1822
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
Transactions Papers Modeling Fading Channel-Estimation Errors in Pilot-Symbol-Assisted Systems, With Application to Turbo Codes Bartosz Mielczarek, Member, IEEE, and Arne Svensson, Fellow, IEEE
Abstract—In this paper, we address the issue of imperfect channel estimation in coded systems on fading channels. Since performance of channel codes is influenced in different ways by different components of channel-estimation errors, we develop a simplified model which separates the estimation errors of a Wiener-filtered received signal into the amplitude error and the phase error. Based on the model, we derive tight bounds on component error variances. Moreover, we prove that the classical Wiener filter results in a biased estimate of the channel amplitude. We also show that the probability of having a phase-estimation error large enough to cause decision errors in the receiver is significant. Using our model, we derive an approximate upper limit on the optimum pilot-symbol spacing and approximate lower limit on bit-error rate performance of coded systems with a given pilot-symbol separation. The proposed model and derivations are confirmed by extensive simulations. Index Terms—Channel-estimation error, phase offset, Rayleigh channel, turbo codes, Wiener filter.
I. INTRODUCTION
C
HANNEL estimation is one of the most basic issues in communication theory, since all existing receiver algorithms depend on some information of the current channel state to successfully decode transmitted information. There is a large body of research devoted to channel estimation and channel coding, but the majority of the papers deal with either channel estimation in uncoded systems or operation of channel codes on fading channels with perfect synchronization and fading compensation. Such simplifications facilitate the analysis (which may not be tractable otherwise), but do not give full insight into the effects of imperfect channel knowledge. The problem needs Paper approved by R. D. Wesel, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received November 21, 2002; revised February 24, 2004 and August 26, 2004. This work was supported by the Swedish Foundation for Strategic Research under the Personal Computing and Communication Program. B. Mielczarek was with the Communication Systems Group, Department of Signals and Systems, Chalmers University of Technology, Göteborg, Sweden. He is now with TRLabs, Edmonton, AB T6G 2V4, Canada (e-mail:
[email protected]). A. Svensson is with the Communication Systems Group, Department of Signals and Systems, Chalmers University of Technology, 412-96 Göteborg, Sweden (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCOMM.2005.858669
particular attention in turbo-coded systems [1], which operate at very low signal-to-noise ratios (SNRs) where the quality of the estimated channel and synchronization parameters may be severely impaired by a large noise variance. The existing literature dealing with both channel estimation and coding (in particular, turbo coding) can be generally divided into two groups. The first group combines the classical solutions for channel estimation and channel decoding by performing their respective tasks serially, and exchanging information in an iterative fashion. An example of such a paper is [2], where intersymbol interference is reduced by exchanging soft information between the channel equalizer and the channel decoder. In the context of channel estimation on fading channels, [3] shows the basic structure for the iterative Wiener filtering of a signal estimated initially with the help of pilot symbols [a method called pilot-symbol-assisted modulation (PSAM)]. After each decoding iteration, the tentative soft decisions are used to improve the initial estimate. This method is further improved in [4], where an additional Viterbi decoder is used to change the starting point of the iterative channel-estimation process, which results in a lower error floor. Another method for iterative channel estimation is shown in [5], where, based on the assumption that fading amplitudes are highly correlated for several consecutive symbols, blockwise maximum a posteriori (MAP) sequence estimation is used to improve the performance of the Wiener filtering approach. Yet another approach is presented in [6], where an iterative soft information-exchange algorithm is used in a convolutionally encoded differential phase-shift keying (DPSK) system. Recently, it was shown in [7] that such a serial operation can be interpreted as an instance of the well-known expectation-maximization (EM) algorithm. The second group of papers attempt to account for the fadingchannel dynamics by modifying the structure of the decoder. In its simplest form, only the decoder metrics are modified, as in [8], where receivers for channels with known amplitude but unknown phase are discussed; in [9], where the opposite case with unknown amplitude but known phase is presented; and in [10], where the general case of noisy channel estimates for both phase and amplitude is described. All the papers show that the inclusion of a statistical description of non-Gaussian channels in the decoder metrics can increase the quality of decoding. The
0090-6778/$20.00 © 2005 IEEE
MIELCZAREK AND SVENSSON: MODELING FADING CHANNEL-ESTIMATION ERRORS IN PILOT-SYMBOL-ASSISTED SYSTEMS
1823
Fig. 1. System model.
more comprehensive papers attempt to modify the structure of the decoder in order to improve channel estimates on fading channels. In particular, changing the structure of the decoder to incorporate the channel memory has received much attention. In extensive work presented in [11]–[13], different structures of iterative detectors for channels with parametric uncertainties are analyzed. The majority of the presented solutions involve extending the decoder trellis by incorporating the channel as an additional inner code with known statistical properties (which increases the complexity of the decoder). In such a context, the model of choice seems to be Markov chains, as in [14], [15], and [16], where additional states representing unknown phase values are incorporated in the code trellis. A similar approach for improved phase-noise estimation has been presented in [17] and [18] where, in addition to the extended trellis state space, the component decoders exchange phase information. A general overview of such iterative techniques can be found in [19]. There are few papers that attempt to model the actual channelestimation errors and try to predict the performance of the coded systems based on given system parameters. The most common practice is to use simulations as means of obtaining the optimum pilot spacing and predict the performance of the PSAM system, as in [3]. We are aware of two papers, [20] and [21], that attempt to deal with this problem by using the density-evolution technique. Our work attempts to provide some insight into imperfect channel estimation using a Wiener filter. We propose a general model that can be used to analyze coded PSAM systems without resorting to lengthy simulations to predict their performance. In the first sections of the paper, we present a general approach for modeling the separate components of the channel-estimation errors. We use this model to obtain tight bounds on the mean value and variance of the amplitude- and phase-estimation errors. Such a distinction is important in systems which require full channel-state information, where amplitude- and phaseestimation errors influence system behavior in different ways. We then use the introduced model to calculate an approximation to the optimum pilot spacing in the transmitter, and to predict the performance of a channel decoder in a semianalytical way. With this method, no extensive simulations are needed to design coded systems using the PSAM. After that, we briefly discuss the operation of a turbo decoder and present a set of assumptions used in our analysis of such decoders. In the last part of the paper, we apply the proposed algorithm to a turbo-coded system and confirm its validity by empirical studies of the actual system performance. The paper is concluded with a summary and suggestions for future work.
II. SYSTEM MODEL The system studied in this paper is shown in Fig. 1. In the following sections, we will present the transmitter, channel, and channel estimator. The decoder structure will be discussed in Sections IV and V. A. Transmitter and Channel Model informaThe binary channel encoder encodes a block of with rate into the sequence of coded tion bits bits , which are then channel-interleaved. In the next step, pilot bits (denoted henceforth as ) are inserted into the sequence of interleaved coded bits. The pilot-insertion positions bits are speciin the resulting sequence of fied by a set containing index values of pilots (between 0 and ). The modified sequence of bits is then converted to binary phase-shift keying (BPSK) symbols , which per symbol.1 The SNR per inare transmitted using energy formation bit for this system is normalized as (1) which allows a fair comparison with systems with different numbers of pilot symbols. The BPSK signal is then transmitted over a flat Rayleigh fading channel (modeled as a sequence of correlated complex amplitudes ) with normalized Doppler (where is the maximum Doppler frequency frequency and is the symbol period). Finally, complex white Gaussian noise samples with variance per dimension are added to the signal. , the At the receiver, after normalizing with a factor of received complex baseband signal can be expressed as (2) where is a vector of length containing the received symbols, is a diagonal matrix with BPSK symbols on the diagonal, is a vector of complex fading gains, and is and a complex noise vector whose component vectors are samples of real-valued, independent, uncorrelated Gaussian . processes with variance In this paper, we use the common model of the fading process based on Clark’s spectrum [22]. We define the vector of correlated complex fading gains with real vectors and 1Even though, to simplify the discussion, we provide analysis for BPSK modulation, it is straightforward to extend our results in the next section to higher order modulations. This can be done, without a loss of generality, by assuming that the pilot symbols p are real and have fixed energy E .
1824
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
of length , which are samples of two independent, multivariate Gaussian processes with zero mean and identical probability densities defined as
(3) as
After defining the correlation between the complex gains
(4) where by
is the zero-order Bessel function of the first kind, the covariance matrix in (3) is given by
Based on the estimates signal as
, the receiver demodulates the (11)
The demodulated signal vector is then stripped of the pilot noisy coded symbols are symbols, and the remaining channel deinterleaved and fed to the channel decoder which attempts to recover the original coded sequence. III. CHANNEL-ESTIMATION ERRORS The variance of the channel-estimation error using the Wiener filter described in the previous section is given as (see [23])
(5) Note that the fading process is assumed to have unit power. B. Channel Estimation With a Wiener Filter After receiving the signal, the first task of the receiver is channel estimation, where the complex fading amplitudes are prior to deinterleaving and decoding. The clasestimated as sical linear approach to channel estimation is to use a Wiener filter, which minimizes the mean square error (MSE) of the esfor timates in the presence of noise. The filter coefficients the Wiener filter are given by
(6) where is a by diagonal matrix containing the known data symbols, called henceforth the pilot matrix. Note that the filter coefficients depend on the position in the block. For the initial channel estimation, we define the pilot matrix as (7)
otherwise.
The role of this matrix is to remove the dependence of the received signal on the transmitted data symbols, which allows for the estimation of the fading amplitudes . To model the influence of the pilot-symbol reliability, we introduce a function defined as (8) The values of function will be either 0 or 1, with 0 corresponding to unknown symbols, and 1 corresponding to perfectly known symbols (as pilot symbols). The well-known Wiener–Hopf equations can be used to calculate the filter coefficients in (6) as (9) where , and the autocorrelation matrix is given (see Appendix I) as otherwise where
(10)
.2
2We use the notation [r ] to denote the position n in vector r , which is specified for position k in the block.
(12) The value of the variance in (12) depends on the position in the block of received symbols, and is the smallest in its middle. At the beginning and at the end of the block, the filter uses a smaller number of symbols whose corresponding fading values are highly correlated, and the resulting edge effects cause increased estimation-error variance at these points. While it is sometimes possible to alleviate this problem by extending the channel estimation over several blocks [3], our assumption is that each block is independently transmitted and that the receiver’s estimation algorithm has to rely only on the information from the given block. This model reflects a realistic situation, when the transmitter sends information at random time instants, and allows us to consider the worst-case scenario. Unfortunately, it is not easy to use (12) to analyze the actual loss of performance of a channel-coded system. The largest losses of performance result usually from phase-estimation errors [since they cause loss of energy in the demodulated signal, as defined in (11)], and the classification of these errors is not readily obtainable from (12). Since some decoding algorithms (for example, Bahl–Cocke–Jelinek–Raviv (BCJR) [24]) require knowledge of the SNR of the signal, the amplitude-estimation error can also cause losses of performance. In the following sections, we derive lower bounds on the variances and mean values of the channel-estimation error components. A. Proposed Model In order to calculate the statistics of the separate components of the channel-estimation errors, we define the channel amplitude-estimation error as (13) and the channel phase-estimation error as (14) Using the Wiener filter defined in (6), the estimate of as
is given
(15) where we assumed without loss of generality that all the pilot symbols in are ones. The term is a diagonal matrix with . Note that even though, entries corresponding to function in general, and are not identity matrices, they can be treated as such, since the positions on their diagonals which are
MIELCZAREK AND SVENSSON: MODELING FADING CHANNEL-ESTIMATION ERRORS IN PILOT-SYMBOL-ASSISTED SYSTEMS
1825
equal to zero correspond to zero values in the coefficient vector .3 This allows us to ignore them in the following discussion. By separating vectors and into real and imaginary parts and rearranging (15), we obtain (16) which will be the starting point of our discussion. A straightforward calculation of the properties of the phaseand amplitude-estimation error from (16) would involve complicated averaging over four partially correlated Gaussian prosamples each. In the following paragraphs and cesses with in Appendix II, we will show a method of modeling the estimation errors using only four univariate, independent Gaussian processes, which greatly simplifies the problem. We start by introducing a model showing the approximate correspondence (preserving the autocorrelation properties) between the th and th elements in vectors and as (17) so that the all elements in vectors and are approximated only and , which are indeby scaled values of the elements pendent samples of Gaussian random processes with zero mean and and variance equal to 1/2. We will now select elements so that applying (17) in (16) will produce values , with error variance equal to a lower bound on the estimation-error variances for all elements in the transmitted signal block. In general, position , at which channel estimates are characterized by the minimum mean-square estimation error, corresponds to the position of a pilot symbol closest to the middle of the block (this can be easily confirmed by evaluating (12) for all positions ). We state that (18) for which . where is the index closest to Inserting (17) into (16) and performing summations over the and results in uncorrelated Gaussian noise processes
Fig. 2. Mean value of the amplitude-estimation error. 320; r = 1=3.
f
T
= 0:01; N =
end of the whole sequence. The total number of pilot symbols is, thus, , and the pilot index set is . Moreover, we assume defined as that the Wiener filter spans the entire block and uses a different set of coefficients for estimating fading amplitude of each received symbol (the edge effects are accounted for), which leads to the optimal MSE implementation when the fading is assumed independent between blocks. In practice, probably only one set of filter coefficients would be implemented for all symbols, and the filter length would be shorter than the whole block. However, to keep the discussion independent of implementation, we chose the canonical version of the filter. Any configuration of pilot symbols and Wiener filter implementation can be used in . our method simply by modifying vector C. Amplitude-Estimation Error Inserting (20) in (13) results in
(19) where , and and are real-valued, independent, identically distributed Gaussian variables with zero mean . In and variance Appendix II, we show that channel-estimation error variance is indeed the lower bound on the channel-estimation variance. To simplify notation, in the following sections, we will omit and . Moreover, we will introduce paramsubscript in eters and so that and (20) B. Assumptions In the following sections, we assume that single pilot symbols are placed in the beginning of each block consisting of coded bits, and, in addition, one pilot symbol is located at the 3When some entries in the diagonal of matrix P are equal to zero, the rank of the matrix R is reduced. Equation (9) is solved in such a case by setting 8 w = 0 (the estimator simply ignores unknown bits).
(21) The first parameter of interest is the mean of the amplitudeestimation error. After some derivations (see Appendix III), it can be shown that
(22) where is the confluent hypergeometric function. Fig. 2 shows the results from (22), and the corresponding simulated values for different SNRs and pilot spacings (the simulated values represent the averaged amplitude-estimation errors, where averaging is performed at position for different realizations of 10 000 simulated blocks). One can see that the amplitude estimates are biased, and the demodulation in (11) results, on average, in underestimation of the channel amplitude (the effects of channel amplitude underestimation are discussed in [25]). To calculate the variance of the amplitude-estimation
1826
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
Fig. 3. Amplitude-estimation error variance. f T = 0:01; N = 320; r = 1=3.
error, we first calculate the second moment of by
, which is given
(23) The bound on amplitude-estimation variance from (18), , and the corresponding simulated values (calculated as above) can be seen in Fig. 3. The derived bounds are very close to the actual values, which confirms the suitability of the chosen model. D. Phase-Estimation Error
Fig. 4.
Phase-estimation error variance. f T = 0:01; N = 320; r = 1=3.
The bounds from (25) and the corresponding simulated values (based on 10 000 simulated blocks and averaged as in the amplitude error case) are shown in Fig. 4. The proposed bounds are very close to the actual simulated values, and get better with increasing SNR. In general, it can be stated that the proposed bounds will improve with higher SNRs, lower normalized Doppler spread, and lower pilot-symbol spacing. E. Probability of Phase Reversal The most severe distortion of the received signal samples occurs when the phase error is so large that . This will be referred to as phase reversal. Such a large phase shift occurs with probability defined as
Inserting (20) in (14) yields (26)
(24) Unfortunately, simplifying (24) is very difficult, due function and the resulting to the presence of the terms. Therefore, we chose to present the bound in integral form. It ,4 so the estimate is unbiased. Thus, can be shown that the variance of the phase-estimation error is given as
(25) where are Gaussian probability density functions (pdfs) with zero mean and variances , and , respectively. The numerical calculation of the phase-estimation error variance in (25) involves a quadruple integral, but it is facilitated by the independence of the four Gaussian variables. Moreover, the complexity of the computation is much lower than a full simulation of the estimator.
1
+
#n
0
is an antisymmetric function, and (#n n )=( + # + ) can be shown to have a symmetric unimodal distribution with
4arctan( )
n
zero-mean value.
After some derivations (see Appendix III), we obtain (27) is the generalized Marcum function. where Examples of the phase-reversal probability curves are shown in Fig. 5. One can see that, even for the relatively large SNR, a few percent of all samples will be reversed. Since (27) is based on (20) (which is shown to give a lower bound on estimationerror variances in Appendix II), we also have in this case that the derived values function as lower bounds on the actual phasereversal probability. IV. PILOT-SPACING LIMIT IN CODED SYSTEMS Assuming that the fading process is a bandlimited process, the minimum frequency of samples needed to represent it without any loss of information has to fulfill the Nyquist sampling theorem, i.e., the insertion rate of the pilot symbols must be at least twice as large as the Doppler bandwidth. Unfortunately, the actual pilot-insertion frequency in coded applications operating on very noisy channels must be much higher than the theoretical
MIELCZAREK AND SVENSSON: MODELING FADING CHANNEL-ESTIMATION ERRORS IN PILOT-SYMBOL-ASSISTED SYSTEMS
1827
B. Assumptions
Fig. 5. Probability of the phase reversal. f
T
= 0:01; N = 320; r = 1=3.
Nyquist limit, which is valid only when the samples are noiseless and their number approaches infinity. In the following sections, we develop a new approximate upper limit for the optimum pilot spacing based on our model, which we apply using the principle of soft decoding with the BCJR algorithm. A. Soft-Input Soft-Output Decoder We consider a binary trellis code with base rate , producing coded bits for each input bit . The rate of the code can be adjusted to by puncturing some coded bits. The optimum method of decoding of such a code, in the symbol-wise MAP sense, is to maximize magnitudes of loglikelihood ratio (LLR) values for information bits , conditioned on the received, scaled, and rotated noisy codeword . In case of trellis codes operating on memoryless channels (or perfectly interleaved channels), the solution to the maximization problem is the BCJR algorithm [24] which calculates LLR values as (28)
Since the BCJR algorithm minimizes the bit-error probability, the analysis of such decoders is quite difficult, due to the presence of forward and backward passes of the algorithm. It is, however, possible to analyze its performance by considering the elementary factor that influences the quality of BCJR decoding, which is the probability of the decoder assigning the to a wrong state in the code trellis highest probability [26] during forward transitions of the BCJR algorithm.5 If such a situation occurs, the hard decision on the output of (28) may be erroneous, increasing the bit-error rate (BER). Since we aim at evaluating the upper limit on the pilot spacing, we consider the best-case situation, when we assume of the that the decoder has perfectly identified the state in the forward pass. After signal trellis at the position may be so distorted by noise and/or that, the signal symbol channel-estimation errors that a wrong state is assigned the highest probability in the position . Unfortunately, it is difficult to quantify analytically how the probability of such a detour will determine the performance of the decoding process, but regardless of the type of decoding algorithm, the increased probability of such a detour will cause increased BERs. In [26] and [27], for example, it was shown how to semianalytically estimate the bit-error performance of a code with a given channel-estimation error based on the known bit-error performance of the same code with perfect channel estimation, simply by comparing their detour probabilities. We will evaluate the detour probability with the following assumptions. • The all-zero codeword is transmitted . • Channel interleaving is perfect (fading values corresponding to encoded bits are independent). • Only the th code symbol is considered, and . The first two assumptions are standard in analysis of linear codes, and the last one stems from our attempt to quantify th step of the BCJR the best possible outcome of the forward pass (the decoder has perfectly identified the all-zero ). state in step C. Upper Limit on Pilot Separation
where the values are the probabilities of arriving at values code trellis state with forward transitions, the are the probabilities of arriving at code trellis state with backward transitions, and values are the extrinsic probabilities of transitions between states and . The summations state transitions corresponding to data are performed over , and state transitions corresponding to data bit bit , respectively. The likelihood of a transition is defined as (29) where
is the channel reliability factor [1].
Using the above assumptions in (29), we define the probability of detour as
(30) where we omitted index to simplify the notation, and used noto denote the valid state following the all-zero state tation after transmission of 1 (for details, see [26]). The dependence stems from of the probability of detour on pilot spacing relation in (1) and the dependence of terms on the quality of 5An
identical method can be used for the backward BCJR transitions.
1828
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
channel estimation, which is directly linked to the pilot-insertion , the funcfrequency. Under the assumption of constant is a convex function of .6 Therefore, tion finding the positive integer value that minimizes in (30) is equivalent to finding an approximation to the optimum pilot and . spacing for the block under given Evaluating (30) requires, in general, simulating the whole system. To avoid this, once again we resort to our model of channel-estimation errors. Assuming no puncturing and applying (11) and (20) in (30), an approximation to can be given as
culate an approximation to the lower bound on the BER performance of a system using any given pilot spacing. This can be done by relating the BER of the considered system (denoted as BER) to the BER of the system with a perfectly estimated channel (BER ) as BER
BER
(35)
The above equation links the empirical knowledge of the code with the analytically derived channel estimation quality (see [26]). V. ANALYSIS FOR TURBO CODES
(31) The variables denote independent complex Gaussian samples per dimension, and the with zero mean and variance pairs are modeled independently, as in Section III-A. Due to the application of lower bounds on channel-estimation appears as a slightly errors, function which is shifted toward the higher modified values of . Therefore, we conjecture that the optimum spacing will be lower than , which minimizes (31). We define the approximate upper limit on pilot spacing as (32) where represents the set of positive integers. Note that if puncturing is used in the encoder, it is straightforward to extend our model by deleting the terms corresponding to punctured bits from (31). D. Equivalent Energy Method Since, as we have shown, phase-estimation errors are rather large, the energy of the signal will be significantly reduced by the imperfect compensation of the phase offset of the channel samples. We quantify the problem by comparing the probability in (31) with the probability of detour, under the assumption that the channel was estimated perfectly. We define the probability of the detour corresponding to the case of perfectly estimated channel as
(33) where all the variables are defined as in (31). We then define , for the equivalent energy of the estimated signal as the which
A. Assumptions We will present the results of our analysis for the special case of turbo-coded PSAM. The system studied in this part of the paper consists of a rate-1/3 parallel turbo encoder with two identical-rate recursive systematic convolutional (RSC) enconnected by an interleaver of size . coders To limit the decoding complexity, the turbo-encoded signal is usually decoded using some version of an iterative algorithm which exchanges soft information between separate MAP decoders operating on the component RSC codes [1]. Such an approach is suboptimal, but allows significantly limiting the complexity of the turbo decoder (introduction of the interleaver makes the number of states in the trellis of the concatenated code many orders of magnitude larger than the number of trellis states of component codes). B. Iterative Channel Estimation and Decoding As shown in previous sections, the number of pilot symbols has a crucial role on the quality of the estimation process. However, since the turbo decoder is iterative, it is possible to include the partially decoded soft bits in the estimation process. This can be done after each full iteration by calculating the soft-bit values (see [28]) (36) where LLR values are calculated for each encoded bit using a straightforward modification of (28), as in [3]. Soft bits can be then interleaved and inserted into the pilot matrix as additional pilot symbols (see the dashed feedback loop in Fig. 1). If the iterative channel estimation is used, the matrix in (7) has to be redefined for each iteration as (37) otherwise.
(34) quantifies the loss of energy incurred by where imperfect channel estimation. Based on (34), it is possible to cal-
M
6Increasing gives more energy to the codeword, but increases the channelestimation error, which eventually results in higher probability of detour (if , the decoder has no channel estimates at all). If decreases, channel estimation becomes better, but the energy spent on coded bits decreases (if 1, all energy is spent on pilot symbols).
M !1 M!
M
in (8) will now be redefined as , and in (10) will now become . The problem with modeling estimation errors in iterative channel estimation and decoding stems from the fact that the function is very difficult to obtain analytically and must be approximated [26]. However, as Fig. 6 shows, already after The function
MIELCZAREK AND SVENSSON: MODELING FADING CHANNEL-ESTIMATION ERRORS IN PILOT-SYMBOL-ASSISTED SYSTEMS
Fig. 6. Phase-estimation error variance in consecutive iterations of the iterative estimation and decoding algorithm. f T = 0:05; N = 320; (7; 5) ; r = 1=3; M = 6.
1829
Fig. 7. Normalized value of for different values of E =N for one-shot estimation and perfect DD estimation.
A. One-Shot Estimation one iteration for most of the cases, the phase-estimation error variance approaches the bound calculated assuming (corresponding to the limiting case, when all the coded bits are known at the receiver: perfect data-directed (DD) estimation), which means that modeling is not necessary to calculate the lower bound on the channel-estimation errors. This is especially valid for larger SNR values. C. Analytic Analysis of Channel Estimation With Turbo Codes The iterative turbo decoding, while attractive from a complexity point of view, makes direct application of results from Section IV very difficult. All equations in that section assume that the code may be decoded in a single pass of the BCJR algorithm, and provide no model for the soft-information exchange between component decoders. To facilitate the analytical calculations in (30), (31), and (33), we will assume that the turbo code decoder is capable of applying the BCJR algorithm from (28) and (29) to the whole state space of the concatenated code (in other words, without separate decoding of component codes). Equations (30), (31), and (33) without any provision for subopare then evaluated for timal exchange of soft bits between the component codes. Obviously, such an approach will result in overestimating the performance of the classical turbo decoder (with soft-bit exchange), which will make our approximate limits on pilot spacing and BERs slightly looser. VI. SIMULATIONS All simulations were conducted using a turbo code . The inof rate 1/3 and a turbo interleaver size of terleaver structure was created according to the wideband codedivision multiple-access (WCDMA) standard [29]. The channel block interleaver, interleaving was performed using the . All and the normalized Doppler shift was set to BER calculations were based on data from 100 000 transmitted blocks with independent, randomly generated parameters. The presented BERs are calculated after 10 iterations.
The graphical representation of numerically evaluated in (31) is shown in Fig. 7, and the corresponding BER evaluations are presented in Fig. 8(a). As one can see, the maximum pilot , which is equal to the optimal spacing from (32) is pilot spacing in the simulated BER. Note that according to the Nyquist theorem, the maximum pilot spacing for this set of parameters is equal to 10. This confirms that our model is more useful in calculations of the optimum pilot spacing. B. Iterative Channel Estimation Unfortunately, iterative channel estimation and decoding makes the calculation of (32) difficult. If only the initial estimation is considered, (32) will give too pessimistic values. If, on the other hand, (31) is calculated assuming perfect knowledge of the all the data bits in the block (perfect DD estimation), the resulting approximate maximum will be too optimistic, as can be seen in Fig. 7. In practice, the optimum spacing will be slightly sparser than in the case of one-shot estimation. This can be seen in Fig. 8(b), where the simulated BERs are shown for different . The optimum spacing of the pilot symbols is values of according to (32). about 6–7, as compared with C. BER Approximations Fig. 9 shows the approximations to the lower bounds on BER performance of the estimated systems as given by (35). One can see that the derived limits are approximately 0.5 dB away from the actual values, and get better for larger SNRs. Only for large , the approximations are not very good, pilot spacing of due to the proximity to the Nyquist limit. VII. CONCLUSIONS AND FUTURE WORK In this paper, we introduced a method for modeling the channel-estimation errors resulting from using a Wiener filter to estimate flat Rayleigh fading channel coefficients. We derived new, tight bounds on amplitude- and phase-estimation variances, and showed that the channel estimation using a
1830
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
Fig. 9. Approximate lower bounds and the simulated BER values for different pilot spacing M .
tical implementation, which would most likely depend on some form of initial pilot-symbol-assisted channel estimation. Another interesting area of research is the design of a new class of nonlinear channel-estimator filters which attempt to minimize the phase-estimation error, as opposed to the amplitude error. APPENDIX I In this section, we outline the derivation of the matrix in (10). We start by evaluating the usual criterion in the Wiener filter (6) using (15) as
Fig. 8. Simulated BER values for different values of E =N and pilot spacing. (a) BER with one-shot estimation. (b) BER with iterative channel estimation.
(38) Wiener filter results in an underestimated amplitude. These results can be used in variety of channel-estimation applications, and their extension to other linear channel-estimation methods is straightforward. The proposed method, together with a novel approach of modeling the behavior of MAP trellis decoders, allowed us to calculate an approximate upper limit on the optimal pilot-symbol spacing in trellis-coded systems. The complexity of this method is considerably lower than the usual approach that involves simulations of coded systems with different pilot-spacing parameters. An equivalent energy method for calculating an approximate lower bound on the performance of a turbo code under a given pilot-symbol spacing was also presented. We showed that this approach can provide quite good prediction on the turbo-code performance without a need for extensive simulations. Future work should concentrate on creating improved models for the phase-estimation error. In particular, the derivation of the pdf of phase errors would be very useful, since its incorporation into turbo decoding algorithms could yield significant improvements in performance. This is particularly interesting in prac-
To find the filter coefficients that minimize the expected value (38), it is necessary to solve a set of linear equations that fulfill (39) Inserting (38) into (39), after some manipulations, we get
(40) For
, (40) reduces to
(41) For
, (40) reduces to
(42)
MIELCZAREK AND SVENSSON: MODELING FADING CHANNEL-ESTIMATION ERRORS IN PILOT-SYMBOL-ASSISTED SYSTEMS
Following the notations and , one obtains for
where [22]. (43)
1831
is the confluent hypergeometric function
B. Amplitude-Error Variance Expanding the quadratic term in (21), we obtain
and for (44) It is now straightforward to arrive at (9) and (10). APPENDIX II Here we outline the proof that our assumption from (17) allows calculating the lower bound on the Wiener-estimation variance. Recall from (4), (12), and Section III-A that for the classical Wiener filter, the minimum MSE of the estimation is (45) Using assumption (17) and employing (20), it follows that the MSE using the proposed simplification is
(49) Here we averaged over three terms having chi-square, noncentral chi-square, and Rice distributions, respectively. C. Probability of the Phase Reversal
(46) Using numeric calculations (due to the complex form of and ), it is now possible to show that for any combination of practically interesting filter structures, Doppler spread, and noise variance. From the above statement, which is valid for the MSE of the complex amplitude estimation, we conjecture and prove empirically that the same is valid for the phase- and amplitude-error variances. APPENDIX III In this section, we outline the derivations of bounds in (22), (23), and (27). We start by defining parameter as (47) and are samples of independent Since the parameters Gaussian with zero mean and variance 1/2, the distribution of is exponential with pdf .
Rewriting (26), we obtain
(50) Here we define parameter
as (51)
are independent Gaussian variables, it folSince both and lows that has exponential distribution with probability density . given as Examining (50), it is easy to see that the sum of two quadratic terms on the left side of the inequality follows noncentral chiconsquare distribution, with noncentrality parameter ditioned on . It follows that
A. Mean Amplitude Error Analyzing (22), one can see that the expected value will be a difference between a mean of the Rayleigh distribution (with base random variables and ) and the Ricean distribution (with base random variables and ), with noncentrality parameter (conditioned on from (47)). Thus
(48)
(52) where
is the generalized Marcum
function [22].
REFERENCES [1] C. Berrou and A. Glavieux, “Near-optimum error-correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, no. 10, pp. 1261–1271, Oct. 1996. [2] C. Douillard, M. Jezequel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, “Iterative correction of intersymbol interference: Turbo equalization,” Eur. Trans. Telecommun., pp. 507–511, Sep. 1995. [3] M. C. Valenti and B. D. Woerner, “Iterative channel estimation and decoding of pilot symbol assisted turbo codes over flat-fading channels,” IEEE J. Sel. Areas Commun., vol. 19, pp. 1697–1705, Sep. 2001. [4] B. Mielczarek and A. Svensson, “Improved iterative channel estimation and turbo decoding over flat-fading channels,” in Proc. IEEE Veh. Technol. Conf., Vancouver, BC, Canada, 2002, pp. 975–980.
1832
[5] Q. Li, C. N. Georghiades, and X. Wang, “An iterative receiver for turbocoded pilot-assisted modulation in fading channels,” IEEE Commun. Lett., vol. 5, no. 4, pp. 145–147, Apr. 2001. [6] P. Hoeher and J. Lodge, “Turbo DPSK: Iterative differential PSK demodulation and channel decoding,” IEEE Trans. Commun., vol. 47, no. 6, pp. 837–843, Jun. 1999. [7] N. Noels, C. Herzet, A. Dejonghe, V. Lottici, H. Steendam, M. Moeneclaey, M. Luise, and L. Vandendorpe, “Turbo synchronization: An EM algorithm interpretation,” in Proc. IEEE Int. Conf. Commun., Anchorage, AK, May 2003, pp. 2933–2937. [8] E. K. Hall and S. G. Wilson, “Turbo codes for noncoherent channels,” in Proc. IEEE Commun. Theory Mini-Conf., Phoenix, AZ, Nov. 1997, pp. 66–70. , “Design and analysis of turbo codes on Rayleigh fading chan[9] nels,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 160–174, Feb. 1998. [10] P. Frenger, “Turbo decoding on Rayleigh fading channels with noisy channel estimates,” in Proc. IEEE Veh. Technol. Conf., Houston, TX, May 1999, pp. 884–888. [11] A. Anastasopoulos and K. M. Chugg, “Adaptive soft-input soft-output algorithms for iterative detection with parametric uncertainty,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1638–1649, Oct. 2000. [12] K. M. Chugg, A. Anastasopoulos, and X. Chen, Iterative Detection: Adaptivity, Complexity Reduction and Applications. Norwell, MA: Kluwer, 2001. [13] A. Anastasopoulos and K. M. Chugg, “Adaptive iterative detection for phase tracking in turbo-coded systems,” IEEE Trans. Commun., vol. 49, no. 12, pp. 2135–2144, Dec. 2001. [14] J. P. Seymour and M. P. Fitz, “Near-optimal symbol-by-symbol detection schemes for flat Rayleigh fading,” IEEE Trans. Commun., vol. 43, no. 2, pp. 1525–1533, Feb. 1995. [15] E. Baccarelli and R. Cusani, “Combined channel estimation and data detection using soft statistics for frequency-selective fast-fading digital links,” IEEE Trans. Commun., vol. 46, no. 4, pp. 424–427, Apr. 1998. [16] C. Komninakis and R. D. Wesel, “Joint iterative channel estimation and decoding in flat correlated Rayleigh fading,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1706–1717, Sep. 2001. [17] B. Mielczarek and A. Svensson, “Phase offset estimation using enhanced turbo decoders,” in Proc. IEEE Int. Conf. Commun., vol. 3, New York, NY, Apr. 2002, pp. 1536–1540. [18] J. Vainappel, E. Hardy, and D. Raphaeli, “Noncoherent turbo decoding,” in Proc. IEEE Globecom, vol. 2, San Antonio, TX, 2001, pp. 952–956. [19] A. P. Worthen and W. E. Stark, “Unified design of iterative receivers using factor graphs,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 843–849, Feb. 2001. [20] K. Fu and A. Anastasopoulos, “Performance analysis of LDPC codes for time-selective complex fading channels,” in Proc. IEEE Globecom, Taipei, Taiwan, Nov. 2002, pp. 1279–1283. [21] R. Nuriyev and A. Anastasopoulos, “Pilot-symbol-assisted coded transmission over the block-noncoherent AWGN channel,” IEEE Trans. Commun., vol. 51, no. 6, pp. 953–963, Jun. 2003. [22] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2000. [23] M. H. Hayes, Statistical Digital Signal Processing and Modeling. New York: Wiley, 1996. [24] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, no. 2, pp. 284–287, Mar. 1974. [25] T. Summers and S. G. Wilson, “SNR mismatch and online estimation in turbo decoding,” IEEE Trans. Commun., vol. 46, no. 4, pp. 421–423, Apr. 1998. [26] B. Mielczarek and A. Svensson, “Timing error recovery in turbo-coded systems on AWGN channels,” IEEE Trans. Commun., vol. 50, no. 10, pp. 1584–1592, Oct. 2002. [27] B. Mielczarek, “Turbo codes and channel estimation in wireless systems,” Ph.D. dissertation, Dept. Signals and Syst., Chalmers Univ. Technol., Göteborg, Sweden, 2002. [28] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 429–445, Mar. 1996. [29] Universal Mobile Telecommunications System UMTS: Multiplexing and Channel Coding (FDD), 3GPP TS 125.212 ver. 3.4.0, Sep. 2000.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 11, NOVEMBER 2005
Bartosz Mielczarek (S’98–M’04) was born in Łódz´, Poland, on April 8, 1973. He received the M.Sc. degree with distinction in electrical engineering from Chalmers University of Technology, Göteborg, Sweden, in 1997, the M.Sc. degree with distinction from the Technical University of Łódz´, Łódz´, Poland, in 1998, and the Dr. Ing. (Teknisk Licentiat) and Ph.D. (Teknisk Doktor) degrees in 2000 and 2002, respectively, from Chalmers University of Technology. Currently, he is with TRLabs in Edmonton, AB, Canada, where he works on packet-data-access solutions for heterogeneous multiuser multiple-antenna broadband systems. From 1997 to 1998, he was with Ericsson Telecom AB, Stockholm, Sweden, studying routing issues for multicast and ad-hoc networks. In 2000, he spent seven months as a Visiting Researcher at the California Institute of Technology and the Jet Propulsion Laboratory, Pasadena, CA, working on synchronization algorithms for deep-space probes. In 2003, he worked on the ULTRAWAVES project, studying the physical layer of ultra wideband radio-access technology. Since then, he has been a Postdoctoral Fellow at the University of Alberta/TRLabs, Edmonton, AB, Canada. His current interests include MIMO-OFDM systems with multiuser diversity, turbo codes, and downlink scheduling algorithms. Dr. Mielczarek received scholarships from Telefonaktiebolaget LM Ericsson’s Stiftelsen för Främjande av Elektroteknisk Forskning, Stiftelsen för Internationalisering av Högre Utbildning och Forskning (STINT), and Alice och Lars Erik Landahl Stipediefond in 2000. In 2003, he was awarded a fellowship from the Alberta Ingenuity Fund, and in 2005, an industrial research fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC). He is a member of the IEEE Communications Society and a reviewer for IEEE, IEE, EURASIP, and JCN journals and numerous conferences. He is also listed in Who’s Who in the World (New Providence, NJ: Marquis).
Arne Svensson (S’82–M’84–SM’90–F’01) was born in Vedåkra, Sweden, on October 22, 1955. He received the M.Sc. (Civilingenjör) degree in electrical engineering in 1979, and the Dr.Ing. (Teknisk Licentiat) and Dr. Techn. (Teknisk Doktor) degrees in 1982 and 1984, respectively, all from the University of Lund, Lund, Sweden. Currently, he is a Professor with the Department of Signal and Systems, School of Electrical Engineering, Chalmers University of Technology, Göteborg, Sweden, and is also Director of Studies in a national research school in personal computing and communications. Before 1985, he held various teaching and research positions at the University of Lund. From April 1985 to July 1987, he was a Research Professor (Docent) at the Department of Telecommunication Theory, University of Lund. In August 1987, he joined the Airborne Electronics Division, Ericsson Radio Systems AB, Mölndal, Sweden. In January 1988, he joined by Ericsson Radar Electronics AB, where he first was a member of the New Projects Group at the Airborne Electronics Division, and then from September 1990 to December 1994, a member of the Mobile Telephone Systems Group at the Microwave Communications Division. His consulting company BOCOM is involved in studies and gives courses in the areas of error-control methods, modulation and demodulation techniques, spread-spectrum and CDMA systems, and computer simulation methods for communication systems. His current interests include channel coding and decoding, digital modulation methods, channel estimation, data detection, multiuser detection, digital satellite systems, wireless IP-based systems, CDMA and spread-spectrum systems, personal communication networks and ultra-wideband systems. He has also published 4 book chapters, 34 journal papers/letters, and more than 150 conference papers. He is a coauthor of Coded Modulation Systems (Norwell, MA: Kluwer Academic/Plenum, 2003). Dr. Svensson received the IEEE Vehicular Technology Society Paper of the Year Award in 1986, and in 1984, the Young Scientists Award from the International Union of Radio Science, URSI. He was an Editor of the Wireless Communication Series of the IEEE JOURNAL OF SELECTED AREAS IN CoMMUNICATIONS until 2001, and is now an Editor for IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He is a member of the IEEE Communications Society, IEEE Signal Processing Society, IEEE Information Theory Society, IEEE Vehicular Technology Society, and the Swedish URSI committee (SNRV, Svenska Nationalkommitten för Radiovetenskap). He was a member of the council of SER (Svenska Elektrooch Dataingenjörers Riksförening) from 1998 to 2002, and is currently a member of the council of NRS (Nordiska Radiosamfundet). He is listed in Who’s Who in Finance and Industry, Who’s Who in the World, and the European Biographical Directory.
