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G FT[x(t)] and E[x] cor[x(t)j(t++)]

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO 3, MARCH 1996

N x N coefficient matnx Founer Transform of x(t) and average of x, respecuvely autocodahon funcoon of x(t)

Commonalities of Several Modulation Techniques Lei Wei

Abstract-In this letter, we study commonalities of new band- width efficient and power efficient quadrature-quadrature phase shift keying (Q’PSK), constant envelope Q’PSK (CEQ’PSK), ro- tative QPSK (RQPSK), frequency and phase shift keying (FPSK), expurgated FPSK (E-FPSK), and the well-known orthogonal frequency division multiplexing (OFDM) (or mdticarrier modu- lation) and L-orthogonal (LO) signaling. In this letter, “identical” systems are defined as systems that have the same bit error rate (BER) versus the bit energy to noise ratio function and the same power spectral density. The results show that: a) Q’PSK, FPSK, and OFDM are “identical,” and b) CEQ’PSK, E4 -FPSK and LO are “identical.” Also, a mistake in [4] is corrected.

I. INTRODUCTION ECENTLY, many new bandwidth efficient and power efficient modulation techniques, such as

quadrature-quadrature phase shift keying (Q2PSK) [ 11, constant envelope Q’PSK (CEQ’PSK) [ 11, rotative QPSK (RQPSK) [2], frequency and phase shift keying (FPSK) [3], and expurgated FPSK (E-FPSK) [3], have been proposed and analyzed. It is noted that these schemes have some similarities with well-known modulation schemes such as, orthogonal frequency division multiplexing (OFDM) (or multicanier modulation) [5], [6], and L-orthogonal (LO) signaling ([7, pp. 235-2401). In this letter, these similarities are explored, which will help us to better understand these new schemes. In Table I several notations used in this letter are defined. It is well known that N sequences of (MPSK or QAM) symbols of duration T, can be transmitted simultaneously and independently by employing waveforms p ( t ) , that satisfy the no cross-symbol interference (CSI) and no intersymbol interference (ISI) conditions

where 6 is the energy per symbol. In th s letter, we assume that the waveforms, { p k , %(t), for k = 1, 2, i = 1, 2 , . . . , N } , have the same energy.

Two communication systems can be constructed by using two waveform vectors, pl(t) and p ~ ( t ) .

Lemmn I : If GGH = IN and pl(t) satisfy (l), then the bit error rates (BER) of the above two systems are identical for optimal detection. If D also satisfies that E[DHD] = AIN, where A is a constant, then the average power spectral densities (PSD) of both systems are identical.

Paper approved by N C Beaulieu, the Editor for Wireless Communication Theory of the IEEE Communications Society Manuscnpt received December 21, 1993, revised May 2, 1994, July 4, 1994, and December 20, 1994 This work was supported in part by Telecom Australia

The author is with the Telecommunications Engineenng Research School of Information Sciences and Engineenng, The Australian Nahonal University, Canberra ACT 0200, Australia

Publisher Item Identifier S 0090-6778(96)01774-6

TABLE I DEFINITION OF NOTATION

Notahon Definruon

M X N matnx of all possible transrmtted bit (or symbol) patterns of length N uh row of D and i=1,2, ~ M

The transmitted signal of system k is 00

%(t> = D’,pl,(t - hTs). (3) &=-a

The sufficient statistics of the matched filter receiver sampled at tT, + Ts/2 for system k are

co

rk = 1, s k ( t ) p ~ ( t - G) d t + n

=SDg+n (4)

where n is an independent Gaussian noise vector with co- variance matrix E[nnT] = 6 1 ~ 2 . Obviously, rl = r’, thus, the BER’s of both systems are identical for optimal detection or decoding. The average PSD of system k is FT{cor [(Dpk(tl))H, Dpk(tl +.)I}. For E[DHD] = AIN, we have

FT{cor [(DP2(tl))H, DP,(tl + 41) = FT{cor [(DP,(tl))H, DPl(t1 + .)I>. (5)

Thus, the average PSD’s of the two systems are identical. Q.E.D.

We define two systems with the same BER and average PSD as “identical” systems. We explore some commonalities of Q’PSK [l], FPSK [3], and OFDM [6] in Section 11, of CEQ2PSK [l], RQPSK [2], E4-FPSK [3], and LO [7] in Section III. In Section IV, a comment on [4] is given.

II. COMMONALITIES OF Q’PSK, FPSK, AND OFDM Let

p,(t) = exp [jw,t + j7r(2n - 1 - N) t /T , ]u~ , ( t ) for n = 1, 2, .” , N (6)

where wps(t) = 1 for [-T,/2, T s / 2 ) and zero, otherwise.

0090-6778/96$05.00 0 1996 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 3, MARCH 1996 299

If d%,n = exp(j@.,,,) and @ 2 , n is an independent p-ary symbol from the set {27rk/p: k = 0, 1, . . . , p - l}, then the ith signal of the total A 4 = p N signals on -Ts/2 5 t < Ts/2 is

N

sZ(t) = Re {exp (jw,t + j 4 2 n - 1 - N ) n=l

’ t/Ts + j @ z , n ) } U T , ( t ) . (7)

Equation (7) is exactly the same as (1) of [3] with h = 1, and also the same as (1) of [6] with pPSK. Thus OFDM [6] and FPSK [3] are not just identical according to the previous definition, but essentially, exactly the same.

Letting N = 2, we can rewrite (7) in the form

the first null bandwidth ([0, 1.5/Ts)) with no ISI. In [l] and [8], the definition of the bandwidth efficiency of PSK and Q2PSK was based on the bandwidth containing either 90% or 99% power of the spectrum. If not bandlimited or bandlimited by the same Butterworth filter [l], [8], Q2PSK can achieve twice the transmission rate of QPSK [I]. For a fair comparison, however, we should compare it with PSK bandlimited by a RNF. Thus, from the results of [l] and [8], QPSK with a RNF is still one of the most bandwidth efficient schemes.

111. COMMONALITIES OF CEQ2PSK, RQPSK, E-FPSK, AND LO

The ith of a total of eight waveforms for CEQ2PSK given by (9a) and (35) of [l], E4-FPSK [3], and LO given by (5-93) on 17, p. 2351 with two frequencies and four phases, are

(t) si(t) = ~ q q p ~ ~ ~ ~ ~ q q p s k (8)

where (9) and (10) (shown at the bottom of this page). Since the ith waveform of Q2PSK is sZ(t) =

(l), from Lemma 1, it is known that for optimal detection, both Q2PSK and 4FPSK of [3] with h = 1 and N = 2 have the same performance on AWGN channels. The performance is the same as that of QPSK [l], although the upper bound of 4FPSK is 0.3 dB larger than that of QPSK [3]. Since E [ ( D q q p s k ) H ( D q q p s k ) ] = 0.514 for equiprobable st(t), from Lemma 1, both 4FPSK and Q2PSK have exactly the same average PSD. One can also obtain the average PSD of 4FPSK by averaging (6) of [3], and it is the same as that of Q2PSK given in (22) of [l]. For binary communications, the Nyquist rate (2 b/Hz/s) can only be achieved by QPSK with an ideal Root Nyquist filter (RNF with roll-off rate ,B = 0), by Q2PSK with ideal pulse pairs [1] or by OFDM with N + m [6]. However, [3] claimed that the bandwidth efficiency of 4FPSK is 33% higher than that of QPSK. This problem is due to the definition of the bandwidth efficiency. In [3] the bandwidth efficiency was based on the first null bandwidth, only suitable for PSK, but neither FPSK, Q2PSK, or EFPSK. This is because the spectrum of PSK can be limited within the first null bandwidth ([0, l/Ts)) with no IS1 by the ideal RNF, and there is no filter that can limit the spectrum of either FPSK, EFPSK, or Q2PSK within

D;qPSk p qqpsk (t) [l], GIG? = 14, and p q q p s k ( t ) satisfies

respectively, where

D c e q q P s k

ceqqPsk DceqqPsk , . . . , ~ r q q p s k ] T = [Dl 7 2

1 -1 -1 1 -1 1 1 -1

-1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1

(14)

1 1 - 1 1 1 -1 1

G 2 = + [-; ; 1 :I 1 -1

to 1 1 01

D g q p s k = [ D q q p s k D q q p s k . . . D q q P ” ] T 1 , 2 , , 1 6

-1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 Jz -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 2 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1

-1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1

=- [ 1 1 - 1 1 1 - 1 1 1 1 -1 -1 -1 1 1 1 - 1

, and pqqpsk(t) = fi

7rt cos(w,t) cos -

T T f

cos(w,t) sin - TS Tt

sin(w,t) cos - T T f

sin(w,t) sin - Ts

1 1 l l T

-1 :;J (9)

1 -1 1

300 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 3, MARCH 1996

Since G2GF = G3GF = 14 andE[(DceqqPSk)HDce~qPsk I = 0.514 for equiprobable sz, from Lemma 1, all three systems are identical. Therefore, CEQ2PSK outperforms QPSK by 1.3 dB, because 3 bit LO requires Eb/N, = 8.3 dB to achieve BER = l o p 5 on AWGN channels [7] which agrees with the upper bound of E4-FPSK [3]. It is interesting to note that both CEQ2PSK and LO have a constant envelope, while E4-FPSK of [3] does not.

pqqpsk(t) is a set of equal energy orthogonal waveforms. Let

p”“”(t) = Jz

7rth T,

cos(w,t) cos - 7rth cos(w,t) sin - T S nt h

sin(w,t) cos - TS

7rth sin (wet) sin 7

1 s

For 0 < h < 1, the waveforms of p”“”(t) are still orthogonal, but have unequal energy. Replacing pqqpsk(t) with pneW (t) in (S), we obtain exactly (1) of [3] with general h, and in (13), we have the transmitted signal

(17) 7rht T s

s ( t ) = A COS w,t + b __ + 4

where b is a binary symbol from the set {+l, -1} and 4 is a four-ary phase symbol. Equation (17) with h = 0.5 is exactly the same as (1) of [2]. Thus, both LO and RQPSK are special cases of (17). Expression (17) is very close to that of binary continuous phase frequency shift keying (CPFSK). CPFSK does not utilize the phase to transmit information, while the phase of (17) carries 2 bits. The results of [ 11 and [9] have shown that (17), with an appropriate bandlimiting filter, has higher power and bandwidth efficiencies than MSK and higher power efficiency than QPSK.

Iv. COMMENTON [4]

In [4] the MF MSK signal, that produces a constant envelope continuous phase signal set, has been proposed. For two frequencies ( N = a), it can transmit eight bi-orthogonal signals using the following four orthogonal waveforms:

where

2 7rt p1( t ) = - COS - cos (% - $) t U T , ( t ) dz Ts

2 p2( t ) = - sin 5 sin ( w, - $) tuT, ( t - %) a Ts

2 7rt

2 7T-t .

p3(t) = - cos - cos (% + g) tu*,(t) d z Ts

d z T s p4(t) = - sin - sin (w. + g) tuTs (i - 2). (19)

Reference [4] claims that (19), with h = 1, is an orthogonal set over any interval [CT,, 1Ts + 0.5Ts). One can easily show, however, that for h, any integer except -1, 0, and 1, (19) is an orthogonal set. For h = 2, the performance of MF MSK remains the same, but the bandwidth efficiency is only half of that presented in [4].

V. CONCLUSION

In this letter, some commonalities of Q~PSK, CEQ~PSK, RQPSK, FPSK, E-FPSK, MF MSK, OFDM, and LO signaling, have been studied on AWGN channels. The results show that: a) Q’PSK, FPSK and OFDM are identical, and b) CEQ’PSK, E4-FPSK, and LO are identical. Thus, techniques developed for one scheme may be adapted to the others. For example, if one of E-FPSK’s is identical to LO, we can use not only the simple upper bound in [3], but also the upper and lower bounds given by (5-104) and (5-105) of [7], to analyze the performance. We should be aware, however, that these identical signals may respond differently on other channels, such as, bandlimited or multipath channels. An example can be found in [lo]. Also, a mistake in [4] was corrected.

ACKNOWLEDGMENT The author would like to thank Dr. I. Kom for several valu-

able discussions and to thank the editor and the anonymous reviewers for their helpful suggestions.

REFERENCES

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[Z] J. Liu, J. Kim, S . C. Kwatra, and G. H. Stevens, “Rotative quadrature phase shift keying,” Electron. Left., vol. 28, pp. 1095-1097, June 1992.

[3] R. A. Khalona, G. E. Atkin, and J. L. LoCicero, “On the performance of a hybird frequency and phase shift keying modulation technique,” IEEE Trans. Commun., vol. 41, pp. 655-659, May 1993.

[4] S. M. Reisher and S. Qu, “Multifrequency minimum shift keying,” IEEE J. Select. Areas Commun., vol. 10, no. 8, pp. 1243-1253, Oct. 1992.

[5] R. W. Chang and R. A. Gibby, “A theoretical study of performance of an orthogonal multiplexing data transmission scheme,” IEEE Trans. Commun., vol. COM-16, pp. 529-540, Aug. 1968.

[6] L. J. Cimini, Jr., “Analysis and simulation of a digital mobile channel us- ing orthogonal frequency division multiplexing,” IEEE Trans. Commun., vol. COM-33, pp. 665-675, July 1985.

[7] W. C. Lindsey and M. K. Simon, Telecommunication System Engineer- ing.

[8] I. Kom and L. Wei, “Q2DPSK in the satellite mobile channel with IS1 and CSI,” IEEE Trans Veh. Technol., vol. 34, pp. 69-78, Feb. 1994.

[9] L. Wei and 1. Kom, “Combined frequency and differential phase shift keying with noncoherent detection in the satellite mobile channel,” to appear in IEEE Trans. Veh. Technol.

[lo] L. Wei, “Q2DPSK with new pulse sharing pair in the satellite mobile channel,” Electron. Lett., vol. 29, no. 23, pp. 2014-2015, Nov. 11, 1993.

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