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Applied Acoustics, Vol. 49, No. 4, pp. 307-320, 1996 0 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0003-682X/96/$15.00+ .OO

PII: SOOO3-682X(96)00028-X

A Comparison of the Frequency Modulation Transfer Function with the Modulation Transfer Function in a Room

Leon Rutkowski

Institute of Acoustics, A. Mickiewicz University, Matejki 48/49, PL 60-769, Poznali, Poland

(Received 13 December 1995; revised version received 6 June 1996; accepted 30 July 1996)

ABSTRACT

Room transmission properties for the frequency modulated (FM) band of noise have been investigated and compared with the calculated amplitude modulation (AM) transfer function. Room modulation attenuation for both kinds of modulation is taken into account. The comparison of the calculated average values of the modulation transfer shows that the mean modulation attenuation is comparable for the AA4 and FM modulation and in a similar way depends on the reverberation time and the modulation frequency. Selected experimental results are also presented to explain room transmission proper- ties for FM signals. 0 1997 Elsevier Science Ltd. All rights reserved

Keywords: Frequency modulated sound transmission, distortions in a fre-

quency domain.

INTRODUCTION

Natural sound such as speech and music can (from an elementary point of view) be treated as sound with simultaneously varying envelope and frequency. In order for a sound waveform to be transmitted in a room accu- rately, not only should the signal envelope changes be reproduced correctly, but also the sound frequency changes should be preserved. Fulfilling these requirements in a room is impossible and not really required because the room (with properly designed acoustics) introduces positive effects. Some sound changes in a room are not desirable and are classified as distortions. Knowledge concerning the sound envelope as influenced by the room and also frequency distortions are very important for improving room transmis- sion properties.

307

308 L. Rutkowskt

The modulation transfer function (MTF) as defined by Schroeder’ deter- mines the amplitude modulation (AM) depth changes caused by room transmission properties. For an ideal room with exponential sound intensity decay and no interfering noise the MTF may be calculated from

MTF=lOlog I+ [ (2;:-:,;]-“2

where jk is the amplitude modulation frequency and T denotes the room reverberation time for a 60 dB sound decay. The MTF function for an ideal room decreases with increasing frequency jl, of intensity changes. It means that the higher the AM modulation frequency the lower the modulation depth (the envelope is smoothed). Thus the room response for envelope changes is similar to low-pass filtering with a cut-off frequency dependent on the room reverberation time.

So far, much less attention has been focused on sound frequency change distortion in a room. Neglecting the simultinarity of these two kinds of sound changes (that is envelope and frequency), it is interesting to see how sound with constant envelope and only a frequency change is transmitted in a room. Previous investigations *J indicated significant distortions of the original frequency modulation (FM) signal frequency changes transmitted into a room. The main aim of this paper is to compare the room transmis- sion for AM and FM described by the calculated MTF with the experimental FMTF (frequency modulation transfer function). It is supposed that signal changes, independent of their character, depend in a similar way on already known room transmission properties. Some factors influencing the FM sig- nal distortions in a room are also mentioned and explained.

THE FM SIGNAL DISTORTIONS IN A ROOM

The FM signal distortions may be analysed on the basis of superposition of the direct and reflected waves with different amplitudes (amplitude ratios) and time delays (relative phase shifts). *J Let us consider the FM signal in a room (direct or reflected) with frequency changes of the form

.f(t) =.fb + Aj cos (27tj;t) (2)

wherejo is a carrier frequency, Ajis a frequency deviation and j; is a mod- ulation frequency. The same signal delayed by t will have frequency changes

.f(t - r) =.f;, + 4f' cos (2?rf;n(t - 5)) (3)

Comparison of the FMTF with the MTF 309

The instantaneous frequency difference (IFD) for both signals is

IFD(t) =JTt - r) -f(t) = 2Af sin (2~& G) sin [2nfm (t - I>1 (4)

It can be seen from eqn (4) that only for signals with in-phase frequency is there a change in the resultant, and the original signal frequency changes will not differ. For the opposite extreme case, that is for signals with momentary frequency changes in anti-phase, the value of the momentary frequency dif- ference can reach values over twice the frequency deviation of the FM signal and will periodically change in time. In reality intermediate phase shifts between signals are most probable. The momentary frequency differences and amplitude ratios for reflections are the main reason for instantaneous frequency changes in a room. As a result of superposition of direct and single reflected waves, effects similar to beating will occur with specific amplitude envelope and instantaneous frequency changes.3 The difference between beating and the FM signals lies in the constant frequency difference and amplitude ratio for beating and their variability for the FM signals. Thus, generally the distorted FM signal measured in a room can be treated as a real narrow-band signal with varying envelope and phase

r(t) = e(t) cos (q(t)) = Re [e(t) exp (jdt))] (5)

where e(t) is an envelope, and q(t) is an instantaneous phase. The instanta- neous frequency (IF) is defined as the phase time derivative

IF(t) = 2ndt 1 dd0 LHzl

Equation (5) can be rewritten in the following form

In ]e(t)] + j2n IF(t) dt s >I

where the argument of the exponent is the complex instantaneous phase. Now, the complex instantaneous frequency CIF(t) equals

CIF(t) = iL!%.o! + jIF(t) 2n le(t)j dt 1 [Hz] (6)

Note that the real part of the CIF depends only on the envelope changes and the imaginary part is the instantaneous frequency IF. For the FM sig- nals it is convenient to refer the frequency changes to the constant carrier

3 10 L. Rutkowki

frequency jb. For this assumption IF(t) =jb +j(t), where j(t) is a function describing the frequency changes only. Therefore the magnitude of the com- plex instantaneous frequency is given by

ICIF(t)l = 1 1 d/e(t)] 2

--~ 2n le(t)l dt 1 +h +AO12 W4 (7)

The ICIF(t)l is a real function and depends on the envelope and phase changes of the real signal (eqn (5)). For signals with constant envelope, as the FM signal transmitted into a room, the ICIF(t)l equals the non-distorted IF(t) function. It may be proved that for the narrow-band signals, the value of the magnitude in eqn (7) depends mainly on the IF, i.e. iCIF(t)l z IIF(t)l, and the influence of envelope changes can be neglected. From the experi- mental point of view the above also means that a good approximation of the IF changes are reciprocals of the successive zero-crossings of the signal in the same direction.

Thus, according to eqn (7) for the narrow-band signal, iCIF(t)l% IF(t) =f;, +f(t). Usually ,f(t) = AfF’(t), where Af is a value of frequency devia- tion (the extent of frequency changes) and F(t) is a slow changing modulat- ing function. Thus, it can be seen that all required information about the signal frequency variability depends on the frequency deviation and the course of the modulating function. Generally, even for the quoted simplest sinusoidal frequency changes of the signal transmitted into a room, the received one will be complex.3.4

Figure 1 shows instantaneous frequency changes recorded in a real room (in semi-free sound field) and a histogram illustrating the instantaneous fre- quency distributions. The signal carrier frequency jb equalled 500 Hz, the frequency deviation .Af’= ZIZ 70 Hz and the modulation frequency,j;,, = 0.8 Hz. Original sinusoidal frequency changes in the range from 430 to 570 Hz for a transmitted signal are disturbed by additional changes introduced by a room. These changes have a periodic character. The specific distortions appear in the modulation period interval and have similar individual values. In Fig. l(b) the normalized histograms for transmitted (dotted line) and recorded (solid line) frequency changes are presented. The frequency interval had the width of 5 Hz, so this number of histogram classes depends on the extent of frequency changes. The normalization was done by dividing the number of counts in all the histogram classes by the maximum value of counts. The frequency distributions in Fig. l(b) are slightly different for transmitted and received signals. It proves that distortion caused by a room may be neglected in such conditions.

Figure 2 is similar to Fig. 1 but with a modulation frequency equal to 3.6 Hz. Extreme frequency changes significantly differ from the original and

Comparison of the FMTF with the MTF 311

800 t

y 700

- 600 ^ E 500

E 400 i

300 c

(a) ‘1

0 I

1

Periods 2

I I I I I 1

1.0 - (b) - 0.8 -

0.6 - a

0.4 -

0.2 -

0.0 - I I 1 I I I

300 400 500 600 700 800

ICIF(t) I [Hz1

Fig. 1. Sine-by-sine FM modulation distortions for semi-free field in a room: (a) complex instantaneous frequency changes, (b) normalized histograms for these changes (solid line) and for an original signal transmitted into a room (dotted line). (The carrier frequency: 500 Hz,

frequency deviation: f 70 Hz, modulation frequency: 0.8 Hz.)

exceed the maximum IFD(t) value. Such frequency deflection appears as a result of superposition of waves with slightly different amplitudes.2 The original modulating sinusoidal function F(t) takes a complex periodic form. The recorded signal frequency change has a wider range than the changes of the original signal. Also the path of the recorded signal frequency distribution (Fig. 2b) is quite different from the sinusoidal frequency changes distribu- tion. According to the distortion introduced by a room, the frequency deviation for modulation frequency fm will be different from its original value. The frequency interval between two maximum values of recorded signal frequency distribution equals approximately l 66 Hz. It gives the frequency deviation of most probable frequency changes equal * 33 Hz and it is about two times lower than the original signal frequency deviation. The results similar to those presented in Figs 1 and 2 including computer calcu- lations are also reported in a forthcoming paper.5 They reflect the fact that the effect of modulation frequency increase is one of the main factors deter- mining the deviation attenuation in a room.

312 L. Rutkowski

The response for a frequency change is one of the most interesting features of sound transmission in a room.3 A good example of the measured specific response of the room for a step frequency change is shown in Fig. 3. The frequency changes its value from 500 to 375 Hz at the moment assigned as zero on the time axis. The parameter 6 is the amplitude ratio for signals with frequencies 375 and 500 Hz measured in a steady sound state. After a

;;; 700- I - 600- x z 500-

z 400- -

300 - 1 I 1 0 I 2

Periods

0.0 0 I / I I 1 I

300 400 500 600 700 800

ICIF(t)l kIz1

Fig. 2. As Fig. I, but for reverberation sound field and modulation frequency 3.6 Hz.

z 600- Af=-125Hz

- 500- ;; 5 400-

z 300-

200 - , / / I 0 I I

-50 0 50 100 150 200 250

Time [msl

Fig. 3. Example of the room response for step frequency change. (The initial frequency: .f; = 500 Hz. the final frequency: b = 375 Hz, 4f= f2-fi, 6-an amplitude ratio-see text.)

Comparison of the FMTF with the MTF 313

frequency change the amplitude envelope for the initial frequency 500 Hz decreases. At the same time the amplitude envelope of the signal with the final frequency 375 Hz increases. Omitting some details explained in Ref. 3, let us notice the oscillating character of instantaneous frequency changes. In the initial phase of changes these oscillations take place around the initial frequency 500 Hz. Beginning from the moment of the envelope’s equaliza- tion, the oscillations take place around the final frequency 375 Hz and decay to the final frequency value. Thus, the final value of the frequency change is not immediately reached in a room and requires a time interval equal to the room reverberation time.

SIGNAL PARAMETERS AND METHOD

The experimental set-up and method were similar to those reported in Ref. 4 except for the excitation signal. Now the excitation signal has been prepared as a FM band of noise. The carrier signal was a band of noise and the modulating signal was sinusoidal. The main advantage of this kind of testing signal in comparison with sine-by-sine frequency modulation is smaller dependence of the results on the microphone and source position in a room. The excitation signal preparation set-up is shown in Fig. 4(a). A band of noise signal created in the array processor card (2) has been sent to a 16-bit digital-to-analogue converter DAC2 (5) and played with a FM sampling frequency. The external sampling signal from a square wave generator DACl (4) was fed to the EXT.CLKIN input of the DAC2. Frequency change of this signal isf CLK1v(f) =f& + Afs sin(2rrf,t), wheref& is a carrier of the sampling frequency, Afs is its frequency deviation and fm denotes a modulation frequency. In such a way on the DAC2 output a FM band of noise has appeared.

The test signal parameters were as follows:

l ‘carrier’ frequencies: 25&4000 Hz in octave order; l modulation frequencies: 1, 2, 4, 8, 16 and 32 Hz; l frequency deviations: always equal to 10% of the noise band centre

frequency; l noise bandwidth: one-third octave.

Such a wide range of the test signal parameters does not completely fulfil the narrow band condition for all the testing signals. It seems important for the carrier frequency of 250 Hz and higher modulating frequencies. How- ever, the main factor determining the narrow band condition that is the frequency deviation chosen as 10% of carrier frequency, seems to be sufficient. Owing to the equality Afs = O.lfos, the required value of signal frequency deviations was guaranteed.

314 L. Rutkowskl

Testing signals with 30 specified parameter combinations have been recor- ded on a DAT recorder (6). Next, a signal from prepared cassette records was transmitted into the room and recorded on a separate DAT recorder. The signals recorded in the room were frequency demodulated on a special demodulation card (@-Fig. 4b), then digitally low-pass filtered and ana- lysed with the fast Fourier transform (FFT) algorithm. Simultaneously, the signal from the DAT recorder (6) was converted to a digital form by means

(4

(b)

i

r

12.bit DACl

, EXT CLKIN r

Array Processor Card . with 50 MHz DSP32C . 16-bit DAC2

2 '. 5 L

PC 466 DX2 1

1 DAT RECORDER

Fiber-optic cables 6 \

._~_______ ~~____

Fiber optic cables

16.bit ADC Array Processor Card with 50 MHz DSP32C

2 PC 406 DX2 -

1 DAT RECORDER FREQUENCY -

4 DEMODULATION CARD

6 8

Fig. 4. The block diagrams of the experimental set-up: (a) set-up for excitation signals pro- cessing and registration, (b) set-up for signal analysis. (I, 3-IBM PC computers, 2-array processor card, 4. 5-Digital-to-analogue converters, 6-DAT recorder type PCM 2700A, 7--

analogue-to-digital converter, 8--frequency demodulation card.)

Comparison of the FMTF with the MTF 315

of an analogue-to-digital converter ADC (7). At this stage of the experiment this signal was used to trigger the analysis process.

The main evaluated parameter was the average amplitude of the FFT spectral component for the given modulating frequency &, which corre- sponds to the average frequency deviation of the received signal. For the testing signal, the amplitude of this component equals the frequency devia- tion of the stimulus.

Next the FMTF as a relative measure of the FM signal distortion caused by a room was calculated from

FMTF&) = 10 log # [dB] t 111

where Afr is a frequency deviation of the received signal, Aft is a frequency deviation of the signal transmitted into the room. So the defined FMTF function equals zero if no distortions are introduced in a room. The negative FMTF value means a reduced frequency deviation. Note that only one component of the complex function describing frequency changes, that cor- responds to modulation frequency fm (cf. Fig. 2) was taken into account.

The measurements were carried out in two different rooms in a diffuse sound field. The reverberant conditions were evaluated with the Briiel & Kjaer Modular Precision Sound Level Meter type 2231 programmed with the Room Acoustic Module type BZ 7109. Ail the measurements including the reverberation time and room transmission properties for the FM signal were performed in the same selected positions. The first room (ROOMl) has dimensions 5.5 mx5.5 mx3.5 m, and the second (ROOM2) 33 mx2 mx3.5 m. From the measured reverberation times, the MTF functions could be simply calculated from eqn (1). Next the FMTF functions were calculated MTF functions.

RESULTS AND DISCUSSION

compared with the

The experimental FMTF (plotted as open circles) and the calculated MTF (plotted as asterisks) functions for the two rooms examined are shown in Fig. 5. Each single pair of relations concerns the specified centre frequency of the noise bandfc. The measured value of the reverberation time T for these centre frequencies is also given within each diagram.

Consider first the results for the FMTF functions. As can be seen from Fig. 5 frequency changes may be transmitted in a room almost without dis- tortion only for the lowest modulation frequency. Generally, the FMTF functions decrease with modulation frequency increase, however some

f ,=250Hz -_-- 7

h ‘\i

T=0.58s

/ . *.*1.1+ .+

r"P

,.,4d * ~, T=1.45s *

L. Rutkowki

f ,=500Hz f,=1000Hz f ,=2000Hz f ,=4000Hz v

,- . ~._---l

I, T=0.76s T=0.715

ROOM 2

1 2 4 8 1632 I 2 4 8 1632 I 2 4 8 1632 I 2 4 8 1632 1 2 4 8 1632 Modulation frequency [Hz]

M FMTF H MTF

Fig. 5. Experimental FMTF functions (open circles) and calculated MTF functions (asterisks) for ROOM I and ROOM2 in the diffuse sound field. (J-band of noise centre frequency, T-

reverberation time.)

irregularities can be observed especially for higher values of the modulation frequency. The FMTF function decrease may be partly explained on the basis of the distortions shown in Fig. 2. Some additionally appearing instantaneous frequency changes destroy the original frequency changes decreasing the value of the signal deviation. To find the main factors influ- encing the observed FMTF function changes we must take into account the introduced by the room phase changes. Kuttrup has already argued that a room introduces the signal phase randomization. The standard deviation of the random phase shifts in a room increases with the distance from a signal source. This phase randomization may also change the original instanta- neous frequency changes and their distributions (cf. Ref. 4 and Figs 1 and 2). An instantaneous frequency as a phase time derivative is strongly sensitive to phase distortions.

On the other hand, knowing the room response for frequency changes it is possible to draw (Fig. 3) conclusions concerning the rate of frequency chan- ges. The final frequency of frequency change in a room is reached after the time period equals the room reverberation time. If the period of a frequency change is greater than the reverberation time, the instantaneous frequency change (distortion) decays to the final value before a new change appears. For a faster frequency change, i.e. for a shorter period of frequency change, earlier frequency changes (distortions) overlap the current changes causing an additional distortion. For a greater value of reverberation time the rate of

Comparison of the FMTF with the MTF 317

frequency change, for which overlapping appears, will be lower. It may be expected that the reverberation time is one of the very significant factors which determine the FMTF versus modulation frequency slope.

For a higher frequency of modulation in ROOM2 the FMTF function increases. This increase is not observed for ROOMl, although for the 250 Hz centre frequency a slight increase exceeding an error bar takes place. Two reasons for these changes may be considered. The first one implies that they are connected with the existence of strong reflection with in-phase fre- quency changes related to a direct wave (see Section 2). The other one results from not quite satisfied ‘narrow-bandness’, although the modulation fre- quencies 16 Hz and also 32 Hz seem relatively small forfo = 250 Hz. On the other hand, the existence of strong reflections is most probable for low acoustic frequencies due to low acoustic absorption and it causes the clear periodicity in the FMTF changes. 2 Including these facts and taking into account the geometry of ROOM2, the considered increase in the FMTF function seems to be reasonable.

The minimum value of the FMTF function (i.e. the maximum measured deformation) for the tested rooms exceeds - 15 dB. For example the - 15 dB attenuation of the FMTF for ROOM2, f0 = 1000 Hz and the modulation frequency 32 Hz (Fig. 5) means that f 100 Hz of the original signal devia- tion value is transformed to f 3.2 Hz in the room! It seems that such a large deviation change (over 30 times) can be certainly perceived, and really exists.

The individual FMTF functions for different carrier frequencies have a specific path which does not clearly correspond to the measured reverbera- tion time values, although the reverberation time addtionally does not strongly depend on frequency for the both tested rooms.

Some similarities between the MTF and FMTF functions can be seen in Fig. 5. It should be remembered that the MTF functions, calculated from the reverberation times measured in the rooms assume some idealization of the exponential intensity decay in a room. To compare overall room transmis- sion properties the logarithmic mean values of the MT for all carriers and all modulation frequencies were separately calculated for the MTF and FMTF functions. Calculations were performed on the basis of the formula

MT = 10 log $ I WI

where N= 30 is the total number of the MT values for a given kind of mod- ulation and i is the index of carrier and j is the index of modulation fre- quency. The individual MT values correspond to the MTF or FMTF values for the AM or FM signal in question. The results of these calculations are shown in Table 1.

318 L. Rutkowski

TABLE 1 Average MT for selected rooms and AM and FM modulation

Room

ROOM I ROOM2

Average MT (dB) MTF FMTF

-2.8 -2.6 4.4 --3.9

The average MT values are similar for the MTF and FMTF functions. The similarity is preserved for two rooms. For ROOM1 the difference between the average MT for the FMTF and MTF equals 0.2 dB. The same difference for ROOM2 equals 0.5 dB. The average MT for the MTF functions is by 1.6 dB lower for ROOM2. Similarly, the average MT for the FMTF func- tion is by 1.3 dB lower for ROOM2. Let us notice that the mean reverbera- tion time for ROOM2 was 2.1 times ( z 3.2 dB) greater than for ROOM1 .

Taking the results for all ‘carrier’ and modulation frequencies it may be concluded that ROOM1 has better transmission properties for both the AM and FM signals. The slightly different average MT values for both kinds of modulation show a comparable extent of room generated distortions. Although the distortions are qualitatively different for the types of modula- tion considered it may be expected that they depend in a similar way on room reverberation time and frequency of modulation.

CONCLUSIONS

Before conclusions are reached, it is important to note that continuous envelope and/or frequency changes in a room cause a permanently existing non-steady-state. Generally, it seems that a room responds in a similar way to a change in signal, independently of which parameter, envelope or fre- quency, is changing. To explain signal distortion in a room for the AM and FM signals in terms of a steady- or non-steady-state in the room, it has to be assumed that for such signals a steady-state may be theoretically reached for the highest modulation frequency. For this assumption the envelope has to be completely smoothed and the frequency deviation zero. The modulation attenuation is a common feature of AM and FM signal distortion in a room. It strongly depends on room reverberation time. The similarity between the MTF and FMTF functions results from the similar room response for rapid envelope and frequency change. Although the changes for the AM and FM signal are qualitatively different, they are similar: the steady-state of the room response appears after a time equal to the room reverberation time.

Comparison of the FMTF with the MTF 319

The similar dependence of the MTF and FMTF functions on modulation frequency and on reverberation time is also important from the perceptual point of view. So far the MTF function has played a fundamental role in the study of speech intelligibility in a room (ST1 and RASTI methods7). Intro- duced in a room envelope smoothing gives the poorest speech intelligibility. Unfortunately there are no papers concerning the audibility of FM signal distortions in a room. The problem of the audibility ot the instantaneous frequency distortion in a room is complex because of simultaneous intensity changes as an additional distortion effect. 2,3 The author’s preliminary listen- ing tests for a band of noise FM signals recorded in a room show clearly noticeable attenuation of the frequency extent (deviation) for higher mod- ulation frequency. So far it has been difficult to indicate the commonly known sound feature which is affected by the instantaneous frequency dis- tortion in a room. In the author’s earlier paper* the instantaneous frequency and envelope changes observed in a room were referred to pitch changes.

It would be very interesting to know what the FMTF function relation to the room impulse response is (as for Schroeder’s MTF function). Some cru- cial experiments are also needed to explain the perceptual importance of the instantaneous frequency distortions. It is to be hoped that in the future those problems will become more clearly explained.

ACKNOWLEDGEMENTS

The main content of the paper was partly presented in 1995, Saarbrticken, Germany. The author expresses reviewer for his useful comments on the manuscript.

5.

6.

REFERENCES

DAGA ‘95, March, his thanks for the

Schroeder, M. R., Modulation transfer functions: definition and measurement. Acustica, 1981,49, 179-182. Ozimek, E. and Rutkowski, L., Deformation of frequency modulated (FM) signals propagating in a room. Appl. Acoustics, 1989, 26, 217-230. Rutkowski, L. and Ozimek, E., Linear and jump frequency changes of a signal in a room. Arch. Acoustics, 1995, 20(2), 115-138. Rutkowski, L. and Ozimek, E., Distributions of instantaneous frequency chan- ges of FM signal propagating in a room. Proceedings of DAGA ‘94, Part B, Dresden 1994, pp. 261-264. Rutkowski L. and Ozimek E., Linear and sinusoidal frequency changes of a signal in a room. AcusticalActa Acustica, 1997, in press. Kuttruff, H., On the audibility of phase distortions in rooms and its significance for sound reproduction and digital simulation in room acoustics. Acustica, 199 1, 74, 3-7.

320 L. Rutkowki

7. Houtgast, T. and Steeneken, H. J., M, the modulation transfer function in room acoustics as a predictor of speech intelligibility. Acustica, 1973, 28, 66-73.

8. Rutkowski, L., Room response to frequency change and its relation to the pitch changes. Arch. Acoustics, 1996, 21(2), 201-214.