Hybrid Digital-Analog Transmission Taking Into Account D/A and A/D Conversion

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1 Hybrid -Analog Transmission Taking Into Accont and Conversion Matthias Rüngeler and Peter Vary Institte of Commnication Systems and Data Processing ( ) RWTH Aachen University, Germany {rengeler vary}@ind.rwth-aachen.de Abstract Efficient digital transmission of continosamplitde signals reqires sorce coding which comes at the price of navoidable qantization errors. Ths, even in clear channel conditions, the qality of the decoded signal is limited de to these sorce coding errors (qantization). Hybrid -Analog (HDA) codes circmvent this limitation by additionally transmitting the sorce coding error with qasianalog methods (discrete-time, qasi-continos-amplitde) with neither increasing the total transmission power, nor the occpied freqency bandwidth on the radio channel. So far, for HDA transmission, the potential distortion additionally introdced by and conversion of the analog signal has not been considered. In this paper, the effect of clipping and limited resoltion of this conversion on the performance of HDA transmission is evalated. For random variables and speech signals, simlations verify that even with poor and conversion (3 bit resoltion) the HDA concept otperforms conventional prely digital transmission systems at all channel qalities while additionally eliminating the qality limitation effect. Index Terms Hybrid -Analog (HDA), clipping and limited resoltion of and converters I. INTRODUCTION Conventional digital transmission systems are based on the sorce-channel separation theorem. The digitized analog speech, adio, or video signal is processed by a sorce encoder which delivers a seqence of qantized sorce parameters. Thereafter, the bit stream of the sorce parameters is protected by a channel encoder. These systems are sally optimized for the expected worst-case channel condition. If a feedback channel is not available, adaptive mlti rate (AMR) techniqes cannot be implemented. Then, even if the receiver experiences a better channel qality (csnr) the signal-tonoise ratio (psnr) of the decoded sorce parameters satrates. This satration is cased by the inherent errors introdced by the digital sorce encoder, even in the case of errorfree transmission of the coded bits. The satration effect can be circmvented by adding continos-amplitde processing and transmission. With these Hybrid- Analog (HDA) transmission systems, the transmission fidelity improves for increasing channel qalities withot the need to adapt the coding scheme. Skoglnd et al. [1] have proposed a HDA system heavily relying on nmerical optimizations and exhastive search in the decoding algorithms. In [], [3] a more efficient design for HDA transmission benefiting from the power and flexibility of conventional digital channel codes is presented. In the literatre, the analog error symbols are distorted only by the noisy analog channel. Additionally, the analog signal is distorted by the navoidable and conversion when sing digital signal processing. This conversion sally comes with limited resoltion and even clipping. This effect shold not be neglected: For a compressing Archimedes spiral, an experimental evalation with software defined radios showed a satration of the qality at arond db which is de to the 16 bit converters [4]. In this paper, the performance of HDA transmission with and converters with limited resoltion and clipping is considered. II. HYBRID DIGITAL-ANALOG TRANSMISSION AND PURELY DIGITAL TRANSMISSION Fig. 1 shows a conventional digital transmission system. The sorce vector consists of M continos-amplitde and discrete-time symbols. It is sorce encoded with F D bits per sorce symbol to in total l vd sorce bits, yielding the bit vector v D. Sbseqently, a digital channel code followed by digital modlation transforms the sorce bits into N realvaled symbols forming the channel inpt y D. Here, channel coding and modlation is combined in one step (ccm), ths, the ratio between the nmber of bits l vd and the nmber of real symbols N is denoted by the coding-modlation rate. When considering an AWGN channel, the noise vector n distrbs the channel symbols, thereby yielding the received symbols z D. The channel signal to noise ratio is rd ccm = lv D N csnr = E{ y D } E{ n }. (1) After demodlation, channel decoding, and reconstrction of qantized vales, û D gives an estimate of the initial sorce symbols. Fig. illstrates the corresponding HDA transmission system []. The general idea is to se a conventional digital transmission system for and to additionally transmit the sorce coding error a H by sing continos-amplitde (psedo-analog) discrete-time processing. The pper branch of the hybrid encoder and is referred to as the digital branch and the lower branch as the analog branch. All operations, also in the analog branch, are condcted by digital signal processing. The continos-amplitde symbols are floating or fixed point variables with a precision depending on the digital processor. All variables in the figres are already digitized. Here, the necessary and conversion is assmed to have a sfficiently high qality to not affect the reslts. The digital branch is a prely digital transmission system with D real channel per M sorce symbols. The analog branch takes Achannel ses. Ths, the nmber of channel ses per HDA frame is N = D + A and the coding-modlation rate in the digital branch is r ccm H = lv H N A = lv H D. In the IEEE Workshop on Signal Processing Advances in Wireless Commnications (SPAWC), Stockholm, Sweden, DOI: 1.119/SPAWC / 5

2 n encoder F D v D 1 l vd Channel encoder & modlator r ccm D y D 1 N z D Demodlator & channel ˆv D û D Fig. 1. Prely digital transmission system. hybrid encoder, the sorce encoder converts to the bit vector v H with F H bits per sorce symbol. The sorce bis are decoded locally in the transmitter to d H while the sorce coding error a H = d H is then transmitted via the analog branch. The analog mapper ses the continos-amplitde fnction yh a = f(a H ) to map the entries of a H to the entries of yh a with length A and average power 1 A E{ ya H } = 1. The ratio between the inpt and the otpt dimensions of the block is r mapp H = M A. This mapping f( ) cold, e.g., be a linear amplification or a nonlinear fnction with a rate of r mapp H = 1 or in case of a mapping yielding one complex symbol for one real inpt symbol r mapp H = 1. After mltiplexing the symbols from the digital and the analog branch and transmission over the AWGN channel, the received symbols are demltiplexed and conveyed to the digital and analog decoding branches. The analog demapper then gives û a H as the estimate of a H which can be facilitated sing several alternative methods sch as maximm likelihood (ML), minimm mean sqare error (MMSE) and linear minimm mean sqare error (LMMSE) estimators. The ML estimator applies the inverse fnction of f( ) whereas the LMMSE estimator additionally weights the csnr received symbols with (1+cSNR) before applying the inverse fnction of f( ) [5]. In this paper, a linear analog mapper and LMMSE estimation is employed. The otpts of the analog and digital branches are added, whereby û H gives an estimate of the initial sorce symbols. Finally, the end-to-end parameter signal to noise ratio for both systems is described by psnr = E{ } E{ û }. () For a fair comparison between the prely digital and the HDA transmission systems, the transmission power as well as the nmber of channel ses N mst be eqal. Ths, the nmber of channel ses D for the digital branch has to be lowered to D = N A. This can be achieved by lowering the bit rate of the sorce encoder in the HDA system. Then fewer bits need to be protected against channel noise and ths the same or even stronger channel coding can be applied (rd ccm rh ccm ) as in the prely digital case. However, the bit rate of the HDA system cannot be lowered too mch, since then the loss in psnr de to coarser qantization cannot be compensated anymore by the analog branch. In [], it is stated by how mch the bit rate can be lowered to design a sperior HDA system for any prely digital system. If additionally an LMMSE estimator is sed as the analog demapper, the performance of the HDA system is sperior or eqal to the prely digital transmission system at all channel qalities for AWGN and even fading channels []. Especially in the context of nknown radio channel qalities, or with channel qalities which may be higher than expected while designing the system, the HDA system exhibits the very desirable property to increase the end-to-end psnr with rising channel qalities. III. AND CONVERSION IN ANALOG BRANCH In Fig., it is assmed that the interface to the real world (the actal sorce samples and the channel symbols) is perfect and no additional distortion is introdced. For a real-world system (Fig. 3), the effect of and conversion also have to be considered. The captred analog (discrete-time, continos-amplitde) inpt signal ũ has to be converted (qantizer 1 ) to the digital domain () before frther processing. Also the analog symbols yh a and the potentially non-binary modlation symbols yh d 4 of the HDA encoder are passed throgh a convertor at the transmitter (ỹh a and ỹh d ) and also the receiver employs converters ( 5 & 3 ). The impact of the and 3 conversion in the analog branch, which are boxed with a bold line, are considered here. The other and converters also appear in a prely digital transmission system and do, if properly designed, not impact the performance difference between prely digital and HDA transmission. The and converters ( & 3 ) are modeled by a symmetric niform mid-rise qantizer. Two parameters are relevant: First, F TX and F RX define the nmber of bits of the qantizers at the transmitter and the receiver, respectively. Second, the range of the qantizer is relevant (( C TX, C TX ) and ( C RX, C RX )). The reconstrction vales of the niform qantizers are in this range and all vales with greater absolte vale are clipped. A. at Transmitter The aim of the parametrization of the converter at the transmitter is to minimize the introdced distortion. There encoder F H v H 1 l vh d H a H 1 M Chan. encoder & modlator r ccm H Analog mapper f( ) y d H 1 D y a H 1 A Mx y H 1 N r mapp H Fig.. HDA transmission system withot consideration of and conversion. n z H Demx z d H z a H Demodlator & chan. dec. Analog demapper ˆv H û a H û d H û H IEEE Workshop on Signal Processing Advances in Wireless Commnications (SPAWC), Stockholm, Sweden, DOI: 1.119/SPAWC / 5

3 ũ 1 HDA encoder y d H y a H 4 ỹ d H 1 D ỹ a H 1 A AWGN branch Analog branch AWGN C TX, F TX C RX, F RX Fig. 3. HDA transmission system with and conversion. The properties of and conversion in the analog branch are considered. The variables C TX and C RX describe the maximm range of the qantization and F TX and F RX the nmber of bits of the niform qantizers. are two effects which are considered here. First the distortion MSE C introdced by the limited range (C TX ) of the qantizer, i.e., the clipping, and second the distortion MSE Q introdced by qantization noise. For a given clipping level C TX and the nmber of bits F TX of the qantizer, the nmber of qantization levels is L = FTX, the step height of the niform qantizer is h = CTX L and the reconstrction vale with the largest vale is given as m = C TX 1 h = C TX (1 L) 1. All vales otside of ( C TX, C TX ) are mapped to the greatest/smallest reconstrction vale ±m of the qantizer. Ths, with the pdf p(yh a ) of the entries ya H of the vector ya H to be transmitted, the inflence of clipping is calclated as [6]: MSE C = (y a H m) p(y a H) dy a H + C TX CTX (y a H + m) p(y a H) dy a H. Vales within the range are mapped to the nearest reconstrction vale of the qantizer. The introdced distortion MSE Q by qantization is: MSE Q = L i=1 C TX+ i h C TX+ (i 1) h ( y a H C TX ( i 1 L 1 )) p(yh) a dyh. a The overall introdced distortion MSE TX,HW = MSE Q +MSE C is the sm of both effects. For a given pdf p(yh a ) and a given nmber of qantization bits F TX, the clipping vale C TX minimizing MSE TX,HW has to be fond. Greater vales lead to a smaller distortion de to clipping, bt the step height h of the qantizer grows. Ths, the distortion de to qantization increases. Smaller vales decrease the qantization noise, bt increase the impact of clipping. The optimal vale is a tradeoff which has to be fond nmerically for a given F TX and p(yh a ). If, as a simple example, the sorce samples ũ follow a niform distribtion and as sorce encoding, niform qantization is sed, the pdf p(yh a ) of the analog otpt of the HDA encoder also follows a niform distribtion. After the analog mapper (Fig. ), the average power 1 A E{ ya H } is 1 and ths its range is ( 3, 3). The nmerical optimization of the clipping vale yields C TX = 3 which is at the same time the smallest possible vale withot clipping. For Gassian sorce symbols and Lloyd-Max qantization in the HDA encoder, the pdf of yh a is not niform anymore. Then, the optimal C TX has to be fond nmerically. For all permtations of the nmber of bits F H per sorce symbol in the HDA encoder and the word length F TX of the converter in the transmitter, the optimal clipping vales C TX are stated in Table I. For rising F TX, the qantization noise MSE Q z d H z a H 5 3 z d H z a H HDA û d H û a H û H F H \F TX TABLE I OPTIMAL C TX FOR CONVERSION IN ANALOG BRANCH FOR GAUSSIAN SOURCE SYMBOLS QUANTIZED WITH F H BITS IN HDA ENCODER. decreases and de to the above described tradeoff, the clipping vale has to be increased. This is reflected in the table. A change of F H leads to different pdfs p( a H ) of the analog otpt of the HDA encoder which also affects the optimal C TX. 4 6 ûh HDA, F TX=6, C TX=9.16 HDA, F TX=6, C TX=3.99 HDA, F TX=4, C TX=3 HDA, F TX=4, C TX=5 HDA, F TX=4, C TX=3.99 Fig. 4. HDA transmission with different converters in the transmitter, an ideal converter 3 in the receiver, and a Gassian sorce with F H =4. To assess the performance of the HDA transmission system, monte carlo simlations with a sorce block length of M = 8 are condcted. First, for Gassian sorce symbols, the sorce encoder is a Lloyd-Max qantizer (LMQ). Here, F D = 5 bits are sed in the prely digital case, while for the HDA case the qantizer in the sorce coder employs one bit less (F H = 4). The otpt bits of the sorce encoder are channel encoded sing convoltional coding with a rate- 1 recrsive systematic convoltional code with the generator polynomial {1, 15/13} 8 the same code which is sed as a component code in the UMTS-LTE Trbo code. Then, BPSK modlation is employed. To garantee a fair comparison between all systems, pnctring is applied to the codewords to always yield N = 56 channel ses. In the HDA case, D = 48 channel ses are allocated to the digital branch and A = M = 8 channel ses to the analog branch. Figre 4 IEEE Workshop on Signal Processing Advances in Wireless Commnications (SPAWC), Stockholm, Sweden, DOI: 1.119/SPAWC / 5

4 shows the simlation reslts for different converters at the transmitter for Gassian sorce symbols. The crve labeled does not consider any distortion introdced by or conversion in the analog branch. The other crves employ an ideal converter at the receiver bt consider the distortion at the transmitter. For F TX = 4 the optimal clipping vale is C TX = Varying the vale to higher C TX = 5 or lower C TX = 3 vales jst slightly decreases the performance. For F TX = 6, a different C TX = 9.16 is chosen which leads to an improved performance. Choosing the vale which was optimal for the lower word length (C TX = 3.99) drastically decreases the performance. Ths, the shold be parametrized wisely. For very low word lengths, i.e., F TX = 1 another effect arises. Since the reconstrction vales of the qantizers are ± , the power of the transmitted vales ỹh a is lower than the targeted power of 1 by 1.9 db. Ths, for a fair comparison, the power shold be increased by this vale which wold lead to an improvement of the performance in psnr of at least 1.9 db with LMMSE estimation. Since for increasing word lengths, e.g., for F TX = 4, this loss decreases to jst. db, this loss is not compensated in the simlations. B. 3 at Receiver The aim at the receiver is to minimize the overall distortion introdced by both, the and the converter. Ths, not the received symbols have to be captred as good as possible, bt the otpt of the receiver converter shold describe the sent reconstrction levels of the qantizer as close as possible. The analogy to a digital demodlation system is qite close. At best, the receiver converter denoises the sent reconstrction levels. This can be achieved by sing the same parametrization at the receiver as in the transmitter, especially when sing the same clipping vales (C RX = C TX ). In [7] it is shown by information theoretic means, that is is sfficient to se the same nmber of qantization levels at the receiver as discrete levels at the transmitter to achieve capacity HDA, F TX=4, F RX=4 HDA, F TX=4, F RX HDA, F TX=4, F RX=8 Fig. 5. HDA transmission with a fixed converter at the transmitter with different converters 3 at the receiver. Gassian sorce pdf, F H =4, C RX =C TX =3.99. Figre 5 shows simlation reslts with one fixed converter at the transmitter and different converters at the receiver. The receivers se the same clipping vales as the transmitter (C RX = C TX ), bt the word length is varied. The system with the same word length (F RX = F TX = 4) achieves the best performance, especially for channel qalities jst below the satration of the system (csnr 5 db). Here, noisy symbols are mapped to the exact sent vales. Systems with a higher world length (F RX = 8) or even withot any distortion by the receiver show an eqal performance which is slightly better at lower channel qalities (csnr 13 db). Bt they cannot denoise the received symbols at csnr 5 db and ths exhibit a big performance gap to F RX = 4. In case of F RX = F TX, the pdf of the effective noise z a H ya H is a convoltion between the qantization noise of the at the transmitter, which is niform, and a series of diracs which emerge when choosing the wrong qantization level from the noisy received symbols. For a small channel noise power, the qantization levels are correct and ths, jst the qantization noise remains. In case of an ideal receiver, the niform qantization noise is convolved with the Gassian channel noise pdf. For small channel noise powers, the reslt is always greater than for F RX = F TX, bt for higher channel noise powers (csnr 18 db) the diracs have a greater power than the channel noise. IV. SIMULATION RESULTS A. Coding by Scalar Qantization Figre 6 shows the performance of prely digital and HDA transmission systems with sorce symbols following a niform pdf. The transmitter and receiver converters are designed with the same parameters (C RX = C TX, F RX = F TX ). With all configrations, the performance of the HDA system increases with rising channel qalities ntil a certain maximm psnr is reached. This limit rises with higher word lengths of the converters obeying the 6dB-per-bit rle, i.e., for a qantizer with one bit more, the limit increases by 6 db. This means that already with F TX =, the performance of a prely digital system is sperseded by HDA transmission. A linear transmission system with LMMSE estimation and and conversion is shown for comparison which exhibits generally a lower performance and satrates at 36 db de to the 6 bit qantizers. 6 4 HDA, F TX=F RX=6 HDA, F TX=F RX=4 HDA, F TX=F RX= Linear, F TX=F RX=6 Fig. 6. HDA and prely digital transmission for a niform sorce pdf. The and conversion in the transmitter and receiver 3 in the analog branch have the same design: C RX =C TX = 3, F RX =F TX, F H =4, F D =5. Prely linear system with LMMSE estimation and and conversion for comparison. Figre 7 shows the same behavior for Gassian sorce symbols. Also here, for rising channel qalities, the psnr increases ntil a certain limit. Here, the 6dB-per-bit rle is not obeyed. An increase from F TX = 6 to 8 bits leads to a gain of jst 8 db instead of 1 db. When sing a converter with a higher word length, the range, i.e, the clipping vale C TX also has to be increased which redces the gain. A IEEE Workshop on Signal Processing Advances in Wireless Commnications (SPAWC), Stockholm, Sweden, DOI: 1.119/SPAWC / 5

5 4 HDA, F TX=F RX=8, C RX=C TX=14.66 HDA, F TX=F RX=6, C RX=C TX=9.16 HDA, F TX=F RX=4, C RX=C TX=3.99 HDA, F TX=F RX=3, C RX=C TX=.85 Fig. 7. HDA and prely digital transmission for a Gassian sorce pdf. C RX =C TX, F RX =F TX, F H =4, F D =5. ftre improvement cold be a nonlinear preprocessing prior to the transmission via the analog branch to sffer less from distortion de to clipping. B. Coding by ADPCM Speech Coding In Fig. 8 a lattice ADPCM speech coder [8] is employed. The speech signal are 96 sentences from the NTT database (16 khz sampling rate) in English which sm p to a total dration of 1 mintes and 48 seconds. The same block lengths and channel encoding parameters as in the previos simlations are employed. The only change to the system depicted in Fig. is in the analog mapper. Here, since the statistics of the speech signal is nknown a-priori, for each frame a power normalization factor is calclated, digitized and mltiplexed with the otpt bits of the speech encoder to be transmitted via the digital branch. The optimal clipping vales are chosen according to the pdf of yh a. Also for this system employing real-world speech signals and speech encoding, HDA transmission is sperior to prely digital transmission. A resoltion of only 3 bits for the and conversion in the analog branch sffices to achieve a sperior performance which can be frther increased with conversion with more bits. 6 4 HDA, F TX=F RX=8, C RX=C TX=5.43 HDA, F TX=F RX=6, C RX=C TX=4 HDA, F TX=F RX=4, C RX=C TX=.93 HDA, F TX=F RX=3, C RX=C TX=.46 Fig. 8. HDA and prely digital transmission for speech sing a lattice ADPCM speech coder [8]. C RX =C TX, F RX =F TX, F H =4, F D =5. Nevertheless, the reqirements for and conversion to achieve great performance of the analog branch of the HDA systems are comparatively low. and conversion with jst 8 bit lead to a performance gain of 4 db in psnr in case of sorce symbols with a niform pdf, to db in case of a Gassian sorce and to 31 db in case of speech coding. The overall psnr of the HDA system is the sm of the performance of the digital branch and the analog branch. Ths, Hybrid -Analog transmission enables high psnrs since the basis is laid with a digital system and the performance of the analog branch is on top. This is an otstanding property compared to systems which only rely on continos-amplitde transmission as, e.g., analog modlo block codes [9] or Archimedes spirals [4], [1]. All in all, it can be shown that the reslts in [] also hold for HDA systems with and conversion, as long as their word lengths are above F TX 3. Ths, also with and conversion with limited resoltion in the analog branch, HDA systems can be designed which exhibit a sperior performance than prely digital transmission for all channel qalities. V. CONCLUSION In this contribtion, Hybrid -Analog transmission with and conversion in the analog branch is considered. Their impact on the overall performance is elaborated and it is shown how to optimally parametrize them. Simlations are condcted for random variables with niform and Gassian pdf employing scalar qantization as sorce coding, as well as lattice ADPCM speech coding for speech signals. Even for conversion with jst 3 bits, the HDA systems always exhibit a better performance than prely digital transmission for all channel qalities and therefore the design rles in [] still hold. The big advantage of HDA transmission over systems with only continos-amplitde transmission is that the performance of the analog branch is added on top to the performance of the digital branch. Ths, very high overall psnr vales can be reached. REFERENCES [1] M. Skoglnd, N. Phamdo, and F. Alajaji, Design and Performance of VQ-based Hybrid -Analog Joint -Channel Codes, IEEE Transactions on Information Theory, vol. 48, no. 3, Mar.. [] M. Rüngeler, J. Bnte, and P. Vary, Design and Evalation of Hybrid -Analog Transmission Otperforming Prely Concepts, Commnications, IEEE Transactions on, vol. 6, no. 11, Nov 14. [3] M. Rüngeler and P. Vary, Hybrid Analog Transform Coding, in 13 IEEE International Conference on Acostics, Speech and Signal Processing (ICASSP). Vancover: IEEE, May 13. [4] J. Garcia-Naya, O. Fresnedo, F. Vazqez-Arajo, M. Gonzalez-Lopez, L. Castedo, and J. Garcia-Frias, Experimental evalation of analog joint sorce-channel coding in indoor environments, in Commnications (ICC), 11 IEEE International Conference on, Jne 11. [5] O. Fresnedo, F. Vazqez-Arajo, L. Castedo, and J. Garcia-Frias, Low- Complexity Near-Optimal Decoding for Analog Joint Channel Coding Using Space-Filling Crves, IEEE Commnications Letters, vol. 17, no. 4, 13. [6] P. Vary and R. Martin, Speech Transmission - Enhancement, Coding & Error Concealment. Wiley, Jan. 6. [7] J. Singh, O. Dabeer, and U. Madhow, On the limits of commnication with low-precision analog-to-digital conversion at the receiver, Commnications, IEEE Transactions on, vol. 57, no. 1, Dec. 9. [8] M. Dietrich, Performance and implementation of a robst ADPCM algorithm for wideband speech coding with 64 kbit/s, in Proc. of Intl. Zürich Seminar on Commnications, Zürich, Switzerland, [9] T. Schmitz, M. Rüngeler, and P. Vary, Analysis of analog modlo block codes, in Proceedings of International ITG Conference on Systems, Commnications and Coding (SCC). IEEE, Feb. 15. [1] P. Floor and T. Ramstad, Noise Analysis for Dimension Expanding Mappings in -Channel Coding, IEEE 7th Workshop on Signal Processing Advances in Wireless Commnications, 6. SPAWC 6. IEEE Workshop on Signal Processing Advances in Wireless Commnications (SPAWC), Stockholm, Sweden, DOI: 1.119/SPAWC / 5