on the capacity of the blockwise incoherent mpsk channel

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 5, MAY 1998 603

On the Capacity of the BlockwiseIncoherent MPSK Channel

Michael Peleg,Senior Member, IEEE, and Shlomo Shamai (Shitz),Fellow, IEEE

Abstract—The capacity ofM -ary phase-shift keying (MPSK)over an additive white Gaussian noise (AWGN) channel withcarrier phase unknown but constant overL symbols is inves-tigated. It is shown that capacity-achieving channel inputs areuniformly distributed and independent MPSK symbols. Capacityover a range of signal-to-noise ratio (SNR) andL is presentedfor binary phase-shift keying (BPSK) and quaternary phase-shiftkeying (QPSK). Upper and lower easy-to-compute bounds oncapacity are derived. It is proven that for large L the coherentcapacity is approached. An analytic asymptotic expression for lowL � SNR is derived exhibiting the expected quadratic dependenceon the SNR.

Index Terms—AWGN, blockwise constant phase, channel ca-pacity, MPSK, noncoherent.

I. INTRODUCTION

I N MANY communication scenarios it is difficult or impos-sible to acquire the phase of the received signal because of

phase noise, fading or frequency hopping. In recent years newnoncoherent communication methods operating with unknownphase were developed, the performance of which approaches inmany cases that of coherent communication systems, utilizingthe limitation on variation of the unknown phase over timeinterval in the range of two to twenty symbols. Multiple-symbol differential detection (MSDD) assuming a constantcarrier phase over consecutive symbols was investigatedwith and without coding by Divsalar and Simon [1], [2];Makrakis, Bouras, and Mathiopoulos [3]; Kofman, Zehavi, andShamai [4]; Raphaeli [5]; Adachi [6]; Peleg and Shamai [7];and references therein. Phase trellis methods assuming smallphase variation between adjacent symbols were investigated byNassar and Soleymani [8]. More recently the iterative “turbo”decoding was utilized by Peleg and Shamai [9] to approximatejoint decoding of MSDD and convolutional codes. The successof combining the iterative procedure for decoding and de-modulating the unknown phase channel ([9] and Chayat [10])and the fading channel (Gertsman and Lodge [11]) make itconceivable to assume that iterative turbo decoding of MSDDand turbo codes (Berrou and Glavieux [12]) will essentiallyachieve performance limited mainly by the channel capacityas was demonstrated in [12] and other works in the coherent

Paper approved by T. Aulin, the Editor for Coding and CommunicationTheory of the IEEE Communications Society. Manuscript received August27, 1997; revised January 7, 1998. This work was supported by the SamuelNeaman Institute for Advanced Studies in Science and Technology of theTechnion.

The authors are with the Department of Electrical Engineering,Technion—Israel Institute of Technology, Haifa 32000, Israel (e-mail:[email protected]; [email protected]).

Publisher Item Identifier S 0090-6778(98)03865-3.

case. Peleg, Shamai, and Galan [13] achieved a performanceof 1.3 dB from capacity over the channel analyzed in thispaper using such an approach. This is in contrast to standardclassical convolutional codes, the performance of which isgoverned mainly by cutoff rate [14]. It is of theoretical andpractical interest, then, to evaluate the capacity of channelwith unknown phase.

In this paper we evaluate the capacity of an additive whiteGaussian noise (AWGN) -ary phase-shift keying (MPSK)-modulated channel, the phase of which remains constantover blocks of symbols, and find a capacity achievingdistribution at the channel input. The cutoff rate of thischannel was derived previously by Kofman, Zehavi, andShamai [15] and by Kaplan and Shamai [16]. Cutoff rateof related channels was derived by Viterbi and Jacobs [17]and Jordan [18]. Capacity of the unknown phase channelbut employing an orthogonal multiple frequency-shift keying(MFSK) modulation was derived by Butman, Bar-David,Levitt, Lyon, and Klass [19]. Capacity of a noncoherentchannel was further investigated for spread spectrum systemsby Chayat [10] with orthogonal modulation and by Cheunand Stark [20] in relation to multiple access. Capacity ofnoncoherent differential MPSK (DMPSK) with wasderived by Chen and Fuja [21] over the Rayleigh-fadingchannel. Zhou, Mei, Xu, and Yao [22] derived the capacityof a blockwise fading channel, the input distribution of whichis Gaussian (not necessarily optimal).

II. M ODEL FORMULATION

The transmitted signals are blocks oflength of MPSK-modulated symbols, the complex repre-sentation of which is . The information-carryingphases are with the constellation size and

an integer. Since the received initial phaseis unknown atthe receiver, we choose arbitrarily with no loss of generality1

and define the vector as thechannel input. The block of corresponding symbols matched,filtered, and sampled at the output of an AWGN channelis , where the complex representation ofsymbol is

(1a)

(1b)

where the unknown phase is constant for each block andis a continuous random variable (RV), uniformly distributed

1The initial phase�o absorbs the initial information-carrying phase�o, ifsuch is used.

0090–6778/98$10.00 1998 IEEE

604 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 5, MAY 1998

over . The complex Gaussian noise is designated by,where the variance of its real and imaginary component is.The symbol signal-to-noise ratio (SNR) is given then by

(1c)

We denote by the average mutual information (AMI)of our channel

Two definitions of per-symbol capacity are of interest.In the case of a frequency-hopping system with hop timeof symbols and with no phase acquisition the propernormalization yields

(2a)

In the case of approximating a noncoherent channel by theassumption of constant phase over blocks ofsymbols withoverlapping of one symbol between the blocks as in Raphaeli[23], Divsalar [1], and Knopp and Leib [24], and in Peleg andShamai [9] the appropriate normalization is

(2b)

This form of is of relevance for MSDD which exploitsthe dependencies between the symbols in consecutive blocksof symbols with an overlapping of one symbol. The expres-sion (2b) of holds for those MSDD-based systems whichremove the dependency between consecutive blocks arisingfrom the one symbol overlapping by a large interleaver insertedbetween the MSDD detector and the following decoders asexplained for differential phase-shift keying (DPSK) in [16]and [7]. In systems using iterative decoding such as [9]the interblock dependency is dispersed by interleaver andunexploited by the decoders; therefore (2b) is expected to yielda bound on performance which is equivalent to capacity.

The AMI expression investigated here constitutes the ul-timate coding capacity as the channel defined by-blockinput/output is memoryless or approximated by such andtherefore standard coding theorems [25] apply.

When presenting numerical results in this paper we adhereto the normalization (2b) as this facilitates a straightforwardcomparison of the results on capacity of standard coherentMPSK over an AWGN channel as evaluated in Blahut [26]and denoted here as .

A hypothetical system communicating at coding rateinformation bits to transmitted symbol equal to capacity

will require information bit energy over noisespectral density ratio of

III. CAPACITY

First we seek the distribution of the channel inputthatmaximizes , thus achieving capacity.

Motivated by (1b), we define as the AMI of a virtualchannel which is identical to our channel except thatisviewed as an additional channel input, thus

(3)

We denote bythe distribution of which is determined by that

of . Clearly

(4)

The virtual channel is defined by its inputs, by itsoutputs , and by (1b). Using the standard mutual informationdecomposition [25] we have

(5)

and hence

(6)

We seek the distribution of which achieves capacity andmaximizes . Since in (6) is independentof distribution of , we can find optimal distribution bymaximizing

The virtual channel with input and output is memo-ryless as defined by Mc-Eliece in [27, eq. (1.16)] due to our(1b). Mc-Eliece [27, proof of Th. 1.9] derived the differencebetween the blockwise and the symbolwise AMI when block

of possibly dependent symbols is transmitted over a mem-oryless channel, the output of which is Substituting for

and for , we have

(7)

The distribution of which maximizes alsoachieves capacity. Since is uniformly distributed, ,

, and do not depend on ; thus, we have todetermine which maximizes only

which is the entropy of the vector .We introduce now a random vector ,

where are uniformly, independently, and identically dis-tributed (u.i.i.d.) over the discrete values

, and define , where designateshere a addition. Let be defined as in (1a)with replacing .

Since a constant phase shift of some symbols representedby a known does not change , we have for anydistribution of

PELEG AND SHAMAI (SHITZ): BLOCKWISE INCOHERENT MPSK CHANNEL 605

Standard information theoretic inequalities[25] yield, therefore,

(8)

The probability function describes clearly a u.i.i.d.sequence , and since choosing u.i.i.d. maximizes ,yielding an equality in (8), this distribution achieves capacity.The same result holds also for distributed discretely anduniformly over the same number of equispaced phasesas

or an integer multiple thereof. Therefore, the capacity ofour channel is achieved by u.i.i.d. modulation phases. Withthe capacity achieving distribution of known, the capacity

in (2b) is evaluated using the formula

(9)

and the Monte Carlo estimation of , the statistical expec-tation with respect to and , which seems to be preferablefor relatively large over multiple integration. The probability

is given in [1, eq. (9)]

(10)

is defined as a function of in the first paragraph of SectionII, and is the modified Bessel function of the first kind oforder zero. Since is u.i.i.d.

(11)

with , where is the number of possible valuesof . The exponential term in (10) is independent ofandtherefore cancels out in (11) and (9).

Since the evaluation of is rather complex for largeand , bounds are useful. For continuously or discretelydistributed using (3), we have

Substituting this in (6) gives

(12)

The first term is AMI over the coherent chan-nel while indicates the degradationdue to the unknown . The relation in (12) demonstrates thatthe here proven fact that u.i.i.d. achieves capacity is notobvious beforehand. That is, since , being thecoherent AMI, is maximized by maxentropic u.i.i.d., theother AMI expression is maximized by zero entropy

, while the expression is independent of . Asshown before, the optimal selection of maximizes not only

but also the sum andtherefore also (12).

An upper bound on is derived by evaluating (12)for discretely and uniformly distributed over phases(possessing the same distribution as). Discretization of is

equivalent to providing the receiver with the side information, where is the nearest discrete value

of below the real ; thus, the corresponding AMI upperbounds . In this case the second term of (12)is coherent AMI of MPSK [26] with SNR increased by a factorof , as for given the calculation of the conditioned AMIis equivalent to having independent measurements of.The third term is , the coherent AMI, as

is independent of becauseare u.i.i.d. and, therefore, only the coordinate carries

information on .With overlapping, we then have

(13)

A lower bound is derived by evaluating (12), incorporatingthe inequality

and evaluating as the capacity of the coherent contin-uous input phase modulated channel (Wyner [28]). (We usedMonte Carlo evaluation instead of the integral formulas in [28,eq. (12)]). The second term of our (12) is evaluated likewisebut with SNR increased by a factor of.

Now we examine the capacity for very large. The capacityis clearly upper bounded by the coherent capacity as seen from(13). On the other hand, the above lower bound can be furtherdecreased by omitting the third term in (12)

(14)

The first term in (14) is the coherent AMI and it increaseslinearly with ; the second term is upper bounded by thecapacity of the Gaussian channel at SNR enlarged by a factorof , i.e., .2 Therefore, for large thelower bound on as given by (14) when normalized bywill tend to the coherent capacity and, hence, the capacity ofthe unknown phase channel will also approach the coherentcapacity for large .

Next we present numerical results. The capacity for binaryphase-shift keying (BPSK) is presented in Fig. 1 as a functionof SNR for different values of . The capacity raises with asexpected. The needed to communicate at coding rateequal to capacity (see the end of Section II) is presented in Fig.2. From both figures it is evident that there is a large gap (3.6dB at code rate of 0.5) between capacity of coherent BPSK andcapacity of a noncoherent channel with which describesapproximatelyinterleaved DPSK [16] noncoherently detectedwith the conventional observation interval of . This gapcan be narrowed to about 1.1 dB for code rates above 0.5 byusing larger than ten. For up to ten there is a substantialdifference between the coherent and noncoherent capacity.The needed to communicate at code ratedecreasesmonotonically with decreasing for coherent MPSK, whilefor noncoherent MPSK it achieves a minimum atand increases substantially for lower. This indicates that

2In fact, by [26, eq. (13)] the behavior forL�SNR� 1 is 1=2 log2( 4�eL�

SNR) bits/block.


Fig. 1. Capacity of the noncoherently and coherently detected AWGNchannel with constant phase and BPSK modulation. Solid lines: noncoherentdetection over observation interval ofL = 2; 3; 4; 6; 8; 10 symbols with onesymbol overlapping. Dashed line: coherent detection—known phase.

Fig. 2. TheEb=No required to communicate at rate equal to capacityover the noncoherently and coherently detected BPSK-modulated AWGNchannel. Solid lines: noncoherent detection over observation interval ofL = 2; 3; 4; 6; 8; 10 symbols with one symbol overlapping. Dashed line:coherent detection.

noncoherent methods with a moderate observation intervalare not well suited for communicating at low code rates, whichaccounts also for spread spectrum systems. The lower andupper bounds for larger (8, 16, 32) are presented in Fig. 3.The bounds show that capacity is not far from that of coherentPSK for over a wide range of SNR as the upper boundis close to the capacity of coherent PSK and the lower boundis about 0.1 bits/symbol below it.

Similar results are presented for quarternary phase-shiftkeying (QPSK) in Figs. 4–6. The improvement for the same

is somewhat smaller than for BPSK; however, the capacityof QPSK is always larger than that of BPSK as expected.

The upper bound on capacity in (13) was derived usingdiscretely distributed unknown phase. Since the u.i.i.d.

Fig. 3. Bounds on capacity of the unknown and constant-phase AWGNchannel with BPSK modulation. Observation interval—x:L = 2, �: L = 8,�: L = 16, and+: L = 32. Dotted line: noncoherent capacity,L = 8 only.Dash-dot lines: lower bound. Solid lines: upper bound which equals alsothe capacity of the discretely unknown phase channel. Dashed line: coherentcapacity (known phase).

Fig. 4. Capacity of the noncoherently and coherently detected AWGNchannel with constant phase and QPSK modulation. Solid lines: noncoherentdetection over observation interval ofL = 2; 3; 4; 5; 6; 7; 9 symbols with onesymbol overlapping. Dashed line: coherent detection—known phase.

distribution of maximizes capacity also for the discretedistribution of , our upper bound is the actual capacity ofan AWGN channel with discretely unknown phase, takingon values of , uniformlydistributed. Such channels frequently rise when the phaseis extracted by means of Costas loops or other pilotlessphase estimation methods. The capacity of this channel iscompared in Fig. 7 for QPSK with the capacity of ourincoherent channel and with that of a coherent channel.We found that for discretely distributed , as above, the

required to achieve capacity is not better by morethan 1.1 dB than that of our incoherent channel with thesame over the following ranges of capacity: 0.2–0.95bits/symbol with BPSK, and 0.2–1.5 bits/symbol with QPSK.


Fig. 5. TheEb=No required to communicate at rate equal to capacityover the noncoherently and coherently detected QPSK-modulated AWGNchannel. Solid lines: noncoherent detection over observation interval ofL = 2; 3; 4; 5; 6; 7; 9 symbols with one symbol overlapping. Dashed line:coherent detection.

Fig. 6. Bounds on capacity of the unknown and constant-phase AWGNchannel with QPSK modulation. Observation interval—�: L = 5, �: L = 16,and+: L = 32. Dotted line: noncoherent capacity,L = 5 only. Dash-dotline: lower bound. Solid lines: upper bound. Dashed line: coherent capacity(known phase).

The capacity gap between the discretely unknown phasechannel with the standard observation interval of andthe coherent channel exceeds 2 dB for coding rates lower than0.6 bits/channel bit for QPSK and BPSK, leaving considerableroom for improvement by various known methods such aspilot symbols, MSDD [1], [6], [9] with metrics modifiedaccording to the statistics of the discretely unknown phaseand appropriate coding such as in Alles and Pasupathy [29],and its references.

In some works the use of noncoherent demodulation isconsidered in spread spectrum and/or repetitive coded systemsin a chipwise fashion [30], [31], demonstrating a significantdegradation in terms of required due to noncoherency.We will show the limitation of this approach in the case

Fig. 7. Capacity of QPSK-modulated AWGN channel detected over obser-vation interval ofL symbols with differenta priori phase information. Solidlines: unknown constant and continuously distributed phase. Dash-dot lines:unknown, constant, and discretely distributed phase over four values. Dashedline: known phase (coherent channel). Observation interval—�: L = 2,�: L = 3, and+: L = 5.

of small by evaluating capacity behavior in ournoncoherent channel for low values of .

Define

(15)

Substituting the two-term Taylor series

to (9)–(11) yields

(16)

The sums over are carried over all the possible values ofthe vector as in (11).

Since for low SNR we have and the followingderivation will show that (16) includes a term , all theterms of powers of higher than four can be omitted. Forthe same reason, the two-term expansion above is sufficient.Following the derivation in Appendix A, we obtain by (A.6)

(17)


and the corresponding per-symbol capacity is

(18)

Thus, at extremely low the capacity is not dependenton , i.e., it is quite insensitive to the modulation scheme aslong as it is symmetric. Furthermore, if we definewe have from (17)

which resembles the low SNR capacity of the same channelwith MFSK modulation as derived in [19, eq. (24)], except that

here replaces the number of frequencies in the result of [19],i.e., our is equal to that of [19] for an equal bandwidth.

The quadratic dependence of capacity on SNR renders usingspread spectrum at low inefficient in respect to therequired .

IV. CONCLUSION

A noncoherent MPSK AWGN channel with the carrier phaseunknown but constant over blocks of symbols is investi-gated. We prove that the capacity achieving input modulationphases are u.i.i.d. Possible efficient capacity-approaching cod-ing schemes should therefore mimic u.i.i.d. inputs to thechannel (Shamai and Verdu [32]), which in practice canperhaps be achieved by means of turbo codes and interleaving.Differential modulation does not destroy the u.i.i.d. propertyand therefore its insertion before the channel input for obviouspractical reasons should not impair system performance. Ca-pacity calculations are presented for BPSK and QPSK modula-tion. Although the capacity approaches that of a coherent chan-nel for large values of , there is a significant capacity loss for

, motivating the use of MSDD jointly decoded with theerror correcting code or equivalent techniques. For lowthe capacity is quadratic in SNR in contrast to the linearbehavior for coherent communication; therefore, for

this channel is not suitable for spread spectrum applications.

APPENDIX

Here we derive for low SNR from (16). We use the shortnotation . By discarding powers of higher thanfour, (16) yields

(A.1)

(A.2)

(A.3)

To evaluate we use (1b), (15), and

(A.4)

where is the complex noise rotated by .

The above sum is a complex Gaussian RV with meanandvariance . Thus, is a noncentral chi-square RV withtwo degrees of freedom. Therefore, and,discarding terms smaller than , we have

To evaluate we start with

but

where stands for the Kronecker delta function.Hence

(A.5)

which is a noncentral chi-square RV with degrees offreedom. Using this and (1b), we have

and, by using the variance of chi-square RV and discardingterms smaller than

To derive we can omit the signal terms due tothe associated denominator of in (A.3) and evaluate thenoise terms only. Since those do not depend onwe find

Adding all the terms into (A.1), (A.2), and (A.3) yields

(A.6)

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[31] Y. E. Dallal and S. Shamai (Shitz), “Generalized cut-off rate of DPSKnoisy-phase channels,”Eur. Trans. Telecommun., vol. 6, no. 5, pp.601–608, Sept./Oct. 1995.

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Michael Peleg (M’87–SM’98) received the B.Sc.and M.Sc. degrees from the Technion—Israel Insti-tute of Technology, Haifa, Israel, in 1978 and 1986,respectively.

He is currently on sabbatical with the Departmentof Electrical Engineering, Technion, Haifa, Israel.From 1978 to 1996 he was with the CommunicationResearch Facilities, Israel Ministry of Defense. Hisresearch interests include wireless digital communi-cations and adaptive antenna arrays.

Shlomo Shamai (Shitz)(S’80–M’82–SM’88–F’94)received the B.Sc., M.Sc., and Ph.D. degrees inelectrical engineering from the Technion—IsraelInstitute of Technology, Haifa, Israel, in 1975, 1981,and 1986, respectively.

During 1975–1985 he was with the Signal CorpsResearch Laboratories (Israel Defense Forces) asa Senior Research Engineer. Since 1986 he hasbeen with the Department of Electrical Engineering,Technion, Haifa, Israel, where he is currently aProfessor. His research interests include topics in

information theory and digital and analog communications, especially theoret-ical limits in communication with practical constraints, digital communicationin optical channels, information-theoretic models for multiuser cellular radiosystems and magnetic recording, channel coding, combined modulation andcoding, turbo coding, and digital spectrally efficient modulation methodsemploying coherent and noncoherent detection.

Dr. Shamai is a member of the Union Radio Scientifique Internationale(URSI). He serves as Associate Editor for the Shannon Theory of the IEEETRANSACTIONS ONINFORMATION THEORY, and since 1995 he has served on theBoard of Governors of the IEEE Information Theory Society.

on the capacity of the blockwise incoherent mpsk channel

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