Alternative Encoding Techniques for Digital Loudspeaker Arrays

Transcription

1 Alternative Encoding Techniques for Digital Loudspeaer Arrays Fotios Kontoichos, Nicolas Alexander Tatlas, and John Mourjopoulos Audio and Acoustic Technology Group, Wire Counications Laboratory, Electrical Engineering and Coputer Engineering Departent, University of Patras, Patras, 65 Greece ABSTRACT Recent developents in Digital Loudspeaers have resulted in the introduction of Digital Transducer Arrays (DTA). In ost ipleentations, DTA loudspeaers are driven by PCM encoded audio signals, usually resapled and requantised to an appropriate nuber of bits, in accordance to the nuber of the transducers constituting the DTA topology. However, given that DTAs generally increase haronic distortion, especially for off-axis listening positions, optiization in signal encoding and bit-to-transducer assignent, is necessary. Here, a nuber of novel, alternative strategies are exained, concerning the input signal encoding via PCM-to-PWM conversion, as well as techniques for bit-assignent on the transducers of a DTA. These tests are supported by siulation results and coparisons between these alternative ethods, for different operating paraeters.. INTRODUCTION Current efforts on the study and developent of digital loudspeaers have led to the eergence of two viable architectures: (a) the Digital Transducer Arrays (DTAs) [] and (b) the Multiple Voice Coil Digital Loudspeaers (MVCDLs) []. In both cases, the digital to analogue audio signal conversion taes place over the air due to the overlapping effect between the acoustic wave reproduction of each bit which constitutes the PCM saple. Although MVCDL offers a vital solution for direct adaptation of current loudspeaer technology in digital audio signals, this paper focuses in DTA technology which is very proising for future digital loudspeaer developent. The Digital Transducer Arrays are further divided into two categories: (a) Binary bit-grouped or one-bit-pertransducer DTAs, where every bit has different significance lie for exaple in PCM encoded signals and (b) Unary DTAs, where every bit has the sae significance, as in one-bit signals lie SDM and PWM encoded signals. In this wor, alternative topologies and encoding strategies are exained, where the DTAs driven with both binary and unary sinusoidal input signals, for various frequencies. A coparison between the proposed novel alternative ethods of bit assignent on the array will be presented, using either -bit PWM signals, or ulti-bit PCM encoded signals, regarding the level of total haronic distortion of the spectru of the generated acoustic wave.. THEORY.. Signal Pre-processing The input signals used for driving the DTA siulation were PCM encoded digital audio sinusoidal saples, converted either into unary PCM, or into PWM using the appropriate apping technique [5].... PCM input signal The PCM input strea s is represented here as a N M atrix, where N is the nuber of bits which constitute the PCM saple word and M, the total nuber of saples (possibly infinite). b,... b = b,... b s... bn,... b,, n, ( ) Any DTA would act as an acoustic Digital-to-Analogue Converter (DAC) on such an input signal. As it is

2 nown, the -th saple instantaneous output of any such DAC would be: s N, N- N A = b ( ) n, + b, bn, Thus fro eq () the total pressure generated by a DTA which operates as a D/AC is derived as: P = P A (3 ) LSB where P LSB is the pressure which corresponds to the Least Significant Bit of the PCM word. PCM words (N bits) Bit N Bit Bit s, s, gain transducers In order to express eq () in a decial for appropriate for DTA conversion, the decial weight (significance) of each bit can be contained in a N vector, which in atrix for will generate a apped signal s to the DTA, as: b,... b, b,... b, s ' = (4 ).... n bn,... bn, In traditional one-bit per transducer assignent DTAs [6], the resulting weighted bit values are apped as variable gains before feeding the transducers as shown in Figure. Alternatively in traditional bit-grouped DTAs [3], the weight is expressed as the nuber of active transducer eleents. Given the practical liitations and nown low effectiveness of realizing the one-bit per transducer schee with variable gains the present wor will exaine alternative novel encoding techniques ainly based on the second option. Due to the nown path-length distortions generated by linear DTAs [4, ], the present wor will only exaine -D circular arrangeent of transducers with optiized spatial allocation of bits to the array eleents. Figure : Traditional one-bit per transducer assignent PCM word apping on the array... Alternative bit assignent ethod The strategy followed in this wor is based on a novel representation of a PCM saple. According to this, the conversion of the N-bit word, into its decial counterpart dictates the nuber of the transducers which should be activated with a bit equal to, in order to reproduce the digital saple, whereas the total nuber of eleents required is N -. N- group group N group Figure : PCM word apping on the bit-grouped array Thus every PCM saple is transfored into a ( N -) vector I which consists of A eleents equal to and ( N -) A eleents equal to. Hence:

3 b b I =. (5 ). N b The vector I is then fed to a counter which allocates the exact value of A (bits equal to ) and subsequently the sae nuber of eleents are fed with a pulse, as shown in Figure 3. The counter controls the N - switches, in order to open the A -th switch for every saple. Figure 4: Bit grouped array reproducing PCM word. (a): eleent bit-group nubers (=LSB, 5=MSB), (b): eleent bit assignent On the other hand, according to the proposed ethod, the assignent topology will have a different activation pattern shown in Figure 5. It is obvious that in this case, the appropriate nuber of transducers is actuated fro the centre of the array towards its circuference and as a result a ore hoogeneous eission will be achieved. The left figure this tie indicates the decial index nuber of each eleent. N - Bit N - Bit Bit N - N - bit= counter Figure 5: Novel array topology reproducing PCM word. (a): eleent index nubers, (b): eleent bit assignent..3. PWM input signal Figure 3: Scheatic analogy of the novel PCM word apping on the array The significance of each bit is representated by the nuber of activated transducers, as it would also happen for the case of bit-grouped DTAs. However in the proposed novel apping assignent, the activated eleents are not located in the strict positions dictated by the bit-grouped schee. For exaple, if the 5-bit PCM word fed a traditional bit-grouped array, then the transducers would reproduce the signals as shown in the Figure 4. The nubers on the left figure indicate the nuber of the bit-group that every eleent belongs to, where corresponds to the LSB and 5 to the MSB, while the right figure shows the bit value which every transducer reproduces. As it is typical in ost PWM ipleentations, the initial PCM input signal is oversapled R ties, noiseshaped and re-quantised resulting in the signal: b,... b, ' b,... b, ' s '' = Q[ s] = (6 )... bn ',... bn ', ' which constitutes an N Μ atrix where M =R M and N the new bit resolution per saple, where N <N. The s signal is then odulated in order to derive the PWM signal which constitutes a -bit strea. The vector of the PWM signal is: PWM [ b b b ] ( s''), n '',, b n'', ' = (7 ) It is well-nown that for PWM bitstreas, every successive group of ( N -) bits constitutes a frae, since it corresponds to the value of the input PCM saple [5]. In this case, the nuber of the transducers on the array should be equal to the length of this PWM frae. As it is obvious, a PWM-based DTA will require

4 twice the nuber of eleents in order to reproduce equivalent bit resolution signal to a PCM-based DTA. As proposed previously, the bits of the frae with value equal to constitute the counter input signal which in turn feeds the appropriate nuber of transducers, in the sae way as with the transfored PCM signal, in order to generate the output acoustic wave (Figure 6). 3. IMPLEMENTATION-SIMULATION PARAMETERS 3.. Input signals In order to exaine DTAs of reasonable physical diensions and nuber of eleents, the digital input signal resolution was ept low. Hence the input signals are sinusoidal, encoded either in 5-bit PCM for the PCM-DTA, or in 4-bit PCM and converted into -bit PWM encoding, for the PWM-DTA. The resulting acoustic wave is calculated at the listening point according to its distance fro every transducer. The siulations are carried out for frequencies of 5Hz to 6 Hz with octave step. The original PCM sinusoidal digital signals have a sapling rate equal to 96KHz. The sapling rate of the PWM-encoded signal is 3 ties greater (equal to ( N -), where N=4) copared to the original PCM, which is approxiately.8mhz. 3.. DTA schee/topologies The DTA exained here consists of 3 transducers, in a two-diensional arrangeent. The horizontal angles of observation vary between (on-axis) to 9 degrees with a 3 degree step, in order to derive the DTA perforance in ters of frequency response and its directivity as a function of frequency. Furtherore, total haronic distortion for each case is also calculated. According to the discussion in section, two types of topologies were exained and copared: (a) the traditional bit-grouped DTA, in which the transducers are grouped according to the significance of the bit which are assigned to reproduce and (b) the novel DTA in which all the transducers are considered equal and are given an index nuber (<<3) for the PCM input signal or (<<3) for the PWM input signal. The PCM word is converted to decial with value PCM and feeds the transducers to PCM with a bit equal to while the reaining transducers are fed with a bit equal to. Siilarly, for the case of the PWM frae, the PWM bits equal feed the transducers to PWM with a digital pulse. Many alternative -D topologies were exained, of which, the ost efficient are illustrated in Figure 8. N - Figure 7: Traditional bit-grouped 5-bit PCM array N - N - bit= counter T T Figure 6: Scheatic analogy of the PWM strea apping on the array

5 e S p( n) = C cos[ ω( n n r = = ) + arg( S )] ( ) Figure 8: Novel eleent assignent DTAs for 5-bit resolution unary PCM or 4-bit resolution PWM (topologies T-). The siulations were perfored considering that the transducers are realistic iniature electrodynaic loudspeaers (also used in [9], []), by taing into account their easured ipulse response, shown in Figure 9. Various bit assignent techniques were tested, in order to achieve optiized response. which represents the total su of the sound pressure generated by each of the e transducers. 4. RESULTS 4.. Frequency response The agnitude response of the pressure wavefors reproduced by the siulation at a distance of r= (far field easureent) for a 4KHz input sinusoidal signal encoded in either ordinary 5-bit PCM, unary PCM, or PWM are shown in Figure. Figure 9: Far-field transducer on axis agnitude response (third octave soothed) Analysis ethod (a) The spectru of each bitstrea over the whole nuber of periods is calculated for the case of every transducer on the array [4]. The Fourier transfor of each strea is: S ( e jω ) = n= s ( n) e jωn (8 ) the sound pressure contribution of each transducer at a distance r is calculated as: S p ( n) = C cos[ ω ( n n ) + arg( S )](9 ) r = (b) where S and arg( S ) are the agnitude and phase of the Fourier transfor of the strea and the wavenuber is chosen so that all the desirable frequencies are included. The total sound pressure produced at the listening point is calculated as:

6 Figure : On-axis frequency response, and level of first and second haronics, topology, (a) unary PCM input, (b) PWM input 4.. Haronic distortion A variety of siulations have taen place before concluding to an efficient unary assignent topology. In the following figures the perforance of soe of these efforts is shown, concerning the level of total haronic distortion at various angles of observation. Figure : Pressure spectru, f in =4KHz, r=, φ=3 ο (a) 5-bit PCM (bit grouped), (b) novel PCM, topology T, (c) PWM, topology T, R=3 (c) Fro the results it is observed that the novel techniques proposed for both encoding of the digital signal and for bit assignent, provide iproved results concerning the DTA total haronic distortion perforance, especially for off-axis positions. Total on-axis frequency response of topology with either unary PCM or PWM input is shown in Figure. Along with the fundaental frequency, the level of the first and second haronics are also shown. - Magnitude (db-fs) fo fo 3fo -8 (a) - - Magnitude (db-fs) fo fo 3fo -7-8 (b)

7 T Bit gr. 3 5 T T Bit gr. (a) T (a) (b) (b) T Bit gr T 3 (c) (c) T Bit gr T 3 (d) (d) Figure : THD+N vs frequency for 5-bit PCM input, coparison between topologies T,, and bit grouped array, angles of observation: (a) φ= ο, (b) φ=3 ο, (c) φ=6 ο, (d) φ=9 ο Figure 3: THD+N vs frequency for PWM input, coparison between topologies T-, angles of observation: (a) φ= ο, (b) φ=3 ο, (c) φ=6 ο, (d) φ=9 ο As it is obvious fro Figure the best overall perforance is achieved by the assignent schee. Generally the use of topology with unary PCM signals results in lower total haronic distortion copared to the bit-grouped assignent strategy in

8 ost exained cases. A siilar coparison taes place also in Figure 3 for PWM input signal, where in ost occasions, the topology achieves the best perforance. It is obvious that the level of haronic distortion rises when PWM signal is used, which is expected given that the PWM signal is obtained fro a PCM 4-bit signal in order to retain the sae nuber of eleents as a 5-bit PCM DTA. This one bit resolution drop, coparing to the unary PCM input signals adds inevitably an aount of haronic distortion to the resulting signal at the listening position PCM bit gr. PCM PWM In Figure 4, the coparison of THD+N values between the unary topology, fed either with PCM or PWMencoded signals, and ordinary PCM bit-grouped assignent technique is shown. Although PWM input signal introduces higher haronic distortion, as was entioned above, it provides better results than the bitgrouped schee if the fundaental frequency equals to KHz. Figure 5 shows the coparison of the levels of THD+N vs frequency for 6 o angle of observation, between bit-grouped assignent tecnhique, topology under unary PCM input and topology fed with PWM input signal PCM bit gr. PCM PWM Angle (degrees) Figure 4: THD+N vs angle of observation, f in =KHz, coparison between PCM (bit-grouped), unary PCM topology and PWM topology Figure 5: THD+N vs frequency, φ=6 ο, coparison between PCM (bit-grouped), unary PCM topology and PWM topology 5. CONCLUSIONS It is well-docuented that the ain source of distortions for the DTA acoustic eission is certainly the different path lengths between each eleent on the array and the listening position [, 4]. On-axis listening positions have the advantage of delay cancellation particularly when a syetric topology is chosen. Another restriction of the present wor was the drop in frequency response of the iniature loudspeaers eployed, at low frequencies. However, the object of this paper is to exaine possible optiization for practical array schees and not cases with ideal eleent response. Thus, a significant constraint which relates to our ipleentation is the low bit resolution (5 or 4 bits) of the input signal, which increases the total haronic distortion at the listening position. Although the bit-grouped DTA approach is widely accepted as the ost efficient bit assignent topology, it suffers fro large haronic distortion especially at off-axis listening positions. The present wor shows that there are alternative encoding and assignent techniques for optiizing the DTA perforance, as was shown in [4], for S/D and PWM -bit signals. The novel bit assignent technique proposed here, taes advantage of syetry in order to iniize tie-delay error distortions. According to this, the input PCM signal is apped into unary for and feeds a nuber of eleents on the array equal to the total nuber of s contained in the saple. Siilarly, a PWM input signal is split into fraes of appropriate length so that they correspond with a saple word of the original 4-bit

9 PCM signal. Priority is generally given to the eleents placed near the center of the array. The novel assignent strategy proposed here, generally introduces lower haronic distortion than the traditional bit-grouped DTAs, especially for large off-axis angles and id-range frequencies. It is also liely that nearfield results for the proposed ethod will be better fro bit-grouped DTAs. Although PWM input induces a bigger aount of haronic distortion because it is derived fro a 4-bit PCM signal in order to be allocated to a coparable nuber of DTA eleents, in soe cases it provides acceptable DTA perforance copared to bit-grouped DTAs, as shown in Figure 4. Thus it is evident that PWM-based DTAs using the proposed alternative structure, achieve lower levels of haronic distortion with respect to traditional PWM bit apping [4], but nevertheless will require approxiately twice the nuber of eleents and thereby twice DTA size, in order to achieve coparable perforance to PCM-based DTAs. It is also clear that saller transducer diensions will eventually lead to better DTA perforance, in ters of haronic distortion, because the arrays ay approach the acoustic eission of a point source. In such a case, ore eleents ay be allocated within a given DTA surface, leading to better than 5-bit resolution, eployed in this wor. Furtherore, future wor will exaine ore optiized topologies than the ones presented here. 6. ACKNOWLEDGEMENTS The authors would lie to than Mr Don Keele Principal Engineer with Haran Intrernational, for providing the Haran Kardon Odyssey speaers, used in this wor. [4] N.A. Tatlas, J. Mourjopoulos, Digital Loudspeaer Arrays driven by -bit signals, Audio Eng. Soc. 6th Convention, May 4. [5] A.C. Floros, J. N. Mourjopoulos, Analytic Derivation of Audio PWM Signals and Spectra, J. Audio Eng. Soc., Vol. 46, No. 7/8, pp , July/Aug [6] J. Mendoza-Lopez, S. C. Busbridge, P. A. Fryer, Direct Acoustic Digital to Analogue Conversion with Digital Transducer Array Loudspeaers, Audio Eng. Soc. 8th Convention, May 5. [7] N. A. Tatlas, A. Floros, P. Hatziantoniou and J. Mourjopoulos, Towards the All-Digital Audio/Acoustic Chain: Challenges and Solutions, proceedings of the Audio Engineering Society 3d Conference, Elsinore, May 3, p.9 p.4. [8] M.O.J. Hawsford, Sart Digital Loudspeaer Arrays, J. Audio Eng. Soc., Vol. 5, No., 3 Deceber. [9] D.B. Keele, Practical Ipleentation of Constant Beawidth Transducer (CBT) Loudspeaer Circular-Arc Line Arrays, presented at the 5 th AES Convention, New Yor October 3. [] J. Mendoza-López, S. C. Busbridge and P. A. Fryer, Sound Field Characterisation in Audio Reproduction with the Bit-Grouped Digital, Transducer Array, presented at the th AES Convention, May 6, Paris, France. 7. REFERENCES [] Y.Huang, S. C. Busbridge, D. S. Gill, Distortion and Directivity in a Digital Transducer Array Loudspeaer, J. Audio Eng. Soc., Vol. 49, No. 5, May. [] Y. Huang, S. C. Busbridge, and P. A. Fryer, Interactions in a Multiple-Voiced-Coil Digital Loudspeaer, J. Audio Eng. Soc. (Engineering Reports), vol.48, pp ( June). [3] S. C. Busbridge, P. A. Fryer, Y. Huang, Digital Loudspeaer Technology: Current State and Future Developents, Audio Eng Soc. th Convention, May.