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397

Blind Equalization of Turbo Trellis-Coded Partial-Response Continuous-Phase Modulation Signaling Over Narrow-Band Rician Fading Channels Onur Osman and Osman N. Ucan

Abstract—In this paper, a blind maximum-likelihood channel estimation algorithm is developed for turbo trellis-coded/continuous-phase modulation (TTC/CPM) signals propagating through additive white Gaussian noise (AWGN) and Rician fading environments. We present CPM for TTC signals, since it provides low spectral occupancy and is suitable for power- and bandwidth-limited channels. Here, the Baum–Welch (BW) algorithm is modified to estimate the channel parameters. We investigate the performance of TTC/CPM for 16-CPFSK over AWGN and Rician channels for different frame sizes, in the case of ideal channel state information (CSI), no CSI, and BW estimated CSI. Index Terms—Continuous-phase modulation (CPM), MAP estimation, parameter estimation, Rician channel.

I. INTRODUCTION

B

ERROU et al. introduced turbo codes in 1993 [1], which achieved high performance close to the Shannon limit. Subsequently, Blackert and Wilson concatenated turbo coding and trellis-coded modulation for multilevel signals in 1996. As a result, turbo trellis-coded (TTC) modulation was developed as in [2]. In band-limited channels, such as deep space and satellite communications, continuous phase modulation (CPM) has explicit advantages, since it has low spectral occupancy property. CPM is composed of continuous phase encoder (CPE) and memoryless mapper (MM). CPE is a convolutional encoder producing codeword sequences that are mapped onto waveforms by the MM, creating a continuous phase signal. CPE-related schemes exhibit better performance than systems using the traditional approach for a given number of trellis states, since they increase Euclidean distance. Once the memory structure of CPM is assigned, it is possible to design a single joint convolutional code, composed of trellis and convolutionally coded CPM systems as in [3]–[5]. To improve error performance and bandwidth efficiency, we combine CPM and TTC, and thus we introduce turbo trelliscoded/continuous-phase modulation (TTC/CPM). The new encoder includes a combined structure of a convolutional encoder (CE) and CPE. Since providing the continuity of the signal for punctured codes is not possible when encoders are considered separately, the solution is to make a suitable connection between

Manuscript received May 21, 2001; revised November 13, 2002, November 21, 2003; accepted December 18, 2003. The editor coordinating the review of this paper and approving it for publication is L. Hanzo. O. Osman is with the Istanbul Commerce University, 34378 Eminonu, Istanbul, Turkey (e-mail: [email protected]). O. N. Ucan is with the Electrical and Electronics Department, Istanbul University, 34850 Avcilar, Istanbul, Turkey (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2004.840214

the encoders. In this paper, we introduce a new combined encoder structure called “continuous phase convolutional encoder (CPCE).” In our proposed system, we can use different dimensional continuous phase frequency shift keying (CPFSK). As the number of dimension increases, it influences the Euclidian distance and therefore the bit error ratio is improved. In many cases, digital transmission is subject to intersymbol interference (ISI) caused by time dispersion of the multipath channels. In general, an adaptive channel estimator (e.g., based on the least mean square or recursive least-square algorithms) is implemented in parallel to a trellis-based equalizer [e.g., maximum likelihood (ML) sequence estimator]. This approach estimates both channel parameters and input data jointly [6]. A further development is the introduction of a per-survivor processing technique [7], where each survivor employs its own channel estimator and no tentative decision delay is afforded. In order to exploit properties of multipath fading and track time-varying channels, model-fitting algorithms are used in [8]–[10]. The ML joint data and channel estimator in the absence of training is presented in [11]. While all of these trellis-based techniques are applied for coherent detection, noncoherent blind equalization techniques are an interesting alternative area of current researches [12]–[14]. In this paper, a new, yet efficient TTC structure is presented to generate multidimensional continuous-phase signals that we call TTC/CPM, and a general ML blind method is investigated to estimate the Rician channel parameters for these signals. We concentrate on ML, since these estimation algorithms perform satisfactorily, even if there is only a short data record available [15]. However, the blind parameter is assumed to be constant in this process. However, implementation of the ML method directly for the blind problems results in a computational burden. The finite memory of the TTC/CPM along with the independent identical distribution (i.i.d.) and finite alphabet structure of the input data allows us to model the system response as a Markov chain [16]. However, the Markov chain is hidden, since it can only be inferred from the noisy observation [17]. We, therefore consider the Hidden Markov Model (HMM) formulation and employ the computationally efficient Baum–Welch (BW) algorithm [16] in order to estimate Rician channel parameters. This paper is organized as follows. In Section II, we briefly describe CPFSK. In Section III, we explain the TTC/CPM encoder, the channel capacity of CPFSK and TTC/CPM signals, and the decoder structure. In Section IV, blind equalization is explained. The simulation results are given in Section V. Section VI concludes the paper.

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Fig. 1.

TTC/CPM encoder structure for M-CPFSK with any h value.

II. CONTINUOUS-PHASE FREQUENCY-SHIFT KEYING Here, we study on -ary CPFSK (M-CPFSK), which is an -dimensional form of CPM. Rimoldi derived the tilted-phase representation of CPM in [3], with the information bearing phase given by (1) , where and are The modulation index is equal to relatively prime integers. is an input sequence of -ary sym. is the channel symbol period. bols, is generally chosen as 1 and the modulation index appears in . is a number that can be calculated the form of as 2 to the power of , the number of memories in CPE. The is a continuous and monotonically phase response function increasing function subject to the constraints (2) where is an integer. The phase response is usually defined in of duration , terms of the integral of a frequency pulse i.e., . For full response signaling, equals 1, and for partial response systems is greater than 1. Finally, is given by the transmitted signal (3) is the asymmetric carrier frequency as and is the carrier frequency. is the enis the initial carrier phase. We ergy per channel symbol and is an integer; this condition leads to a simpliassume that fication when using the equivalent representation of the CPM waveform. where

III. TTC/CPM In this section, we explain encoder and decoder structures of TTC/CPM. The most critical point is to keep the continuity of the signals in spite of the discontinuity property of puncturer. A. Encoder Structure In (3), corresponds to the initial angle of the instant signal and the final angle of the previous signal. We must know the final angle of the previous signal to provide the continuity. If

there is no puncturer block at the transmitter, the trellis states of the joint structure indicate the final and initial angles. Since our transmitter includes a puncturer, previous information of the encoder is lost, and thus we cannot keep the continuity property. This is due to the fact that previous information of the considered branch cannot be used because the puncturer switches to the other branch in the following coding step. Another drawback is the de-interleaver block at the output of the lower branch of the TTCM encoder as in [1]. First of all, we remove the de-interleaver at the transmitter side and then change the interleaver structure in the decoder to reassemble the data. In the TTCM encoder, there are two parallel identical encoders, but only one of them is used instantly. The criterion behind choosing the signal is based on the next state of other encoder’s CPE. Thus, we use the inputs of the other encoder’s CPE memory units as inputs of the MM in addition to the encoder outputs, so the continuity is provided. Fig. 1 shows the general structure of the TTC/CPM encoder. We can use any number of memory units in CPE, but we have to form connections from the input of the memories to the MM, and this results in the variation of the modulation index and signal space. A new approach of this paper is to offer a concrete solution for continuous phase signaling by not using extra memories for CPE. We use some of the CE memories for CPE as common and, thus, the TTCM encoder structure complexity is kept constant. This new structure that comprises CE and CPE is called the continuous phase convolutional encoder (CPCE). This combined structure is used not only to code the input data but also to provide the continuity. As an example, we show the encoder structure and input–output table for 16-CPFSK, while there are two data streams as inputs. In Fig. 2, the TTC/CPM and structure is shown for 16-CPFSK. Here, input bit . In this encoder structure, there is only one code rate memory unit for CPE. Thus, the modulation index is designated as 1/2, while there is one additional input to MM. In our example, the last memory unit is also used as a CPE memory, since there is a close relationship between the signal phase state and CPE memory. If the memory unit value is 0, the signal phase will be 0, and if the memory is 1, the signal phase will be 1. If there is only one memory unit for the CPE, there should be only two phase states, and, if there are two phase states, . If we use two memory units for CPE, we have four , , ), then becomes 1/4. But, if we use two phases (0, memories, there are five outputs. Therefore, modulation must be . This means that there is no alternative 32-CPFSK for

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Fig. 2. TTC/CPM encoder structure for 16-CPFSK and h = 1=2.

for 16-CPFSK other than . Moreover, there is no alfor the proposed ternative for 32-CPFSK other than system, while . In our example (Fig. 2), at the upper encoder, CE1 consists of three memory units and CPE1 uses the last memory unit of CE1. At the lower encoder, CE2 consists of three memory units and CPE2 uses the last memory unit of CE2. When it is the upper encoder’s turn, we use CE1 of the upper and CPE2 of the lower encoder. This combined structure (CE1 + CPE2) is defined as CPCE1. If it is lower encoder’s turn, we use CE2 of the lower and CPE1 of the upper encoder. This combined structure (CE2 + CPE1) is defined as CPCE2. In Table I, input–output data and signal constellations are given. Here, “o1” and “o2” systematic bits show the first two outputs and refer to the input bits “i1” and “i2,” respectively. “o3” is the coded bit and “o4” is the input data of the third memory of the other encoder, which is used for CPFSK. “o4” shows us at which phase the signal will start at the next coding interval. If “o4” is 0, the end phase of the instant signal and the starting angle of the next signal phase is 0. If “o4” is 1, the end phase of the instant signal and the starting angle of the next signal phase is . If “o3” is 0, the signal is positive and, if “o3” is 1, the signal is negative. Only if these conditions are met is continuity granted. According to Fig. 3 and Table I, the transmitted signals at the phase transitions are as follows: from 0 phase to 0 ; from 0 phase to phase, ; from phase, phase to 0 phase, and from phase to phase, .

TABLE I INPUT–OUTPUT AND SIGNAL CONSTELLATION FOR 16-CPFSK

B. Channel Capacity for CPFSK and TTC/CPM It is very important to know the channel capacity of the system for different values of the CPFSK parameter . First dimensions, of all, we deal with M-CPFSK only. If there are the number of encoder outputs and the number of inputs of . In general, except MM, , can be calculated as for TTC/CPM, the signal constellation consists of signals, but, for TTC/CPM, as shown in Table I; there for 16-CPFSK. This are 16 different signals

decrease in the number of signal sets causes a slight decrease in channel capacity. The output of the additive white Gaussian noise (AWGN) channel is (4) where denotes a real valued discrete channel signal transmitted , and is an independent normally disat modulation time

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Fig. 3. 16-CPFSK h = 1=2 signal phase diagram.

tributed noise sample with zero mean and variance each dimension. The average SNR is defined as

along

(5) Normalized average signal power is assumed. Channel capacity can be calculated from the formula which was given in [20], if the codes with equiprobable occurrence of channel input signals are of interest

(6) in b/T. Using a Gaussian random number generator, has been evaluated by Monte Carlo averaging of (6). In Fig. 4, channel capacity is plotted as a function of SNR for different dimensional CPFSK and TTC/CPM for 16-CPFSK. Here, it can be easily seen that the systems with CPFSK have quite high channel capacities compared to 2-D and one-dimensional (1-D) related modulation schemes such as MPSK, QAM, QPSK, and BPSK. In Fig. 4, the dotted line shows the channel capacity . Maxfor TTC/CPM in the case of 16-CPFSK imum channel capacity is 4, because and . In our case, TTC/CPM for 16-CPFSK and for the transmission of 2 b/T, and the theoretical limit is 5.1 dB. For the low channel capacities and with some error assumptions, channel capacity can be of the modulation generalized according to the dimension as follows: (7) C. Decoder Structure Here, in the turbo trellis decoder, the symbol-based decoder is equalized. We do not use a de-interleaver in the encoder, thus the symbol interleaver of the decoder only interleaves the odd indexed symbols, which are created at the upper branch of the encoder. The even indexed symbols are interleaved in the encoder. In the encoder, the memory structure is kept constant, but signal constellation is changed. In the decoding process, we use input data and parity data (o1 o2 o3) as mentioned in the example before; the “o4” is used only to provide continuity. It can be easily seen that there are different signals which have

Fig. 4. Channel capacity C of band-limited AWGN channels for TTC/CPM signals and different dimensional CPFSK signals. (Dotted line shows TTC/CPM for 16-CPFSK, h = 1=2. Bold lines show various M-CPFSK, and thin lines show various asymptotic curves.)

same input and parity data. As an example, in Table I, first and third rows have the same “o1 o2 o3,” but “o4” makes the first row “s0” and third row “s1”, and “o4” has no role in the decoding process, because the inputs of the two CEs are different when originating from the interleaver. CPM demodulator finds dimensional signal from orthonormal functions for the every observed signal. In metric calculation, the probability of dimensional signal is every observed demodulated computed according to Gaussian distribution. and indicate transmitted and received data, respectively. In general, there are different signals, among which some have the same input and parity data. The most probable one (MPO) chooses the most probable metric in the signal set, of which the input bits and parity bit are the same. The formulation of the MPO is given by

(8) is a vector that contains elements for every observed signal. is the number of the states of the CE (here, , there are memories, thus ) and for is a set of metrics, which has the same “o1 o2 o3.” The number depends on the modulation index or of elements in the number of memories in the CPE. There are elements in each . Therefore, is obtained from elements. (In our example, and , .) The output of the metric and MPO block is the input of the S-b-S decoder, which is explained in [3] in detail. At the decoder, metric-s calculation is only used in the first decoding step. Because of the structure and modulation of the TTCM encoder, systematic bits and parity bits are mapped to one signal, and, at the receiver, parity data cannot be separated from the received signal. Hence, in the first half of the first decoding for the S-b-S MAP decoder-1, a priori information has

OSMAN AND UCAN: BLIND EQUALIZATION OF TTC PARTIAL RESPONSE CPM SIGNALING OVER NARROW-BAND RICIAN FADING CHANNELS

Fig. 5.

401

TTC/CPM decoder structure.

not been generated. Metric-s calculates the a priori information from the even index in the received sequence. The next stage in the decoding process is an ordinary MAP algorithm. In Fig. 5, asterisk “ ” shows the switch position when the even indexed signals pass. IV. BLIND EQUALIZATION Consider the block diagram of the TTC/CPM scheme shown in Fig. 2. At the th epoch, the -bit information vector is fed to the TTC/CPM system. The . If we consider the output of this system is denoted as transmission of TTC/CPM signals over a nondispersive narrowband fading channel, the received signal has the form of (9) . The following assumptions are imposed where on the received signal model. AS1) is a zero-mean i.i.d. AWGN sequence with the variance of . is an -digit binary equiprobable information seAS2) quence input to the TTC/CPM scheme. represents a normalized fading magnitude having AS3) a Rician pdf. AS4) Slowly varying nondispersive fading channels are considered.

The basic issue of concern is the estimation of only the nondispersive Rician fading channel parameters from a set of observations. Due to its asymptotic efficiency, we put our emphasis on the ML approach [17] for the problem at hand. Based on the assumption AS1), the ML metric for is proportional to the conditional pdf of the received data sequence of length (given ) and is in the form of

(10) , is the number of signals where which have the same “o1 o2 o3” output values, and is a function of MM of the state of the CPCE and the value from the other encoder. As an example, in Fig. 2, there is only one connection from the other encoder. Thus, is 2 and there are two functions: the first one is used while “o4” is zero and the second is used while “o4” is one. Since the transmitted data sequence is not available at the receiver, the ML metric to be optimized over all poscan be obtained by taking the average of sible transmitted sequences , which is expressed as . Then, the ML estimator of is the one that maximizes the average ML metric . In this process, remains constant during the observation interval according to AS4). The metric for can be obtained by taking

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the average of (10) over all possible values of mulated as

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, which is for-

(11) Since the direct minimization of (11) requires the evaluation of (10) over all possible transmitted sequences, the detection complexity increases exponentially with data length . Fortunately, a general iterative approach to the computation of ML estimation, known as the Baum–Welch algorithm, can be used to significantly reduce the computational burden of an exhaustive search. Although only a narrow-band Rician channel is considered, equalizing the effects of partially response signaling is necessary. Baum–Welch Algorithm: The BW algorithm is an iterative ML estimation procedure, which was originally developed to estimate the parameters of a probabilistic function of a Markov chain [16]. The evaluation of the estimates in the iterative BW algorithm starts with an initial guess . At the th iteration, it takes estimate from the th iteration as an initial by maximizing over value and reestimates as follows: (12) This procedure is repeated until the parameters converge to a stable point. It can be shown that the explicit form of the auxiliary function at the th observation interval is in the form of

(13) is the joint likelihood of where data at state at time and is a constant. Using the definiand forward variables, we obtain tion of backward

(14)

Fig. 6. Simulation results of TTC/CPM for 16-CPFSK, h = 1=2, frame size = 256.

is the initial probability of the signals and equals . The can be probabilities of the partial observation sequences at recursively obtained as follows [15]–[18], given the state time : (19) The iterative estimation formulas can be obtained by setting the , gradient of the auxiliary function to zero . Then, the following formulas are applied until the stable point is reached:

(20)

(21) V. SIMULATION RESULTS OF TTC/CPM FOR 16-CPFSK

(15) where is the state transition probability of the CPCE and for our model. is the pdf of the observation is always and has a Gaussian nature (16) is the probable observed signal and is the variance of the Gaussian noise. The initial conditions of and are (17) (18)

In our simulation, we assume that we have two inputs and we design the TTC/CPM system for 16-CPFSK, as shown in Fig. 2. Modulation index is chosen as 1/2. Fading channel parameters are assumed to be constant during every ten symbols and Doppler frequency is not taken into consideration. In Figs. 6 and 7, the bit error performances of TTC/CPM for 16-CPFSK are 256 and evaluated for the first and fourth iterations while 1024, respectively. The channel model is chosen as AWGN and 4 and 10 dB in the case of ideal CSI, no CSI, and Rician for BW channel estimation. It is very interesting that the error probat a 1.5-dB SNR value, but ability of TTC/CPM is the similar BER is observed at 7.2 dB in TTCM for in the AWGN channel as in [19]. This performance improvement can also be carried out for various channel models and different frame sizes. These results show the high performance

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channel models. Finally, the results of TTC/CPM preprocessed with the BW algorithm and the results in the case of no CSI are found to be extraordinary. REFERENCES

Fig. 7.

Simulation results of TTC/CPM for 16-CPFSK, h = 1=2, frame size

= 1024.

of our proposed TTC/CPM compared to the classical schemes in various satellite channels. In the case of BW channel parameter estimation, the BERs of TTC/CPM are found to be very close to those in ideal cases. VI. CONCLUSION In this paper, a new yet very efficient coding method denoted turbo trellis coded/continuous phase modulation (TTC/CPM) is explained. Channel capacity of the proposed system and asymptotic curves of multidimensional modulations are given. According to the channel capacity curve of 16-CPFSK for TTC/CPM, error-free transmission of 2 b/T is theoretically 5.1 dB. TTC/CPM simulation results for possible at SNR 1024 and BER is only 3.6 dB worse than the theoretical channel capacity. At the decoder, the BW channel parameter estimation algorithm is used. The combined structure has more bandwidth efficiency, since it has CPM characteristics. When using common memory units for both convolutional and continuous phase encoders, we state that memory size and trellis state kept constant. The combined structure has a considerably better error performance compared to TTCM. According to the channel capacity of M-CPFSK modulation, the multidimensional property of CPM can be considered as the main reasons of high performance improvement. We sketch the results for different SNRs, iteration numbers, frame sizes, and

[1] C. Berrou, A. Glavieux, and P. Thitimasjshima, “Near Shannonlimit error correcting coding and decoding: turbo codes,” in Proc. ICC’93, May 1993, pp. 1064–1070. [2] W. Blackert and S. Wilson, “Turbo trellis coded modulation,” in Proc. CISS’96, Mar. 1996. [3] B. E. Rimoldi, “A decomposition approach to CPM,” IEEE Trans. Inf. Theory, vol. 34, no. 2, pp. 260–270, Mar. 1988. , “Design of coded CPFSK modulation systems for band-width and [4] energy efficiency,” IEEE Trans. Commun, vol. 37, no. 9, pp. 897–905, Sep. 1989. [5] R. H. H. Yang and D. P. Taylor, “Trellis-coded continuous-phase frequency-shift keying with ring convolutional codes,” IEEE Trans. Inf. Theory, vol. 40, no. 4, pp. 1057–1067, Jul. 1994. [6] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [7] R. Raheli, A. Polydoros, and C. K. Txou, “PSP: a general approach to MLSE in uncertain environments,” IEEE Trans. Commun., vol. 43, no. 2, pp. 354–364, Feb./Mar./Apr. 1995. [8] Q. Dai and E. Shwedyk, “Detection of bandlimited signals over frequency selective Rayleigh fading channels,” IEEE Trans. Commun., vol. 42, no. 2, pp. 941–950, Feb./Mar./Apr. 1994. [9] Y. Zhang, M. P. Fitz, and S. B. Gelfand, “Soft output demodulation on frequency-selective Rayleigh fading channels using AR models,” in Proc. IEEE GLOBCOM, vol. 1, Nov. 1997, pp. 327–330. [10] L. M. Davis and I. B. Collings, “DPSK versus pilot-aided PSK MAP equalization for fast-fading channels,” IEEE Trans. Commun., vol. 49, no. 2, pp. 226–228, Feb. 2001. [11] N. Seshadri, “Joint data and channel estimation using blind trellis search techniques,” IEEE Trans. Commun., vol. 42, no. 2, pp. 1000–1011, Feb. 1994. [12] X. Yu and S. Pasupathy, “Innovations-based MLSE for Rayleigh fading channels,” IEEE Trans. Commun., vol. 43, no. 11, pp. 1534–1544, Nov. 1995. [13] A. M. Fred and S. Pasupathy, “Nonlinear equalization of multi-path fading channels with noncoherent demodulation,” IEEE J. Sel. Areas Commun., vol. 14, no. 4, pp. 512–520, Apr. 1996. [14] G. Colavolpe and R. Raheli, “Noncoherent sequence detection,” IEEE Trans. Commun., vol. 47, no. 9, pp. 1376–1385, Sep. 1999. [15] M. Erkurt and J. G. Proakis, “Joint data detection and channel estimation for rapidly fading channels,” in Proc. IEEE Globecom, Dec. 1992, pp. 910–914. [16] L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains,” Ann. Math. Stat., vol. 41, pp. 164–171, 1970. [17] L. R. Rabiner, “A tutorial on hidden Markov models and selected applications in speech recognition,” Proc. IEEE, vol. 77, no. 2, pp. 257–274, Feb. 1989. [18] G. K. Kaleh and R. Vallet, “Joint parameter estimation and symbol detection for linear or non linear unknown channels,” IEEE Trans. Commun., vol. 42, no. 7, pp. 2406–2413, Jul. 1994. [19] P. Robertson and T. Worz, “Bandwidth-efficient Turbo Trellis-coded modulation using punctured component codes,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 206–218, Feb. 1998. [20] G. Ungerboeck, “Channel coding with multi level/phase signals,” IEEE Trans. Inf. Theory, vol. IT-28, no. 1, pp. 55–67, Jan. 1982.

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Blind Equalization of Turbo Trellis-Coded Partial-Response Continuous-Phase Modulation Signaling Over Narrow-Band Rician Fading Channels Onur Osman and Osman N. Ucan

Abstract—In this paper, a blind maximum-likelihood channel estimation algorithm is developed for turbo trellis-coded/continuous-phase modulation (TTC/CPM) signals propagating through additive white Gaussian noise (AWGN) and Rician fading environments. We present CPM for TTC signals, since it provides low spectral occupancy and is suitable for power- and bandwidth-limited channels. Here, the Baum–Welch (BW) algorithm is modified to estimate the channel parameters. We investigate the performance of TTC/CPM for 16-CPFSK over AWGN and Rician channels for different frame sizes, in the case of ideal channel state information (CSI), no CSI, and BW estimated CSI. Index Terms—Continuous-phase modulation (CPM), MAP estimation, parameter estimation, Rician channel.

I. INTRODUCTION

B

ERROU et al. introduced turbo codes in 1993 [1], which achieved high performance close to the Shannon limit. Subsequently, Blackert and Wilson concatenated turbo coding and trellis-coded modulation for multilevel signals in 1996. As a result, turbo trellis-coded (TTC) modulation was developed as in [2]. In band-limited channels, such as deep space and satellite communications, continuous phase modulation (CPM) has explicit advantages, since it has low spectral occupancy property. CPM is composed of continuous phase encoder (CPE) and memoryless mapper (MM). CPE is a convolutional encoder producing codeword sequences that are mapped onto waveforms by the MM, creating a continuous phase signal. CPE-related schemes exhibit better performance than systems using the traditional approach for a given number of trellis states, since they increase Euclidean distance. Once the memory structure of CPM is assigned, it is possible to design a single joint convolutional code, composed of trellis and convolutionally coded CPM systems as in [3]–[5]. To improve error performance and bandwidth efficiency, we combine CPM and TTC, and thus we introduce turbo trelliscoded/continuous-phase modulation (TTC/CPM). The new encoder includes a combined structure of a convolutional encoder (CE) and CPE. Since providing the continuity of the signal for punctured codes is not possible when encoders are considered separately, the solution is to make a suitable connection between

Manuscript received May 21, 2001; revised November 13, 2002, November 21, 2003; accepted December 18, 2003. The editor coordinating the review of this paper and approving it for publication is L. Hanzo. O. Osman is with the Istanbul Commerce University, 34378 Eminonu, Istanbul, Turkey (e-mail: [email protected]). O. N. Ucan is with the Electrical and Electronics Department, Istanbul University, 34850 Avcilar, Istanbul, Turkey (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2004.840214

the encoders. In this paper, we introduce a new combined encoder structure called “continuous phase convolutional encoder (CPCE).” In our proposed system, we can use different dimensional continuous phase frequency shift keying (CPFSK). As the number of dimension increases, it influences the Euclidian distance and therefore the bit error ratio is improved. In many cases, digital transmission is subject to intersymbol interference (ISI) caused by time dispersion of the multipath channels. In general, an adaptive channel estimator (e.g., based on the least mean square or recursive least-square algorithms) is implemented in parallel to a trellis-based equalizer [e.g., maximum likelihood (ML) sequence estimator]. This approach estimates both channel parameters and input data jointly [6]. A further development is the introduction of a per-survivor processing technique [7], where each survivor employs its own channel estimator and no tentative decision delay is afforded. In order to exploit properties of multipath fading and track time-varying channels, model-fitting algorithms are used in [8]–[10]. The ML joint data and channel estimator in the absence of training is presented in [11]. While all of these trellis-based techniques are applied for coherent detection, noncoherent blind equalization techniques are an interesting alternative area of current researches [12]–[14]. In this paper, a new, yet efficient TTC structure is presented to generate multidimensional continuous-phase signals that we call TTC/CPM, and a general ML blind method is investigated to estimate the Rician channel parameters for these signals. We concentrate on ML, since these estimation algorithms perform satisfactorily, even if there is only a short data record available [15]. However, the blind parameter is assumed to be constant in this process. However, implementation of the ML method directly for the blind problems results in a computational burden. The finite memory of the TTC/CPM along with the independent identical distribution (i.i.d.) and finite alphabet structure of the input data allows us to model the system response as a Markov chain [16]. However, the Markov chain is hidden, since it can only be inferred from the noisy observation [17]. We, therefore consider the Hidden Markov Model (HMM) formulation and employ the computationally efficient Baum–Welch (BW) algorithm [16] in order to estimate Rician channel parameters. This paper is organized as follows. In Section II, we briefly describe CPFSK. In Section III, we explain the TTC/CPM encoder, the channel capacity of CPFSK and TTC/CPM signals, and the decoder structure. In Section IV, blind equalization is explained. The simulation results are given in Section V. Section VI concludes the paper.

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Fig. 1.

TTC/CPM encoder structure for M-CPFSK with any h value.

II. CONTINUOUS-PHASE FREQUENCY-SHIFT KEYING Here, we study on -ary CPFSK (M-CPFSK), which is an -dimensional form of CPM. Rimoldi derived the tilted-phase representation of CPM in [3], with the information bearing phase given by (1) , where and are The modulation index is equal to relatively prime integers. is an input sequence of -ary sym. is the channel symbol period. bols, is generally chosen as 1 and the modulation index appears in . is a number that can be calculated the form of as 2 to the power of , the number of memories in CPE. The is a continuous and monotonically phase response function increasing function subject to the constraints (2) where is an integer. The phase response is usually defined in of duration , terms of the integral of a frequency pulse i.e., . For full response signaling, equals 1, and for partial response systems is greater than 1. Finally, is given by the transmitted signal (3) is the asymmetric carrier frequency as and is the carrier frequency. is the enis the initial carrier phase. We ergy per channel symbol and is an integer; this condition leads to a simpliassume that fication when using the equivalent representation of the CPM waveform. where

III. TTC/CPM In this section, we explain encoder and decoder structures of TTC/CPM. The most critical point is to keep the continuity of the signals in spite of the discontinuity property of puncturer. A. Encoder Structure In (3), corresponds to the initial angle of the instant signal and the final angle of the previous signal. We must know the final angle of the previous signal to provide the continuity. If

there is no puncturer block at the transmitter, the trellis states of the joint structure indicate the final and initial angles. Since our transmitter includes a puncturer, previous information of the encoder is lost, and thus we cannot keep the continuity property. This is due to the fact that previous information of the considered branch cannot be used because the puncturer switches to the other branch in the following coding step. Another drawback is the de-interleaver block at the output of the lower branch of the TTCM encoder as in [1]. First of all, we remove the de-interleaver at the transmitter side and then change the interleaver structure in the decoder to reassemble the data. In the TTCM encoder, there are two parallel identical encoders, but only one of them is used instantly. The criterion behind choosing the signal is based on the next state of other encoder’s CPE. Thus, we use the inputs of the other encoder’s CPE memory units as inputs of the MM in addition to the encoder outputs, so the continuity is provided. Fig. 1 shows the general structure of the TTC/CPM encoder. We can use any number of memory units in CPE, but we have to form connections from the input of the memories to the MM, and this results in the variation of the modulation index and signal space. A new approach of this paper is to offer a concrete solution for continuous phase signaling by not using extra memories for CPE. We use some of the CE memories for CPE as common and, thus, the TTCM encoder structure complexity is kept constant. This new structure that comprises CE and CPE is called the continuous phase convolutional encoder (CPCE). This combined structure is used not only to code the input data but also to provide the continuity. As an example, we show the encoder structure and input–output table for 16-CPFSK, while there are two data streams as inputs. In Fig. 2, the TTC/CPM and structure is shown for 16-CPFSK. Here, input bit . In this encoder structure, there is only one code rate memory unit for CPE. Thus, the modulation index is designated as 1/2, while there is one additional input to MM. In our example, the last memory unit is also used as a CPE memory, since there is a close relationship between the signal phase state and CPE memory. If the memory unit value is 0, the signal phase will be 0, and if the memory is 1, the signal phase will be 1. If there is only one memory unit for the CPE, there should be only two phase states, and, if there are two phase states, . If we use two memory units for CPE, we have four , , ), then becomes 1/4. But, if we use two phases (0, memories, there are five outputs. Therefore, modulation must be . This means that there is no alternative 32-CPFSK for

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Fig. 2. TTC/CPM encoder structure for 16-CPFSK and h = 1=2.

for 16-CPFSK other than . Moreover, there is no alfor the proposed ternative for 32-CPFSK other than system, while . In our example (Fig. 2), at the upper encoder, CE1 consists of three memory units and CPE1 uses the last memory unit of CE1. At the lower encoder, CE2 consists of three memory units and CPE2 uses the last memory unit of CE2. When it is the upper encoder’s turn, we use CE1 of the upper and CPE2 of the lower encoder. This combined structure (CE1 + CPE2) is defined as CPCE1. If it is lower encoder’s turn, we use CE2 of the lower and CPE1 of the upper encoder. This combined structure (CE2 + CPE1) is defined as CPCE2. In Table I, input–output data and signal constellations are given. Here, “o1” and “o2” systematic bits show the first two outputs and refer to the input bits “i1” and “i2,” respectively. “o3” is the coded bit and “o4” is the input data of the third memory of the other encoder, which is used for CPFSK. “o4” shows us at which phase the signal will start at the next coding interval. If “o4” is 0, the end phase of the instant signal and the starting angle of the next signal phase is 0. If “o4” is 1, the end phase of the instant signal and the starting angle of the next signal phase is . If “o3” is 0, the signal is positive and, if “o3” is 1, the signal is negative. Only if these conditions are met is continuity granted. According to Fig. 3 and Table I, the transmitted signals at the phase transitions are as follows: from 0 phase to 0 ; from 0 phase to phase, ; from phase, phase to 0 phase, and from phase to phase, .

TABLE I INPUT–OUTPUT AND SIGNAL CONSTELLATION FOR 16-CPFSK

B. Channel Capacity for CPFSK and TTC/CPM It is very important to know the channel capacity of the system for different values of the CPFSK parameter . First dimensions, of all, we deal with M-CPFSK only. If there are the number of encoder outputs and the number of inputs of . In general, except MM, , can be calculated as for TTC/CPM, the signal constellation consists of signals, but, for TTC/CPM, as shown in Table I; there for 16-CPFSK. This are 16 different signals

decrease in the number of signal sets causes a slight decrease in channel capacity. The output of the additive white Gaussian noise (AWGN) channel is (4) where denotes a real valued discrete channel signal transmitted , and is an independent normally disat modulation time

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Fig. 3. 16-CPFSK h = 1=2 signal phase diagram.

tributed noise sample with zero mean and variance each dimension. The average SNR is defined as

along

(5) Normalized average signal power is assumed. Channel capacity can be calculated from the formula which was given in [20], if the codes with equiprobable occurrence of channel input signals are of interest

(6) in b/T. Using a Gaussian random number generator, has been evaluated by Monte Carlo averaging of (6). In Fig. 4, channel capacity is plotted as a function of SNR for different dimensional CPFSK and TTC/CPM for 16-CPFSK. Here, it can be easily seen that the systems with CPFSK have quite high channel capacities compared to 2-D and one-dimensional (1-D) related modulation schemes such as MPSK, QAM, QPSK, and BPSK. In Fig. 4, the dotted line shows the channel capacity . Maxfor TTC/CPM in the case of 16-CPFSK imum channel capacity is 4, because and . In our case, TTC/CPM for 16-CPFSK and for the transmission of 2 b/T, and the theoretical limit is 5.1 dB. For the low channel capacities and with some error assumptions, channel capacity can be of the modulation generalized according to the dimension as follows: (7) C. Decoder Structure Here, in the turbo trellis decoder, the symbol-based decoder is equalized. We do not use a de-interleaver in the encoder, thus the symbol interleaver of the decoder only interleaves the odd indexed symbols, which are created at the upper branch of the encoder. The even indexed symbols are interleaved in the encoder. In the encoder, the memory structure is kept constant, but signal constellation is changed. In the decoding process, we use input data and parity data (o1 o2 o3) as mentioned in the example before; the “o4” is used only to provide continuity. It can be easily seen that there are different signals which have

Fig. 4. Channel capacity C of band-limited AWGN channels for TTC/CPM signals and different dimensional CPFSK signals. (Dotted line shows TTC/CPM for 16-CPFSK, h = 1=2. Bold lines show various M-CPFSK, and thin lines show various asymptotic curves.)

same input and parity data. As an example, in Table I, first and third rows have the same “o1 o2 o3,” but “o4” makes the first row “s0” and third row “s1”, and “o4” has no role in the decoding process, because the inputs of the two CEs are different when originating from the interleaver. CPM demodulator finds dimensional signal from orthonormal functions for the every observed signal. In metric calculation, the probability of dimensional signal is every observed demodulated computed according to Gaussian distribution. and indicate transmitted and received data, respectively. In general, there are different signals, among which some have the same input and parity data. The most probable one (MPO) chooses the most probable metric in the signal set, of which the input bits and parity bit are the same. The formulation of the MPO is given by

(8) is a vector that contains elements for every observed signal. is the number of the states of the CE (here, , there are memories, thus ) and for is a set of metrics, which has the same “o1 o2 o3.” The number depends on the modulation index or of elements in the number of memories in the CPE. There are elements in each . Therefore, is obtained from elements. (In our example, and , .) The output of the metric and MPO block is the input of the S-b-S decoder, which is explained in [3] in detail. At the decoder, metric-s calculation is only used in the first decoding step. Because of the structure and modulation of the TTCM encoder, systematic bits and parity bits are mapped to one signal, and, at the receiver, parity data cannot be separated from the received signal. Hence, in the first half of the first decoding for the S-b-S MAP decoder-1, a priori information has

OSMAN AND UCAN: BLIND EQUALIZATION OF TTC PARTIAL RESPONSE CPM SIGNALING OVER NARROW-BAND RICIAN FADING CHANNELS

Fig. 5.

401

TTC/CPM decoder structure.

not been generated. Metric-s calculates the a priori information from the even index in the received sequence. The next stage in the decoding process is an ordinary MAP algorithm. In Fig. 5, asterisk “ ” shows the switch position when the even indexed signals pass. IV. BLIND EQUALIZATION Consider the block diagram of the TTC/CPM scheme shown in Fig. 2. At the th epoch, the -bit information vector is fed to the TTC/CPM system. The . If we consider the output of this system is denoted as transmission of TTC/CPM signals over a nondispersive narrowband fading channel, the received signal has the form of (9) . The following assumptions are imposed where on the received signal model. AS1) is a zero-mean i.i.d. AWGN sequence with the variance of . is an -digit binary equiprobable information seAS2) quence input to the TTC/CPM scheme. represents a normalized fading magnitude having AS3) a Rician pdf. AS4) Slowly varying nondispersive fading channels are considered.

The basic issue of concern is the estimation of only the nondispersive Rician fading channel parameters from a set of observations. Due to its asymptotic efficiency, we put our emphasis on the ML approach [17] for the problem at hand. Based on the assumption AS1), the ML metric for is proportional to the conditional pdf of the received data sequence of length (given ) and is in the form of

(10) , is the number of signals where which have the same “o1 o2 o3” output values, and is a function of MM of the state of the CPCE and the value from the other encoder. As an example, in Fig. 2, there is only one connection from the other encoder. Thus, is 2 and there are two functions: the first one is used while “o4” is zero and the second is used while “o4” is one. Since the transmitted data sequence is not available at the receiver, the ML metric to be optimized over all poscan be obtained by taking the average of sible transmitted sequences , which is expressed as . Then, the ML estimator of is the one that maximizes the average ML metric . In this process, remains constant during the observation interval according to AS4). The metric for can be obtained by taking

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the average of (10) over all possible values of mulated as

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, which is for-

(11) Since the direct minimization of (11) requires the evaluation of (10) over all possible transmitted sequences, the detection complexity increases exponentially with data length . Fortunately, a general iterative approach to the computation of ML estimation, known as the Baum–Welch algorithm, can be used to significantly reduce the computational burden of an exhaustive search. Although only a narrow-band Rician channel is considered, equalizing the effects of partially response signaling is necessary. Baum–Welch Algorithm: The BW algorithm is an iterative ML estimation procedure, which was originally developed to estimate the parameters of a probabilistic function of a Markov chain [16]. The evaluation of the estimates in the iterative BW algorithm starts with an initial guess . At the th iteration, it takes estimate from the th iteration as an initial by maximizing over value and reestimates as follows: (12) This procedure is repeated until the parameters converge to a stable point. It can be shown that the explicit form of the auxiliary function at the th observation interval is in the form of

(13) is the joint likelihood of where data at state at time and is a constant. Using the definiand forward variables, we obtain tion of backward

(14)

Fig. 6. Simulation results of TTC/CPM for 16-CPFSK, h = 1=2, frame size = 256.

is the initial probability of the signals and equals . The can be probabilities of the partial observation sequences at recursively obtained as follows [15]–[18], given the state time : (19) The iterative estimation formulas can be obtained by setting the , gradient of the auxiliary function to zero . Then, the following formulas are applied until the stable point is reached:

(20)

(21) V. SIMULATION RESULTS OF TTC/CPM FOR 16-CPFSK

(15) where is the state transition probability of the CPCE and for our model. is the pdf of the observation is always and has a Gaussian nature (16) is the probable observed signal and is the variance of the Gaussian noise. The initial conditions of and are (17) (18)

In our simulation, we assume that we have two inputs and we design the TTC/CPM system for 16-CPFSK, as shown in Fig. 2. Modulation index is chosen as 1/2. Fading channel parameters are assumed to be constant during every ten symbols and Doppler frequency is not taken into consideration. In Figs. 6 and 7, the bit error performances of TTC/CPM for 16-CPFSK are 256 and evaluated for the first and fourth iterations while 1024, respectively. The channel model is chosen as AWGN and 4 and 10 dB in the case of ideal CSI, no CSI, and Rician for BW channel estimation. It is very interesting that the error probat a 1.5-dB SNR value, but ability of TTC/CPM is the similar BER is observed at 7.2 dB in TTCM for in the AWGN channel as in [19]. This performance improvement can also be carried out for various channel models and different frame sizes. These results show the high performance

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channel models. Finally, the results of TTC/CPM preprocessed with the BW algorithm and the results in the case of no CSI are found to be extraordinary. REFERENCES

Fig. 7.

Simulation results of TTC/CPM for 16-CPFSK, h = 1=2, frame size

= 1024.

of our proposed TTC/CPM compared to the classical schemes in various satellite channels. In the case of BW channel parameter estimation, the BERs of TTC/CPM are found to be very close to those in ideal cases. VI. CONCLUSION In this paper, a new yet very efficient coding method denoted turbo trellis coded/continuous phase modulation (TTC/CPM) is explained. Channel capacity of the proposed system and asymptotic curves of multidimensional modulations are given. According to the channel capacity curve of 16-CPFSK for TTC/CPM, error-free transmission of 2 b/T is theoretically 5.1 dB. TTC/CPM simulation results for possible at SNR 1024 and BER is only 3.6 dB worse than the theoretical channel capacity. At the decoder, the BW channel parameter estimation algorithm is used. The combined structure has more bandwidth efficiency, since it has CPM characteristics. When using common memory units for both convolutional and continuous phase encoders, we state that memory size and trellis state kept constant. The combined structure has a considerably better error performance compared to TTCM. According to the channel capacity of M-CPFSK modulation, the multidimensional property of CPM can be considered as the main reasons of high performance improvement. We sketch the results for different SNRs, iteration numbers, frame sizes, and

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