Multiplexing Two Information Sources over Fading. Channels: A Cross-layer Design Perspective

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1 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER 2005 Mutipexing Two Information Sources over Fading Channes: A Cross-ayer Design Perspective Zhiyu Yang and Lang Tong Abstract We consider the transmission over an unknown frequency-seective channe of two independent sources with different appication-ayer characteristics: one source such as voice has a ow information rate with a strict deay constraint; the other such as data has a high rate but without any deay constraints. We proposed a system structure that jointy considers the different decoding requirements of the appication ayer and the unknown fading nature of the physica channe. In the proposed communication system, piot symbos are not present and the ow-rate information is decoded non-coherenty first. The decoded ow-rate codewords are then used for channe estimation to faciitate coherent decoding of the high-rate source. For a fixed detection error probabiity of the ow-rate source, we derive achievabe rate expressions for the high-rate source. We demonstrate a convergence behavior of the achievabe rate of the high-rate source as the decision error probabiity of the ow-rate source goes to zero. Numerica resuts show that the achievabe rate of the high-rate source converges to that achievabe by a training-based scheme at moderate decision error eves. Index Terms Achievabe rates, cross-ayer design, decision-directed receivers, fading channes, mixed-rate transmission, orthogona-frequency-division-mutipexing OFDM. Corresponding author Z. Yang and L. Tong are with the Schoo of Eectrica and Computer Engineering, Corne University, Ithaca, NY 4853, USA emai: {zy26, tong}@ece.corne.edu. This work was supported in part by the Army Research Laboratory CTA on Communications and Networks under Grant DAAD , the Mutidiscipinary University Research Initiative MURI under the Office of Nava Research Contract N , and the Nationa Science Foundation under Contract CCR Part of this work has been presented at ISIT, Yokohama, Japan, June 2003.

2 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER I. INTRODUCTION A. Motivation We consider the transmission of a high-rate source mixed with a deay-sensitive ow-rate source over an unknown frequency-seective wireess channe. One exampe is transmitting ow-rate voice and high-rate data over the same ink. Another is mutipexing protoco information aong with high-rate data. In digita video broadcasting ], for exampe, contro information such as frame sequence numbers and moduation schemes is transmitted with video data. We assume that there is a deay constraint imposed on the ow-rate source, such as in the case of voice and other rea-time traffic. Therefore the codeword ength for the ow-rate source cannot be made arbitrariy ong. The ow-rate messages are aowed to be decoded with a certain error probabiity. For the high-rate source such as data traffic, on the other hand, our goa is to provide the highest rate at which reiabe communication is possibe. Therefore, we aow its codeword ength to be sufficienty ong to satisfy any predetermined error probabiity. We assume a time-varying ergodic bock fading channe, unknown at both transmitting and receiving ends. The optima approach in the information theoretic setting is to encode the two sources jointy across fading bocks and to perform non-coherent decoding that fuy expoits the channe statistics. Such an approach, despite its theoretica significance, provides itte guidance in practica impementation. A ayered, standard practica approach is to incude piot symbos in data packets. The channe acquisition in the physica ayer is made using the piot symbos and possiby data symbos as we within each bock and the decoder performs coherent decoding using the estimated channe. The demutipexing of the two sources is carried out in a higher ayer. Such a ayered approach, despite its simpicity, requires very high signa-to-noise ratio SNR in the physica ayer in order to satisfy simutaneousy the requirements of the appication ayer: the stringent deay constraint of the ow-rate source and the ow decoding error requirement of the high-rate source. Such an approach treats the two types of sources equay without taking into account the different decoding requirements set by the appication ayer. Furthermore, the use of piot symbos reduces the number of channe uses avaiabe for data transmission and is in genera suboptima. Can one expoit the decoding requirements from the appication ayer for a better physica

3 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER channe utiization? Can piot symbos be removed to save the bandwidth? One possibe approach is to use bind channe estimation techniques that acquire the common channe without requiring piot symbos. However, the anaysis of bind channe estimation couped with coherent detection is intractabe, and therefore, its performance unknown. In this paper, from a cross-ayer perspective, we propose an approach that expicity considers the decoding constraints from the appication ayer for a better physica ayer efficiency. B. Summary of Resuts We propose a system structure in which the two sources are encoded independenty in the physica ayer. We use a sequentia decision-directed receiver where a non-coherent detection is first used to decode the ow-rate source. The decoded codewords are then used as piots to estimate the high-rate channe response. The idea is to expoit the high redundancy in the coded ow-rate source to cope with the channe uncertainty. Once the codewords of the ow-rate source are obtained, they are amost as good as training for obtaining accurate channe estimates for the coherent detection of the high-rate source. Such an approach, jointy considering the appication ayer requirements and the physica channe uncertainty, requires ower SNR and has a higher channe utiization than the ayered approach. The probem woud be trivia, of course, if we assumed that the ow-rate source were aways decoded correcty. In this paper, we make an expicit assumption that decoding errors may be present for the ow-rate source and focus on the achievabe rate for the high-rate source. We demonstrate that when the ow-rate source has an error rate beow 0 3, which is a moderate error probabiity requirement for voice appications, the performance of the system for the high-rate source is cose to that when the ow-rate source is substituted by piot symbos. We first provide a ower bound on the achievabe rate for the high-rate source assuming that the decoded ow-rate source with errors provides partia channe state information. This ower bound generaizes a channe capacity bounding technique first popuarized by Medard 2]. Next, we provide an expicit evauation of the ower bound when the owrate source uses orthogona signaing a natura choice for non-coherent detection. We then consider specific ow-rate decoders and evauate the performance of the high-rate source under different decoding error probabiities of the ow-rate source. Numerica exampes are incuded to iustrate the effect of the decision error on the achievabe rate for the high-rate source.

4 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER C. Reated Work In the framework of transmitting mixed-rate sources as considered in this paper, one appication is to mode expicity the protoco information such as headers and other contro information as the ow-rate source. The idea of considering protoco information in transmission was first considered by Gaager 3], where the amount of protoco information needed aside from data to reconstruct random arriva sources is studied from an information theoretica perspective. A practica soution that expicity uses the ow-rate source for channe estimation and equaization was considered in 4], where it was assumed that the ow-rate source was decoded without error. In this work, the ow-rate decoding error is taken into consideration in the anaysis. The fading channe capacity has been a popuar research topic. A commony used bock fading channe mode is given by Y i j = H i k X k j + W i j where the subscripts denote the dimension of the matrices, and Y i j is the reception matrix, H i k the channe matrix, X k j the input matrix, W i j the noise matrix. H i k and W i j are ergodic and independent, and independent of the input X k j. In the mutipe-input-mutipeoutput MIMO fat fading case see, e.g., 5], i is the number of receivers, j the coherence time during which the channe stays constant, and k the number of transmitters. In the singeinput-singe-output SISO frequency seective case see, e.g., 6], j =, i is the receive vector ength, and k the transmit vector ength. In the iterature, extensive research has been done on the fading channe capacity for different assumptions on the knowedge of the fading channe H i k. Teatar obtained a cosed-form expression for the capacity of the Gaussian channe with Rayeigh fading assuming prefect fading knowedge at the receiver 7]. For the reaization of H i k unknown at the receiver, 8] and 9] address the properties of the capacity achieving input distribution. However, there are no cosed-form expressions for the channe capacity. Between the two extreme assumptions on the knowedge of the channe state, other work assumes that the receiver has partia information on the channe response, either provided by imperfect channe estimation and tracking or by some genie. The capacity metric with partia side information is difficut to obtain anayticay. As a resut, ower bounds on the channe capacity achievabe rates are often used as the performance measure of a system with partia side information. Most of the work assumes Gaussian codebooks and an ML

5 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER decoder to compute an achievabe rate e.g., see 5], 6], 0] 8]. In their anaysis, it is assumed that EH i k Ĥi k ] = Ĥi k where Ĥi k is the channe estimate side information. The above assumption is vaid if the MMSE channe estimator is used, which has a simpe form if the channe is estimated based ony on the outputs due to known piot symbos. In this work, we propose a channe estimation structure that utiizes the decoded ow-rate messages as piots for channe estimation. The resuted channe estimate does not satisfy. To obtain an achievabe rate expression, one can appy 9, Theorem 4.0.2], which is more genera and does not require the side information to satisfy. In this work, we obtain an achievabe rate expression that is arger than or equa to 9, Theorem 4.0.2] and use it to evauate the performance of the decision-directed receiver structure. D. Notations For convenience, denote the probabiity density function p.d.f. of a circuary symmetric compex Gaussian vector z of ength L with mean a and covariance matrix Q by N z a, Q π L Q e z ah Q z a. When there is no ambiguity, we use N a, Q to denote N z a, Q. Let diagz be the diagona matrix with the diagona entries equa to the entries of vector z. Let and H conjugate and conjugate transpose, respectivey. denote The paper is organized as foows. The system mode is introduced in Section II. In Section III, we derive a ower bound on the mutua information with Gaussian codebooks and side information and appy the ower bound to obtain an achievabe rate expression for the high-rate source by the decision-directed structure. Under the assumptions of binary ow-rate orthogona signaing, the computationa compexity of the achievabe rate is further reduced to that of a rea doube integra. Numerica resuts are presented in Section IV and the paper is concuded in Section V. II. SYSTEM MODEL A. Channe Mode We assume an ISI channe of ength m and it experiences identica independent distributed i.i.d. Rayeigh bock fading with coherence interva T c. The channe response, unknown at

6 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER either the transmitter or the receiver, remains constant during one coherence interva T c and changes to an independent vaue in the next interva. Denote h k = h k, h k 2,..., h k m] T the channe response at the k-th coherence interva. The channe responses h, h 2,... are i.i.d. N 0, m I. We assume that the system uses orthogona-frequency-division-mutipexing OFDM moduation with inverse-discrete-fourier-transform IDFT bock size T and cycic prefix ength T c T. Further assume T c T m. The channe is then converted into T parae fat fading channes: y k g k 0 0 x k y2 k 0 g k 2 x k 2 = +w..... k. y k T }{{} y k x k T } 0 {{ gt k }}{{} G k x k where the superscript k represents the OFDM bock index. x k and y k are the input and output vectors in the k-th bock, respectivey. We assume a power constraint on each input position of x k : E x k i 2] P. The white additive noise w k is i.i.d. N 0, I across bocks. The i-th diagonaized channe response g k i Fourier-transform DFT of h k, i.e., matrix of size T m with B. System Structure F] i = exp is the i-th Fourier coefficient of the T -point discrete- ] g k,..., gt k T = Fh k, where F is the truncated DFT j2πi. T We assume that there are two independent information sources to be communicated over the unknown frequency-seective fading channe. One source has a ow information rate R and an encoding deay constraint of N bocks, the other a high information rate R h and no deay constraints. The ow-rate source with the deay constraint may be voice or protoco information that is deay-sensitive. The high rate source without deay constraints may be deay-insensitive data. Since the two sources have different deay requirements, we assume that the two sources are encoded separatey and transmitted over dedicated tones Fig.. Suppose that T tones are aocated for the ow-rate transmission and T h tones for the high-rate transmission. Denote Π {0, } T T and Π h {0, } T h T the tone seection matrices for the ow-rate and highrate transmissions respectivey, where a tone seection matrix is part of a permutation matrix

7 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER that connects each input position to a distinct tone. Thus we have x k = Π x k and x k h = Π hx k as the channe inputs for the two sources respectivey. And the channe is given by y k = G k x k + w k 2 yh k = G k hx k h + wh k 3 where G k = diagπ Fh k and G k h = diagπ hfh k. M i M h N Enc n Enc h x x h Π Π h x g... g T Π T Π T h w + w h + y y h Dec Dec h Fig.. The system structure. Because of the different deay requirements, the information bit fows from the two sources are coected as messages and encoded into codewords based on different time scaes. Thus, mutipe ow-rate messages are sent during the transmission of one high-rate message, which, of course, has much more information bits packed due to a higher rate and a onger duration. Suppose that the ow-rate and high-rate codewords have bock engths N and n, respectivey. Denote M h {,..., 2 nr h } the high-rate message and M i {,..., 2 NR }, i n N, the i-th ow-rate message during the transmission of M h. Let {z} j i denote zi,..., z j. Then M i is encoded into codeword {x } in +i N and M h is encoded into codeword {x h } n Fig. 2. The ow-rate symbos x s and the high-rate symbos x h s are mutipexed through the tone seection matrices Π and Π h, obtaining the input sequence {x} n to the OFDM moduation. On the receiver side, due to the deay constraint, we assume that M i is decoded based on {y } in +i N ony. Without any deay constraints, the optima high-rate receiver decodes M h based on both {y h } n and {y } n with a maximum ikeihood ML decoder. When the ow-rate transmission is predetermined, i.e., known piot symbos are transmitted, Fig. reduces to a training-based scheme. C. Decision-directed Receiver Due to the compexity of the optima high-rate receiver, in this paper, we consider a receiver structure with a decision-directed channe estimator Fig. 3. For simpicity, we assume that

8 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER M M 2 x x 2 x N x x x 2 x n T Π x x h x Π h x h x 2 h x n h T h Fig. 2. The encoding structure. the ow-rate encoder uses binary constant moduus moduation, i.e., x k {u, u 2 } where PSfrag u i,j 2 repacements = P for i =, 2 and j =,..., T. Rewrite the ow-rate channe 2 as The symbo vector x k y k = diagx k Π F }{{} h k + w k. 4 S k {u, u 2 } has a one-to-one correspondence with the symbo matrix S k {U, U 2 }, where U i = diagu i Π F for i =, 2. {y } Det { S} ˆM ˆM,k Dec y k M.M. Est Ŝ k Enc ĥ k y k h Dec h Fig. 3. The decision-directed high-rate receiver. In the modified version, ˆM,k is decoded based on { S} k ony. As shown in Fig. 3, the ow-rate receiver first performs the symbo ML detection, and then decodes the ow-rate messages. The decoded ow-rate messages are re-encoded to faciitate the channe estimation, and the channe estimates are used to assist the high-rate decoding.

9 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER More specificay, the j-th symbo matrix S j, j n, is detected by S j = arg max p j y S 5 S {U,U 2 } where p j y S is the conditiona p.d.f. of y j given S j. The message M i, i n N, is then decoded based on { S} in +i N. Denote ˆM i the decoded i-th ow-rate message. To estimate the channe in the k-th bock h k, where + i N k in for some i, the decoded ow-rate message ˆM i is re-encoded to obtain Ŝk, the k-th symbo decision after error correction. The channe output y k and the decision Ŝk are pugged into the mismatched minimum-mean-squared-error MMSE estimator to obtain the k-th channe estimate ĥk, which assumes the decision Ŝk is aways correct, ĥ k Ŝk, y k = E h k S k = Ŝk, y k ] = Ŝ k H Ŝ k Ŝk H + mi y k. 6 The high-rate message M h is then decoded based on {y h } n and {ĥ}n. Due to the deay constraint, the decoding error of the ow-rate source is ower bounded from zero. However, if the information rate of the high-rate source is beow some threshod, the decoding error of the high-rate source can be made arbitrariy sma by appying arbitrariy ong high-rate codewords. In the next section, we study the rate at which the high-rate source can be reiaby communicated with the decision-directed receiver structure. III. HIGH-RATE CHANNEL ACHIEVABLE RATE We consider the achievabe rate for the high-rate channe in this section. We first prove a ower bound on the mutua information with Gaussian codebooks and side information. The ower bound is appied to obtain an expression for the achievabe rate with a Gaussian codebook and the decision-directed receiver. Under the assumption of binary orthogona owrate signaing and a modified channe estimator, the achievabe rate is further reduced to an expression that invoves ony a rea doube integra of an agebraic function. A. Lower Bound on Mutua Information with Gaussian Codebooks for the High-rate Source Since the ow-rate messages, unknown to the receiver, are encoded into codewords of ength N, we consider the N-th extension of the channe and omit the ow-rate message index i. Let {z} denote {z} N. For any high-rate input distribution p{x h }, the rate I{x N h}; {y h }, {ĥ} is achievabe by the decision-directed receiver structure. The optima input distribution p{x h }, however, is difficut to obtain. In this paper, we are interested in the achievabe rate by

10 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER an i.i.d. Gaussian codebook, i.e., when x h,..., xn h are i.i.d. N 0, P I, and denote the mutua information when using a Gaussian codebook as I g ;. The mutua information I N g{x h }; {y h }, {ĥ}, however, is sti difficut to evauate. In this section, we derive a simpe achievabe rate by ower bounding the mutua information. For convenience, define a function F on the domain D {c, r, P : c C, r R, r c 2, P R + }: P c 2 F c, r, P og P r c 2 Lemma : Let x N 0, P and et w C with zero mean and unit variance be independent of x. Let g C and G SI beongs to an arbitrary domain, and et g, G SI be independent of x, w. Let y = gx + w. Let f be a measurabe function of G SI, and z = fg SI. Then I g x; y, G SI E z F Eg z], E g 2 z ], P ]. 8 The ower bound is maximized by etting z = G SI. Proof: See Appendix I. A ower bound expression on the mutua information with Gaussian codebooks and side information was obtained in 9] by using z = Eg GSI ]. However, 8 is more genera and the maxima is achieved when z = G SI. Let x k h,i and yk h,i denote the i-th entries of xk h and yk h respectivey. Let gk h,i denote the i-th diagona entry of the matrix G k h = diagπ hfh k. Since x k h,i, i T h and k N, are i.i.d., the Gaussian mutua information is ower bounded as foows, N I g {xh }; {y h }, {ĥ} = N h {x h } N h{x h} {y h }, {ĥ} = N T h hx k N h,i N h {x h } {y h }, {ĥ} N = N N k= i= N T h hx k h,i hx k h,i y k h,i, ĥk k= i= N T h I g x k h,i ; yh,i, k ĥk k= T h i= i= N k= Eĥk 9 F Egh,i k ĥ k ], E gh,i k 2 ĥ k], P ] 0 where 9 is because conditioning reduces entropy, 0 by viewing ĥk as the side information and appying Lemma.

11 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER 2005 Thus we have obtained a ower bound on N I g {xh }; {y h }, {ĥ}. Denote p k ĥ h the conditiona p.d.f. of ĥk given h k. If we know p k ĥ h for a k {,..., N}, it is possibe to evauate 0. However, in some situations, we may ony be abe to access the average of p k ĥ h, pĥ h N N p k ĥ h. The next proposition summarizes an achievabe rate expression that does not depend on individua p k ĥ h s. k= Since x k h, hk, wh k are i.i.d. across bock index k {,..., N}, there is no ost of generaity in considering k = for the next anaysis. Introduce a new random vector ĥ with pĥ h as the p.d.f. conditioning on h. Let h, ĥ, w h, and x h be independent. Proposition 2: The Gaussian mutua information is ower bounded by N I g {xh }; {y h }, {ĥ} T h i= Proof: See Appendix II. Eĥ F Egh,i ĥ], E gh,i 2 ĥ ], P ] R d. 2 B. Achievabe Rate under Binary Orthogona Low-rate Signaing Since gh,i is a function of h, if ph, ĥ is known, it is possibe to evauate the achievabe rate R d by 2. However, the computation invoves an m-dimensiona compex integration and the cacuations of the conditiona mean E gh,i ] ĥ and the conditiona second moment E gh,i 2 ĥ ] for a ĥ. Next, under the assumptions of binary orthogona ow-rate signaing and a modified channe estimator, we reduce R d to an expression that invoves ony a doube integra. We assume that T 2m. The binary ow-rate signaing set {u, u 2 } and the ow-rate tone seection matrix Π are designed such that for i, j =, 2, P T U H I if i = j i U j = 0 if i j where U i = diagu i Π F. Exampes of orthogona signaing designs incude equay spaced ow-rate tones and {u, u 2 } being coumns of a Fourier transform matrix. For instance, if T = qt where q is an integer, aocate to the ow-rate channe the tones {p, p+q,..., p+qt } where p q. Let u i, = exp j2πi m T U and U 2 have the orthogona property 3. 3 for i =, 2 and =,..., T. The resuting

12 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER Because of the orthogonaity, we have Ŝk H Ŝ k = P T I. Thus I + mŝk Ŝk H = I m + P T Ŝ k Ŝk H. 4 Therefore, the channe estimator 6 reduces to ĥ k Ŝk = H y k m + P T 5 Ŝk = H S k h k + w k, m + P T 6 where 6 is obtained by substituting the ow-rate channe 4 into 5. In the anaysis, we consider a modified channe estimator Fig. 3. Consider estimating the k-th channe h k, k N. Let {z} k denote {z,..., z k, z k+,..., z N }. In the modified version, the ow-rate message is decoded based on the observation { S} k ony. Denote ˆM,k the output of the modified ow-rate decoder when S k is discarded. The rest of the operation of the modified estimator remains the same as the origina version: obtain Ŝk ; Ŝk and y k ˆM,k is re-encoded to are pugged into the mismatched MMSE estimator 5 to obtain the estimate ĥk. Since one of the inputs to the ow-rate decoder is discarded in the modified estimator, we expect that the origina version has a better performance. As described above, Ŝk in the modified estimator is obtained based on the observation of {y } k, a function of {h} k, {w } k, {S} k. Since h k, w k is independent of {h}k, {w } k, {S}, h k, w k is independent of Sk, Ŝk. Therefore, from 6 and the orthogonaity of the owrate symbos 3, the conditiona p.d.f. of ĥk conditioning on h k, S k and Ŝk is given by ĥ Nĥ P T P T p k h, S, Ŝ m+p T = h, m+p T I if S = Ŝ 2 Nĥ 0, P T m+p T I if S Ŝ. 2 Denote P k S, Ŝ the probabiity mass function p.m.f. of Sk, Ŝk. Appy the independence of S k, Ŝk and h k and rewrite as pĥ h = N P k S, N Ŝpkĥ h, S, Ŝ k= S,Ŝ = θnĥ P T m + P T h, P T m + P T I + 2 θnĥ 0, where the average decision error probabiity θ N P r S k N Ŝk. k= P T m + P T I, 7 2 It is worth noting that the decision error probabiity at different bocks P r S k Ŝk may not be the same, depending on the ow-rate codebook. However, pĥ h depends ony on θ.

13 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER Appying 7 to 2 gives the foowing proposition: Proposition 3: Under the assumption of the orthogona ow-rate signaing and the modified channe estimator, the achievabe rate is given by R d = T h θf P T P T + θf, 8 mm + P T m + P T 2 where fλ v m 2 2 exp v +v 2 /2 2 m m 2! f exp v /2 λ f 2 v 2, λ2 v 2 2 mλ 2 v, λ2 v +v 2 dv 2 2 dv 2 if m 2, dv if m =, f a 2, b 2 P f 2 a 2, b 2 og + + P f 3 a 2, b 2 f 2 a 2, b 2, f 2 a 2, b 2 a 2 f 4 b 2 θf 5 b 2 P T 2,, mm + P T f 3 a 2, b 2 θf f 4 b 2 5 b 2 P T m, + a 2 + θf 5 b 2, mm + P T m + P T f 4 b 2 θf 5 b 2 P T, + θf 5 b 2 P T,, mm + P T m + P T 2 f 5 b 2, c 2 exp b2 /c 2 πc 2 m. Proof: See Appendix III. P T m + P T 2, Equation 8 invoves ony a doube integra function, which can be readiy numerica evauated. Appying Dominant Convergence Theorem to 8 gives the foowing coroary: Coroary 4: The achievabe rate R d has the foowing convergence, im R d = T h e z P 2 T z og + dz R t. 9 θ 0 0 m + P m + T The rate R t is achievabe by the training-based scheme 6]. Coroary 4 shows that the achievabe rate by the decision-directed receiver converges to that by the training-based scheme as the decision error probabiity goes to zero. C. Low-rate Decoder As shown in 8, the achievabe rate R d is affected by the ow-rate decision error θ. Here we study the performance of the modified ow-rate decoder. We first derive the crossover probabiity of the equivaent binary symmetric channe BSC, and then obtain the error probabiity of the modified decoder with a bock code or a random code. Rewrite the ML ow-rate symbo detector 5 as S j = arg max S {U,U 2 } N y j 0, m SSH + I

14 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER y = arg min y j H S {U,U 2 } m SSH j + I 20 = arg max S H y j 2 2 S {U,U 2 } where 2 is because of 4 and 20 because of the foowing equaity, m + P T m. det m SSH + I = det m SH S + I = m The ML detector 2 converts the ow-rate channe into a BSC. The crossover probabiity of the BSC, ɛ P r S j S j, is given by ɛ = m i=0 m + i! m!i!. 22 m+p T + i + m+p T m m The derivation of 22 is given in Appendix IV. The decision error probabiity θ depends on the error correction performance of the owrate code. We first consider using an N, k, t code for the ow-rate source, where N, k, t denotes bock codes with ength N, k input bits, and t bits error correction capabiity. Reca that in the modified high-rate receiver Fig. 3, m ˆM,k is decoded based on { S} k ony. We repace S k with an independent fair coin fip output and use the decoder that corrects up to t bits of error. The decision error probabiity is then bounded by θ N P r N ˆM,k M 23 k= = N N ɛ i ɛ N i + N N ɛ i ɛ N i 24 2 i 2 i i=t+ where 23 is because {Ŝk S k } impies { ˆM,k M }, 24 due to the random repacement. Next, we consider a random code for the ow-rate source. Denote C the random codebook, with 2 NR messages and N symbos for each message, generated with equa probabiity over the binary aphabet. By the random coding argument 20], the modified ow-rate decoder that discards S k has the ensembe error probabiity, averaged over the codebook ensembe, i=t E C P r ˆM ],k M C 2 N Er N N R,ɛ 25 where the random coding exponent E r R, ɛ = max ρ + ρ og 2 ɛ +ρ + ɛ +ρ 0 ρ Since the right hand side of 25 does not depend on k, we have N ] E C P r N ˆM,k M C 2 N Er N N R,ɛ, k= ρr.

15 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER which, together with 23, impies that there exists a codebook such that θ 2 N Er N N R,ɛ. 26 D. Optima Decoding In this subsection, we investigate the rate achievabe by decoders without the expicit channe estimation structure. We derive an achievabe rate for the optima decoding structure, expressed as a function of the ow-rate decoding error probabiity. We compare the achievabe rates by the decision-directed and the optima-decoding receivers in Section IV and iustrate the sub-optimaity of the decision-directed structure at arge decision error probabiities. Denote γ P r ˆM M the ow-rate decoding error probabiity. When the expicit channe estimation constraint is removed, the rate N I {x h }; {y h }, {y } is achievabe. By the chain rue of mutua information, I {x h }; {y h }, {y }, {x } I {x h }; {y h }, {y } = I {x h }; {x } {y h }, {y } H {x } {y h }, {y } 27 H {x } {y } 28 Hγ + γnr 29 where 27 is due to the positivity of entropy, 28 due to the fact that conditioning decreases entropy, and 29 due to Fano s inequaity. The achievabe rate by the optima decoding structure is therefore ower bounded by N I {x h }; {y h }, {y } N I {x h }; {y h }, {y }, {x } Hγ N γr. 30 As γ goes to zero, the optima receiver achieves the performance of the training based scheme I {x N h }; {y h }, {y }, {x }. Even if γ is arge, the difference of the two achievabe rates is ess than N + R, which is sma since R is sma. have Since R t 9 is achievabe by the training-based scheme with a Gaussian codebook, we N I g {xh }; {y h }, {y }, {x } R t. 3 Appying 3 to 30 gives the achievabe rate with the optima decoder and a Gaussian codebook N I g {xh }; {y h }, {y } R t Hγ N γr R o. 32

16 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER IV. NUMERICAL RESULTS In the first exampe, we assume that the ow-rate channe uses the binary orthogona signaing and an N, k, t error correction code. The ow-rate detector converts the ow-rate channe into a BSC with crossover probabiity ɛ 22. The decision error probabiity θ is bounded by 24. We assume the channe ength m = 4, the OFDM bock size T = 64, the tone aocation T, T h = 8, 56, and the 7, 4, Hamming code for ow-rate error correction. Fig. 4 shows the achievabe rate versus P, the signa-to-noise ratio SNR: R d 8 for the decisiondirected structure, R t 9 for the training-based scheme, and R o 32 for the optima-decoding structure. Aso shown in Fig. 4 is the decision error probabiity θ. The SNR vaue of the vertica ine that a θ vaue points at is the SNR vaue at which the decision error probabiity equas that θ vaue. In Fig. 4 and the foowing figures, achievabe rates are normaized by the OFDM bock size T..5 Normaized Achievabe Rate vs. SNR: T,T h =8,56, N,k,t=7,4,, m=4 Normaized Achievabe Rate bits/symbo/tone 0.5 θ=0 θ=0 2 Decision Directed Training Based Optima Decoding SNR P db Fig. 4. Normaized achievabe rate versus SNR. As shown in Fig. 4, when SNR increases, the ow-rate decoding error probabiity decreases, and the achievabe rate by the decision-directed receiver converges to the achievabe rate by the training-based scheme. The convergence happens at moderate decision error probabiity eves, around θ = The achievabe rate by the optima-decoding receiver, on

17 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER the other hand, is very cose to that by the training-based scheme regardess of the SNR eve. In the second exampe, we fix SNR and the ow-rate information rate R and vary the ow-rate code ength N. The decision error probabiity θ decreases as the code ength N increases. Fig. 5 shows the achievabe rates versus the decision error probabiity θ when T, T h = 8, 56 and m = 4 under two SNR eves, P = 5dB and P = 7dB. When the decision error probabiity θ goes to zero, the achievabe rate R d for the decision-directed structure converges to R t for the training-based scheme. The convergence happens at moderate decision error probabiities, around θ = 0 3. The reationship between θ and N is given by the random coding bound 26. Fig. 6 shows the achievabe rate R d versus the coding ength N when T, T h = 8, 56, m = 4, and R = /2 bits per OFDM bock under two SNR eves, P = 5dB and P = 7dB. Fig. 6 is very simiar to the mirror image of Fig. 5 aong the vertica axis. Fig. 6 shows how ong the ow-rate message shoud be encoded with a random code in order to consider the ow-rate decoding is error free. In Fig. 6, the curves converge when N 20. Aso shown in Fig. 5 and Fig. 6 is the achievabe rate by the optimadecoding receiver, which is very cose to that by the training-based scheme, indicating the sub-optimaity of the decision-directed scheme when the decision error probabiity is arge. 3.5 Normaized Achievabe Rate vs. θ: T,T h =8,56, m=4 3 Normaized Achievabe Rate bits/symbo/tone Decision Directed, P=5dB Training Based, P=5dB Optima Decoding, P=5dB Decision Directed, P=7dB Training Based, P=7dB Optima Decoding, P=7dB Decision Error Probabiity θ Fig. 5. Normaized achievabe rate versus decision-error probabiity.

18 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER Normaized Achievabe Rate vs. N: T,T h =8,56, m=4, R =/2bits/symbo 3 Normaized Achievabe Rate bits/symbo/tone Decision Directed, P=5dB Training Based, P=5dB Optima Decoding, P=5dB Decision Directed, P=7dB Training Based, P=7dB Optima Decoding, P=7dB Low Rate Coding Length N Fig. 6. Normaized achievabe rate versus ow-rate coding ength. V. SUMMARY We have studied the communication of two independent sources over unknown fading channes. We have proposed, and anayzed the performance of, a system structure Fig. that jointy considers the different decoding requirements of the appication ayer and the unknown fading nature of the physica channe for a better channe utiization. Two receiver structures are considered. In the decision-directed structure, the ow-rate transmission is utiized to estimate the channe response, which assists the high-rate decoding. In the optima decoding structure, the expicit channe estimation constraint is removed. We have derived a ower bound on the mutua information with Gaussian codebooks and side information. The ower bound generaizes previous pubished resuts 2], 5], 6], 0] 9] in that it does not require the side information to satisfy and it is tighter than the resut in 9]. We have used the ower bound to obtain high-rate achievabe rate expressions for both receiver structures. The cacuation of the achievabe rate by the decision-directed receiver is reduced to a doube integra under the assumption of binary orthogona ow-rate signaing. We have shown that the achievabe rates by the decision-directed receiver and the optima receiver converge to that by the training-based scheme as the ow-rate decision error

19 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER probabiity goes to zero. By simuation, the convergence happens at moderate ow-rate error probabiities. Compared with the training-based scheme, the rate achievabe using optima decoding is ower by at most N + R regardess of the ow-rate error probabiity. When the ow-rate error probabiity is arge, the decision-directed structure is suboptima. When the ow-rate decoding error probabiity is sma, both decoding schemes provide an additiona ow-rate channe whie maintaining the performance of the high-rate channe cose to that of the training-based system. APPENDIX I PROOF OF LEMMA Reca y = gx + w. We have Exy z] = P Eg z] and E y 2 z ] = P E g 2 z ] +. Defining covv u E v Ev] 2 u ], we have cov x Exy z] E y 2 z ]y z = E x 2 z ] Exy z] 2 E y 2 z ] = P P 2 The Gaussian mutua information is ower bounded as foows, Eg z] 2 P E g 2 z ] I g x; y, G SI I g x; y, z 34 = hx hx y, z = hx h x Exy z] E y 2 z ]y y, z hx h x Exy z] E y 2 z ]y z hx E z og πe cov x Exy z] E y 2 z ]y z = E z F Eg z], E g 2 z ], P ] 38 where 34 is because of the data processing inequaity, 35 because yexy z]/e y 2 z ] is a function of y, z, 36 because conditioning reduces entropy, 37 because the Gaussian distribution maximizes entropy among distributions with the same covariance, 38 because of 33 and the fact that x is compex Gaussian distributed with variance P. Thus we have proved the first part of Lemma. Next, we show the optimaity of z = G SI. Lemma 5: For r, r 2, r 3 R and r > r r 2 3, the function og r /r r 2 2 r 2 3 is convex in the tripe r, r 2, r 3. Proof: matrix with Let h denote the function and et 2 hr, r 2, r 3 R 3 3 denote the Hessian 2 h r i r j as the entry in the i-th row and the j-th coumn. Since the domain r >

20 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER r r 2 3 is convex, we ony need to show that 2 hr, r 2, r 3 is non-negative definite in the domain r > r r 2 3. Equivaenty, we need to show that the matrix A = r 2 r r 2 2 r hr, r 2, r 3 is non-negative definite. Some agebra reveas that detλi A = λ 2 αλ + βλ γ, α = 2r 2 r + r r r r 2 32r r 2 2 r 2 3, β = 2r 2 r r r r 2 2 r 2 3, γ = 2r 2 r r 2 2 r 2 3. Since r > r r 2 3, we have α, β, γ 0. Therefore, the three eigenvaues of the symmetric matrix A are non-negative, proving the non-negative definiteness of A. Coroary 6: For a fixed P > 0, F c, r, P is convex in the pair r, c in the domain D. Proof: Suppose c = c r + jc i where c r, c i R. Let r 0 = r + /P. We have F = og r 0 /r 0 c 2 r c 2 i and r 0 > c 2 r + c 2 i. By Lemma 5, F is convex in r 0, c r, c i, which is a inear function of r, c, thus proving the convexity of F in r, c. Since z is a function of G SI, we have E g z g] = E GSI ze g GSI,zg] = E GSI ze g GSI g]. Simiary, E g z g 2 ] = E GSI ze g GSI g 2 ]. Therefore, F E g z g], E g z g 2 ], P = F E GSI ze g GSI g], E GSI ze g GSI g 2 ], P E GSI z F E g GSI g], E g GSI g 2 ], P ] 39 where 39 is due to the convexity of F Coroary 6 and the fact that E g GSI g 2 ] E g GSI g] 2. Taking expectation over both sides of 39 concudes the proof of the optimaity of z = G SI. APPENDIX II PROOF OF PROPOSITION 2 Consider a fixed i {,..., T h }. Denote p k ĥ, g the p.d.f. of ĥk, gh,i k for k N. Let K {,..., N} be uniform distributed and ĥ, g have conditiona p.d.f. pĥ, g K = k = p k ĥ, g. Thus, Eg k h,i ĥ k = h 0 ] = Eg ĥ = h 0, K = k], 40 E g k h,i 2 ĥ k = h 0 ] = E g 2 ĥ = h0, K = k ]. 4

21 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER The i-th term of 0 is then ower bounded as foows N Eĥk F Egh,i k ĥ k ], E g k N h,i 2 ĥ k], P ] k= = E K Eĥ K F Eg ĥ, K], E g 2 ĥ, K ], P ] 42 = EĥE K ĥ F Eg ĥ, K], E g 2 ĥ, K ], P ] Eĥ F Eg ĥ], E g 2 ĥ ], P ] 43 = Eĥ F Egh,i ĥ], E gh,i 2 ĥ ], P ] 44 where 42 appies 40 and 4, 43 because of the convexity of F see Coroary 6 in Appendix I, 44 because ĥ, g h,i has the same distribution as ĥ, g. Summing over i on both sides of 44 and appying 0 concude the proof. APPENDIX III PROOF OF PROPOSITION 3 Reca G h = diagπ hfh. Denoting f H i the i-th row of Π h F, we have g h,i = f H i h. Since h N 0, m I and the p.d.f. of ĥ conditioning on h, pĥ h, is given by 7, it can be shown that the p.d.f. of ĥ and the conditiona p.d.f. of g h,i P T 0, 0, pĥĥ = θn ĥ p g ĥg ĥ = h,i pĥĥ θnĥ + θnĥ 0, From 46, some cacuations revea that Eg h,i Therefore, it is cear that E f f H i mm + P T I + θnĥ 0, P T mm + P T I N g fi H P T m + P T I N 2 g 0, ĥ] 2 = f 2 f H i ĥ 2, ĥ 2, E g h,i 2 ĥ ] = f 3 f H i ĥ 2, ĥ 2. Next, we consider the evauation of E f f H i given ĥ are P T m + P T I, 45 2 ĥ, m m + P T. 46 ĥ 2, ĥ 2 ] is the i-th term in the summation of 2. r 2, r 2 ] where r has ength m and distribution N 0, λ 2 I. Let s = Ur where U is a unitary matrix and the first row of U is equa to f H i / m. Denote s j the j-th entry of s, j m. We have s N 0, λ 2 I, s 2 = r 2, and f H i r 2 = m s 2. Define v 2 λ 2 s 2 and v 2 2 λ 2 m i=2 s i 2. When m 2, since s,..., s m are independent compex Gaussians, v and v 2 are independent gamma random variabes

22 TO APPEAR IN EURASIP SIGNAL PROCESSING JOURNAL, 4TH QUARTER with p.d.f. p v v = e v/2 and p 2 v2 v = vm 2 e v/2, respectivey 2]. When m =, we have 2 m m 2! p v v = e v/2 and p 2 v2 v = δv. Therefore, E f f H i r 2, r 2] = E f m s 2, s 2] mλ 2 v = E v,v 2 f, λ2 v + v 2 ] 2 2 = fλ 2. Since ĥ is a Gaussian mixture 45, we have E f f H i ĥ 2, ĥ 2] P T = θf mm + P T Reca that E f f H i gives the desired resut. + θf P T. m + P T 2 ĥ 2, ĥ 2 ] is the i-th term in the summation of 2. Summing over i APPENDIX IV DETECTION ERROR PROBABILITY In the derivation, we drop the symbo index j in the ML detector 2. Let r i = U H i y and v i = r i 2 for i =, 2. Suppose S = U is transmitted. The detection error probabiity is ɛ = P r v v 2 S = U. 47 Reca y = Sh + w, h N 0, m I, w N 0, I, and U, U 2 are orthogona 3. Given S = U, r and r 2 are Gaussians with p.d.f. N 0, P T m+p T m respectivey. Because of the orthogonaity of U and U 2 3, we have E r r H 2 I and N 0, P T I, ] S = U = 0. Therefore, given S = U, r and r 2 are independent zero mean Gaussian random vectors. Hence, given S = U, v and v 2 are independenty centra chi-square distributed with v m exp vm v P T S = U = m+p T mm 48! p v S P T m+p T m P r v2 v m S = U = e v P T i=0 v i 49 i! P T 2]. By the conditiona independence of v and v 2, the detection error 47 is expressed as ɛ = 0 P r v2 v S = U pv S v S = U dv. 50 Substituting 48 and 49 into 50, switching the order of the integration and the summation, and integrating produce the formua 22.

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