We investigate reliable communication over quantized channels from an information theoretical point of view. People seldom consider the effect of quantization in conventional coded modulation systems since Analog-to-Digital (AD) converters used in these systems always have high resolution, e.g. 2/3 source bits are often quantized into 10/12 bits. However, AD converters with a high resolution are power consuming. In this paper, we present a scheme to design an optimum quantizer with low resolution which can be used to communicate over the quantized  channel. Moreover, we show that reliable transmission over the Additive White Gaussian Noise (AWGN) channel at a rate of R bit/use is possible with R + 1 or R + 2 quantized bits.\u

Cronie, Harm S.

Shao, Xiaoying

English

We investigate reliable communication over quantized channels from an information theoretical point of view. People seldom consider the effect of quantization in conventional coded modulation systems since Analog-to-Digital (AD) converters used in these systems always have high resolution, e.g. 2/3 source bits are often quantized into 10/12 bits. However, AD converters with a high resolution are power consuming. In this paper, we present a scheme to design an optimum quantizer with low resolution which can be used to communicate over the quantized channel. Moreover, we show that reliable transmission over the Additive White Gaussian Noise (AWGN) channel at a rate of R bit/use is possible with R + 1 or R + 2 quantized bits

Shao, X.

Cronie, H.S.

University of Twente Research Information

MODULATION AND CODING FOR QUANTIZED CHANNELSXiaoying Shao and Harm S. Croniex.shao@ewi.utwente.nl, h.s.cronie@ewi.utwente.nlUniversity of Twente, Faculty of EEMCS, Signals and Systems GroupP.O. box 217 - 7500 AE Enschede - The NetherlandsABSTRACTWe investigate reliable communication over quantized channelsfrom an information theoretical point of view. People seldomconsider the effect of quantization in conventional coded modu-lation systems since Analog-to-Digital (AD) converters used inthese systems always have high resolution, e.g. 2/3 source bitsare often quantized into 10/12 bits. However, AD converterswith a high resolution are power consuming. In this paper, wepresent a scheme to design an optimum quantizer with low res-olution which can be used to communicate over the quantizedchannel. Moreover, we show that reliable transmission over theAdditive White Gaussian Noise (AWGN) channel at a rate ofRbit/use is possible withR + 1 or R + 2 quantized bits.Key words: quantized channels, quantization, coded modula-tion system, AD converter, AWGN channel1. INTRODUCTIONIn this paper we investigate reliable and bandwidth effi-cient communication over quantized channels. Quantizedchannels arise in practical communication systems whereAD converters are used to sample analog signals corre-sponding to the transmitted data. Conventional coded mod-ulation systems usually do not take quantization into ac-count, which is reasonable since a large number of quan-tization levels are used. In this case the difference be-tween the quantized and unquantized channel can be ne-glected. However, for e.g. mobile communication sys-tems the power consumption at the receiver is proportionalto the resolution of the AD converter. Hence it is of inter-est to lower the resolution of the AD converter.In this paper we consider quantized channels from an in-formation theoretical point of view. We investigate infor-mation theoretical limits of transmission over quantizedchannels. The main question we try to answer is as fol-lows. If we wish to transmit reliably at a rate ofR bit/use,do we actually require more thanR quantization bits? Inthis paper we show that for the AWGN channel, reliabletransmission at a rate ofR bit/use is possible withR + 1or R+2 quantization bits without sacrificing transmissionpower.The organization of the paper is as follows. First, we in-troduce the system model in section 2. Second, we inves-Bit source Channel Q(y))x,...,x( d1fX Y ZFigure 1: The system modeltigate the effect of quantization from an information theo-retical point of view in section 3. Furthermore, we presenta scheme which can be used to communicate over thequantized channel in this section. In addition, we find outan optimum quantization scheme which makes the theo-retical limits of the quantized AWGN channel as close tothe limits of the unquantized AWGN channel as possible.We end with conclusions in section 4.2. SYSTEM MODELWe consider communication over the memoryless time-discrete AWGN channel which is defined by:Y = X + N (1)where the channel inputX is disturbed by additive noiseN which has a Gaussian distribution with varianceσ2. Tocommunicate over this channel we map a sequence ofdbitsX1, . . . ,Xd to a channel input symbolX:X = φ(X1, . . . ,Xd) (2)whereφ(· · · ) is defined as:φ : GF (2)d → R (3)We refer toφ as themodulation map and it defines the sig-nal constellationS and mapping from bits to constellationsymbols:S = {x ∈ R : x = φ(X1, . . . ,Xd)} (4)The energy expended per channel use is defined as themathematical expectation ofX2:Es = E[X2] (5)Proceedings of SPS-DARTS 2007 (the 2007 The third annual IEEE BENELUX/DSP Valley Signal Processing Symposium)179The channel output is quantized by a quantization mapQwhich is defined as:Q : R → A (6)whereA ⊂ N and represents the output of the quantizedchannel. As we will see later it is not necessary to asso-ciate the elements ofA with the real numbers. The ele-ments just represent the channel outputs. A quantizationmap is surjective andQ also defines an inverse quantiza-tion map which is defined as the set functionQ−1:Q−1(i) = {x ∈ R : Q(x) = i, i ∈ A} (7)We restrict ourselves to quantizers where eachQ−1(i) isof the formIi = (a, b] for a, b ∈ R. In this case{Ii}partitionsR and the quantizer is defined by the set of in-tervals{Ii}. The goal is to make|A| as small as possibleand still achieve a reasonable mutual information betweenthe input bits and the quantized channel output. With thesedefinitions the quantized channel is defined as:Z = Q(φ(X1, . . . ,Xd) + N) (8)whereZ is the quantized channel output and takes valuesfrom A. Figure 1 shows an overview of the system modelwe have defined so far. In the next section we study thissystem from an information theoretical point of view andderive a scheme to communicate reliable over the quan-tized channel.3. QUANTIZATION AND MUTUALINFORMATIONIn this section we study the effect of quantization froman information theoretical point of view. Moreover, inthis section we consider the AWGN channel as an ex-ample where we restrict ourselves to uniform quantizersand conventional pulse amplitude modulation (PAM) sig-nal constellations. Also, we consider the case of design-ing non-uniform quantizers for the communication overthe AWGN channel with conventional PAM modulation.In later section we find out which is the optimum designof the quantization scheme.We are interested in the mutual information between(X1,. . . ,Xd) andZ. First, note that the following sequence ofrandom variables forms a Markov chain:(X1, . . . ,Xd) → X → Y → Z (9)We can expressI((X1, . . . ,Xd),X;Z) as:I((X1, . . . ,Xd),X;Z) =I(X;Z) + I((X1, . . . ,Xd);Z|X) =I((X1, . . . ,Xd);Z) + I(X;Z|(X1, . . . ,Xd)) (10)where we have used the chain rule of mutual information.Since(X1, . . . ,Xd) → X → Z forms a Markov chain aswell it follows that:I((X1, . . . ,Xd);Z|X) = 0 (11)Moreover,X is a function of(X1, . . . ,Xd) which impliesthat:I(X;Z|(X1, . . . ,Xd)) = 0 (12)With (10) we have the following equality:I((X1, . . . ,Xd);Z) = I(X;Z) (13)which shows that the mutual information of interest isfully defined by the signal constellation and the distribu-tion of the quantized symbols. Next, we consider the mu-tual information betweenX andZ. From the chain ruleof mutual information it follows that:I((Y,Z);X) = I(Y ;X) + I(Z;X|Y )= I(X;Z) + I(Y ;X|Z) (14)SinceX → Y → Z forms a Markov chainI(Z;X|Y ) =0 from which follows that:I(X;Z) = I(Y ;X) − I(Y ;X|Z) (15)Given a channel there exists a distribution which achievesthe maximum valueC of I(Y ;X). For the AWGN chan-nel this distribution is the Gaussian distribution. A par-ticular choice of modulation leads to a certainI(Y ;X).Moreover, when the output of the channel is quantized,the quantity of interest isI(X;Z) which is upperboundedI(X;Y ) by (15). In this paper we assume the transmit-ted signal is PAM-modulated and consider the design ofquantization schemes for the AWGN channel for whichI(X;Z) is as close to the capacity of the AWGN channelas possible. Furthermore, the number of quantization bitsis only slightly larger thanR, whereR is the rate at whichwe transmit.3.1. Uniform Quantization for AWGN with PAM Con-stellationsNow we consider uniform quantization for the AWGNchannel where conventional PAM constellations are used.A PAM constellation with2d constellation symbols is de-fined as:SPAM2d = {−(2n − 1), . . . ,−(2i − 1), . . . ,−1, 1, . . . , 2i − 1, . . . , 2n − 1} (16)wherei = 1, 2, . . . , n andn = 2d−1. In this case theconstellation symbols are selected with equal probabilityandEs is given by:Es =4d − 13(17)Proceedings of SPS-DARTS 2007 (the 2007 The third annual IEEE BENELUX/DSP Valley Signal Processing Symposium)180One can normalize these constellation such thatEs = 1.We define a uniform interval quantizer with2m levels andspacingq as an interval quantizer for which the set ofquantization levels is given by:{Ii : i = 1 . . . 2m} ={(−∞,−nq], . . . , (−iq,−(i − 1)q], . . . , (−q, 0],(0, q], . . . , ((i − 1)q, iq], . . . , (nq,+∞)} (18)wherei = 1, 2, . . . , n andn = 2m−1 − 1.Given these definitions we can expressed the mutual infor-mation between the channel input and the quantized chan-nel output as:I(X;Z) =∑x∈S∑i∈AP (Z = i,X = x) log2P (Z = i|X = x)∑x′∈S P (Z = i,X = x′)(19)whereP (Z = i,X = x) is given by:P (Z = i,X = x) =P (Z = i|X = x)P (X = x) = 2−dP (Z = i|X = x) (20)andP (Z = i|X = x) can be computed as follows:P (Z = i|X = x) = P (y ∈ Q−1(i)|X = x)=∫Q−1(i)dyfY |X(y|x) (21)wherefY |X(y|x) is the channel transition probability den-sity function (pdf).Consider the case thatEs/σ2 is equal to 5 dB and wetransmit source bits over the AWGN channel using a PAM-4 constellation which is defined asS = {−3,−1, 1, 3}.Suppose that we use a uniform interval quantizer to quan-tize the channel output into2m levels wherem is equal to2, 3 or 4. Figure 2 shows a plot ofI(X;Z) as a functionof q for these values ofm and the case where no quanti-zation is used. The figure also shows the capacity of theAWGN channel. We observe that there is a loss comparedto the case where no quantization is used. However, witha proper spacingq and anm slightly larger thand the lossin rate can be made small. Furthermore, one does not haveto use a uniform interval quantizer. The quantization lev-els can be chosen in such a way to give a higher value ofI(X;Z).3.2. Non-uniform Quantization for AWGN with PAMConstellationsHere we consider non-uniform quantization for the AWGNchannel and conventional PAM constellations. In this case,we define a non-uniform interval quantizer with2m levels0 1 2 3 4 50.550.60.650.70.750.80.850.90.9511.05qrateUnquant limitm=4m=3m=2Shannon limitFigure 2:Uniform quantized interval q with d = 2, PAMconstellation and SNR=5dBand non-equal spacing as an interval quantizer for whichthe set of quantization levels is given by:{Ii : i = 1 . . . 2m} ={(−∞,−qn], . . . , (−qi,−qi−1], . . . , (−q1, 0],(0, q1], . . . , (qi−1, qi], . . . , (qn,+∞)} (22)wherei = 1, 2, . . . , n andn = 2m−1 − 1.From (19)∼ (21), we see thatI(X;Z) determined by(22). In other words, we can optimize the{q1, q2, . . . , qn}which makeI(X;Z) as close toI(X;Y ) as possible. Wecan optimize the values ofqi by a numerical maximizationof the mutual information between the channel input andthe quantized channel output.As an example, we assume thatEs/σ2 is equal to 13dB and we transmit bits over the AWGN channel using aPAM-8 constellation which is defined asS = {−7,−5,−3,− 1, 1, 3, 5, 7}. Suppose that we use a non-uniform inter-val quantizer to quantize the channel output into2m levelswherem is equal to 4 or 5. The result of the optimizationis as follows:{qi} = {0.75, 1.58, 2.55, 3.52, 4.5, 5.44, 6.55}for m = 4 (23){qi} = {0.55, 1.11, 1.51, 1.87, 2.28, 2.73, 3.42,3.72, 4.4, 5.05, 5.39, 5.91, 6.47, 7.07, 7.34}for m = 5 (24)Figure 3 shows the quadrature signal constellation and op-timum non-uniform quantization scheme which are gener-ated by using each dimension independently.3.3. ComparisonsIn this section, we show the quantization effect from aninformation theoretical point of view and find out the opti-Proceedings of SPS-DARTS 2007 (the 2007 The third annual IEEE BENELUX/DSP Valley Signal Processing Symposium)181mum quantization scheme by comparing the uniform quan-tization scheme and the non-uniform quantization scheme.We map the source bits into a conventional PAM signalconstellation symbol as defined in (16). We do the com-parison in two situations as follows. First, we choosed = 2 and SNR = 10 dB, and optimize the set of quan-tization levels for the two schemes as described in section3.1 and 3.1 whenm is equal to 3 or 4. The optimum spac-ing q for the uniform quantization is:q = {1 for m = 30.5 for m = 4(25)and the optimum set of quantization levels{qi} for thenon-uniform quantization is:{qi} = {0.64, 1.71, 2.37} for m = 3 (26){qi} = {0.31, 0.79, 1.29,1.75, 2.06, 2.38, 2.87} for m = 4 (27)For the second case, we choosed = 3 and SNR = 13dB and find out the optimum set of quantization levels foreach scheme form = 4 or 5. The optimum results for theuniform quantization whend = 3 are:q = {1 for m = 40.5 for m = 5(28)and the optimum set for the non-uniform quantization isshown in (23).Suppose the channel in figure 1 is an AWGN channel. Byusing the quantization scheme we define in section 3.1 and3.1, we calculate the theoretical channel capacity limitsfor the quantized channel, which are shown in figure 4.This figure shows the constrained capacity of the quan-tized channelI(X;Z) as a function of the bit signal-to-noise ratioEbN0of the channel for eachd, m and quantiza-tion scheme. The figure also shows the capacity limit ofthe AWGN channel and the Shannon limit.Figure 4 shows the non-uniform quantization scheme worksslightly better than the uniform one, especially in the lowerm. In this figure, we observe that there is a loss in capac-ity limit and transmitting power due to the quantization,but the loss is small comparing to the unquantized chan-nel. From an information theoretical point view, the ratedifference between the quantized channel and the unquan-tized channel can be neglected form = d + 2. So, for theAWGN channel reliable transmission at a rate ofR bit/userequires at mostR + 2 quantized bits.4. CONCLUSIONSWe have investigated the quantized channel from an in-formation theoretical point of view. We have shown anoptimum quantized scheme which can be used in the re-liable communication over the quantized channel. Fur-thermore, we have shown thatR + 2 quantized bits isalready enough for reliable communication at rate ofRbit/use over the AWGN channel. Therefore, it is possibleto lower the resolution of AD converter, which means thepower consumption at the receiver can be lowered.5. REFERENCES[1] T.Richardson and R.Urbanke,Modern CodingTheory, http://lthcwww.epfl.ch/mct/index.php.[2] T.S. Rappaport,Wireless Communications Principlesand Practice, Prenitce Hall[3] H. Imai and S.Hirakawa,A new multilevel codingmethod using error-correcting codes, IEEE trans. In-form. Theory, vol.23, pp.371-377, May 1977.[4] U. Wachsmann, R.F.H.Fischer and J.B.Huber,Multi-level codes: theoretical concepts and practical designrules, IEEE Trans. Inform. Theory, vol.45, pp.1361-1391, July 1999.Proceedings of SPS-DARTS 2007 (the 2007 The third annual IEEE BENELUX/DSP Valley Signal Processing Symposium)182−8 −6 −4 −2 0 2 4 6 8−8−6−4−202468QuadratureIn−PhaseScatter plot(a) m = 4−8 −6 −4 −2 0 2 4 6 8−8−6−4−202468QuadratureIn−PhaseScatter plot(b) m = 5Figure 3: The quadrature PAM signal constellation with64 symbols and optimum non-uniform quantization withd = 3 and SNR = 13 dB.0 2 4 6 8 10 12 14 16 18 2000.511.522.533.544.5Eb/N0 (dB)RateShannon limitUnquant limitNon−uni quant limit with m=4Uni quant limit with m=4Non−uni quant limit with m=3Uni quant limit with m=3(a) d=20 2 4 6 8 10 12 14 16 1800.511.522.533.544.5Eb/N0 (dB)RateShannon limitUnquant limitNon−uni quant limit with m=5Uni quant limit with m=5Non−uni quant limit with m=4Uni quant limit with m=4(b) d=3Figure 4:The capacity limit of the quantized channel withPAM constellationProceedings of SPS-DARTS 2007 (the 2007 The third annual IEEE BENELUX/DSP Valley Signal Processing Symposium)183

Modulation and coding for quantized channels

Q u a d r a t u r e In−Phase Scatter plot (a) m

R.F.H.Fischer and J.B.Huber, Multilevel codes: theoretical concepts and practical design rules,

Wireless Communications Principles and Practice, Prenitce Hall [3] H. Imai and S.Hirakawa, A new multilevel coding method using error-correcting codes,

http://purl.utwente.nl/publications/64258

Modulation and coding for quantized channels

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