A Union Bound on the Error Probability of Binary
Codes Over Block-Fading Channels
Salam A. Zummo, Member, IEEE, Ping-Cheng Yeh, Member, IEEE, and Wayne E. Stark, Fellow, IEEE
Abstract—Block-fading is a popular channel model that approx-imates the behavior of different wireless communication systems. In this paper, a union bound on the error probability of binary-coded systems over block-fading channels is proposed. The bound is based on uniform interleaving of the coded sequence prior to transmission over the channel. The distribution of error bits over the fading blocks is computed. For a specific distribution pattern, the pairwise error probability is derived. Block-fading channels modeled as Rician and Nakagami distributions are studied. We consider coherent receivers with perfect and imperfect channel side information (SI) as well as noncoherent receivers employ-ing square-law combinemploy-ing. Throughout the paper, imperfect SI is obtained using pilot-aided estimation. A lower bound on the per-formance of iterative receivers that perform joint decoding and channel estimation is obtained assuming the receiver knows the correct data and uses them as pilots. From this, the tradeoff be-tween channel diversity and channel estimation is investigated and the optimal channel memory is approximated analytically. Fur-thermore, the optimal energy allocation for pilot signals is found for different channel memory lengths.
Index Terms—Block fading, block interference, channel esti-mation, convolutional codes, interleaving, Nakagami, pilot-aided, Rayleigh, Rician, union bound.
I. INTRODUCTION
A
SERIOUS challenge to having good communication qual-ity in wireless communication systems is the time-varying multipath fading environment, which causes the signal to be attenuated randomly and, consequently, the received signal-to-noise ratio (SNR) to vary severely. The fading distribution varies according to the propagation environment. For example, if a line-of-sight exists between the transmitter and receiver in ad-dition to the multipath reception, the fading process is modeled by a Rician distribution [1]. Another popular model for the fad-ing process is the Nakagami distribution [2], which provides a family of distributions that are well matched to measurements under different propagation environments [3], [4].Error-correcting codes and diversity techniques are standard approaches to mitigate multipath fading. In these techniques, an attempt to providing the receiver with independent fading Manuscript received October 28, 2003; revised December 20, 2004. This work was supported in part by King Fahd University of Petroleum and Minerals, National Taiwan University, and the Office of Naval Research under Grant N00014-03-1-0232. The review of this paper was coordinated by Prof. J Shea.
S. A. Zummo is with the Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (e-mail: zummo@kfupm.edu.sa).
P. Yeh is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: pcyeh@ntu.edu.tw).
W. E. Stark is with the Department of Electrical Engineering and Com-puter Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: stark@eecs.umich.edu).
Digital Object Identifier 10.1109/TVT.2005.858173
realizations of the channel is made. In coded systems, the channel diversity is defined as the number of independent fading realizations available to decode a codeword. Typically, the coded bits are interleaved prior to transmission over the channel in order to distribute errors resulting from consecutive deeply faded bits evenly over the codeword. The performance of coded systems over infinitely interleaved fading channels is commonly analyzed using the union bound as in [5] and [6]. In [3] and [7], the performance of noncoherent and coherent diversity systems over multipath Nakagami fading channels was derived assuming infinite interleaving.
In delay-sensitive applications, infinite interleaving becomes an impractical assumption. Therefore, channel models that exhibit memory such as the block-fading channel [8] are used to model wireless systems including frequency-hopped spread-spectrum (FH-SS) [9], time-division multiplexing (TDM) sys-tems, and orthogonal frequency division multiplexing (OFDM). In this model, a frame consists of blocks of bits that undergo independent fading, where the fading is constant for signals within each block. As an effort to optimize codes for block-fading channels, Leung et al. [10] proposed a class of codes suitable for block-fading channels referred to as multifrequency trellis codes. These codes are not practical for block-fading channels with large number of fading blocks. The perfor-mance of multifrequency codes was analyzed by Malkamaki et al. [11]. Also, several trellis and block codes suitable for block-fading channels with a small number of fading blocks were presented in [12]. In [13], convolutional codes were optimized for block-fading channels. The performance of incremental redundancy convolutional codes over block-fading channels was derived in [14].
In coherent receivers, channel side information (SI) is needed for decoding, i.e., the fading phase and amplitude. If the receiver knows the channel SI perfectly, large channel diversity improves the system performance. Clearly, this is a hypothetical case only. In practice, the receiver needs to estimate the channel. In this case, long channel memory permits better channel estimation. Therefore, longer channel memory improves the performance if the frame size is infinite. However, if the frame size is fi-nite, there exists a fundamental tradeoff between the channel diversity and channel estimation [15]. As the channel memory length increases, the channel diversity is reduced, but the chan-nel estimation becomes more accurate. On the other hand, short channel memory increases the channel diversity, which enables the decoder to average out the channel behavior at the cost of less accurate channel estimation.
In [15], Worthen et al. used the error exponent to find the optimal channel memory length of a coded system over some 0018-9545/$20.00 © 2005 IEEE
simple block memory channels. However, a method to analyze the performance of specific codes over block-fading channels with arbitrary frame size and channel memory length is needed. Such a method is crucial in optimizing the channel memory of a coded system employing practical channel estimation tech-niques. In this paper, we derive a union bound on the perfor-mance of binary convolutional codes over block-fading chan-nels. We assume uniform interleaving of the coded sequence prior to transmission over the channel. Based on this assump-tion, the distribution of error bits over the fading blocks is de-rived and the corresponding pairwise error probability is dede-rived under different channel SI assumptions. The proposed bound is used with pilot-aided channel estimation to optimize the chan-nel memory at which the system should operate. One issue that determines the performance of pilot-aided channel estimation is the fraction of energy devoted to the pilot signals. This problem was investigated for different communication systems (see, for example, [16]–[19]). In this paper, we investigate analytically the optimal pilot energy for pilot-aided channel estimation in block-fading channels.
The paper is organized as follows. In Section II, the coded system model is described. Then, the union bound on the error probability of convolutional codes over block-fading channels is derived in Section III. In Sections IV and V, the pairwise error probabilities are derived for Rician and Nakagami fading distributions, respectively, and results are discussed therein. The main conclusions are discussed in Section VI.
II. SYSTEMMODEL
The general block diagram of a binary coded system over block-fading channels is shown in Fig. 1. The transmitter con-sists of a convolutional encoder, a random interleaver, and a modulator. Time is divided into frames of duration N T sec-onds, where T is the transmission interval of a bit and N is the number of bits transmitted in a frame. In each time interval of duration KT , a rate-Rcencoder maps K information bits into
N coded bits, where Rc = K/N is the code rate. Each coded bit is modulated to generate a signal using either binary phase-shift keying (BPSK) or binary frequency-phase-shift keying (BFSK). The channel we adopt is a block-fading channel in which each frame is subject to F independent fading realizations, resulting in a block of m =N/F signals being affected by the same fading realization. If the channel coherence time is longer than the transmission duration of each block, the channel is reason-ably assumed to be constant [20]. The coded bits are interleaved prior to transmission over the channel in order to spread out burst errors in the decoder, which result from low instantaneous SNR at the the demodulator output due to fading.
Coherent or noncoherent detection can be employed at the receiver. In coherent receivers, the matched filter sampled output at time lT in the f th fading block is given by
yf ,l =
Eshfsf ,l+ zf ,l (1) where Es is the average received signal energy, sf ,l = (−1)cf , l, where c
f ,l is the corresponding coded bit, and zf ,l
Fig. 1. Structure of a binary coded system.
is an AWGN sample with a complex normalCN (0, N0)
distri-bution. The coefficient hf is the channel gain in fading block
f modeled asCN (0, 1). The channel gain can be written as hf = afexp(jθf), where θf is a uniformly distributed phase and af is the amplitude, which is assumed in the paper to have a Rician or a Nakagami distribution.
The receiver employs maximum likelihood (ML) sequence decoding which minimizes the frame error probability. In this rule, the decoder chooses the codeword S ={sf ,l, f = 1, . . . , F, l = 1, . . . , m} that maximizes the likelihood func-tion p(Y| S), where Y = {yf ,l, f = 1, . . . , F, l = 1, . . . , m}. If perfect SI is available at the receiver, the decoder chooses the codeword S that maximizes the metric
m(Y, S) = F f =1 m l=1 Re{y∗f ,lhfsf ,l} (2)
where Re{·} represents the real part of a complex number. Note that the perfect SI is a hypothetical assumption that is used to predict the best performance of the code. In practice, the channel is estimated at the receiver as will be discussed in Section IV-B.
In noncoherent systems, BFSK is used where the carrier fre-quency of the modulated signal is set to one of two frequencies according to whether the coded bit is c = 0, 1. For each received signal, square-law combining [21] is employed, whose outputs for c = 0, 1 are represented by
r(I ,c)f ,l =Esafδ(cf ,l, c) cos(θf) + η(I ,c)f ,l
r(Q ,c)f ,l =Esafδ(cf ,l, c) sin(θf) + ηf ,l(Q ,c) (3) where rf ,l(I ,c)and r(Q ,c)f ,l for c = 0, 1 are respectively the cor-relation of the received signal with the in-phase and quadra-ture dimensions of the signal corresponding to a coded bit c. In (3), θf is the unknown phase of the received signals in block f, δ(x, y) = 1 if x = y and δ(x, y) = 0, otherwise, and ηf ,l(I ,0), ηf ,l(Q ,0)ηf ,l(I ,1)and ηf ,l(Q ,1)are independent random variables with normal distribution, i.e.,N (0, N0/2) distribution. The
de-coder chooses the codeword S that maximizes
m(R, S) = F f =1 m l=1 r(I ,c)f ,l 2 + r(Q ,c)f ,l 2 (4)
where R ={rf ,l(I ,c), rf ,l(Q ,c), f = 1, . . . , F, l = 1, . . . , m}. Note that this decoder makes no use of channel SI in decoding, and is suboptimal with respect to minimizing the frame error prob-ability.
III. PERFORMANCEANALYSIS
In this section, a union bound on the bit error probability of convolutional codes over block-fading channels is derived. Throughout the paper, the subscripts c, u, and b are used to denote conditional, unconditional, and bit error probabilities, respectively. For linear convolutional codes with k input bits, the bit error probability is upper bounded [5] as
Pb ≤ 1 k N d=dmin wdPu(d) (5)
where dminis the minimum distance of the code, Pu(d) is the unconditional pairwise error probability defined as the prob-ability of decoding a received sequence as a weight-d code-word given that the all-zero codecode-word was transmitted. In (5), wd =
N
i=1iAi,d is the number of codewords with out-put weight d, where Ai,d is the number of codewords with output weight d and input weight i. The weight distribution {wd}Nd=dminis obtained directly from the weight enumerator of
the code [5].
A. Union Bound for Block-Fading Channels
In block-fading channels, Pu(d) in (5) is a function of the distribution of the d nonzero bits over the F fading blocks. This distribution is quantified assuming uniform channel interleaving of the coded bits over the fading blocks in a frame. Denote the number of fading blocks with weight v by fv and define
w = min(m, d), then the fading blocks are distributed according to the pattern f ={fv}wv =0if F = w v =0 fv, d = w v =1 vfv. (6)
Denote by L = F− f0 the number of fading blocks with
nonzero weights. Then, Pu(d) is determined by averaging over all possible fading block patterns as
Pu(d) = d L =d/m L1 f1=0 L2 f2=0 . . . Lw fw=0 Pu(d| f)pd(f ) (7) where Lv = min L− v−1 r =1 fr, d−vr =1−1rfr v , 1≤ v ≤ w. (8)
The probability of a fading block pattern for a specific codeword weight d is computed using combinatorics as
pd(f ) = m 1 f1 m 2 f2 . . . m w fw mF d · F ! f0!f1! . . . fw! . (9)
The left factor of pd(f ) in (9) is the probability of distributing
d nonzero bits over F blocks with fv blocks having v bits for possible values of v. The right term of pd(f ) is the number of combinations of f ={fv}wv =0 among the F fading blocks. Using (7)–(9), the union bound on the bit error probability of convolutional codes over a block fading channels is found by substituting (7) in (5).
It should be noted that carefully designed interleavers may outperform the uniform interleaver. However, analyzing coded systems with specific interleavers is much more complicated. Note that the number of summations involved in computing Pu(d) in (7) increases as the channel memory length increases. This makes the computation of (5) when summing over all d≤ N for long channel memory a time consuming task. A good approximation to the union bound is obtained by truncating (5) to a small value of dmax< N . This results in an approximation
to the error probability rather than an upper bound. However, it is well known that the low-weight terms in the union bound dominate the performance at high SNR values, where the bound is more useful. For low SNR values, simulations can always be performed easily to check the performance. However, at high SNR the bound becomes more accurate because of the domination of the low-weight terms. Thus the bound truncation does not affect the result at high SNR values for which the performance analysis is most important.
B. Pairwise Error Probability
The conditional pairwise error probability Pc(d| f) is de-fined as the probability of decoding a received sequence Y as a weight-d codeword ˆS given that the all-zero codeword S was
transmitted and conditioned on the channel fading gains and the fading block pattern f . It is given by
Pc(d| f) = Pr(m(Y, S) − m(Y, ˆS) < 0 | H, S, f) (10) where H ={hf}Ff =1. For a specific receiver, the unconditional pairwise error probability Pu(d| f) is found by substituting the corresponding decoding metric in (10) and then averaging over the fading statistics. The rest of the paper is devoted to deriving expressions for Pu(d| f) of coded systems over block-fading channels with different receivers and fading distributions.
IV. PAIRWISEERRORPROBABILITY FORRICIANFADING In this section, we derive the pairwise error probability of coded systems over Rician block-fading channels. Rician fading arises if there is a line-of-sight between the transmitter and the receiver [1]. In this model, the received signal is composed of two signal-dependent components, namely, the specular and diffuse components. The specular component is due to the
line-of-sight reception, whereas the diffuse component results from multipath reception. In this case, the channel gain in each fading block hf is modeled as a complex Gaussian variable with CN (b, 1) distribution, where b represents the specular component of the channel. Thus the amplitude af has a Rician distribution with a normalized density function [6] given by
faf(a) = 2a(1 + K)exp[−K − a
2(1 + K)] × I0 2aK(1 + K) , a≥ 0 (11)
where K = b2 is the energy of the specular component, and I0(· ) is the zero-order modified Bessel function of the first
kind. In this context, K denotes the ratio of the specular com-ponent energy to the diffuse comcom-ponent energy. When K = 0, the specular component is zero, resulting in the well-known Rayleigh fading distribution. In the following, Pu(d| f) is derived for coherent detection with perfect and imperfect SI available at the receiver. Furthermore, a square-law combining receiver is considered with Rayleigh fading.
A. Coherent Detection—Perfect SI
Recall that the received signal over a block-fading channel is given by (1) and the corresponding ML decoding rule is given by (2). Substituting the metric (2) in (10), the conditional pairwise error probability for coherent detection with perfect SI is given by Pc(d| f) = Pr L f =1 af m l=1 Re{yf ,l} < 0 H, S, f . (12) The distribution of Re{yf ,l} conditioned on af is Gaussian with mean√Esafsf ,land variance N0. Thus, Pc(d|f) simplifies to
Pc(d| f) = Q 2Rcγb w v =1 v fv i=1 a2 i (13)
where γb = Eb/N0 is the SNR per information bit. Note that
the average energy per bit is given by Eb = RcEs, where
Rc is the code rate. To find the unconditional pairwise error probability Pu(d| f), (13) is averaged over the statistics of the fading amplitudes in (11). An exact expression of the pairwise error probability is found using the integral form of the Q-function, Q(x) = 1ππ2 0 e(−x 2/2 sin2θ ) dθ [22], resulting in Pu(d| f) = 1 πEA π 2 0 exp −Rcγb sin2θ w v =1 v fv i=1 a2i dθ = 1 π π 2 0 w v =1 1 + K 1 + K + vRcγb/ sin2θ fv × exp − KvfvRcγb/ sin2θ 1 + K + vRcγb/ sin2θ dθ (14)
where the product results from the independence of the fading variables in different fading blocks. Since the integral in (14)
Fig. 2. Bit error probability of a rate-1/2 (23,35) convolutional code with perfect SI and a frame size N = 1024 for channel memory lengths m = 1, 8, 16, 32, 64 (solid: approximation using the union bound, dash: simulation).
is definite, its computation is straightforward using standard numerical integration packages.
Throughout the paper, the union bound was evaluated for a rate-1/2 (23,35) convolutional code with a frame size of N = 1024 coded bits. As discussed in Section III-A, the union bound is truncated to sum over codewords with a distance dmax≤ 12 in order to reduce the computational complexity.
The bound is compared to simulation results, in which the chan-nel interleaver is chosen randomly and is changed every ten frames to simulate the effect of the uniform interleaver. This re-sults in the performance of the average-performing interleaver over block-fading channels. In coherent systems, BPSK signal-ing is employed, and (14) is used in (7) and (5) to compute the union bound.
Fig. 2 shows the results for Rayleigh fading channels with perfect SI and different channel memory lengths. Since sim-ulating very low error rates is too difficult, we plot simulation curves down to error rates around Pb = 10−6. However, the ana-lytical curves are shown for all SNR values. We observe that the approximation is very close to the simulation curves for a wide range of channel memory lengths. Also, the approximation starts to be loose as the SNR decreases. It is well established that the union bound diverges at SNR values lower than the cutoff rate of the channel [23]. However, the value of the analytical results is more interesting for high SNR values, where simulation is dif-ficult to conduct. In the rest of the paper, analytical results are shown for high SNR values to make the presentation more clear.
Fig. 3 shows the SNR required for the convolutional code to achieve Pb = 10−4versus the specular-to-diffuse ratio K of a Rician fading channel. We observe that increasing the energy of the line-of-site component of the channel reduces the effect of the diversity provided by the independent fading blocks. This is expected, since increasing K causes the channel to become less random, which reduces the need for diversity at the decoder. We conclude that when perfect SI is available at the receiver,
Fig. 3. SNR required for a rate-1/2 (23, 35) convolutional code to achieve
Pb= 10−4versus the specular-to-diffuse ratio K (linear scale) for memory lengths m = 8, 16, 32, 64.
smaller specular-to-diffuse ratio makes the performance more sensitive to the lack of channel diversity.
B. Coherent Detection—Imperfect SI
For coherent detection with imperfect SI, it is necessary to estimate the channel SI. This is achieved by transmitting a pilot signal with energy Epin each fading block. The corresponding received signal is given by
yf ,p =
Ephf + zf ,p. (15) The ML estimator for hf is given by ˆhf = yf ,p/
Ep =
hf + ef, where ef = zf ,p/
Ep is the estimation error. The distribution of ef isCN (0, σ2e), where σ2e = N0/Ep. The cor-relation coefficient between the actual channel gain and its esti-mate is given by µ = E[(hf − b)(ˆhf − b) ∗] Var(hf)Var(ˆhf) = 1 1 + σ2 e . (16)
In order implement an ML sequence decoding rule, the like-lihood function of the channel observations (received and pilot signals) conditioned on the transmitted codeword p(Y, ˆH|S)
should be maximized. In [24], the ML decoding rule was shown to be difficult to implement in a Viterbi receiver. Therefore, a suboptimal decoding metric that maximizes the likelihood func-tion p(Y| ˆH, S) is used. It is given by choosing a codeword S
that maximizes the metric
m(Y, S) = F f =1 m l=1 Re{yf ,l∗ ˆhfsf ,l}. (17)
Substituting (17) in (10), the conditional pairwise error proba-bility for the suboptimal decoder becomes
Pc(d|, f) = Pr L f =1 m l=1 Re{yf ,l∗ ˆhf} < 0 H, S, fˆ . (18) In order to find (18), we need to find the distribution of yf ,l con-ditioned on ˆhf, which is a complex Gaussian random variable with a mean√Essf ,lE[h| ˆh] and a variance N0+ (1− µ2)Es, where E[h| ˆh] = µ/σ(ˆhf − b) + b and σ2= Var(ˆhf) = 1 +
σ2e. Thus, the conditional pairwise error probability for the sub-optimal decoder is given by
Pc(d| f) = Q 2Es F f =1df µσ(ˆhf − b) + b 2 N0+ (1− µ2)Es (19) where df is the number of nonzero error bits in fading block
f . Define the normalized complex Gaussian random variable ζf = (ˆhf − b)/σ + b/µ with a distribution CN (b/µ, 1). Then, the conditional pairwise error probability simplifies to
Pc(d| f) = Q 2µ2R cγb w v =1v fv i=1|ζi|2 1 + Rcγb(1− µ2) . (20)
Comparing (13) and (20), we conclude that the pairwise error probability for the case of imperfect SI is given by (14), with γb and K being replaced by ˆγb = µ2γb/(1 + Rcγb(1− µ2)) and
K/µ2, respectively.
Two scenarios can be considered for the channel estimation using pilot signals with Ep = Es. The first case results from only pilot estimation (OPE) with an estimation error variance of σ2
e = N0/Es. The second case considers a lower bound on the performance of receivers employing iterative joint decoding and channel estimation. In such receivers, the decoding results are used to improve the channel estimates, which are used to improve the decoding results. This process is repeated itera-tively. In general, the more reliable the decoding results, the more accurate the channel estimation. A lower bound on the performance of iterative receivers is obtained if the signals in each fading block are known with probability one. In this case they can be considered as pilot signals resulting in an estimation error of variance σe2= N0/(mEs). This case is referred to as correct data estimation (CDE). Similar channel estimation sce-narios were used in [17] and [18] for channel estimation with LDPC codes.
In simulating systems with pilot-aided channel estimation, one coded bit is punctured every m coded bits to account for the rate reduction resulting from inserting a pilot signal every m− 1 signals. This affects the whole distance distribution of the resulting code and may reduce the minimum distance of the code. The resultant code rate after puncturing is given by
˜ Rc=
mRc
m− 1. (21)
Table I shows the code rates ˜Rc and the minimum distances of the punctured codes for different channel memory lengths. Also,
TABLE I
RATES, MINIMUMDISTANCES ANDPUNCTURINGLOCATIONSWITHINEACH FADINGBLOCK OF THEPUNCTUREDRATE-1/2 (23,35)
CONVOLUTIONALCODE
Fig. 4. Bit error probability of a rate-1/2 (23,35) convolutional code with imperfect SI (OPE receiver with Ep= Es) and a frame size N = 1024 for
channel memory lengths m = 4, 8, 16, 32, and 64.
the puncturing pattern is presented in the table. According to the table, the code rate increases with reduced channel memory length, which decreases the error correction capability of the code. Thus, systems with short channel memory are expected to have more channel diversity at the cost of lower minimum distance and worse channel estimation quality. On the other hand, longer channel memory results in more powerful codes, as well as better channel estimation, at the cost of less channel diversity.
Note that optimizing the channel memory is under the control of the system designer. As mentioned in the introduction, the block-fading model is used frequently to model communication systems such as FH-SS, TDM, and OFDM systems. In these systems, the channel memory is under the control of the sys-tem designer. More specifically, the designer can optimize the memory length by finding the most appropriate number of hops that the transmitter should hop within a codeword in FH-SS sys-tems, and the number of time slots in a TDM frame over which a codeword is transmitted in TDM systems.
Fig. 4 shows the results for the rate-1/2 (23,35) convolutional code with imperfect SI with an OPE receiver over Rayleigh fading channels. Note that the energy of the pilot is taken into account in the SNR axis. Unlike perfect SI, imperfect SI results in a clear tradeoff between channel diversity and estimation. From the figure, the cases of m = 16 and m = 32 are the best performing systems, where the former becomes better than the
Fig. 5. Approximation of the bit error probability of a rate-1/2 (23,35) convolutional code over a Rician fading channel with K = 1, 10 dB, im-perfect SI (OPE receiver), and a frame size N = 1024 for memory lengths
m = 8, 16, 32, and 64.
later for an SNR values exceeding 14 dB. This suggests that the optimal channel memory length is between m = 16 and m = 32. Also, the case of m = 8 is worse than the case of m = 64 at low SNR, and starts to improve as the SNR increases. This is because the resulting code for m = 8 is less powerful than the code for m = 64 but has larger amount of channel diversity. Although the case of m = 64 has the best code and channel estimation quality, it lacks enough channel diversity to perform better than the other cases. This is expected since the number of transmitted pilot signals is reduced as the channel memory gets longer, and hence, the system becomes more energy efficient and the code become stronger.
Fig. 5 shows the results of imperfect SI with an OPE receiver over different Rician fading channels. We observe that systems with long channel memory perform better as the energy of the specular component of the channel increases. This is because as K increases, the channel becomes less faded, which reduces the need for the decoder to average over the statistics of the channel. Therefore, the channel diversity becomes less crucial, causing systems with long channel memory to outperform systems with short memory. Another reason for this is the larger fraction of energy spent on pilot signals in systems with short channel memory lengths than in systems with long memory. This is obvious for the case of K = 10 dB, where the performance of m = 64 is nearly optimal for most of the SNR values. On the other hand, the case of m = 8 is the worst everywhere when K = 10 dB, where it outperforms the case of m = 64 when K = 1 dB. From Fig. 3, the optimal channel memory value for an OPE receiver with Ep = Es at a bit error rate of 10−4 is
m = 32 for a Rayleigh fading channel; i.e., K = 0, where it is m = 64 for a Rician channel with K = 10. Also, note that the case of m = 8 outperforms the case of m = 64 when the channel is more faded, where the inverse occurs for channels that are less faded, i.e., larger values of K.
Fig. 6. Approximation of the bit error probability of a rate-1/2 (23,35) con-volutional code over a Rician fading channel with K = 1, 10 dB, frame size
N = 1024, and memory lengths m = 8, 32 using perfect and imperfect SI with Ep= Es(solid: m = 8, dash: m = 32).
Fig. 6 shows a comparison of systems employing perfect SI, OPE, and CDE assumptions with channel memory lengths m = 8 and m = 32 over Rician fading channels with K = 1, 10 dB. It is clear that as the channel memory gets longer, the SNR degradation due to imperfection in the channel SI reduces. This is because long channel memory causes less penalty in the rate and energy than short channel memory does, as well as an im-proved channel SI under the CDE assumption. In general, the optimal memory tends to increase under the CDE assumption compared to the OPE receiver due to the improved channel es-timation [24]. Moreover, the SNR loss in OPE receivers with long channel memory increases with increased energy of the specular component of the channel. When the channel is esti-mated using a pilot signal, the channel estimation error adds a fading component to the channel gain at the decoder. The effect of this new fading component increases as the energy of the specular component increases of the channel, which degrades the performance of OPE receivers more as K increases.
When the energy allocated for the pilot signal is varied, the performance of an OPE receiver is expected to change as a function of the channel memory. The energy per information bit can be written as
Eb =
(m− 1)Es+ Ep
mRc
. (22)
Thus, for a fixed channel memory, there exists an optimal value for pilot energy. This is illustrated in Fig. 7, where the SNR required for the coded system to achieve a bit error probability of Pb = 10−4 is shown versus the pilot-to-signal energy ratio
Ep/Es in decibels. We observe that as the channel memory length increases the optimal value for Ep/Esincreases. This is expected since longer channel memory permits the allocation of more energy to the pilot signal. On the other hand, when the channel memory is short, a wiser usage of the available energy seems to transmit the information signals rather than to estimate
Fig. 7. SNR required for a rate-1/2 (23, 35) convolutional code to achieve
Pb= 10−4versus Ep/Es for the OPE receiver with Ep= Es and memory
lengths m = 16 and 32.
the channel. We conclude from Fig. 7 that optimizing the pilot energy results in an SNR gain slightly less than 1 dB over the case, where Ep = Es. This SNR gain increases as the channel memory increases, since longer memory increases the amount of energy that can be devoted to channel estimation, which improves the overall performance. Moreover, the optimal pilot energy is almost independent of the fading nature of the channel, i.e., independent of the energy of the specular component K of the channel. This is because the amount of energy available in each fading block, which can be used in estimating the channel, is the controlling factor of the optimal pilot energy allocation. Clearly, this energy is a function of the channel memory length only. Also, the SNR gain resulting from optimizing the pilot energy is almost independent of the channel fading behavior. Same observations were reported in [17]–[19].
C. Noncoherent Detection
In this section, we derive a Chernoff bound on the pairwise error probability of coded systems employing square-law com-biner over Rayleigh block-fading channels. Recall that the out-puts of the square-law combiner and the corresponding decoding metrics are given by (3) and (4), respectively. Substituting the metric (4) in (10) yields Pc(d| f) = Pr F f =1 df r(I ,1) f 2+ rf(Q ,1) 2 − r(I ,0) f 2 − r(Q ,0) f 2 > 0 H,S,f (23) where df is the number of error bits in the fading block f . The variables{rf(I ,0), r(Q ,0)f } and {rf(I ,1), r(Q ,1)f } are zero mean Gaussian random variables with variances equal to 1/2(Es+
N0) and 1/2N0, respectively. Let|rf(c)|2=|r
(I ,c)
f |2+|r
(Q ,c)
f |2 for c = 0, 1 and define κ =Ff =1df(|rf(1)|2− |r
(0)
Fig. 8. Bit error probability of a rate-1
2 (23, 35) convolutional code with
noncoherent detection and a frame size N = 1024 for channel memory lengths
m = 1, 8, 16, 32, and 64.
the unconditional pairwise error probability is Pu(d| f) =
∞
0
p(κ) dκ≤ Eκ[eλ κ]. (24) Due to the complicated form of the pdf p(κ) [20], the Chernoff bound was used in (24) to upper bound Pu(d| f), where λ > 0 is the Chernoff parameter that should be optimized for the tightest bound. Substituting for κ in the Chernoff bound and collecting terms having the same distance results in
Pu(d| f) ≤ w v =1 E eλ v|r(1)v |2 fv E e−λv|r(0)v |2 fv . (25) The Chernoff parameter λ is optimized as in [21], and the result-ing Chernoff bound for the pairwise error probability simplifies to Pu(d| f) ≤ w v =1 [4Dv(1− Dv)]fv (26) where Dv = 1/(2 + vRcγb). The Chernoff bound approxima-tion for convoluapproxima-tionally encoded BFSK signals with square-law combining is shown in Fig. 8. Note that we show the approxi-mation for high SNR values to avoid curves to overlap resulting in unclear presentation. We observe that the approximation is not as close to the simulation curves as in the case of coherent receivers. This is mainly because of the use of the Chernoff bounding technique. This makes our approximation not as use-ful in predicting the error probability of square-law combining. However, the proposed approximation predicts precisely the performance loss due to channel memory. Note that the union bound for the performance of square-law combiner can be eval-uated more precisely for the case of m = 1, i.e., without the need for the Chernoff bound [20]. Since the Chernoff bound predicts the performance loss due to channel memory, it can be compared with the more precise union bound for m = 1 case in order to predict the performance of the square-law combiner in a Rayleigh block-fading environment.
Fig. 9. SNR required for a rate-1/2 (23, 35) convolutional code with perfect SI to achieve Pb= 10−3versus the Nakagami parameter M for memory lengths m = 1, 8, 16, 32, and 64.
V. PAIRWISEERRORPROBABILITY FORNAKAGAMIFADING In this section, we derive the pairwise error probability of coded systems over Nakagami block-fading channels. In Nak-agami block-fading channels, the fading amplitude in each fad-ing block is Nakagami distributed with a normalized density function [2] given by faf(a) = 2MM Γ(M )ΩMa 2M−1e−M a2/Ω , a≥ 0, M ≥ 0.5 (27) where Ω = E[a2] = 1, M = Ω2/Var[a] is the fading parameter,
and Γ(· ) is the Gamma function. As M increases, the fading becomes less severe and reaches the nonfading scenario when M → ∞, i.e., AWGN channels. The Nakagami distribution cov-ers a wide range of fading distributions including Rayleigh fad-ing when M = 1, and the sfad-ingle-sided Gaussian distribution when M = 0.5. In the following, only coherent detection with perfect SI is considered.
If the channel SI is known perfectly at the receiver, the ML decoder chooses the codeword S that maximizes the metric in (2). As in Section IV, the conditional pairwise error probability is given by (13). Averaging over the statistics of the fading amplitude in (27), an exact expression of Pu(d| f) [25] is found as Pu(d| f) = 1 π π 2 0 w v =1 1 1 + v Rcγb M sin2θ M fv dθ. (28) The union bound for Nakagami block-fading channels is com-puted by substituting (28) in (7). The effect of the fading pa-rameter on the performance of coherent detection with perfect SI is shown in Fig. 9. We observe that as the fading parameter increases, the SNR loss due to channel memory reduces. This is expected, since as the fading parameter increases, the diver-sity becomes less important since the channel approaches the AWGN channel.
VI. CONCLUSION
In this paper, a union bound on the performance of binary-coded systems over block-fading channels was proposed. The bound is based on uniform interleaving of the coded sequence prior to transmission over the channel and was evaluated for block-fading channels with different fading distributions. Co-herent and noncoCo-herent receivers were considered. Results show that the SNR degradation due to channel memory reduces as the channel becomes less fading. Moreover, the interesting situa-tion of imperfect SI obtained from pilot-aided channel estima-tion was considered. Using the CDE assumpestima-tion, a lower bound on the performance of receivers employing iterative decoding and channel estimation was obtained. Moreover, the tradeoff between channel estimation and channel diversity was investi-gated. Results show that the optimal channel memory increases as the channel becomes less fading. Moreover, it was shown that the optimal pilot energy allocation is a function of the channel memory length and does not depend on the channel distribution.
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Salam A. Zummo (M’03) received the B.Sc. and
M.Sc. degrees (with highest honors) in electrical en-gineering from King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia, in 1998 and 1999, respectively. He received the Ph.D. degree from the University of Michigan, Ann Arbor, in June 2003.
He is currently an Assistant Professor of Elec-trical Engineering at KFUPM. His research interests include error control coding, coded modulation, iter-ative receivers, interference modeling and analysis, and cross-layer design of wireless communication networks
Dr. Zummo was the recipient of the British Council Summer Research Fel-lowship in June 2004.
Ping-Cheng Yeh (M’05) received the B.S. degree in
mathematics and the M.S. degree in electrical engi-neering from the National Taiwan University, Taipei, Taiwan, R.O.C., in 1996 and 1998, respectively. He received the Ph.D. degree from the University of Michigan, Ann Arbor, in 2005.
He is currently an Assistant Professor in the De-partment of Electrical Engineering at the National Taiwan University. His research interests include channel coding, coded modulation, directional an-tennas, and cross-layer design in wireless networks.
Wayne E. Stark (F’98) received the B.S. (with
high-est honors), M.S., and Ph.D. degrees in electrical en-gineering from the University of Illinois, Urbana, in 1978, 1979, and 1982, respectively.
Since September 1982, he has been a faculty member in the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, where he is currently Professor. His research interests are in the areas of coding and communica-tion theory, especially for spread-spectrum and wire-less communication networks.
Dr. Stark was Editor for Communication Theory of the IEEE TRANSACTIONS ONCOMMUNICATIONin the area of Spread-Spectrum Communications from 1984 to 1989. He was involved in the planning and organization of the 1986 International Symposium on Information Theory which was held in Ann Arbor, Michigan. He was selected by the National Science Foundation as a 1985 Presi-dential Young Investigator. He was the principal investigator of a Army Research Office Multidisciplinary University Research Initiative (MURI) project on Low Energy Mobile Communications. He received the IEEE Military Communica-tions Conference Technical Achievement Award in 2002. He is a member of Eta Kappa Nu, Phi Kappa Phi and Tau Beta Pi.
