Source-Channel Coding Tradeoff in MultipleAntenna Multiple Access Channels

Ebrahim MolavianJazi and J. Nicholas LanemanDepartment of Electrical Engineering

University of Notre DameNotre Dame, IN 46556

Email: {emolavia,jnl}@nd.edu

Abstract—We investigate channel code rates for communica-

tion of finite-dimensional analog sources over a multiple-antenna

multiple access channel (MAC) so that the average end-to-end

distortions are minimized. Our analysis uses the high-resolution

quantization theory for the sources and the high-SNR diversity-

multiplexing tradeoff for the MAC. We prove that carefully

balanced channel coding rates, usually far from the boundary of

the MAC capacity region, are necessary to achieve the optimal

distortion exponent in a separated architecture. In particular,

for the case of source vectors of equal dimension, we show that

the channel interference from multiple users becomes crucial

in characterizing the optimal channel coding rates if individual

minimization of distortion for each user leads to a heavily loaded

regime for the MAC.

I. INTRODUCTION

The problem of joint source and channel coding is afundamental one in information theory that characterizes howthe encoder should communicate the best possible descriptionof the source through the channel. The traditional source-channel theorems are based on the assumption of asymptoti-cally large source and channel code blocklengths and suggestthat information be separately source- and channel-encoded ata rate just below the channel capacity.

In real-world applications, however, the coding blocklengthsare usually finite due to delay and complexity limitations. Forexample, communication of interactive voice or video overwireless cellular networks have stringent latency requirementsdemanding relatively short source blocklengths; e.g., CDMA-based IS-95 standard advises voice transmission in blocksof roughly 160 samples [1]. In such regimes, the classicalsource-channel analyses do not hold, and one should useother tools, such as source quantization theory and channelerror exponents, to obtain guidelines for system design. Ithas been shown that, under these assumptions, there is afundamental tradeoff between the source coding and channelcoding rates [2]–[4]; for a given (high) end-to-end transmissionrate, the channel coding rate must be large enough to avoidlow-resolution description of the source but small enough toallow sufficient redundancy for combating channel errors. Thechannel coding rate resulting from this balancing act has beenobserved to fall well below the channel capacity.

In this paper, we investigate a multiuser version of this prob-lem, in which several sources are to be communicated over ablock fading multiple access channel. Although not necessarily

optimal in general, we study the separation-based architecturethat is practically appealing due to its modular structure andtractable analysis. We consider multiple antennas for all theparties and focus on the moderate to high SNR regime. Insuch scenarios, the users of the communication system canexploit these multiple antennas to tradeoff their reliability andrate, as characterized by the well-known diversity-multiplexingtradeoff [5], [6]. Our results for the symmetric case showthat the multiple-user interference is effective if the receiver’santennae cannot spatially separate the users’s signals and thesource vector dimensions are greater than a certain threshold.

II. PROBLEM STATEMENT

As illustrated in Figure 1, we are interested in communicat-ing L independent sources of information over a multiple an-tenna multiple access channel. Consider sources Uk1

1 , ..., UkLL

that take values according to probability density functionsf1(uk1), ..., fL(ukL) on compact sets S1 ⊂ Rk1 , ...,SL ⊂RkL , respectively. To communicate these sources over theMAC, we use a concatenation of quantizers (source encoders)and channel encoders distributed across the sources. Eachsource U

kj

j , j = 1, ..., L, is first source coded via an mj-bitquantizer

Qj

�U

kj

j

�=

2mj�

i=1

vkj

j (i)1Aj(i)

�U

kj

j

�,

where {Aj(i)}2mj

i=1 is a partitioning of the set Sj ⊂ Rkj

into disjoint regions, each represented by the source code-word v

kj

j (i). As in [2], a permutation πj is applied to theset of mj-bit indices of these source codewords I

mj

j = ito symmetrize the channel code performance. Finally, forcommunication over the multiple access channel (MAC), thejth transmitter encodes the binary message πj(I

mj

j ) into thechannel codeword Xj using a channel codebook Cj(SNR)of blocklength n, rate mj

n = rj log SNR, and unit averageenergy per transmit antenna per symbol per codeword. Here,we consider M antennas for each transmitter and N antennasfor the receiver, facilitating a spatial multiplexing gain of rj foruser j as SNR →∞. The channel is a frequency-nonselectiveRayleigh block-fading multiple access channel

Y =�

SNRM

L�

j=1

HjXj + W ,

with multi-path fading captured by N ×M matrices {Hj}Lj=1

consisting of independent and identically distributed (i.i.d.)CN (0, 1) complex Gaussian entries and noise captured bythe N × n matrix W with i.i.d. complex Gaussian CN (0, 1)entries. The channel coding blocklength n is large withrespect to noise but short with respect to fading, so thatthe fading matrices, which are known at the receiver butunknown at the transmitters, remain constant over a singlechannel coding blocklength but evolve i.i.d from block toblock. At the receiver, a channel decoder decides jointly onthe L channel codewords, and the inverse permutation π−1

j anddequantization Q−1

j recover the source as Ukj

j = Vkj

j (Ikj

j ).

Fig. 1. Block diagram of communication of L analog sources over a MIMO-MAC.

The performance of this communication system is charac-terized by the average end-to-end distortion of the sources

D(Qj , MAC) = E����U

kj

j − Ukj

j

���p�

= Eπj

2mj�

i=1

2mj�

i=1

p�πj (i)|πj(i)

�Dii(Qj)

,

where � · �p is the p-th power of the Euclidian norm, p(i|i)is the probability of decoding message i as message i, andDii(Qj) is the noiseless-channel distortion of the (lossy)quantizer Qj between the source indices i and i

Dii(Qj) =�

Aj(i)

���ukj

j − vkj

j (i)���

pfj(u

kj

j )dukj

j .

The system design problem is to determine, for given (finite)values of source vector dimensions (k1, ..., kL) and channelcode blocklength n and as SNR →∞, the spatial multiplexinggains (r1, ..., rL) of the users such that the average end-to-enddistortions (D(Q1, MAC), ..., D(QL, MAC)) are minimized.

III. END-TO-END BEHAVIOR IN THE EXPONENT

From high-resolution quantization theory, as mj =nrj log SNR →∞ due to SNR growing asymptotically large,

any good quantizer has an exponentially decaying noiseless-channel distortion [2], [3]

D(Qj) = E����U

kj

j −Qj

�U

kj

j

����p�

=2mj�

i=1

Dii(Qj).= 2−pmj/kj .= SNR−pnrj/kj , (1)

where .= denotes equality in exponent in the limit of highSNR.

The results of Hochwald and Zeger [2] imply that theaverage end-to-end distortion can be decomposed into adistortion term due to source quantization plus a distortionterm due to channel impairments. Specifically, for permutationfunctions πj chosen uniformly at random, we have

D(Qj) + Pej,av ≤ D(Qj , MAC) ≤ D(Qj) + Pej,max , (2)

where Pej,max and Pej,av are the maximal and average errorprobabilities of user j in the MAC, respectively, and ≤ denotesinequality in exponent in the limit of high SNR.

In the moderate to high SNR regime, the well-knowndiversity-multiplexing tradeoff (DMT) [5] is a convenientframework for system analysis and design. For a multipleaccess channel, Tse and Viswanath and Zheng [6] show thatthe probability of a “type S error”, in which the channeldecoder makes error on messages in the set S ⊆ {1, ..., L}of users, behaves as

P (E(S)) .= SNR−d∗|S|M,N (�

i∈S ri) , (3)

where d∗M,N (r) is the well-known DMT function [5]. Al-though the analysis in [6] only considers the average errorprobability, one can verify that the above result also holds forthe maximal error probability: the outage probability is a lowerbound for the average and a fortiori maximal error probability,and the random coding error exponent upper bound usespairwise independence of all codewords of all users and thusapplies to each of the codewords, hence an upper bound forthe maximal error probability; for details, check [5, p.1082]and [6, p.1872].

The equality in the exponent of the average and maximalerror probability in the limit of high SNR along with (2)implies that

D(Qj , MAC) .= D(Qj) + Pej

where (1) asserts D(Qj).= SNR−pnrj/kj and (3) implies that

Pej =P

�

S⊆{1,...,L}j∈S

E(S)

.=

L�

i=1

�

|S|=ij∈S

SNR−d∗iM,N (�

l∈S rl) ,

since the events {E(S) : S ⊆ {1, ...,K}} are disjoint bydefinition. All together, we obtain the following expressionfor the average end-to-end distortion of each source.

D(Qj , MAC) .= SNR−pnrj/kj +L�

i=1

�

|S|=ij∈S

SNR−d∗iM,N (�

l∈S rl) .

(4)

IV. OPTIMIZED ENCODING RATES

In this section, we first give our main results on theoptimal channel coding rates and the corresponding distortionexponents separately for the cases of equal and unequal sourcevector dimensions, and then compare them with those of thetime division multiple access (TDMA) scheme.

A. The case of equal source vector dimensions

If all source vectors have equal dimensions k1 = ... = kL =k, the whole communication system is symmetric. Therefore,the optimal strategy for jointly minimizing the average end-to-end distortions is to set all multiplexing gains equal, i.e.,r1 = ... = rL = r. With this symmetry in (4), all sources willhave distortions that behave as

D(Qj , MAC) .= SNR−pnr/k +L�

i=1

�L− 1i− 1

�SNR−d∗iM,N (ir)

.= SNR−pnr/k +L�

i=1

SNR−d∗iM,N (ir)

.= SNR−pnr/k+SNR−d∗M,N (r)+SNR−d∗LM,N (Lr)

(5)

where the last equality is based on the analysis of [6], whichasserts that the dominant type of error for a symmetric MACis either for a single user’s message (|S| = 1) or for all users’messages (|S| = L).

As SNR → ∞, the asymptotic end-to-end distortion isminimized by equating the exponents of the three terms in (5).Notice that, the function pnr/k is linearly increasing in r,and recall from [5], [6] that the two DMT curves d∗M,N (r)and d∗LM,N (Lr) are piece-wise linearly decreasing in r. Theoptimal r∗ is thus given by

r∗ = argmax0≤r≤min{M,N/L}

min{pnr/k, d∗M,N (r), d∗LM,N (Lr)} .

(6)To characterize this expression, let rsu be the solution tothe single-user equation pnr/k = d∗M,N (r) and rau thesolution to the all-users equation pnr/k = d∗LM,N (Lr).Recalling from [6] that the solution of the equation d∗M,N (r) =d∗LM,N (Lr) is given by r = min(M, N

L+1 ), we concludea similar categorization to that observed for the DMT of asymmetric MAC [6] as follows:

• If rsu < rau < min(M, NL+1 ), then the system is in the

lightly loaded regime, r∗ = rsu, and only the DMT for asingle user being in error matters.

• If rsu > rau > min(M, NL+1 ), then the system is in the

heavily loaded regime, r∗ = rau, and only the DMT forall users being in error matters.

• The case rsu = rau = min(M, NL+1 ) can be seen as a

threshold between these two regimes.Our analysis shows that the multiple-user interference af-

fects the optimal multiplexing gain and the minimal end-to-end distortion if (i) N < (L + 1)M , i.e., the receiver doesnot have enough antennas to spatially separate different users’

signals, and (ii) k > kth, i.e., the source vector dimensions arenot small enough, where

kth =pnN

(L + 1)d∗M,N ( NL+1 )

. (7)

Hence, with sufficiently large number of receive antennas N ≥(L + 1)M or sufficiently small source vector dimensions k ≤kth, multiple-user interference is not dominant and each usercan minimize its individual distortion.

 

Fig. 2. Optimal and single-user multiplexing gains versus source vectordimension for a 2-user MIMO-MAC with M = 2, p = 2 and n = 500.

 

Fig. 3. Optimal and single-user distortion exponents versus source vectordimension for a 2-user MIMO-MAC with M = 2, p = 2 and n = 500.

We make the preceding remarks more concrete throughthe following illustrations for the case of L = 2 userseach with M = 2 transmit antennas. Figure 2 comparesthe optimal multiplexing gain r∗ as a function of the sourcevector dimension k for the case of a MIMO-MAC with that ofthe point-to-point MIMO case without interference [3]. Fromthis plot, one observes that the optimal multiplexing gain isan increasing function of the source vector dimension and,for most values of k, is well below the maximal symmetric

spatial multiplexing gain limk→∞ r∗ = min(M, NL ) of the

MAC, i.e., min(2, 22 ) = 1 for the case of N = 2 and

min(2, 42 ) = min(2, 6

2 ) = 2 for the cases of N = 4 andN = 6. Moreover, for a small number of receive antennas,2 = N < (L + 1)M = 6 and 4 = N < (L + 1)M = 6, athreshold exists as given in (7), kth ≈ 333 and kth ≈ 667respectively, beyond which the optimal r∗ for the MIMO-MAC is less than that for the single-user MIMO case, whilefor sufficiently large number of receive antennas 6 = N ≥(L + 1)M = 6, the optimal r∗ for the point-to-point MIMOand MIMO-MAC is observed to be identical. Figure 3 depictsa similar comparison for the optimal distortion exponent versusthe source vector dimension k. The effect of the number ofreceive antennas on the optimal distortion exponent is similarto that upon the optimal multiplexing gain, except for thefact that the difference in distortions of the MIMO-MAC andthe point-to-point MIMO configurations, if any, appears to bequite small.

B. The case of unequal source vector dimensionsIf the source dimensions are not equal, symmetric multiplex-

ing gains no longer appear to be optimal. However, one canminimize the sum of the end-to-end distortions of all sources:

min(r1,...,rL)

L�

j=1

SNR−pnrj/kj +�

S⊆{1,...,L}

SNR−d∗|S|M,N (�

l∈S rl)

subject to 0 ≤�

l∈S

rl ≤ min(|S|M,N), ∀S ⊆ {1, ..., L} .

Although we can apply the technique of balancing exponents,the above optimization problem does not lend itself to atractable analytical solution. Therefore, we restrict attentionto numerical study for the case of L = 2 users. For this case,the above optimization problem takes the simpler form:

min(r1,r2)

SNR−pnr1/k1 + SNR−pnr2/k2 + SNR−d∗M,N (r1)

+ SNR−d∗M,N (r2) + SNR−d∗2M,N (r1+r2)

subject to 0 ≤ r1, r2 ≤ min(M, N),0 ≤ r1 + r2 ≤ min(2M, N) .

Note that, there is a possibility that not one, but a familyof multiplexing gains (r∗1 , r∗2) lead to the minimal distortionexponent. In such cases, we choose the one which gives themaximal sum-rate.

Figure 4 shows how asymmetric source dimensions lead toasymmetric rate allocations by illustrating the ratio of optimalmultiplexing gains r∗2/r∗1 as a function of the ratio of sourcevector dimensions k2/k1 for several values of k1 selectedin different regimes k1 � n, k1 ≈ n, and k1 � n. Thisfigure suggests that, for a small number of receive antennas,the higher-dimensional source requires a greater multiplexinggain, while for a large number of receive antennas, theasymmetry of the source vector dimensions does not matterand the multiplexing gains are essentially allocated equallybetween the users. Furthermore, Figure 5 gives some sense ofhow far the optimal multiplexing gains are from the boundary

of the MAC capacity region by depicting the ratio r∗1+r∗2min(2M,N)

for different k1 and k2 values as above. This figure suggeststhat the optimal sum multiplexing gain of the two users isusually far below the maximal sum multiplexing gain, and thiseffect is more apparent when at least one of the two sourcedimensions is small.

 

Fig. 4. Ratio r∗2/r∗1 of optimal multiplexing gains versus the ratio k2/k1of source vector dimensions for a 2-user MIMO-MAC with M = 2, p = 2and n = 500.

 

Fig. 5. Ratio r∗1+r∗2min(2M,N) of optimal to maximal sum multiplexing gains

versus the ratio k2/k1 of source vector dimensions for a 2-user MIMO-MACwith M = 2, p = 2 and n = 500.

C. Comparison with TDMAIn this subsection, we compare the performance of the

optimal joint-decoding MAC operation with that of a timedivision multiple access (TDMA) scheme. TDMA is a sub-optimal but simple method for multiple users to access acommon channel by dividing channel uses among users so thatthey access the channel in turn. Accordingly, TDMA suffersfrom smaller rate for users, but benefits from preventing anyinterference to them. Therefore, in our setup, the communi-cation of different sources will decouple and the distortion

of each user will be only affected by an appropriate single-user DMT curve. Focusing on the case of source vectorswith equal dimensions, the best TDMA strategy is to assignsub-blocks of equal length to each user due to symmetry.By the time-sharing argument of [6, p.1869], the channelcoding rate for each user is then given by mTDMA/n =rL log SNR = rTDMA log SNR, while the diversity is expressedas 1

LSNR−d∗M,N (r) .= SNR−d∗M,N (r) = SNR−d∗M,N (LrTDMA). Onthe other hand, the source distortion exponent is expressedas D(Qj)

.= 2−pk mTDMA = SNR−

pnk rTDMA . Therefore, the best

TDMA scheme chooses its multiplexing gain r∗TDMA as thesolution of the equation

pn

krTDMA = d∗M,N (LrTDMA) . (8)

The inferior performance of the TDMA scheme in termsof the average end-to-end distortion is easily implied byobserving that, for all values of r, the scaled single-userDMT curve d∗M,N (Lr) of (8) is strictly below both theoriginal single-user DMT curve d∗M,N (r) and the all-usersDMT curve d∗LM,N (Lr) of (6). Hence, the best TDMA schemechooses a smaller multiplexing gain and thus achieves asmaller distortion exponent than the optimal MIMO-MAC, asillustrated in Figures 6 and 7 for the case of L = 2 users. Oneobserves that the gap between the performance of the TDMAscheme and that of the optimal MAC operation is often mostsignificant for moderate to large values of the source vectordimension.

 

Fig. 6. Optimal and TDMA multiplexing gains versus source vectordimension for a 2-user MIMO-MAC with M = 2, p = 2 and n = 500.

V. CONCLUSION

We have studied the problem of allocating source andchannel coding rates among several information sources tocommunicate them with the minimal end-to-end distortion.Although classical source-channel theorems in the asymp-totically large blocklength regime usually suggest operatingclose to the boundary of the channel capacity region, we haveshown that for transmission of finite-dimensional sources over

 

Fig. 7. Optimal and TDMA distortion exponents versus source vectordimension for a 2-user MIMO-MAC with M = 2, p = 2 and n = 500.

block fading channels in the high SNR regime, one needs toavoid such high rates and tolerate source coding with lowerresolution so that the channel fading effects can be mitigated.Our results suggest that applications with shorter sourcevectors may not be dominated by multiple-user interference,and applications with longer source vectors may require moreattention to it. It is worth noting that the practically interestingseparate architecture studied in this paper may not be optimalin general [7] and other joint techniques such as hybrid digital-analog coding [8] and superposition coding [9] can potentiallyachieve better distortion exponents.

ACKNOWLEDGMENT

This work was supported in part by NSF Grant CCF05-46618.

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