2988 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 11, NOVEMBER 2006

Space Time Codes for CPFSKWith Arbitrary Number of Receive Antennas

Sung Kwon Hong, Member, IEEE, and Jong-Moon Chung, Senior Member, IEEE

Abstract— In this letter, space time (ST) trellis codes are devel-oped for systems with an arbitrary number of receiver antennasin quasi-static fading channels applying M -ary continuous phasefrequency shift keying (CPFSK) modulation with M = 4 and 8 forh = 1/M . The maximizing minimum squared Euclidean distancecriterion (MMSEDC) is applied in the code search algorithmwhen obtaining the ST codes. The results show that a significantperformance gain can be obtained by using the ST codes fromthe MMSEDC code search algorithm compared to the ST codesobtained in [1] for CPFSK modulation systems.

Index Terms— Code search, CPFSK, CPM, optimal code, spacetime code.

I. INTRODUCTION

W IRELESS communication systems using multiple an-tennas are becoming more popular due to several

beneficial features they can provide. Especially, the supe-rior error control coding performance of ST codes is oneof the major attractive features. Therefore, combining STcodes with various digital modulation schemes have beenattempted. CPFSK systems applying ST trellis coding overmultiple antennas can result in an advantageous combinationbased on the characteristics of CPFSK being able to be usedeffectively in systems with low cost nonlinear amplifiers [1].The design process of ST codes for CPFSK systems with anarbitrary number of receiver antennas is more complex thandesigning ST codes for linear-modulation types due to boththe nonlinear characteristics in the frequency and phase ofthe CPFSK signals and the more complex coding metricsof the ST codes. In order to simplify the design process,the authors of [1] use linear decomposition of continuousphase modulation (CPM) signals and apply the rank criteriaavailable in linear modulation to identify some general codeconstruction with CPM. Although, the ST codes obtained in[1] for CPM cannot be considered as optimal codes based onthe perspective of rank theory. Various code search criterionshave been applied in search of ST codes that are optimal underquasi-static fading channel conditions, where in this paper theMMSEDC is applied. For single receiver antenna systems, thedeterminant of the signal matrix becomes the dominant factorin deciding the system performance. Alternatively, for systemswith multiple receiver antennas, the Euclidean distance of thecode becomes the dominant factor [2]-[7]. In this letter, ST

Manuscript received April 20, 2004; revised June 17, 2005; accepted March24, 2006. The associate editor coordinating the review of this letter andapproving it for publication was K. B. Lee. This research was supportedby Yonsei University.

S. K. Hong is with Samsung Electronics, Inc., Suwon, Korea (email:sungkwon.hong@samsung.com).

J.-M. Chung is with the School of Electrical and Electronic Engineering,Yonsei University, Seoul, Korea (email: jmc@yonsei.ac.kr).

Digital Object Identifier 10.1109/TWC.2006.04255

codes for CPFSK receivers (i.e., M = 4, 8 and h = 1/M ) withan arbitrary number of receiver antennas are presented. Basedon the framework of [1], which uses the decomposition ofthe CPFSK system into the continuous phase encoder (CPE)and the memoryless modulator (MM) parts, the MMSEDC isapplied to obtain optimal ST codes.

II. MATHEMATICAL MODEL AND SYSTEM STRUCTURE

This letter applies the mathematical description for spacetime coded CPFSK signals based on the framework of [1].The mobile communication scheme considered is equippedwith Lt transmitter antennas and Lr receiver antennas, wherethe input to the space time encoder consists of Kb informationbits of {�I(k)} and the outputs {Ji(k)}, i = 1, 2, · · · , Lt,are generated. The received signal at each receiver antennais a superposition of the Lt transmitted signals corruptedby Rayleigh fading and zero mean complex additive whiteGaussian noise (AWGN). Applying the decomposition of [1],we can construct a CPFSK transmitter using the CPE and MMas depicted in Fig. 1. In this letter, the channel encoder (CE)is constructed by the structure of the ZM valued convolutionalcode. The ZM valued convolutional code is introduced by [7]as a space and time encoder for linear phase shift keying (PSK)modulation. The CE of the linear space time convolutionalcode over the ZM code can be described as follows. A one-branch feedforward ZM valued shift register with memoryorder ν is used to model the ST encoder with ZM transmitterantennas. At time index k, the information vector {�I(k)}forms the ZM valued input I(K) and I(K) passes throughthe feedforward registers and gets multiplied to the generatorpolynomial vector gi = (g0

i , g1i , g2

i , · · · , g0ν), where for gj

i ∈{0, 1, · · · , M − 1} for i = 1, 2, · · · , Lt, and j = 1, 2, · · · , ν.The symbol assigned to the ith antenna can be represented as

Jki =

( ν∑j=0

I(k − j)gji

)modM, i = 1, 2, · · · , Lt. (1)

The CPFSK signal received by antenna m can be written as,

Ym(t, J) =Lt∑i=1

Ci,mX(t, �Ji) + Nm(t), m = 1, 2, · · · , Lr (2)

and the transmitted signal for each transmit antenna can bewritten as

X(t, �Ji) =

√Es

Texp(jφ(t, �Ji)), i = 1, 2, · · · , Lt (3)

where Ci,m is the channel gain between transmit antenna iand receiver antenna m, and is modeled as an independentcomplex Gaussian random variable with a variance 0.5 perreal complex dimension.

1536-1276/06$20.00 c© 2006 IEEE

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 11, NOVEMBER 2006 2989

SpaceTime

Encoder

D

+-

D

+-

CE part CPE part MM part

Fig. 1. ST transmitter for CPFSK signals (h=1/M).

Additionally, h is the modulation index, Es is the symbolenergy, and Nm(t) is the AWGN at the mth receiver antennawith a single-sided power spectral density N0. Nc is the framelength. The codeword matrix J is defined as

�Ji =

⎛⎜⎜⎜⎝

J1(1) · · · JLt(1)J1(2) · · · JLt(2)

......

...J1(Nc) · · · JLt(Nc)

⎞⎟⎟⎟⎠ = [ �J1 · · · �JLt ] (4)

where Ji(k) has a M -ary integer value (i.e., Ji(k) ∈{0, 1, · · · , M − 1}). The tilted phase and the physical tiltedphase over the kth symbol interval can be obtained from

�φ(τ + kT, �Ji) = R2π

[2πhRp

(k−Lq∑l=0

Ji(l))4π

h

Lq−1∑l=0

Ji(k − i)q(τ + iT ) + W (τ)

], 0 ≤ τ < T (5)

where

W (τ) =πh(M − 1)τ

t− 2πh(M − 1) ·

Lq−1∑l=0

+(L − 1)(M − 1)πh, 0 ≤ τ < T (6)

in which T is the symbol time, Rp is the modulo p operator,and h = n/p (n and p are relatively prime). The term q(t) isthe phase smoothing response function with memory lengthLq. When Lq = 1, h = 1/M , and q(t) is given in the fullresponse case, by introducing the scrambler as in [8], (5) canbe simplified as

�φ(τ + kT, �Ji) = R2π

[2πhJi(k − 1)

(2πh(Ji(k)

−Ji(k − 1))modM)τ

T

], 0 ≤ τ < T, Lq = 1, h = 1/M. (7)

The physical tilted phase can be completely specified by[Vi(k) Ui(k)] where

Vi(k) = Ji(k − 1), Ui(k) = (Ji(k) − Ji(k − 1))modM. (8)

When M = pn and n is an integer, V (k) only depends onthe least significant digit (LSD) and (8) can be redefined as

Vi(k) = Ji(k − 1)(n) (9)

where Ji(k)(n) denotes the nth LSD of Ji(k).

III. CODE DESIGN AND SEARCH RESULTS

When J = jα, �y(t, jα) = {y1(jα), y2(jα), · · · , yLr(jα)}T

denotes the signals received from all antennas and x(t,�jαi)denotes the signals transmitted from the ith transmit antenna.If J = jβ is an erroneously decoded codeword matrix andCi,m = ci,m, from [1], the pairwise error probability (PWEP)can be expressed as

P (jα → jβ) = Pr

[−

Lr∑m=1

∫ NcT

0

∣∣∣Nm(t) +Lt∑i=1

ci,m ·

[x(t,�jαi − x(t,�jβi)

]∣∣∣2dt +Lr∑

m=1

∫ NcT

0

|Nm(t)|2dt

]. (10)

By setting Ωm = (C1,m, · · · , CLt,m), the upper bound of(10) can be derived as

P (jα → jβ) ≤ exp

�− Es

4N0

Lr�m=1

� NcT

0

Lt�i=1

Lt�i′=1

ci,mc̄i′,mΔi(t)Δi′(t)dt

�

= exp

�− Es

4N0

Lr�m=1

Lt�i=1

Lt�i′=1

ci,mc̄i′,m

� NcT

0

Δi(t)Δi′(t)dt

�

= exp

�− Es

4N0

Lr�m=1

ΩmCsΩ∗m

�

= exp

�− Es

4N0

Lr�m=1

Lt�i=1

λi|βi,m|2�

(11)

where βi,m are independent complex Gaussian random vari-ables with variance 0.5 per dimension [6], and Cs is the signalmatrix whose component can be represented as

Ci,i′s =

∫ NcT

0

Δi(t)Δ̄i′ (t)dt, i, i′ = 1, 2, · · · , Lt (12)

in which we define

Δi(t) = x(t,�jαi) − x(t,�jβi) (13)

where x(t,�j) is the transmitted CPFSK signal that is definedby the parameter pair [Vi(k) Ui(k)]. From (7) and (12), wecan obtain a closed expression of the components of

Ci,i′s =

∫ NcT

0

Δi(t)Δ̄i′ (t)dt =Nc−1∑l=0

∫ (l+1)T

lT

Δi(t)Δ̄i′ (t)dt =Nc−1∑l=0

∫ (l+1)T

lT

exp

(j(2pihVik

+2pihUikt

T

))dt =

Nc−1∑l=0

f(k) (14)

2990 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 11, NOVEMBER 2006

TABLE I

THE CODE FROM THE MAXIMIZING MINIMUM SQUARED EUCLIDEAN

CRITERION (h=0.25, M=4)

ν states g1, g2 determinant trace1 8 1 2, 2 1 15.4 81 16 1 1, 1 2 17.6 122 32 1 0 2, 1 1 0 17.6 122 64 1 0 1, 1 1 3 32 16

TABLE II

THE CODE FROM THE MAXIMIZING MINIMUM SQUARED EUCLIDEAN

CRITERION(h=0.125, M=8)

ν states g1, g2 determinant trace1 16 2 3, 3 4 16.01 81 32 1 2, 4 5 16.12 8

where f(k) is defined as,

f(k) =

⎧⎪⎪⎨⎪⎪⎩

exp(2pihj(Vi(k) − Vi′ (k))), Ui(k) = Ui′(k)1

2πh(Ui(k)−Ui′ (k)) (exp(2πhj(Vi(k) − Vi′(k)+Ui(k) − Ui′(k))) − exp(2πhj(Vi(k)−Vi′ (k)))), Ui(k) �= Ui′(k)

(15)

By calculating (14) we can obtain the Cs for each code. From(11), we can get the same PWEP as obtained in [6] from

P (jα → jβ) ≤(

r∏i=1

λi

)−Lr(

Es

4N0

)−rLr

(16)

where r is the rank of Cs and λi represents the nonzeroeigenvalues of the matrix Cs which consists of the complexproducts of the channel symbol differences over the framelength Nc. To achieve the minimum error probability, weshould maximize the minimum rank and the minimum productof all nonzero eigenvalues of Cs. If a full rank is achieved,it is equivalent to maximizing the minimum determinant ofCs. In [3], [4], [5], and [7], the rank and determinant criteriaapply to systems with a single receiver antenna and smallnumber of transmit antennas, and if rLr ≥ 4, the pairwiseerror probability is upperbounded by

P (jα → jβ) ≤ 14exp

(−Lr

Es

N0

r∑i=1

λi

). (17)

The bound of (17) indicates that if we maximize the min-imum squared Euclidean distance between any two differentcodewords then the minimum determinant no longer domi-nates the code performance. The squared Euclidean distancemeans the trace of Cs.

In Table I and Table II, the space time code with twotransmit antennas are provided for respectively the two casesof h = 0.25 M = 4-ary CPFSK signals and h = 0.125M = 8-ary CPFSK signals, where the maximizing minimumsquared Euclidean distance criterion is used for both cases.

The number of states in the coded CPM system depends onthe memory element of the CPE and the constraint length ofthe CE. When M = 2Kb and Lt = 2, the maximum possiblenumber of states in the ST coded CPFSK systems is 2Kb(2+ν),and from [8] the actual number of states can be reduced to2Kb(2+ν)/p = 2Kb(1+ν) when h = n/p = 1/M .

Fig. 2. FER performance comparison of ST code for quarternary CPFSK(h=0.25) with 16 and 32 states.

In case of ν =1, we can not acquire a ST code achievingfull transmit diversity when the number of states is less than22Kb (for Kb = 2 or 3) by applying the structure of [7]. But,by introducing some modifications to the encoder, we can findsome ST codes that achieve full diversity.

When ν =1, the output of the CE is determined by I(k)and I(k − 1), where I(k − 1) is the previous input and itsdigit is Kb. If the number of states is Sν , we introduce themodification as below

I(k − 1) → I(k − 1)/2Kb−Sν . (18)

In Table 1 and II, the codes for the 8 state of 4-ary CPFSKand the 16 and 32 states of 8-ary CPFSK have been obtainedby applying the modification methods of (18).

IV. SIMULATION RESULTS

Fig. 2 and Fig. 3 show the simulation results of the STcode. In the computer simulation based experiments, eachframe consists of 130 transmissions per transmit antenna and8 samples for each symbol are used. Fig. 2 compares theperformance of the code obtained from the delay diversity(DD) to the code resulting from the search algorithm forthe quaternary case with h=0.25. Based on Fig. 2, for the 2antenna receiver case at the frame error rate (FER) of 10−2, thenew codes obtained from the search algorithm show a codinggain exceeding 1.8 dB for the 16 state case and an additionalcoding gain of approximately 1.2 dB is obtained for the 32state case compared to the DD codes. When the receiver has4 antennas, at the FER of 10−2, the codes obtained from thesearch algorithm show a coding gain of 2.9 dB for the 16state case, and an additional coding gain of 0.85 dB can beobtained for the 32 state case. In Fig. 3, for the 8-CPFSKcase with h=0.125, the DD code has 64 states, where thecodes obtained from the code search can have 16 or 32 states.For the 2 antenna receiver case at the FER of 10−2, the new

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 11, NOVEMBER 2006 2991

Fig. 3. FER performance comparison of ST code for 8-ary CPFSK (h=0.125)with 16 and 32 states.

codes obtained from the search algorithm shows a coding gainexceeding 1.89 dB for the 16 state case and an additionalcoding gain of approximately 1.07 dB is obtained for the32 state case compared to the DD codes. When the receiverhas 4 antennas, at the FER of 10−2, the codes obtained fromthe search algorithm show a significant coding gain of 3.51dB for the 16 state case, and an additional coding gain ofapproximately 1 dB can be obtained for the 32 state case.

V. CONCLUSION

In this letter, novel ST trellis codes are developed for 4-ary and 8-ary CPFSK modulation systems with an arbitrarynumber of receiver antennas in quasi-static fading channels.The results of this paper significantly extend the studies of[1] by applying the MMSEDC code search algorithm for ST-CPFSK systems. Based on the simulation results of the 4-

ary CPFSK systems, in comparison to the 16 state DD codesfor the 2 and 4 antenna receiver cases, at the FER of 10−2,the ST codes obtained from the search algorithm providea 1.8−2.9 dB coding gain for the 16 state case, where anadditional coding gain above 0.85 dB can be obtained for the32 state case. For the 8-ary CPFSK systems, based on the2 and 4 antenna receiver case, at the FER of 10−2, the STcodes obtained from the search algorithm provide a 1.89−3.51dB coding gain for the 16 state case where an additionalcoding gain above 1.0 dB can be obtained for the 32 statecase, compared to the 64 state DD codes. In conclusion, byapplying the MMSEDC ST code search algorithm to CPFSKmodulation, a significant performance gain can be obtained incomparison to the ST codes obtained from the delay diversityscheme of [1].

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