persons were aged 20-25 and we may thus discard the effects of high frequency hearing loss due to age.The test sequences were presented through headphones - diotically at an individually adjusted 'pleasant' level. The persons were asked if the level was too low or too high in order to comply with their individually most comfortable levels. The headphones used were AKG K240DF. The reason for the cho ce of headphones nstead of loudspeakers is the desire to minimize disturbing phase contributions.5. R ESULTSSince 10 persons li stened to 36 sequences, the number of judgements are 360. Of those there were 243 votes for a perceivable difference, that is a 'there-is-difference'score of 68%. In the following table the results are given as function of each parameter.Since i n these experi ments we do not expli ci tly know the a pri ori probabi li ty of a 'there-i s-di fference' answer, it is only possible to carry out a statistical test confirming or rejecting the obvious hypothesis: The excess phase contributions are not significantly audible, by accepting a harsh estimate of the 'there-is-difference' propabality based on the exper i ment. Such est i mat i on i s not cons i dered, s i nce rel i able results must presumably imply a larger statistical material.E XPERIMENTAL ISSUE N UMBER OF VOTEStotal there-is-differenceE XPERIMENT NO . 1180 160 89% E XPERIMENT NO . 2180 83 46%R OOM R1240 139 58%R OOM R2120 103 86%R ESPONSE LENGTH 150msec.120 74 62%R ESPONSE LENGTH 300msec.240 168 70%E XCITATION SIGNAL A)120 89 74%E XCITATION SIGNAL B)120 100 83%E XCITATION SIGNAL C)120 55 46%The test persons show excellent consistency. Testing for reverse-sequence consistency shows that of all 'there-is-difference' votes 51% is given on AB sequences and 49% on BA sequences. Hence there is no bias and the votes on AB sequences and BA sequences are equal reliable.6.D ISCUSSIONNow, it is obvious to pose the following questions:•Why are the number of 'there-is-difference' votes in test no. 2 significantly lower than in test no. 1?•Why do the three excitation signals yield different number of 'there-is-difference' votes?•Why does a long impulse response cause greater impact than a short one?•Why are impulse responses from different rooms yielding different results?The results listed in sec. 5 clearly indicate that it is not indifferent whether excess phase is thrown away. Considered as an isolated filter in experiment 1, it affects an anechoic signal sufficiently for the human ear to detect it almost every time. This is not a reali sti c si tuati on however. The excess phase appears as part of the enti re i mpulse response, and when we try to remove it, the audible difference from the entire response i s much less that before, despi te the fact that i n both experi ments the absolute difference in signals is the same. This result indicates that the main part of an impulse response (the minimum phase part which holds the magnitude information) is able in some sense to mask the effects of the excess phase.Another result of the experiments is that when room reverberation time increases,the excess phase effects become more audi ble, and when the length of the i mpulse response is increased the same happens. In both cases it can be explained by the fact that the excess phase part increases in relation to the minimum phase part.We can also see that the audible impacts are most significant when a male voice is used, slightly decreasing effects on female voice, and a remarkable decrease when it comes to guitar music. One possible explanation is that speech signals contain some short moments of almost si lence where the effects of reveberati on (and thus excess phase) will be emphasized.In loudspeaker/room equalization tasks one of the major signal processing issues are whether or not to deal with the excess phase in the response. Since Ohm in the last century clai med that percepti on of sound only depends on i ts frequency domai n magni tude i t has been wi dely di scussed to what extent phase changes are audi ble.Experiment have shown clear audibility and others none at all - the answers depending highly on the specific test parameters.7. C ONCLUSIONSSo far i t has rema i ned unanswered whether or not the excess phase i n a loudspeaker/room transfer function is audible. Our experiments have clearly shown that it is in fact perceivable. We have also accomplished to show which parameters influence the degree of audibility.The conclusion must be: We can not allow the excess phase to be neglected, and we will have to get arround the equalization task in another way. One possible solution is to i ntroduce a delay i n the transfer functi on [12] thus enabli ng the possi bi li ty of noncausal equali zati on transfer functi ons. Another way i s to multi ply the transfer functi on with an unstable allpass filter thus matching the zeros outside the unit circle with equally placed poles [13]. In order to seal the excess phase zeros thi s method requi res strict knowledge on the arrangement of the zeros through a very thorough system identification process.But even if a total equalization could be accomplished we must remember that there are still heavy practical problems left. The fact that these kinds of equalization are based on one unique playback path from loudspeaker to the listener giving one unique transfer function on which to operate is perhaps a too gross simplification of reality.Works deal i ng w i th these pract i cal problems often tend to use some k i nd of adaptab i l i ty of the fi lters [3] - [5] someti mes based on an average of more i mpulseresponses [20] ri si ng from di fferent recei vi ng posi ti ons i n the room. In such cases subtle phase effects can, no matter how important, of course not be taken into account. In headphone reproducti on however a major part of these problems are removed. Magni tude/mi ni mum-phase equalization of headphone transfer functions is possible yielding increased performance. Again, in this situation the excess phase contribution survives and the audible impact must resemble that found in experiment 1.8.R EFERENCES[1] C. Bean and P. Craven, "Loudspeaker and room correction using digital signal processing." Presented atthe 86th Convention 1989, Hamburg. An Audio Engineering Society preprint.[2]J. Mourjopoulos, "Digital equalization methods for audio systems." Presented at the 84th Convention 1988,Paris. An Audio Engineering Society preprint.[3]J. Kuriyama and Y. Furukawa, "Adaptive Loudspeaker System," J. Audio Eng. Soc., Vol 37, No. 11, 1989November.[4]R. P. Genereux, "Adaptive Loudspeaker Systems: Correcting for the Acoustic Environment." Presented atthe AES 8th Int. Conference, Washington D.C. May 1990.[5]P. G. Craven and M. A. Gerzon, "Practical Adaptive Room and Loudspeaker Equaliser for Hi-Fi Use."Presented at the 92nd Convention 1992, Vienna.[6] D. Preis, "Phase Distortion and Phase Equalization in Audio Signal Processing - A Tutorial Review," J.Audio Eng. Soc., Vol 30, No. 11, 1982 November.[7]R. Plomp and H. J. M. Steeneken, "Effect of Phase on the Timbre of Complex Tones," J. Acoust. Soc. Am.46 part 2, 1969.[8] F. A. Bilsen, "On the Influence of the Number and Phase of Harmonics on the Perceptability of the Pitchof Complex Signal," ACUSTICA Vol. 28, 1973.[9]L. G. Johansen and P. Hazell, "Joi nt Ti me-Frequency Analysi s Tools: Appli cabi li ty i n the Desi gn ofLoudspeaker/room nterface Equal zat on F lters." M. Sc. thes s, Aalborg Un vers ty, Inst tute for Electronic Systems 1994.[10]J. A. Pedersen et al., "The Di stri buti on of the Low Frequency Sound Fi eld and i ts Relati on to RoomEqualization." Presented at the 96th Convention 1994 February, Amsterdam.[11] A. V. Oppenheim and R. W. Schafer, "Discrete-Time Signal Processing," Prentice-Hall International Inc.1989.[12]S. T. Neely and J. B. Allen, "Invertibility of a room impulse response," J. Acoust. Soc. Am. 66 (1) July1979.[13] A. Ambardar, "Analog and Digital Signal Processing," PWS Publishing Company, Boston 1995.[14]L. Cohen, "Time-Frequency Distributions - A Review," Proceedings of the IEEE, vol. 77, no. 7, July 1989.[15] C. P. Janse and A. J. M. Kaizer, "Time-Frequency Distributions of Loudspeakers: The Application of theWigner Distribution," J. Audio Eng. Soc., Vol 31, No. 4, 1983 April.[16] F. T. Agerkvist, "Time-Frequency analysis with temporal and spectral resolution as the human auditorysystem." The Acoustics Laboratory, Technical University of Denmark.[17]N. Yen, "Ti me and frequency representati on of acousti c si gnals by means of the Wi gner di stri buti onfunction: Implementation and interpretation," J. Acoust. Soc. Am. 81 (6) June 1987.[18] F. E. Toole, "Listening Tests - Turning Opinion into Fact," J. Audio Eng. Soc., Vol 30, No 6, 1982 June.[19]CCIR - Recommendation 562-2 (1993): Subjective assessment of sound quality.[20]S. J. Elliott and P. A. Nelson, "Multiple-Point Equalization in a Room Using Adaptive Digital Filters,"J. Audio Eng. Soc., Vol 37, No 11 1989 November.Figure 1: The equalizer and loudspeaker/room systemFigure 2: Phase splitting by homomorfic deconvolution。