172 BEE TRANSACTIONS ON COMMwICAJlONS, VOL. 44, NO. 2, FEBRU.4Rk' 1996

Noncoherent Coded Modulation Dan Raphaeli,

Abstract-Tkellis coded modulation with two or multidimen- sional signal constellations, together with coherent maximum- likelihood detection, is considered an attractive solution for com- munications over the additive while Gaussian noise (AWGN) channel. In this paper a new noncoherent communication system is introduced called noncoherent coded modulation (NCM) as an alternative to coherent coded modulation. NCM achieves almmt the same power efficiency, without bandwidth expansion or an extensive increase in complexity. As a noncoherent system, the method does not need carrier phase estimation. Nonetheless, dif- ferential encoding is not required. High performance noncoherent detection is achieved by using multiple symbol observations. Unlike previous approaches, a sliding window for the observa- tions is used, with each observation covering several branches of the trellis, such that the observations are time-overlapped. We define a new type of noncoherent maximum-likelihood sequence estimator (MLSE), and analyze its performance over the AWGN channel by numerical calcullaliun of the union bound. We perform a computerized search and present new codw for noncoherent deteetion with their performance. The new codes cover many useful rates and complexities and achieve higher performance than existing codes for noncoherent detection. The method ean also be used for multiple symbol demodulation of MDPSK with better results than existing methods.

M I. INTRODUCTION

ANY communication channels can be characterized by a time varying phase response together with ad-

ditive white Gaussian noise (AWGN). In many cases it is not desired or is impractical to estimate the phase of the channel [ 11. In such cases, or simply for eliminating the phase estimation overhead, noncoherent modulations are preferred' (see for example [2]). There are many additional advantages of noncoherent detection over coherent detection. There is no acquisition time so that no data would be lost until trackmg is established. There is no acquisition mode and system requirements involved, such as lock detection and mode switching. Fast recovery Prom severe fading occurs in noncoherent modulation. There is no false-lock, phase slipping and loss of lock associated with a PLL circuit. In many cases, coherent modulations suffer high degradation from phase estimation errors. ,Phase estimation is a problem, especially when bandwidth efficient modulations are used like trellis coded modulation (TCM) [3 ] and continuous phase modulation (CPM) [41,, PI.

Paper approved by D. Divsalar, the Editor for codiog Theory and Appli- cations of the IEEE Communications Society. Manuscript received October 6, 1993; revised February 7, 1995; April 27, 1995. This paper was presented in paxt at the 1994 Conference on Information Science and Systems, March 1994, Princeton, NJ USA.

The author is with the Lkpartment of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel.

Publisher Item Identifier S 0090-6778(96)016194. 'The detection techniques that are called sern-coherent are essentially

noucohereut.

009&6778/96$05

Member, IEEE

Currently, there is a gap between the performance of coded noncoherent systems and the coherent ones. For example, there is about 3 dB difference in performance on the AWGN channel between constraint length 7 rate 1/2, coded PSK and DPSK (differential PSK), even when interleaving is used for the DPSK [6]. Moreover, there is not much available trade-off between bandwidth and power efficiency with currently used noncoherent modulations as with coherent ones.

Recently it has been shown that the performance of nonco- herent detection can be improved by using a longer observation time' (one symbol for FSK and two symbols for differential detection). Multiple symbol noncoherent detection of ancoded DPSK is considered in [7], [SI, and [9], and rnulliple symbol noncoherent detection of uncoded CPM is considered in [4], [SI, [lo], and [ l I]. An application to block coded MSK is found in [lo], and a,new block coded MPSK for noncoherent detection is considered in [12]. Multiple-symbol differential detection of trellis coded MDPSK is considcrcd in [I31 and [14]. Multiple-symbol noncoherent detection of trellis coded modulation has been confined to applications which use dif- ferential encoding. Also, no optimization of the code selection has been pedormed. In all of the above contributions, the observations are independent. In the case of uncoded or block coded modulation, one observation was used for each deci- sion. When convolutional coded modulations were decoded, independent observations were used. In some methods, the observations overlap in one symbol and interleaving was used for mak3ng them independent. The interleaving function adds extra Complexity to the system and causes unwanted delay in the received data; although in some applications such as fading channels, interleaving is required. '

We show a new noncoherent decoding scheme whxh can be applied to almost all types of coded modulation, without changing the encoder, and with performance, that in general approaches the cohcrcnt dccoding of thcsc codes as the ob- servation length grows. We also show new codes with high noncoherent performance which are also optimal for coherent decoding (have maximum free distance) and their performance with the efEcient noncoherent decoding.

Unlike the previous approaches; we use maximally over- lapped observations (see Fig. I), which will not and cannot be made independent. We thus utilize the infornlation from all possible (time shifted) observations of length T . When applying this technique to trellis coded modulation, each obser- vation spans several branches of the code trellis. The resulting sequence estimator is derived, and its performance for equal energy symbols over the AWGN channel is found by using the union bound. The bound is computed numerically. We show

2The observation is the time window in which the carrier phase is assumed to be constant

.oO 0 1996 LEEE

RAPHAELI: NONCOHERENT CODED MODULATION 173

+T+ * - T 4

k - T +

+ T 4

+ T 4 Fig. 1. Demonstration of overlapped observations

by examples and prove analytically that the performance of the noncoherent sequence estimator approaches the coherent one as the observation length grows (on a constant phase channel).

Also, we suggest the use of a type of coded modulation scheme, the linear noncoherent coded modulation (LNCM) [16], which exhibits uniform error property (UEP) when used with our sequence estimator. The LNCM uses constant energy symbols, e.g., MPSK. Note that the proposed decoding algorithm can be applied to many other codes and modulations, e.g., CPM [17].

The above efforts lead to a new approach for designing non- coherent systems. We call this coded modulation and detection scheme “noncoherent coded modulation (NCM)” (see Fig. 2). This scheme provides a trade-off between robustness to phase variations (or frequency uncertainty) and power efficiency, by controlling the observation length of the detector. The existence of decoding algorithms whose complexity depend only slightly on the observation length [18], makes this trade- off efficient. NCM may be attractive even in cases where phase synchronization is not a major problem, since its degradation relative to coherent demodulation can be smaller than the degradation caused. by imperfect phase estimation.

In coherent coded modulations, in cases where enough bandwidth is available, better performance can be achieved with lower code rates without considerably increasing the complexity. In the same way, it is possible to trade bandwidth efficiency with power efficiency in NCM. Note that this trade- off is not efficient when using conventional detection of coded DPSK or MFSK.

The paper is organized as follows: In Section 11, the gen- eral problem of detection is discussed, and the Independent Overlapped observations Noncoherent maximum-likelihood sequence estimator (IO-NMLSE) is derived. In Section III, we briefly describe the codes which will be used. Section IV is devoted to the error probability analysis for a coded system employing the IO-NMLSE decoder. The search for NCM codes is described in Section V. Finally, in Section VI, the results for several good codes, and for differentially encoded PSK, are presented.

11. NONCOHEFZENT SEQUENCE ESTIMATION

The solution of optimal noncoherent maximum likelihood sequence estimator (NMLSE) depends on the statistics of the

Slowly varying

AWGN channel -. -. phase and --. Modulator -c Encoder IO-NMLSE -

decoder

Fig. 2. The noncoherent coded modulation system.

time varying carrier phase. When such statistics are unavail- able, the derivation of the NMLSE must start from some broad assumptions. The commonly used assumption is that the carrier phase is constant (but completely unknown) during some observation interval (t , t + T) for any t. When trying to use this assumption for the derivation of the NMLSE, we notice that the problem is ill posed. If the phase is constant over every interval of certain length, then it has to be constant everywhere. It is clear that some phase variation should be allowed over the observation time. To make the optimal NMLSE a well-defined problem, the allowed phase variation over the observation interval, and the way to measure it, should be defined. Then, the worst-case phase random process among those which fit into the constraint can be found and used for deriving the maximum-likelihood function. This approach seems too complex and probably will not lead to any simple implementable solution. Leaving the optimal NMLSE still undefined, we choose to develop a metric which provides good performance for a broad range of channels, but is suboptimal with respect to any specific channel. However, the degradation might be very small.

In previous approaches based on constant phase observa- tions, the observations were either independent or overlapping in one symbol. In our approach, maximally overlapped ob- servations, which make use of the fact that the carrier phase can be assumed to be constant for any observation of length T , are used. Thus, the channel memory is utilized in a more efficient manner. In fact, it will be shown in the results, that the decoding performance can be improved by increasing the overlapping ratio K, (which will be defined shortly). It is further assumed that the observations, even when they overlap in time, are independent, and have independent phases. Note that the observations are not made independent by any means. They are only treated as such for the derivation of the estimator. We call the resulting estimator independent overlapped observations NMLSE (IO-NMLSE). The indepedency also helps in the decoding by facilitating the use of the Viterbi algorithm. Furthermore, it enables fast recovery from deep fade.

In the case of coded DPSK, which uses two-symbols observations, the overlapping is inherent. However, the ob- servations are commonly made independent by interleaving. Recently Kaplan et al. [19] showed that the noninterleaved DPSK can achieve higher performance in terms of cutoff rate and capacity, thus supporting our approach. The importance of using overlapped observations is also demonstrated in example 1.

The IO-NMLSE discriminates between a set of possible transmitted waveforms { ~ ( t ) } by choosing the signal m

174 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 2, FEBRUARY 1996

which maximizes the following metric Q3

rl(x,(t)) = log Qm(ke , h- + T ) (1)

where Qm(Tar Tb) is the ML metric for one observation interval T, 5 t 5 T b , and is found in [I, equation 4c.121, k is the observation number, r is the observations spacing and T is the observation length. The choice of T is a trade- off between maximizing BER performance and minimizing system complexity

k = - c a

=exp [-:I zm(t)q.&(t)dt , ] Io [ a f r(t)f&(t)dt I] (2)

where r ( t ) is the received wavefdrm (both z ( t ) and ~ ( t ) appear in the baseband representation), a is the channel attenuation, and qm(t) is defined in [l].

The overlapping ratio is ti = (T - T ) / T and has values be- tween 0 to l. In the case of IC = 0, we arrive at nonoverlapped observations.

Tn the case of additive white Gaussian noise with one sided spectral density NO, qm ( t ) = xm (t) /NO [I, pp. 3441. Adding the case of equal energy signals, the estimator can as well maximize

For low signal-to-noise ratio [(SNR) small argument] the logIo(z) function is approximated by x2/4, leading to an estimator which maximizes the metric

We have confirmed by simulations that the use of this ap- proximation does not lead to any noticeable performance degradation. In a digital implementation, where xm(t ) is a sequence of symbols of duration T, and each symbol is constmcted using a 2 x D dimensional signal space, the metric can be written as

m

I

where S is the observation length in symbols, I (an integer) is the observations spacing in symbols, and for every symbol i, Fi is a complex vector which assumes the output of D complex matched filters, each for one complex dimension of modulation. The sequence of vectors of dimension D ; d m ) ,

is the signal space representation of a,(t). Let us define L as the number of trellis branches which

are covered by one observation, is., [L = S/nl (assuming 1 is a multiple of n ; or else, in some cases it is necessary to add one), where n is the number of symbols in a trellis branch. L is more important than S since it determines the

complexity of the decoder and also relates more closely to the actual observation time T. Unless stated otherwise, we will use 1 = n for maximal overlapping and convenient decoder implementation.

If the code is not noncoherently catastrophic (see Section ILB), then as S increases (and the allowed phase variations are reduced appropriately), the performance of the IO-NMLSE approaches that of the MLSE with a completely known phase. The proof is given in Appendix A. This provides a trade-off between robustness to phase variations and power efficiency.

Since we want an efficient algorithm for decoding, we are motivated to use the Viterbi Algorithm (VA). From ( 3 , let us try to define the branch metric as q k . It is clear that 171, is a function of the current branch value %k together with the previous branch values (%,&I,. . I , %,&S+l}. Since the tentative decisions made by the VA should not affect the following branch metrics, this choice of metric will not cause optimal operation of the VA as a metric maximization

In order to make the branch metric independent of previous decisions, we can construct a new trellis diagram with NRL-l states, where N is the number of states in the original trellis and R is the number of possible transitions from each state. A state in the new trellis will be assigned to each of the possible sequences of L consecutive states {ao, . . . , U L - ~ } thal can be produced by the original trellis. The original transitions from state ah-1 to state UL are mapped to transitions from state {uo,...,uL-~} to state {al,...,a~} for all possible choices of {UO, . . I , U L } . The corresponding branch value is the sequence of symbols {XO, - , XL-I} which is the output of the path {ao, . . . , a ~ } on the original trellis. When using the new trellis, Ilk is a function of the branch value only, enabling true maximizations of the metric (5) by the VA. Note that having NRL-' states is sufficient but not necessary.

If the complexity is measured in the number of correlations needed, the complexity is similar to that of the multiple symbol differential detector of Divsalar et al. [13]. Remarkably efficient, suboptimal decoding algorithms are presented in [18], in which the dependence of thc complexity on L has linear afiinity.

Example 1: Suppose that we want to discriminate between two possible received signals {q,. . . ,rg!, one is { 1,1,1,1,1,1,1, l} and the other IS

{1,1,1,1; -1, -1, -1, -1). Assume that our observation ignores the absolute phase. By taking the nonoverlapping observation (11, ... , t-4) and {rs! ... , r8) it is clear that the two possible received signals are indistinguishable. By adding the observation (r3, . . . ,1'6} we are able to distinguish between the two hypothesis; for the first case we get {1,1,1, l} and for the second case we get {1,1, -1, -1). This shows the importance of the overlapped observations.

algorithm.

A. The Relation Between Observation Length and Spectral Eficiency

We argue that more bandwidth efficient schemes tend to require longer observations. We can view the noncoherent decoder as one that estimates the phase and then uses its phase

RAF'HAELI: NONCOHERENT CODED MODULATION 175

estimate to coherently detect the same symbols. The quality of the phase estimation is dependent on the SNR per observation, and the last is a function of the length of the observation. As the scheme becomes more bandwidth efficient, it is typically more sensitive to phase reference errors. Thus, the observation length should be increased.

In a more general view, the principle underlying the op- eration of any noncoherent decoder is that the phase noise spectrum is narrower than the modulation spectrum. We will call this spectrum separation. The phase noise bandwidth is inversely related to the maximum time constant of the detector allowed for low degradation. As the modulation spectrum becomes narrower, the phase noise spectrum should become narrower as well to insure the spectral separation property. As a result, the allowed maximum time constant of the detector is increased. If the detector time constant is not increased correspondingly, the degradation cannot stay low. If this is not the case, it leads to a contradiction, since the phase noise spectrum is then allowed to remain wide, disobeying the spectrum separation property, hence impossible. The same argument applies to any receiver, including coherent ones (when the phase tracking circuit is included, the coherent is essentially equivalent to a noncoherent one). Noncoherent or coherent schemes in which the time constant is variable (for the coherent this will be the PLL time constant) can be distinguished by the different spectral separation required for a given loss. The same arguments apply also to time varying flat fading processes. With some more rigorous definition, spectral separation might serve as an engineering tool for comparing different detection schemes.

B. Noncoherently Catastrophic Codes

Usually, catastrophic codes refer to convolutional codes in which an infinite number of decoding errors can he produced as a result of a finite error event. This happens when two infinite length codewords, which have the same output for an infinite number of'symhols, exist and correspond to different input bits.

We define noncoherently catastrophic (NC) codes (the code includes the modulation) as codes in which two infinite length codewords, which have the same output or a constant phase shift from one another for an infinite number of symbols, exist, and correspond to different input bits. NC codes include the usual catastrophic codes as a special case. For example, every transparent (inverting the input of the encoder causes the output to be inverted) binary convolutional code with BPSK modulation is NC.

Rotationally transparent codes (not to confuse with binary transparent codes) are codes in which certain phases of rotation in the channel will not affect the decoding. These codes are not NC and can be used in our noncoherent system. However, they may perform worse due to additional possible error events, as will be explained in Section IV.

Due to the overlapping of the observations, our decoder observes all the phase transitions between consecutive symbols in its decisions. Thus, any non-NC code can be decoded by this decoder. A commonly used method in noncoherent communications with PSK symbols is to use differential

encoder after the channel encoder. However, when using our decoder, there is no need to use this differential encoder.

III. LINEAR NONCOHEWNT CODED MODULATION (LNCM)

We suggest the use of a specific type of trellis coded modula- tion called LNCM in conjunction with the sequence estimator described above. These codes exhibit the uniform error prop- erty (UEP) for noncoherent detection, and are a simple case of the general UEP codes [ 161. UEP means that the error proba- bility does not depend on the specific codeword that was sent. This property simplifies the analysis and helps in the search for good codes. By using this property, we do not have to check all possible pairs of codewords in order to find the decoder error probability when using the union bound. We assume that good codes are more likely to be found within the class of UEP codes. This assumption is based on the fact that good binary codes are found within the class of linear codes, and TCM codes within the class of geometrically uniform codes.

When using coded modulation and a noncoherent metric, linearity of a code does not imply UEP. The general UEP codes indeed satisfy the UEP under noncoherent decoding.

In this paper we limit ourselves to convolutionally coded PSK modulation, without set partitioning. For these codes, the LNCM reduces to the following codes.

Let K be the constraint length of the code, b be the number of input bits per trellis branch, and R = 2b be the number of branches reaching a node and also the number of phases in the MPSK modulation. RK-' is the number of states, and n is the number of PSK symbols in one branch. Let the input group and the output group be the modulo R set of integers Z,, i.e., the input field is not GF(2) for R > 2. The encoder consists of a shift register of K stages, and each stage contains an integer between zero to R - 1. Each output i of the encoder is generated by a modulo R weighted sum of the shift register contents and the generator gi = ai,^, QQ,. . . ai,^), 0 5

The output is mapped to the phase of the PSK signals linearly (not by set partitioning). The rate of the code is b/nb meaning that for every branch b bits going in and nb bits getting out, mapped to n, PSK symbols. Other rates are also possible by having multiple shift registers or using punctured codes.

As shown in [16], any UEP code which is found to be NC, is rotationally invariant, and thus can be made rotationally transparent by differentially encoding its input. For the rest of the paper we will assume that this conversion has been performed on every code which has been found to be NC. Note that the differential encoding can be incorporated in the encoder without increasing the number of states.

" i , j < R.

Iv. ERROR PROBAsILITY ANALYSIS

In this section we evaluate the IO-NMLSE error probability for a coded modulation transmitted over a channel with AWGN and a slowly varying carrier phase. We will treat the case of UEP. The result can be extended to non-UEP systems by using encoder-decoder pair-states trellis (see, for example, [20]) having N2 states. This will greatly increase the computation time requirement.

116 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 2, FEBRUARY 1996

Since for any slow varying phase process, in which the assumption of constant phase over S symbols approximately holds, the metric (5) will give the same result, and we can assume without loss of generality that the phase is constant everywhere. This analysis holds for any arbitrary slowly varying phase process.

The decoder compares the metrics of all the possible paths and chooses the path with the maximal metric. For error probability computation, we assume that a specific codeword is transmitted, namely the all-Z sequence, 6, where the symbol Z denotes the output symbol which corresponds to an akzero input to the encoder. We then find the probability that another path on the trellis will get a larger metric. The following derivation can be easily modified to incorporate any other transmitted sequence.

We call the codewords that happen to be a phase shift of the all-Z sequence "constant sequences." It can be shown that in UEP codes, constant sequences are only found in transparent codes (or NC ones). In this case, paths branching from 6 and reaching a constant sequence are also competitors and should be included as possible error events. Such competitors will have less and less contribution as L increases. This is explained as follows. If the observation L is larger than the error event, some of the observations will cover the error event. Then, as L increases, more symbols in the competitor are different than the transmitted ones over these observations, leading to lower error probability.

We will use the union bound for estimating the bit error probability, which is an upper bound. For simplicity we will assume that the observation length S is an integer multiple of n,, the number of symbols per branch, and that I = n in (5). The solution for the case of I = np , where p is an arbitrary integer, can be similarly obtained. Note that in this case the error events have p different positions relative to the beginning of the observations. Hence, one should average over all the possible positions while computing the union bound.

Unfortunately, the pairwise error probability cannot be ex- pressed in a form suitable for the transfer function method [21]. Instead, we numerically approximate the bit error probability as

1 b

r b E - b(%("))P,(%(")} (6) I x ( m ) l < M

where j i ( " ) are all possible competing sequences, is the length of the error event in d m ) ( d m ) itself has infinite length), and !I(%(")) is the number of nonzero bits in the decoding of d m ) . Pe(dm)} is the p h i s e error probability between 4 and F. M is a sufficiently large number such that the residual contribution of the larger error events can be neglected. Usually, an error event is defined as one starting from state 0 and returning back to state 0 without "visiting" (passing through) state 0. In ( B ) , we should allow the error event to visit state 0 for a maximum of L - 1 consecutive places. The above error events are equivalent to regular error events (those which do not visit the zero state) on the augmented trellis defined in Section II. Nevertheless, in this paper (except for DPSK), we have neglected these special error

events, since they have minimum length of twice the shortest error event and thus have very small contribution to the error probability.

Let us derive P,{f}. Let Fi be the received sequence, and let f be an infinite encoder output sequence containing an error event of length N' symbols. The number of symbols that affect the decision is larger and equal to N = N' + 2(L - 1)n. The sequence of symbols X is formed such that the n(L- 1) symbols { g o , . . . , %,L(L-l)-l} are Z and the symbols {fN--n(L-l), . a a , % N - l , % N , . . .} are equal to either Z or a phase shift of Z . The latter is appropriate for sequences which diverge to a constant sequence. The pairwise error probability is P,(x) = Pr(y2 > y l } , where

Each element, Fi or Xi, is a vector of dimension D. Let us map F, z1 and 4 each to a vector, r, x and 4, of length DN by

- rDi+j = Fi,j , x ~ i + j = Xi, j and d ~ i + j = q$,j = Z,

O l i S N - 1 , O < j < U - l . (8)

(N-S)/Z S-1 D-1

92 = rDkZ+Di+jXLkl+Di+j

( N - S ) / I DkZ+DS-1 Dkl+DS-1

: xpx:r;rq.

Define

w(a,j; IC) = 1, if Dkl 5 i , j 5 Dlcl + DS - 1 ; 0, otherwise. (10)

Then

RAPHAEL1 NONCOHFRENT CODED MODmATION

In the same way we can express y1 as

D N - 1 D N - l

177

v. SEARCHING FOR GOOD CODES

p=o q=o

where ( N - S ) / l

k=O

Let

aP,q = a 2 , P , 4 - aLP,q (N--S)IL

= (x& - z p mod D Z q mod D * ) W(P7 49 k) (14) k=O

and A = {up,*}, a DN x D N matrix, then

y = y2 - y1 = rtAr. (15)

We want to find the probability of y > 0. The vector r is a complex Gaussian random vector of length D N , with mean E[r] = $4 and covariance matrix C = 21 (unit variance in the real and the imaginary parts of each of the vector components), where

and I is the identity matrix. Let us express A as A = QAQ-’, where A is the diagonal eigenvalue matrix, and define Z = Q-lr, then

D N - 1

y = r ~ Q A Q - ~ = Z+AZ = xilzi12. (17) i =O

It is easy to show that E[Z] = Q-lE[r] and that the covari- ance matrix of Z is 21.

The distribution of y is that of a noncentral general quadratic form in Gaussian variables [22]. The general case, which we have, is the most difficult case to treat. Some special cases are easier, among them are the case where A is positive definite and the case where we have a central general quadratic form, defined as E[Z] = 0. {al.p,q} and { Q , ~ , ~ } are each a positive definite matrix but their difference is not.

The characteristic function can be expressed as [22],

D N - 1 .

where pi = E[Zi]. This characteristic function should be inverted in order to find the distribution of y.

There is no known closed form solution for the distribution. We have not found even a convenient numerical method available. We have developed and used a simple and efficient numerical method (see Appendix B). An alternative method that we have developed is given in [23].

Given a specific observation length and code parameters, the optimal or close to optimal code was found by a computer search. The minimum Euclidean distance serves only as a lower bound, thus it is not very useful for NCM selection (unless the observation L is large enough, but then the NCM performance is almost identical to coherent one and no search is required). For the BPSK and QPSK case we have used L = 4, and for the 8PSK case we have used L = 3.

We have used the bit error probability bound (6) as the optimization criterion, but we took care to minimize the maximum error event probability as much as possible. For small K , R, and n, the number of possible codes is not too large, and a one by one search was performed to check all possible codes in order to find the optimal one. For larger values of the parameters, this method is not practical. In this case, the codes to be tested were produced at random. Since there is a large number of good codes, the probability of finding one of these codes is not too small.

With either method, a large number of codes had to be checked. By observing symmetry properties of the codes, we can reduce the number of codes to be checked. For example, a code and its reverse (the generators are reversed) have exactly the same performance, since all the possible error events are reversed. Permutating the generators yields an equivalent code, and the same thing happens if we negate all the coefficients “ i , j .

The exact error probability computation takes too much time to compute for all codes. We have used simple lower bound criteria for fast preselection of codes to speed up the search. The search was performed in two stages; one for small M ( M = K + 2) and one for the final value. In each stage preselection was followed by an exact computation.

The first preselection is based on the computation of the correlation (absolute squared) of the sequence with the all-Z sequence,

The total correlation can be used to compute a lower bound on the error probability. Our decoder cannot be better than the optimal way of deciding between two hypotheses under ran- dom phase conditions. The latter has an error probability (for a constant symbol energy) which is a function of the correlation p and the sequence length N . For the fast preselection, p was compared to a threshold which is a pre-computed function of N . The error probability of binary noncoherent block detection is found in [l, sec. 4.3.1~1. This lower bound is tighter than the one based on the Euclidean distance which is the coherent error probability.

The second preselection rule was ha& on a high SNR approximation. In this approximation, we ignore the n X n term in the expression of (15) and we get a Gaussian r.v. Then, the painvise error probability is

178 lEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 2, FEBRUARY 1996

TABLE I BEST NONCOHERENT COD= FOUND (EACH

REPRESENTS A CLASS OF EQUIVALENT CODES)

* Produced by random search.

VI. RESULTS AND DISCUSSION

We present the performance of some NCM codes using BPSK, QPSK, and 8PSK modulations over the AWGN channel when detected noncoherently with various observation lengths, and coherently for comparison. All the results presented are computed for codes, given in Table I, found through the search for the best codes. Note that each code in this table is a representative of a class of equivalent codes.

Fortunately, except for the case K = 7, rate 112 convolu- tionally coded BPSK, the presented codes also have maximum free distance, thus they are also optimal for coherent detection. The best convolutional codes are tabulated in [l, table 5.311. For the K = 7, rate 112 case the free distance of the best found NCM is nine, while the maximum free distance possible for K = 7, rate 1/2 is 10. For the BPSK and QPSK cases we get a fair comparison between noncoherent and coherent detection. By fair comparison we mean that the best found NCM is &so optimal for coherent detection, thus the performance curves for this code will reflect the best possible for either case. In this comparison we ignore the added complexity in the noncoherent decoder. For the case of 8PSK it is difficult to make fair comparison since to the best of OUT knowledge no optimal rate 3/6 TCM using 8PSK has bcen published. However, the coherent results, for the 8PSK code found are very good (compared to BPSK and QPSK). Thus we get a good comparison between coherent and noncoherent detection also in this case.

We see that as the observation length grows, the perfor- mance of the noncoherent detection approaches that of the coherent (on a constant phase channel), and we observe that the rate of convergence seems to be only slightly dependent on the SNR.

In Fig. 3, a K = 5 rate 1/2 coded BPSK from Table I is evaluated. For comparison, when we decode the same code with nonoverlapping observations, we get poor performance. The computation of the nonoverlapped metric bit error rate is performed in a similar fashion to the derivation of the performance of the overlapped metric treated in Section Iv. We repeat the derivation, assuming I = S in (5).

In Figs. 4 and 5, two rate 2/4 coded QPSK schemes (two bit in and four bit out at each clock tick mapped to two 2 QPSK symbols, n = 2 and b = 2) are evaluated One with K = 3 (16 states) and the other with K = 4 (64 states). We have simulated the code of K = 3 (Fig. 4) using the VA to confirm the results of the analysis, including phase noise as discussed later.

16'

1Ci2

I O3

I 0"l

'b

10 -6

1 o7

I d8

2 3 4 5 6 1 8

EbNO (dB)

Fig. 3. Noncoherent and coherent decoding of rate 1/2 coded BPSK with 16 states.

Fig. 6 presents the performance of a rate 216 coded QPSK (b = 2, n = 3) with K = 4. We use this example to show the pussibility of using larger bandwidth in order to improve the results. This code has a free distance of 16 which is the bound for rate Z 6 codes having 64 states. Note that the bound €or rate 1/3 convolutional code is only 15. An example using 8PSK is given in Fig. 7. In this case b = 3 , n = 2, R = 8, and K = 3 (64 states).

We see that the optimal code for noncoherent detection is very close to the optimal one for the coherent case. Can we reverse the argument and say that the best coherent code will bc the best also if decoded noncoherently? The answer is no. First, a rotationally transparent code will have more sequences as candidates for error events and so will have a higher error probability when decoded noncoherently. This is the case with the cdmmonly used K = 7 rate 1/2 convolutional code with generators in octal 133, 171 which is a noncoherently catastrophic code. Wilh dilferential encoded input it is rotationally transparent. This code has a free distance of 10. The best NCM for the same parameters and L = 4 has free distance of nine and is not rotationally transparent. Second, two codes with equal coherent performances might have a different noncoherent performance. For example, in Fig. 8, two codes with 6 = 2, n = 2, and R = 4. One is 111, 312 and the other is 133, 231 (the generators in base four). Both have the best free Hamming distance of seven. When decoded noncoherently their performance differ considerably; 133, 231 is much better (twice the degradation at L = 4 and

Looking at the performance of the different codes, see Figs. 3-7, we conclude that for a degradation of less than 0.5 dB relative to the coherent case, L = 3 or 4 is sufficient. It is

pb = 10-5).

RAPHAELI: NONCOHERENT CODED MODULATION 179

I o6 I O7

1 O8

I O9

2 3 4 5 6 1 8

16'

'b 1 o4

1 6 '

10 -6

107

1 2 3 4 5 6 7

Eb/NO (dB) (b)

2 3 4 5 6 1 8

Eb/NO (dB)

Fig. 5 . Noncoherent and coherent decoding of a rate 214 coded QPSK with 64 states.

'b

2 3 4 5 6 1 8

Eb/NO (dB)

Fig. 4. Noncoherent and coherent decoding of a 214 coded QPsK with 16 Fig. 6. Noncoherent and coherent decoding of 2/6 coded QPSK with 64 states. (a) Union bound, (b) Simulation results including phase noise. states.

interesting to note that L = 3 in a code with b = 3 is equivalent decoder is equivalent to the conventional decoder for coded in SNR per observation to L = 9 in a code with b = 1. This DPSK. The only difference is that no differential encoding is related to the bandwidth efficiency of the scheme. As the takes place (this only means a different code). We notice the scheme becomes more bandwidth efficient, larger observations slight improvement for the case of I = 1 which correspond to (measured in input bits) are required. higher overlapping ratio K .

In Fig. 9 we compare the cases of two different observation Differentially encoded MPSK (DMPSK) can also be de- spacings, l = 1 a d l = 2. In the case of l = l and S = 2, the coded by the IO-NMLSE. We show the results for the case of

180 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 4 4 , KO. 2, FEBRUARY 1996

EbNO (dB)

Fig. 7. Noncoherent and coherent decoding of 316 coded 8PSK with 64 states.

2 3 4 5 6 7 8

EbMO (dB)

Fig. 8. Comparison of noncoherent perfomancc of two maximal free dis- . tance codes wilh b = 2 , IL = 2 and h' = 3. Code 1=111, 312 and Code

2=133. 231.

M = 4 as an example. We compare our results to the optimal block Multi-Symbol differential detector which was suggested by Divsalar and Simon [7]. In both cases, the performance for S = 2 is equal to the performance of conventional DMPSK as expected. Also in both cases as S ---f 00 the performance is equal to that of coherent detection. Unlike the block detector which utilizes one observation at a time, the IO-NMLSE

-1 10

-2 10

-3 10

4 10

'b

-6 10

-1 10

10 -8

10 -9

2 3 4 5 6 7 8

2 3 4 5 6 7 8

Eb/NO (dB)

(b)

Fig. 9. Comparison between the decoding with 1 = 1 and 1 = 2. (a) The code of Fig. 3 decoded with 1 = 1. (b) Comparison between the decoding with I = 1 and I = 2.

utilizes all possible observations of length S and by this achieves a superior performance for any S > 2, leading to a faster convergence to the coherent case as S increases. The comparison is presented in Fig. 10.

In order to compute the bit error rate of DQPSK with IO-NMLSE, we use the truncated sequence union bound

RAPHAEL1 NONCOHERENT CODED MODULATION 181

10'

1 o2

'b

10 -4

-5 10

-6 10

I I I I I I \ \ \ I

6 7 8 9 10 I I 12 13

EWNO (dB)

(a)

10 -1

10 -2

10 -4

10 -5

10 -6

6 7 8 9 10 11 12 13

J W N O (dB)

@)

Fig. 10. (a) Performance of DQPSK when decoded by the IO-NMSLE. (b) Performance of DQPSK when detected by the block multiple-symbol differential detector [7].

of (6) with M = 3 since the contribution of larger error events is found to be negligible. The main contributors to the error probability for the cases S = 2 and S = 3 are the error events of the form {- . . , +1, +1, +j, + j , . . .} which correspond to a one bit error. The main contributor for larger S is {. . . , +1, +l, + j , +1, +1, - e $}, the same error event that dominates in the coherent case.

For the phase noise model we assume a very simplified model where the phase noise is a first-order Markov process with Gaussian transition probability distribution with variance ~5~. This corresponds to a frequency spectrum that behaves as

A very low degradation occurs for 6 < 5°/symbol even for a large observation as S = 12, where the loss from the coherent case is less than 0.3 dB. We compared this result to a conventional coherent technique. As checked by simulation of a QPSK Costas loop (at Eb/No = 3 dB, rate 1/2 coded QPSK), it is not possible to achieve Stable lock at B > 0.5" / s ym bolt whereas the degradation of the noncoherent scheme is very small even when B = 5'/symbol. This means that the specifications of the allowed phase instability can be relaxed by more than 20 dB when switching from coherent to noncoherent.

Vf2.

VII. CONCLUSION We have introduced a noncoherent coded modulation system

which approaches the performance of the coherent ones over the AWGN without the need for carrier phase estimation, but with increased decoder complexity. The method promise robustness to carrier phase noise and frequency uncertainty, where there is a possible trade-off between robustness and performance by varying the observation length. It also might be appropriate with fading channels, since phase synchroniza- tion is difficult in those environments. Efficient suboptimal decoding algorithms for the NCM are presented in a separate paper by the author [ 181.

We have shown the importance of overlapping observations. We do not need to use interleaving which adds to the system complexity and causes unwanted delay in the received data.

Our method can also be applied to differentially encoded MPSK. We have demonstrated higher performance and faster convergence to coherent detection performance than previ- ously used methods. Our detection method can be applied to other known modulations like continuous phase modulation with promising results.

APPENDIX A PROOF-APPROACHING COHERENT MLSE

PERFORMANCE BY INCREASING S IN IO-NMLSE We will show that the pairwise error probability between

two sequences x and y of the output of a trellis encoder, when IO-NMLSE is used for the decoding, approaches the coherent error probability for S + 00. The encoder is assumed not to be noncoherently catastrophic. Let x be sent with an arbitrary phase shift 6. Let x and y differ in d consecutive places, i E y = { t , t + 1,. a , t + d - l}, and assume d is small, since it is clear that the average error probability is determined mainly by the short error events. Error events (if they exist) which cause y; to be a phase shift of xiVz p t + d will have diminishing contribution for large S (see Section IV). Choose S large enough such that S >> d. Let ri = x + n and

182 B E TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 2, FEBRUARY 1996

then the painvise error probability of deciding y instead of x is P,(s > 0).

s = k = - w f3

k+S-I

rixi + i=k i E y

k+S-1 c i=k i t7

2

r!Yil

+ lkE1 i=k i E 7

, k + S - 1 , 2

k+S-1 k + S - 1

- 2Re [ ( rjxi) .!xi] }. i=k i Br

i=k iE7

For S >> d

- where E = ,[xi. Let

and

For S 2 d, E = E O + (S - d)cl, where

and

Y

‘ + +

Fig. 11. Waveforms used in Appendix B (7 = 0).

APPENDLX B NUMERICAL METHOD FOR EVALUATING CUMULAI‘IVE DISTRIBUTION FUNCTION FROM THE CHARACTERISTIC FUNCTION

For the search for the best codes, a very fast numerical method had to be developed for computing the error probabil-

ity P& 2 0) = p(y)dy, where p(y) is the p.d.f of y. p(y)

is the inverse Fourier transform of @ ( - j w ) defined in (18). The following method requires a very short computation time for a given accuracy, without actually inverting the function. Let w(y) be a square wave of period P which assumes the values zero and one. The period starts with 1 at y = 0 (see Fig. 11). For large enough P and assuming p ( y ) vanishes for large Iyl, we get

a3

0

/ P ( a ) d t = 7 4 Y M Y ) . (28) 0 -oo

The Fourier transform of ~ ( y ) is n=oo 1 27l

n P nS(w) - 2j . -S(w - n-). (29) n=-m n odd

Let

RAPHAELI: NONCOHERENT CODED MODULATION 183

Its Fourier transform is

co

(32)

Since @(&I) is a Fourier transform of a real function, its imaginary part is odd and its real part is even. We also use the fact that @( 0) = 1, and get

2 ” 1

n=l n odd

p ( t ) d t 2 0.5 - - -Im{@(w - n-)}. (33) 27r P n

0

The result is a nicely converging sum which is computationally efficient, when P is appropriately chosen.

ACKNOWLEDGMENT Thanks are due to R. Raphaeli for her help in editing this

manuscript.

[9] F. Edbauer, “Bit error rate of binary and quaternary DPSK signals with multiple differential feedback detection.” IEEE Trans. Commun. vol. 40. io. 3, pp. 457-460, Mar. 1992.

[IO] H. Leib and S . Pasupathy, “Noncoberent block demodulation of MSK with inherent and enhanced encoding,” IEEE Trans. Cummun. vol. 40, no. 9, pp. 1430-1441, Sept. 1992.

[ I l l M. K. Simon and D. Divsalar, “Maximum likelihood block detection of noncoherent continuous phase modulation,” IEEE Trans. Commun., vol.

[12] R. Knopp and H. Leib, “Module-phase codes with noncoherent detec- 41, no. 1, pp. 90-98, Jan. 1993.

[13] D. Divsalar, M. K. Simon, and M. Shahshahani, “The performance tion,” ICC’93 Geneva, pp. 1054-1058, May 1993.

of trellis-coded MDPSK with multiple symbol detection,” IEEE Trans. Commun. vol. 38, no. 9, pp. 1391-1403, Sept. 1990.

[141 K. Yu and P. Ho, “Trellis coded modulation with multiple symbol differential detection,” in ICC’93 Geneva, pp. 1414-1418, May 1993.

[I51 D. Raphaeli, “Improvement of noncoherent orthogonal coding by time overlapping,” in IEE Proc.-Comm., vol. 141, no. 5, pp. 309-311, Oct. 1994.

[16] ~, “Construction of uniform error property codes for generalized decoding,” Commun. Theory Mini-Con&, Singapore, Nov. 1995, pp. 12-16.

[17] D. Raphaeli and D. Divsalar, “Noucoherent decoding of uncoded and convolutionally coded continuous phase. modulation using multiple- symbol overlapped observations,” submitted to IEEE Trans. Veh. Tech- nul.

1181 D. Raphaeli, “Decoding algorithms for the noncoherent coded mod- ulation,” to appear in IEEE Trans. Commun., vol. 44, no. 3, Mar. 1995.

1191 G. Kaplan and S . Shamai, “On the achievable information rate of

[20] M. G. Muligan and S . G. Wilson, “An improved algorithm for evalu- DPSK,” in Proc. Inst. Elec. Eng., vol. 139, no. 3, pp. 31 1-318, 1992.

ating trellis phase codes,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 846-851, Nov. 1984.

[21] A. J. Viterbi and J. K. Omura, Principles ofDigifal Communicarion and Coding. New York McGraw-Hill, 1979.

[22] N. L. Johnson and S . Kotz, Continuous Univariate Distributions, vol. 2. New York: Houghton Mifflin, 1970.

[23] D. Raphaeli, “Distribution of nonccntral indefinite quadratic forms in REFERENCES comp&x normal variables,” submitted to IEEE Trans. Inform. Theory.

J. G. F’roakis, Digital Communications. New York: McGraw-Hill, 1989. S . Rhodes and W. Hagmann, “Signal design for INMARSAT standard C ship earth stations,” in GLOBECOM’82, Miami FL, USA, vol. 3, pp.

G. Ungerboeck, “Trellis-coded modulation with redundant signal sets 1051-1060, Nov. 1982.

part 2: State of the art,” IEEE Comm. Mag., vol. 25, no. 2, pp. 12-21, Feb. 1987. T. Aulin and C. E. Sundberg, “Partially coherent detection of digital full response continuous phase modulated signals,” ZEEE Trans. Commun. vol. COM-30, no. 5, pp. 1096-1117, May 1982. A. Svensson, T. .4ulin, and C. E. Sundberg, “Symbol error probabil- ity behavior for continuous phase modulation with partially coherent detection,” AEU, vol. 40, no. 1, pp. 3745, 1986. G. C. Clark and J. B. Caiu, Error-Correction Coding for Digital

D. Divsalar and M:. K. Simon, “Multiple-symbol differential detection of Communications. New York Plenum, 1981.

MPSK,” ZEEE Trans. Commun. vol. 38, no. 3, pp. 3W308, Mar. 1990. H. Leib and S . Pasupathy, “Optimal noncoherent block demodulation of differential phase shift keying (DPSK),” AEU, vol. 45, no. 5 , pp. 299-305, Sep. 1991.

Dan Raphaeli (”92) was born in Israel in 1967. He received the B.Sc. degree in electrical and com-

rael, in 1986, and lhe M.S. and Ph.D. degrees in puter enginccring from Ben Gurion University, Is-

electrical engineering from the California Institute of Technology, Pasadena, CA, in 1992 and 1994, respectively.

the Electronic Research Institute of the Israel De- From 1986 to 1991 be was a research member at

fense Ministry. From 1992 to 1994 be was research

CA, where he was involved in the design of the communication subsystems of scientist at the Jet Propulsion Laboratory, Pasadena,

future space-crafts, including the Advanced Transponder studies. Since 1994 he has been an assistant professor at the department of Electrical Engineering- Systems, Tel Aviv University, Tel Aviv, Israel. His research subjects include digital modulation and demodulation, coding and decoding algorithms, spread spectrum, mobile communication, satellite communication, synchronimtion, equalization, and digital signal processing.