ABSTRACT 1. INTRODUCTION

Transcription

1 THE EXCESS PHASE IN LOUDSPEAKER/ROOM TRANSFER FUNCTIONS: Can it be ignored in Equalization Tasks? Lars G. Johansen & Per Rubak DSP-group, Dept. of Communication Technology Institute of Electronic Systems, Aalborg University, Denmark {lgj, ABSTRACT The 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 controlled listening tests. That is, it can not be neglected in the design of equalization systems.. INTRODUCTION In high-fidelity reproduction of sound a crucial point is to obtain as little coloration as possible due to the reproduction system. We will define the term coloration as amplitude as well as phase distortion. Whenever signals are recorded, steps are taken to ensure that the recording system itself does not contribute significantly to the recording - at least not in the frequency range where human beings perceive sound. Microphones and analog/digital recording equipment can in general be designed to affect the signals only very little. - -

2 But when the recording is played back coloration effects are certainly imposed. 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 the impacts from the loudspeaker/room system, see fig.. Varoius attempts have been made to incorporate equalization in reproduction systems, see [] - [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 since nonlinear phase changes might give rise to difficulties in equalization tasks. 2. SYSTEM IMPULSE RESPONSES Design of equalization filters requires 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 is recorded by a microphone the output, called c[n], can be regarded as unambiguous 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-parameterized, can be regarded as one generated by a digital filter, and it is quite obvious that the ideal equalization filter must then comply with eq. 2.. (2.) The spectral information in C(z) is remarkably changed if the loudspeaker and microphone positions are varied just slightly []. Furthermore, the human hearing works binaurally. Hence two transfer functions are evoked and for higher frequencies - 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 difficulties will most certainly rise attempting to carry out the division in eq. 2.. Consider C(z) as the ratio of two rational polynomials with M respectively N distinct roots as in eq 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 nondeterminable. Nevertheless this model is resonable. (2.2) When some of the zeros z in C(z), i.e. the roots of B(z) fall outside the unit circle in j the 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 a minimum phase part and an excess phase part according to eq. 2.3 with C ep(z) possibly also containing a pure delay. (2.3) Now, an important feature of the decomposition in eq. 2.3 is that all magnitude information of C(z) is held in C mp(z) whereas the magnitude of C ep(z) always will be unity and thus C ep(z) constitutes an allpass filter. According to the arrangement of zeros of minimum phase functions we can invert C mp(z) but not C ep(z), see [2] and [2], meaning 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 it not being equalized. Since excess phase usually do undertake an extensive part of the entire 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

4 3. HOMOMORPHIC DECONVOLUTION The experiments thus require a method to separate the minimum phase and the excess phase parts of an impulse response. Such separation can be accomplished by employing the techniques of homomorphic deconvolution []. Homomorphic deconvolution takes advantage of a signal analysis domain in which transformed signals 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. Transformation from the time domain can be executed by employing 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 [2]. In mathematical terms it becomes: (3.) The most important property of this transformation is perhaps the introduction of a nonlinear operator - the logarithm. The cepstrum exhibits as a consequence the property of mapping the convolution operation in the time domain into addition through the three relationships of the filtering operation: time domain - convolution / deconvolution frequency domain - multiplication / division cepstral domain - addition / subtraction The extraction of C mp(z) from C(z) involves deconvolution, and now it becomes possible to perform deconvolution simply by subtraction, operating in the cepstral domain. 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 min, leaving only the causal part of the cepstrum, see eq. 3.2, where u[n] is the unit step function. The - 4 -

5 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 (3.2) 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 properties of the Hilbert Transform (HT), see e.g. [3]. This transform links together magnitude and minimum phase, so that minimum phase can be derived unambiguously from the magnitude, see eqs If one is restricted to operate with relatively small lengths of DFT the Hilbert Transform method is more accurate. (3.3) (3.4) (3.5) (3.6) 4. LISTENING TEST CONDITIONS If 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], [4] - [7] but no unambigous connection between these methods and the hearing seems to have been proven. So it is necessary to turn to a subjective analysis method, - the listening test, see e.g. [8]. There are numerous ways to plan and implement such tests, and standards have been made to ensure some kind of objectivity, see e.g. [9]. 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

6 4. Two related experiments The 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 life however the excess phase exists unisolated in conjunction with minimum 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 impulse response c[n]. One in which an unmodified signal is 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 A s[n] B s[n] c [n] eph EXPERIMENT 2 A s[n] c [n] mph B s[n] c mph[n] c eph[n] 4.2 Excitation signals In 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 voice b) A male voice c) A guitar solo All three are recorded anechoically, i.e no significant transfer function colouration imposed, and are sampled from the Compact Disc, "Music for Archimedes", CD B&O 992. 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

7 4.3 Loudspeaker & rooms In figs. 3-5 are shown the characteristics of the loudspeaker, a KEF 7. Its magnitude deviation from db is less than ±2dB in the range 6Hz to 2kHz. It is assumed that the phase contribution from the microphone, being a B&K 433, is negligible in the range 6Hz to 2kHz. We chose for the experiment not only one room impulse response. In fact we tested for three different situations: Two inherently different impulse responses from two different rooms called R and R2, these being of length 3msec. and for room R also a shorter version of 5msec. In figs. 6- are shown the three characteristics in both time and frequency domains. Room R is a standard listening room with reverberation time.45sec. Room R2 is larger and less damped having a reverberation time of.6sec. After 3msec. the response of room R is down 4dB (2dB for 5msec.), and for room R2 it is down 3dB. The minimum- 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 Obtaining impulse responses The impulse responses c[n] are obtained by use of a PC based MLSSA system. That is a Maximum Length Sequence noise burst, denoted a[n], is generated and transmitted 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 colouration. Each response is obtained as the average of 2 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 reaching for thorough determination of the room acoustics but only searching 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 2kHz and downwards to 6Hz, since no significant signal energy appears outside these boundaries. 4.5 Test sequences Each test signal applied to each test object in the two tests yield 8 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 sequences - 7 -

8 are recorded on DAT in random order, and the task for the test persons is after each sequence to judge whether or not they believe in a difference 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 test 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 6Hz to 2kHz, it will not be necessary to evaluate for frequency dependent hearing losses. All test persons were aged 2-25 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 K24DF. The reason for the choice of headphones instead of loudspeakers is the desire to minimize disturbing phase contributions. 5. RESULTS Since persons listened to 36 sequences, the number of judgements are 36. 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 in these experiments we do not explicitly know the a priori probability of a 'there-is-difference' 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 experiment. Such estimation is not considered, since reliable results must presumably imply a larger statistical material. EXPERIMENTAL ISSUE NUMBER OF VOTES total there-is-difference EXPERIMENT NO % EXPERIMENT NO % - 8 -

9 ROOM R % ROOM R % RESPONSE LENGTH 5msec % RESPONSE LENGTH 3msec % EXCITATION SIGNAL A) % EXCITATION SIGNAL B) 2 83% EXCITATION SIGNAL C) % The test persons show excellent consistency. Testing for reverse-sequence consistency shows that of all 'there-is-difference' votes 5% 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. DISCUSSION Now, 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.? 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, it affects an anechoic signal sufficiently for the human ear to detect it almost every time. This is not a realistic situation however. The excess phase appears as part of the entire impulse response, and when we try to remove it, the audible difference from the entire response is much less that before, despite the fact that in both experiments 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

10 Another result of the experiments is that when room reverberation time increases, the excess phase effects become more audible, and when the length of the impulse 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 silence where the effects of reveberation (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 claimed that perception of sound only depends on its frequency domain magnitude it has been widely discussed to what extent phase changes are audible. Experiment have shown clear audibility and others none at all - the answers depending highly on the specific test parameters. 7. CONCLUSIONS So far it has remained unanswered whether or not the excess phase in 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 introduce a delay in the transfer function [2] thus enabling the possibility of noncausal equalization transfer functions. Another way is to multiply the transfer function with an unstable allpass filter thus matching the zeros outside the unit circle with equally placed poles [3]. In order to seal the excess phase zeros this method requires 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 dealing with these practical problems often tend to use some kind of adaptability of the filters [3] - [5] sometimes based on an average of more impulse - -

11 responses [2] rising from different receiving positions in the room. In such cases subtle phase effects can, no matter how important, of course not be taken into account. In headphone reproduction however a major part of these problems are removed. Magnitude/minimum-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. 8. REFERENCES [] C. Bean and P. Craven, "Loudspeaker and room correction using digital signal processing." Presented at the 86th Convention 989, Hamburg. An Audio Engineering Society preprint. [2] J. Mourjopoulos, "Digital equalization methods for audio systems." Presented at the 84th Convention 988, Paris. An Audio Engineering Society preprint. [3] J. Kuriyama and Y. Furukawa, "Adaptive Loudspeaker System," J. Audio Eng. Soc., Vol 37, No., 989 November. [4] R. P. Genereux, "Adaptive Loudspeaker Systems: Correcting for the Acoustic Environment." Presented at the AES 8th Int. Conference, Washington D.C. May 99. [5] P. G. Craven and M. A. Gerzon, "Practical Adaptive Room and Loudspeaker Equaliser for Hi-Fi Use." Presented at the 92nd Convention 992, Vienna. [6] D. Preis, "Phase Distortion and Phase Equalization in Audio Signal Processing - A Tutorial Review," J. Audio Eng. Soc., Vol 3, No., 982 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, 969. [8] F. A. Bilsen, "On the Influence of the Number and Phase of Harmonics on the Perceptability of the Pitch of Complex Signal," ACUSTICA Vol. 28, 973. [9] L. G. Johansen and P. Hazell, "Joint Time-Frequency Analysis Tools: Applicability in the Design of Loudspeaker/room interface Equalization Filters." M. Sc. thesis, Aalborg University, Institute for Electronic Systems 994. [] J. A. Pedersen et al., "The Distribution of the Low Frequency Sound Field and its Relation to Room Equalization." Presented at the 96th Convention 994 February, Amsterdam. [] A. V. Oppenheim and R. W. Schafer, "Discrete-Time Signal Processing," Prentice-Hall International Inc [2] S. T. Neely and J. B. Allen, "Invertibility of a room impulse response," J. Acoust. Soc. Am. 66 () July 979. [3] A. Ambardar, "Analog and Digital Signal Processing," PWS Publishing Company, Boston 995. [4] L. Cohen, "Time-Frequency Distributions - A Review," Proceedings of the IEEE, vol. 77, no. 7, July 989. [5] C. P. Janse and A. J. M. Kaizer, "Time-Frequency Distributions of Loudspeakers: The Application of the Wigner Distribution," J. Audio Eng. Soc., Vol 3, No. 4, 983 April. [6] F. T. Agerkvist, "Time-Frequency analysis with temporal and spectral resolution as the human auditory system." The Acoustics Laboratory, Technical University of Denmark. [7] N. Yen, "Time and frequency representation of acoustic signals by means of the Wigner distribution function: Implementation and interpretation," J. Acoust. Soc. Am. 8 (6) June 987. [8] F. E. Toole, "Listening Tests - Turning Opinion into Fact," J. Audio Eng. Soc., Vol 3, No 6, 982 June. [9] CCIR - Recommendation (993): Subjective assessment of sound quality. [2] S. J. Elliott and P. A. Nelson, "Multiple-Point Equalization in a Room Using Adaptive Digital Filters," J. Audio Eng. Soc., Vol 37, No 989 November. - -

12 Figure : The equalizer and loudspeaker/room system Figure 2: Phase splitting by homomorfic deconvolution - 2 -

13 Loudspeaker impulse response Figure 3: Loudspeaker impulse response 2 Loudspeaker magnitude spectrum Figure 4: Loudspeaker magnitude spectrum 2 Loudspeaker phase spectrum 2 Phase [rad] Figure 5: Loudspeaker phase spectrum - 3 -

14 Entire impulse response 2 Entire magnitude spectrum Figure 6: Room R, 5ms impulse response Figure 9: Room R, 5ms magnitude spectrum Entire impulse response 2 Entire magnitude spectrum Figure 7: Room R, 3ms impulse response Figure : Room R, 3ms magnitude spectrum Entire impulse response 2 Entire magnitude spectrum Figure 8: Room R2, 3ms impulse response Figure : Room R2, 3ms magnitude spectrum - 4 -

15 2 Minimum phase part impulse response.3 Excess phase part impulse response Figure 2: Room R, 5ms Figure 5: Room R, 5ms 2 Minimum phase part magnitude spectrum Excess phase part magnitude spectrum Figure 3: Room R, 5ms Figure 6: Room R, 5ms Minimum phase part phase spectrum Excess phase part phase spectrum 5 Phase [rad] Phase [rad] Figure 4: Room R, 5ms Figure 7: Room R, 5ms - 5 -

16 2 Minimum phase part impulse response.3 Excess phase part impulse response Figure 8: Room R, 3ms Figure 2: Room R, 3ms 2 Minimum phase part magnitude spectrum Excess phase part magnitude spectrum Figure 9: Room R,3ms Figure 22: Room R, 3ms 5 Minimum phase part phase spectrum Excess phase part phase spectrum Phase [rad] Phase [rad] Figure 2: Room R, 3ms Figure 23: Room R, 3ms - 6 -

17 2.5 Minimum phase part impulse response.3 Excess phase part impulse response Figure 24: Room R2, 3ms Figure 27: Room R2, 3ms 2 Minimum phase part magnitude spectrum Excess phase part magnitude spectrum Figure 25: Room R2, 3ms Figure 28: Room R2, 3ms 5 Minimum phase part phase spectrum Excess phase part phase spectrum Phase [rad] Phase [rad] Figure 26: Room R2, 3ms Figure 29: Room R2, 3ms - 7 -