CA2041279A1 - Quadrature biorthogonal modulation - Google Patents

Abstract

Quadrature Biorthogonal Modulation Abstract Digital modulation/demodulation methods utilize multiplexing in quadrature of two N-dimensional biorthogonal signal sets wherein the said signal sets are transmitted on each of the two quadrature carriers cos.omega.ct and sin.omega.ct in order to increase the bandwidth efficiency and/or the power efficiency as compared to known digital transmission methods. One embodiment of the invention utilizes two sets of combined frequency/phase modulated (FPM) signals of the type NFSK/2PSK with adjacent frequencies separation of 1/T Hz as the constituent biorthogonal signal sets, and is referred to as Quadrature Frequency/Phase Modulation (QFPM). A preferred embodiment of the invention referred to as Continuous Phase Quadrature Frequency/Phase Modulation (CP QFPM) is essentially a modified QFPM scheme with sinusoidal shaping and an offset of T/2 in the relative alignment of the baseband modulating waveforms for the two quadrature components of the QFPM
signal set. Another preferred embodiment of the invention, referred to as Continuous Phase Frequency/Phase Modulation (CP FPM), is a particular case of CP QFPM wherein the same frequencies are used in both quadrature components of CP QFPM which yields a constant-envelope signal set.

Description

Specification Background of ~e Invention This invention relates to digital information transmission methods wherein the data transmission rate and/or the bandwidth efficiency are essentially increased as a result of quadrature multiplexing of two N-dimensional biorthogonal signal sets of size L=2N
each, which yields a new 2N-dimensional signal set of size M=(2N)2, with N=211, and ~1 being an integer, wherein the two biorthogonal signal sets are generated as a result of modulation of two srthogonal carriers, cos~ct and sinc~ct, of the same frequency c~c~
with each of the two orthogonal carriers bearing one biorthogonal signal set, and as a result of shaping of the baseband modulating waveforms.
We refer to the invented transmission system as Quadrature Biorthogonal Modulation As it is well known [1], a biorthogonal signal set of size L consists of L/2 pairs of antipodal signals, each pair of which is orthogonal to all other pairs. In particular, a combined frequency/phase modulation (FPM), NFSK/2PSK signal set with a frequencymodulation index h=2fdT-2k, with k an integer, 2fd the spacing between two adjacent frequencies, N=the number of frequencies, and T=the signal interval, belongs to the class of biorthogonal signal sets.
Consequently, one particular embodiment of the invention is a scheme wherein twobiorthogonal signal sets of the type NFSK/2PSK with h=l are transrniKed on two quadrature carriers cos~ct and sinCDct. In the sequel, this embodiment of the invention is referred to as the Quadrature Frequency/Phase Modulation scheme (QFPM), and the two signal sets associated with cosc3ct and sin~ct are referred to as the inphase and quadrature components of QFPM, respectively.
The principle of transmitting on two independently modulated carriers in quadrature is well known and, in particular, is implemented in Quadrature Amplitude Modulation(QAM) scheme. However, in the QAM scheme each of the two quadrature multiplexed signals constitutes a one-dimensional multi-amplitude (ASK) signal set, whereas the underlying principle in the proposed QBOM scheme consists in quadrature multiplexing of N-dimensional biorthogonal signal sets. While QAM belongs to the class of bandwidth-efficient modulation schemes, QBOM in general and QFPM in particular belong to the class of power-efficient modulation schemes. Therefore the power-efficiency of QFPM is always higher than that of QAM, however it will be shown in the sequel that the bandwidth efficiency of QFPM is approaching that of QAM under certain conditions (e.g., in terms of 99%-power containment bandwidth). It is appropriate to compare QFPM with QAM since both yield M-ary nonconstant-envelope signal sets.
However, the QFPM yields a constant-energy signal set whereas the QAM signal set, in general, contains signals at different energy levels.
As compared to power-efficient modulation schemes (e.g., coherently orthogonal MFSK, coherently biorthogonal NFSK/2FSK) the proposed QFPM, with the same power efficiency, yields a higher bandwidth efficiency at the expense of nonconstant envelope and greater complexity of the system.
The 4-ary QAM (and the equivalent QPSK) scheme can be viewed as a degenerate QFPM signal set with N= 1.
It will be shown that the recently proposed Q2PSK signal set [2] can be viewed and, consequently, generated and demodulated as a particular case of QFPM with N=2, rather than as a quadrature-quadrature PSK signal with "... four streams of data pulses ...
having l~redetermined pulse shapes and quadrature pulse phase relationships with respect to one another" [3].
The preferred embodiment of the invention is a modification of QFPM wherein a relative offset of the baseband waveforms corresponding to the inphase and quadrature components of the QFPM signal, as well as sinusoidal shaping of the baseband waveforms, are introduced. This preferred embodiment of the invention, referred to as Continuous Phase Quadrature Frequency/Phase Modulation (CP QFPM), yields significantly improved spectral properties of the QFPM constellation similar to those achieved in MSK relatively to the QPSK format. In fact, MSK can be viewed as a CP QFPM system with N=l. With CP QFPM, the afore-mentioned advantages of QFPM over other modulation schemes are greatly enhanced, at the expense of somewhat increased complexity.
Another preferred embodiment of the invention is a version of CP QFPM wherein the same frequencies are used in both the inphase and quadrature components of the CP QFPM signal. This preferred embodiment of the invention yields a constant-envelope Continuous Phase FrequencytPhase Modulated signal and is referred to asCP FPM. The size of the signal set with CP FPM is reduced to M=4N as compared toM=4N2 with CP QFPM, and accordingly the bandwidth efficiency of CP FPM versus CP QFPM is reduced, with a constant envelope at premium. CP FPM is superior to other constant-envelope modulation schemes in that it yields a higher bandwidth efficiency versus MFSK, NFSK/2PSK, and CP FSK, whereas versus MPSK the CP FPM scheme demonstrates higher power efficiency and, possibly, higher bandwidth efficiency (e. g., in terms of 99%-power containment bandwidth).

Brief Descri~ion of the Draw;ngs Comprehension of the invention is facilitated by reading the following description of the annexed drawings, in which:
Fig. 1 is an illustrative timing diagram showing the waveforms of the various components involved in generating the QFPM signal;
Fig. 2 is a function block and partially schematic representation of a QFPM modulator constructed in accordance with the principles of the invention;
Fig. 3 is a function block and partially schematic representation of a QFPM demodulator constructed in accordance with the invention;
Fig. 4 is a timing diagram illustrating for CP QFPM the offset between the inphase and quadrature components of the baseband waveforms, the sinusoidal shaping of aC(t) and aS(t), and the resulting modulated signals;

Fig. 5 is a function block and partially schematic representation of a CP QFPM
modulator constructed in accordance with the principles of the invention;
Fig. 6 is a function block and partially schematic representation of a CP Q~PM
demodulator constructed in accordance with the invention;
Fig. 7 is a timing diagram showing the waveforms oî the various components involved in generating the CP FPM signal;
Fig. 8 is a function block and partially schematic representation of two versions of a CP FPM modulator constructed in accordance with the invention;
Fig. 9 is a function block and partially schematic representation of a CP FPM
demodulator constructed in accordance with the invention;
Fig. 10 is a graphical illustration helpful in describing the CP FPM demodulator of Fig. 9;
Fig. 11 is a graphical representation of the bit error probabilities of CP FPM with the number of frequencies N=l, 2, 4, 8 arld 16, as a function of Eb/No;
Fig. 12 is a graphical representation of the power spectral density of NFSK/2PSK, h=0.5, with the number of frequencies N=l, 2, 4, 8, 16, and 32, as a function of the normalized frequency fl b;
Fig. 13 is a graphical representation of the power spectral density of QFPM with the number of frequencies N=l, 2, 4, 8, 16, and 32, as a function of the normalized frequency fTb;
Fig. 14 is a graphical representation of the power spectral density of CP QFPM with the number of frequencies N=l, 2, 4 8, 16, and 32, as a fsnction of the normalized frequenCY fTb;
Fig. 15 is a graphical representation of the power spectral density of CP FPM with the number of frequencies N=l, 2, 4, 8, 16, 32, as a function of the normalized frequency fTb;
Fig. 16 is a graphical representation of the out-of-band power containment, in decibels as a function of the normalized bandwidth BTb, for NFSK/2PSK, h=0.5;
Fig. 17 is a graphical representation of the out-of-band power containment, in decibels as a function of the norrnalized bandwidth BTb for QFPM;
Fig. 18 is a graphical representation of the out-of-band power containment, in decibels as a function of the norma~ized bandwidth BTb for CP QFPM;
Fig. 19 is a graphical representation of the out-of-band power containment, in decibels as a function of the normalized bandwidth BTb, for CP FPM, and Fig. 20 is a graphical representation of the out-of-band power containment, in decibels as a function of the normalized bandwidth BTb, for CP FSK.

Detailed Descrip~
Detailed descriptions of an embodiment of the proposed QBOM system, the QFPM
scheme, as well as of the preferred embodiments of the invention, ~e CP QFPM scheme and the CP FPM scheme, follow.

A. QFPM

The (2N)2 -ary QFPM signal can be represented in any symbol interval of constantduration T=Tblog2(2N)2 = 2(1+1Og2N)Tb as s(t) = A{ac(t) cos[c~c + bC(t) todlt + aS(t) sin[c~c -~ bs(t) C~dlt} (1) wherein each of the two sets of signals, ac(t)cos[~oct+bc(t)c~d]t and aS(t)sin [~Ct+bs(t)od]t~ is a biorthogonal set of the type NFSK/2PSK with minimum frequencies spacing l/T, corresponding to ~3dT=7~ and a frequency modulation index h=2fdT=l, wherein A is a constant amplitude, C~C is the carrier frequency with c~cT = m1~ and m an integer, wherein it is assumed that the input binary bit stream {an}, an-+l, n=1,2,... arrives at a rate rb=l/Tb and is separated into two sequences dc and ds of even-numbered and odd-numbered blocks, respectively, each possessing ~=l+log2N bits, wherein aC(t) and aS(t) are binary rectangular pulses of levels il representing the first bits in the even-numbered (dc) and odd-numbered (dS), respectively, blocks of ~ bits, and bC(t) and bS(t) are N-ary rectangular pulses of levels $1, i3, ..., + (N-l) determined in each interval T by the blocks of ~ bits from the sequences dc and ds respectively, in accordance with a coding rule [1] illustrated in Table I and in the example waveforms of Fig. 1 for N=4. Note that in Table I and Fig. 1, dCl(t), dC2(t) and dSl(t), dS2(t~ represent the remaining (~-1) bits (~=3 for N=4) in the even-numbered and odd-numbered blocks, respectively.
The signal (1) in any interval of length T can also be represented as s(t) = A[i cos cl)it + sin c~t], i, j=l, 2, .. , N (2) wherein ~n= c - [N-(2n-l)]~l)d n=i or j (3a) and n = 2 [N+l+b(t)] b(t) =bC(t) or bS(t) (3b) In accordance with (1), (2), the signal s(t) is a sum of two sinusoids in quadrature with constant amplitudes, and in any kth interval (kT, (k+l)T) of length T the frequencies, ~
and ~3j, and the phases (0 or 180) are constant. Each of these constituent sinusoids belongs to a separate N-dimensional biorthogonal NFSK/2PSK set with constant energy and constant envelope.
The QFPM signal s(t) is 2N-dimensional and is constant-energy, but has a non-constant envelope in the general case of i~j.
Indeed, (1) can be expressed as s(t) = A aC(t) {cos(a)c+bc(t)~l)d)t + aS(t)) cos[(~l)c+bs(t)~l)d)t - 7~/2]}
= 2A aC(t)cos[bc(t)2bs(t) codt i ~/4] cos[(~ + bc(t)+bs(t) ~ - 4 where the upper signs correspond to aaC(t)) =1, and the lower signs to aaC(t) = -1. Clearly, the envelope is constant only if bC(t)=bS(t), in which case the (2N)2-ary QFPM signal reduces to a 4N-ary NFSK/4PSK signal.

It is interesting to note that Q2PSK (Quadrature-Quadrature Phase Shift Keying),described in [2] and [3], is identical to QFPM with N=2.
Indeed, with N=2, bC(t) = + 1, bS(t) = i 1, and since oDd=7c/T, (1) can be expressed as s(t) = A {aC(t) [cos( T )cosc~ct - bC(t) sin( T )sin~ct ]
+ aS(t) ~cos( Tt )sincoct + bS(t) sin( Tt )cosc~ct ]} (5) Since aC(t), aS(t), ac(t)bc(t) and as(t)bs(t) are all binary quantities with values ~1, we can use the notations al(t) = ac(t)~ a2(t) = as(t)bs(t)~ a3(t) = aS(t), a4(t) = -ac(t)b~(t) (6) in order to represent QFPM with N=2 in the form of the Q2PSK signal [2, eq.9a]
s(t) = A [al(t) cos(Tt )cos~ct + a2(t) sin(Tt )coscoct + a3(t) cos(T )sinc~ct + a4(t) sin~F )sinc~ct ] (7) Thus, the Q2PSK signal (7) can be generated and demodulated as a QFPM signal with N=2, in accordance with the description given in the sequel (Fig. 2 and Fig. 3~.
The block-diagram of the modulator for a (2N)2-ary QFPM embodiment of the invention, constructed in accordance with the principles of the invention, is presented in Fig. 2. As shown therein, the input stream of binaly data {an} is demultiplexed into two sequences dc and ds wherein the sequence dc contains the even-numbered blocks of bits from the input stream {an}, and the sequence ds contains the odd-numbered blocks of bits from the input stream {an}, each block of ~=(l+log2N)bits.
The two sequences dc and ds are applied to two frequency/phase (biorthogonal) modulators Mc and Ms, respectively, which are well known [1]. The frequencies c~ and ~ are selected in accordance with the coding rule ~able I), and the frst bit in each block selects the phase (sign) of the corresponding constituent NFSK/2PSK signal in the interval of duration T.

Table I. Coding rule (N=4) aS(t) or aC(t) dSl(t) or dCl(t) dS2(t) or d~2(t) bS(t) or bC(t) The coding rule of Table I states that if a block of ~ bits (a, dl, d2, ...d~ l) corresponds to b, then (-a, -dl, -d2, ...-d~. l) also corresponds to b with (see eq.(3b)) b=2n-(N+ 1) n = 1, 2, .. , N (8) where 'a' stands for aC(t) or aS(t), 'b' for bC(t) or bS(t), and (dl, d2, ...d~ l) for the sequence dc or ds of (~-1) bits. The two signals sc(t) and ss(t), orthogonal over the interval T and of constant energy A2T/2 each, are summed to produce the channel signal s(t) of energy E=A2T.
The two biorthogonal signal sets transmitted over cosc3ct and sin~oct can be demodulated separately, since they are orthogonal over an interval of length T, and one can apply the well-known results for coherent reception of biorthogonal signals [1] in order to design a receiver for QFPM, as well as to analyze the performance of the latter.
The receiver for the QFPM signal is represented in Fig. 3a as a combination of two receivers, optimum for biorthogonal signals sc(t) and ss(t) on the ideal AWGN channel.
Each of the demodulators Dc and Ds can be represented by the block-diagram of Fig. 3b.
The demodulation is accomplished in two steps. The outputs of N correlators (matched filters) are sampled at time t =kT, and the largest magnitude without regard to the sign is selected. Having deterrnined which of the signals is the largest, that particular correlator output is examined again and the polarity of the signal is determined.
After a decision has been reached regarding the outputs of both demodulators Dc and Ds, the signals are decoded and a block of 2(1+10g2N) bits is recovered at the output of the receiver (Fig. 3a).
The total number of required correlators for M-ary QFPM is ~M, whereas it is N=M2 for M-ary biorthogonal NFSK/2PSK, and M for orthogonal MFSK.
The performance of the QFPM scheme in terms of bandwidth efficiency o~b and power efficiency Eb/No is evaluated by comparing it to various modulation schemes, power-efficient biorthogonal NFSKt2PSK, and bandwidth-efficient QAM and MPSK. Note that the orthogonal MFSK scheme is inferior to the biorthogonal NFSK/2PSK scheme in bandwidth efficiency while the power efficiency of both is (approximately) the same.
For the coherently biorthogonal NFSK/2PSK the upper bound on the bit error probability on the distortionless AWGN channel is given in [1] as*

Peb < (N-l) Q[~NO (l~log2N)) ] + Q[~ No (l+lg2N) ], (9) where Eb is the energy per bit and No is the one-sided power spectral density (PSD) of the input AWGN.
Exact values of Nb required with biorthogonal signal sets for a given Peb are given in [1, Table 5.2], and are reproduced in Table II for Peb=lO-S and various values of M.
With the proposed (2N)2-aly QFPM scheme, the bit error probability for a given N is the same as for the NFSK/2PSK scheme and is determined by (9), since the two constituent NFSK/2PSK signals can be demodulated indepe~dently. This conclusion is analogousto that for a QAM scheme. It is well known M that the bit error probability for both, the L,ary ASK scheme and the M-ary QAM scheme, M=L2, is approximately given by Peb#~2L Q[l L2 l No ] (10) This result assumes a Gray code bit to symbol mapping and ignores symbol errors that occur in non-adjacent levels.
* Q(x) is the Gaussian probability integral deflned by Q(x)~ I exp(y2/2) dy.
~ 2~x .. . .

The values of Eb required for Peb=l0-s with various modulation systems are given in Table II.

Table II. Bandwid~ and power efficiencies of various modulation schemes tPeb = 10-5, EblNo in dB, o = ~L in bits/sec/Hz~.

EbEb ~ 13b ( )b Eb Eb Ns~ No No -- No No -- NQ
4 1 0 9.6 0.67 9.6 0.67 9.6 0.76 9.6 1.0 9.6 1.0 9.6 8 _ _ _ _ 0.75 8.3 0.83 8.3 _ _ _ 16 133 96 _ 96 067 _ 0.71 74 20 134 2.0 17.4 32 _ _ _ 0.5 6.6 0.52 6.6 _ _ --~
64 1.2 8.3 1.0 8.3 0.33 6.1 0.34 6.1 3 0 17 8 3 0 27.5 256 0.89 7.4 0.8 7 4 0.12 5.3 0.12 5.3 4.0 æ.s 4.0 38.
1024 0.59 6.6 0.55 6 6 0.04 4.8 0 04 4.8 5 0 27 5 5 0 49.1 4096 0.36 6.1 0.35 6.1 0.01 4.4 0.01 4.4 6.0 32.6 6.0 60.2 In Table II the bandwidth efficiencies are also presented, in terms of null-to-null bandwidth Bnn, of various modulation schemes. It is well known that the null-to-null bandwidth of QAM and MPSK is Bnn=2r, with r being the symbol rate. The null-to-null bandwidth for other transmission schemes of interest are derived from the power spectral density curves calculated and presented in Section D. For NFSK/2PSK (h=0.5) and QFPM, the null-to-null bandwid~s can be given as Bnn (NFSK/2PSK) ~ (N+3) r (11~
Bnn (QFPM) = (N+l) r (12) where r=rb/2(1+1Og2N) is the symbol rate for QFPM.
From the data in Table II one may conclude that in tenns of Bnn the bandwidth efficiency of QFPM, for any given N and rb, is better than that of NFSK/2PSK, with the power efficiency being the same (Note that, e.g., N=2 corresponds to M=4 for NFSK/2PSKand M=16 for QFPM). Compared to QAM and MPSK, the bandwidth efficiency tin terms of Bnn) with QFPM is, as expected, much lower while the power efficiency of QFPM is significantly better. However, as it is shown in Section D (e.g., Table IV), in terms of 99%-power containment bandwidth, Bgg, the bandwidth efficiency of QFPM is the same as that of QAM and MPSK. Note that QFPM yields a gain in power versus QAM of 3.8dB with M=16, 9.5dB with M=64, and 15dB with M=256. The gain in power is larger versus MPSK, but MPSK has a constant envelope.

. CP OFPM
A modified version of the QFPM scheme is the (2N)2-ary Continuous Phase Quadrature Frequency/Phase Modulation (CP QFPM) defined as s(t)=A{a~(t) cos(7ct/I) cos[~c+bc(t)~dJt ~ as(t)sin(~VT) sin[c~c+bs(t)~dlt} (13) which is essentially the QFPM signal with an offset in the relative alignm~nts of the sequences dC(t) and dS(t) equal to T/2 secol~ds and with half-sinusoidal shaping of the bit streams aC(t) and aS(t) (Fig. 4). The formation of the CP QFPM signal from the QFPM
signal is similar to that of the MSK signal from the QPSK signal [5].

~~ , .~.

In QFPM, due to the coincident aligmnent of all bit streams ac(t), aS(t), bC(t) and bS(t), phase and frequency changes may occur only once every T second at t=kT, k=1,2,....
From (4) and Table I, it is easy to see that transitions in the data bits may result in phase discontinuities at t =kT.
In the CP QFPM signal (13), due to staggered alignment of the data bits and shaping (Fig.4), the phase of the transmitted signal at transition instants t =(2k-l)T/2 and t=kT
remains continuous.
Indeed, at t=T/2 (13) reduces to s(T/2) = as(T/2) A sin {[c3c+bs(T/2)~dl T/2} (14) Thus at t=T/2 the signal value depends only on aS(t) and bS(t) which do not havetransitions at t = T/2.
Similarly, at t =T the signal is s(T) = ac(T) A cos{[~c + bc(T) a)dlT} (15) i.e. it depends on aC(t) and bC(T) only. However neither aC(t) nor bC(t) may have transitions at t=T and the phase remains continuous.
Clearly with bc(t)=bs(t)=O, i.e. N=l, the CP QFPM signal (13) degenerates into the MSK signal.
The block diagram of the CP QFPM modulator can be obtained from the block diagram of the QFPM modulator (Fig. 2) by introducing an offset alignment of the bit sequence ds relatively to dc and adding sinusoidal shaping of the bit pulses ac~t) and aS(t), as shown in Fig. 5.
The demodulator for the CP QFPM signal (Fig. 6) is a modification of the demodulator for QFPM (Fig. 3) wherein the reference signals cos~it and sinc~t for the correlators are replaced by cos(T--)coscoit and sin(T--)sin~t respectively, and the integration limits and sampling instants for the inphase component of the signal are adjusted accordingly, as shown in Fig. 6.
The error perforrnance of CP QFPM on the AWGN channel is the same as that of QFPM. The values of Nb required for Peb=lO-S are given in Table II. The spectralproperties of CP QFPM are discussed in Section D.

C. CP FPM
The CP FPM version of the CP QFPM scheme is a (4N)-ary constant-envelope Continuous Phase Frequency/Phase Modulation signal set defined as s(t) = A {aC(t) cos(~t/I)cos[c~c+b(t)cddlt + as(t)sin(~t/I')sin[~c+b(t)cod]t }
= A [aC(t) cos(~tlT)cosc~t + as(t)sin(~t/T)sinc~it ] (16) and is essentially a CP QFPM signal with the same frequencies C~i = ~c + b(t)~l)d i=l, 2, .. ,N (17) in both the inphase and the quadrature components, wherein aS(t) = *1, b(t) = ~ 3, ..., i(N-l) and ~3i may have transitions at t=kT only, ac(t)-+l and may have transitions at t=(k-l/2)T only, and b(t) is determined in the same way as bS(t) for CP QFPM. The inphase component sC(t)=ac(t) cos(7~T--)cos~oit and the quadrature component sS(t)=aS(t)sin(~T--)sinc~t are orthogonal over an interval of duration T/2 sec.
The CP FPM signal (16) can also be represented as s(t) = A aC(t) cos[~t - aC(t) aS(t) Tt ] = A aC(t) cos{c3ct + [b(t)- aC(t) aS(t)] Tt } (18) From (18) it is obvious that the envelope is constant since ac(t)=il. The phase of the signal is continuous since a transition in aS(t) or b(t) at t =kT, as well as a transition in aC(t) at t=(k- 1/2)T, leads to a phase change equal to 27~ mod 27~.
In Fig. 7 the CP FPM signal is plotted with N=4, along with its inphase sc(t) and quadrature ss(t) components and the quanti~ies aC(t)cos(Tt ), aS(t) sin(Tt ), and b(t) the same as bS(t) in Fig. 4.
With N=l, the CP FPM signal degenerates into the MSK signal. In general, CP FPM
may be viewed as a Multi-Frequency MSK (MF MSK) signal.
Block-diagrams of two possible implementations of the CP FPM modula~or are given in Fig. 8. Blocks of (~+1)= 2+10g2N input bits are used to generate the (4N)-ary CP FPM
signal in intervals of duration T=(~+l)Tb seconds, wherein the first bit in each block is designated as aC(t), the second bit as as(t)~ and the remaining log2N bits as dsl~ ds2. ---.
ds;~ 1- In Fig. 8a, corresponding to the signal representation (16), aC(t) is used to control the phase of the inphase signal component sc(t), and the remaining ~ bits are used to control the phase of the quadrature signal component ss(t) and the frequency common for both sc(t) and ss(t), in accordance with the coding rule of Table I. The phase of the quadrature signal component sS(t) is controlled by the first bit out of these ~ bits. The block diagram of Fig. 8b corresponds to the signal representation (18) and is self-explanatory.
The demodulator for the CP FPM signal, shown in Fig. 9, consists of two blocks, Ds and Dc. The Ds block, similarly to Ds in Fig. 6, contains N correlators and is used to demodulate the quadrature signal component, in accordance with (16), sS(t)=as(t)sin(T--)sin~it and to recover the ~ data bits associated with aS(t) and b(t). The Dc block also contains N correlators and is used to demodulate the inphase signal component sC(t)=ac(t)cos(~3cosc~it and to recover the remaining data bit associated with ac(t).
In the Dc block, no decisions with regard to the frequen- y are made, and the proper i-th correlator to be sampled at t-kT (see part A of Dc, Fig. 9) is chosen in accordance with the i-th frequency selected in the Ds block. Because of the coincident alignment of the frequency-controlling b(t) with the signal component ss(t), the frequency is constant within intervals (k-l)T<t<kT and may have transitions at t=kT only, (Fig. lOa) (various types of shading correspond to different frequencies); i.e., in the middle of the pulse aC(t)cos(~3 associated with sc(t) (Fig. lOb). Therefore the integration intervals in the correlators of the block Dc must be limited to a duration of T/2 sec only, and the integrators must be discharged every T/2 sec.
For instance, in order to detect the value of aC(t) in the interval (2T' 32--) (Fig. lOb), one shall sample the output of the i-th correlator at ~,=T if the detected frequency in the interval (O,T) is ~. However, the sample at t=kT utilizes only part 1 of the pulse aC(t)cos(T--) in the interval of interest (2-' 2--) The detectability of aC(t) can be enhanced by utilizing part 2 of the pulse as well. This goal is achieved in the demodulator of Fig. 9 at the expense of additional complexity and a T/2-sec delay in detection of a~(t). The sample xil of the i-th integrator output taken at t=T, is delayed by T sec before being applied to the decision circuit A2. At the same time, all integrator outputs corresponding to part 2 of the pulse between T and 3T/2 are delayed by T/2 sec before being sampled at t=2T. At t=2T, the j-th integrator output is sampled, assuming that the frequency COj has been detected for the interval (T, 2T).
The delayed sample xil and the sample xj2 taken from (possibly) different correlators and corresponding to different time instants t=T and t=2T, but to the same value of a~ (t) in the interval (T/2, 3T/2), are summed and applied to the decision circuit A2 at t=2T. This yields a 3dB-gain in SNR at the input of A2 bringing the performance of phase detection in the scheme of Fig. 9 to the maximum attainable level.
The detectability of the frequency may be improved in a similar way. The signal components sS(t) and sc(t), with a common frequency cdi, are independently perturbed by Gaussian noise. The potential gain can be realized by adding the samples taken at t=kT
from the i-th correlators in the block Ds and in the block Dc, as shown in the demodulator of Fig. 9. The implementation of this idea involves additional complexity because a) the phase modulation must be removed from the samples prior to their summation, and b) the integration intervals in the block Dc are restricted to T/2 sec.
Consequently, the samples (iES +nl) taken at the end of integration intervals [(k-l)T, (2k- l)T/2] should be delayed by T/2 sec. The delayed samples and the samples (_ 4S+n2) corresponding to integration intervals [(2k-l)T/2, kT] are added and subtracted in the subblocks Bi, i=l, 2, ..., N yielding lE
uil = l 2s + nl + n2 or nl + n2 lEs ui2= ~ 2 +nl-n2 or nl-n2 (19) Here, Es/4 is the energy of sc(t) over an interval of Tl2 sec with ES=Eb(2+log2N)~ and nl, n2 are independent zero-mean Gaussian random variables with variance c~2. The signal component is present in the quantities uil only if the input frequency is c~i and aC(t) has no transition at t=(2k-l)T/2, while in the quantities ui2 the sigDal component is present only when the input frequency is ~3i and aC(t) has a transition at t=(2k-l)T/2.
At t=kT the sample (i~ + n3) from the output of the i-th correlator in the block Ds is applied, along with the quantities Uil and ui2, to the summators, yielding Vil = iEs+ nl + n2 + n3, or i 2s + nl + n2+ n3, or nl + n2+ n3 vi2 = +Fs- nl - n2 + n3, or i 2 - nl - n2+ n3, or -nl - n2+ n3 vi3 = iEs- nl + n2 + n3, or i 2s nl + n2+ n3, or -nl + n2+ n3 vi4 = iEs+nl-n2 + n3, or i 2s + nl - n2+ n3~ or nl - n2+ n3 (20) Here, Es/2 is the energy of s5(t) (or sc(t)) over an interval of T sec, and n3 is a zero-,, mean Gaussian randorn variable with variance 2~2, independent of nl and n2. If the input frequency is c~i, a signal component is available in one (and only one) of the quantities vjp, p=l, 2, 3, 4, depending on the values of aC(t) and aS(t) in the k-th transmission mterval. If aC(t) has no transition at t=(2k-l)T/2, then either vjl contains Es (as(t)=ac(t)=l) or -Es (as(t)=ac(t)=-l)~ or vj2 contains Es (aS(t) = -ac(t)=l) or -Es (aS(t) =
-ac(t)=-l). If aC(t) has a transition at t=(2k-l)T/2, then either Vj3 contains Es (as(t)=l, transition in aC(t) from -1 to 1) or -~s (as(t)=-l, transition in aC(t) from 1 to -1), or Vj4 contains Es (aS(t)=l, transition in 4(t) from 1 to -1) or-Es (aS(t)=-l, transition in aC(t) from-l to 1).
Thus, to the decision circuit 1~l in Fig.9 is always applied a sample with a signal to noise ratio Es2/4~2, as opposed to an SNR (Esl2)2/2~2 in the case when the frequency detection is based on the quadrature component s5 only. This yields a power gain of 3 dB
in frequency detection, as expected. Note also that no error occurs in detecting the frequency as long as one of the quantities vjp has the largest magnitude.
The symbol and bit error probabilities of CP FPM with the demodulator of Fig. 9 are defined by Pes S 2 (2N-l) Q(~Nb (2+10g2N)) (21,a) Peb 5 (2N-l) Q(~Nb (2+1og2N)) (21,b) The receiver of Fig. 9 is optimum for the CP FPM signal since both the phase and the frequency are detected coherently utilizing all the available signal power. The two-stage detection procedure does not lead to any perforrnance degradation.
The curves of Peb as a function of Nb are plotted in Fig. 11 for N=l, 2, 4, 8 and 16.
Fig. 11 demonstrates that the power efficiency of CP FPM is better than that of MSK
(N=l) or QPSK by up to 3 dB. The values of Eb/No required for Peb=10-5 are given in Table II.
The spectral proper~ies of CP FPM are discussed in the following Section D.

D. Spectral Properties of QFPM. CP QFPM and CP FPM
A realistic evaluation of the spectral properties of QFPM, CP QFPM and CP FPM
requires the knowledge of their respective power spectral densities (PSD's).
The normali~ed equivalent lowpass PSD's for NFSK/2PSK, QFPM, CP QFPM and CP FPM have been derived and are given, respectively, as S(f) = Nb sinc2(~Tb N-l I ) (22) S (f) = N b ~ sinc2 (2~ f Tb + i N-l ) (23) S (f) = Nb ~ [sinc (2~ f Tb - 2 + i) + sinc (2~fTb - 2 + i + 1)]2 (24) S (f) = 2N o {sinc [(1+~) f Tb - 2 + i~ + sinc [( l+~)fTb - 2--+ i + 1] }2 (25) The PSD's S(bf) are plotted in Fig. 12 for NFSK/2PSK, in Fig. 13 for QFPM, in Fig. 14 for CP QFPM and in Fig.15 for CP FPM, w;th N=l, 2, 4, 8, 16 and 32. Notethat the curve with N=l in Fig. 12 represents the PSD of BPSK, the curve with N=l in Fig. 13 represents the PSD of QPSK and the curves with N=l in Figs. 14 and 15 represent the PSD of MSK.
Figs. 13 to 15 demonstrate that the main lobes of the CP QFP~ and CP FPM spectra are wider than that of QFPM, but the PSD's of CP QFPM and CP FPM fall off at a higher rate than that of QFPM.
Important spectral features are explicitly demonstrated by the fractional out-of-band power containment calculated as ll= lOloglo[ 1- B¦2S (f)df] (dB) (26) The curves of fractional out-of-band power containment for NFSK/2PSK, QFPM, CP QFPM, CP FPM and (for comparison purposes) CP FSK(h=0.5) are plotted in Figs. 16 to 20, respectively.
Some results are also presented in Tables m and IV in the form of 9û%-power and 99%-power containment bandwidths for QFPM, CP QFPM, CP FPM, QAM(MPSK), CPFSK and NFSK/2PSK. In the Tables, QFPM and CP QFPM (CP FPM) with M=4 correspond to QPSK and MSK respectively, as indicated above.

Table m Comparison of 90% power containment bandwidth efficiencies PSK) (h=o.S? (h-0.5 ;.16 ;.25 ;.25 1 516 --;.7 1:os4 __ 1 11 32 ~ _ 0 69 ~ - 0.65 64 1.52 1.6 0.42 3.47 0.21-- 041 128 0.24 _ 256 1.07 1.17 4.62 1024 0.6g 0.71 S.78 4096- 0.42 0.43 6.93 Table IV Comparison of 99%-power bandwid~ efficiencies 2 o.PoS5K) ~ho-.83) (ho=.
4 0.1~~~ 0.83 0.83 0.1 0.79 U.I

16 0.2 1.-25 0 81 0.2 0.47 020 64 0.3 1.2 0 37 0.3 - 0-25 128 0.22----256 0.37 0.89 0.4~~-1024 1i~43 0.62 0.5 ~
4096 0.35 0.38 0.6 _ ~, ~

. .

It is seen from Table m that in terms of a bandwidth Bgo that captures 90% of the total power, and with M > 4 the bandwidth efficiency of QFPM and CP QFPM is approximately the same, the bandwidth efficiency of NFSK/2PSK and of CP FPM is lower, and that of QAM(MPSK) is higher. In contrast, in terms of bandwidth Bgg that captures 99% of the total power and with M=4 to 1024, the bandwidth efficiency of QFPM, QAM(MPSK), and NFSK/2PSK is approximately the same, while CP QFPM
has the best bandwidth efficiency. Comparing the bandwidth efficiency of QAM(MPSK) with that of the other modulation schemes one should remember that thepower efficiencies of the latter are better than that of the former.
The demonstrated spectral properties in conjunction with power efficiency considerations suggest that the proposed QFPM, CP QFPM, and CP FPM schemes may have a wide range of applications, in particular in situations where lit~e or no filtering is desired. The CP FPM scheme, combining such features as constant envelope, compact spectrum and good power efficiency lends itself for application on nonlinear channels.
In conclusion, we compare various modulation schemes in terms of fractional power containment within a bandwidth B=1.2 rb (this bandwidth is sometimes referred to in the literature as one containing 99.09% of the total power of an MSK signal). From Table V
it can be seen that for M=4 to 256 the largest fraction of the total power within this bandwidth is captured by CP QFPM.

Table V Signal power (percentage of the total) captured within a bandwidth B=1.2rb M QFPM C:P QFPM CP FP~ QAM CP FSK
~ g9.0g ~ ~
16 95.8 99.80 98 56 95.5g 61.79 32~ 74.91 -96.68 ~ 55625 97.25 ~7lr 256 96.83 99.90~~- 97.96-- ---1024 69.24 74.99 98.31 4096 44.69 45. 98.S6

Claims (8)

1. A modulation method that consists of quadrature multiplexing of two biorthogonal signal sets, wherein the first set is carried on a carrier cos.omega.ct and the second set is carried on a carrier sin.omega.ct wherein both carriers are of the same frequency and the second carrier is in quadrature with respect to the first carrier.

2. An embodiment of claim 1, named QFPM, wherein the two said biorthogonal sets are generated as a result of combined frequency/phase modulation (FPM), wherein the first biorthogonal set, named inphase component of the QFPM composite signal, results from imposing the said modulation on the carrier cos.omega.ct and the second biorthogonal set, named quadrature component of the QFPM composite signal, results from imposing the said modulation on the carrier sin.omega.ct.

3. The modulation method of claim 2, wherein the said FPM is of the type NFSK/2PSK, with N?2 being the number of frequencies, preferably N=2µ
with µ being an integer, and with adjacent frequencies separated by T=2(1+log2N)Tb being the channel symbol duration and being the bit rate of the binary source.

4. The modulation method of claims 2 and 3, wherein prior to performing said steps of NFSK/2PSK modulation and addition in quadrature of two NFSK/2PSK signals, the input stream of rectangular data pulses is divided into two sequences, dc(t) and ds(t), and in each transmission interval of T
seconds blocks of (1+log2N) bits from each of the sequences dc(t) and ds(t) are applied to the NFSK/2PSK modulator Mc and to the NFSK/2PSK
modulator Ms respectively, wherein the demultiplexing of binary data and the implementation of said modulators Mc and Ms are well known and described in the literature.

5. A demodulation method for recovering the data stream, comprising two demodulators, Dc and Ds, for the inphase and quadrature components of the composite QFPM signal respectively, each followed by a decoder, with the output data of two decoders applied to a parallel-to-serial converter, wherein the implementation of the demodulators Dc and Ds, the decoders and the parallel-to-serial converter are well known and described in the literature.

6. A preferred embodiment of claim 1, named Continuous Phase Quadrature Frequency/Phase Modulation (CP QFPM) comprising the elements of claims 2, 3, 4 and 5, with two essential modifications introduced in the modulation part, namely (a) an offset in the relative alignments of the sequences dc and ds by an amount equal to T/2, and (b) half-sinusoidal shaping of the inputs of said NFSK/2PSK modulators, Mc and Ms, and with respective modifications introduced in the demodulation part, namely (a) the reference signals for the correlators are cos()cos.omega.it and sin( )sin.omega.jt rather than cos.omega.it and sin.omega.jt respectively, and (b) the integration limits and sampling instants for the inphase and quadrature components of the CPQFPM signals are adjusted accordingly with the offset introduced in the modulation part.

7. A preferred embodiment of claim 6, named Continuous Phase Frequency/Phase Modulation (CP FPM), wherein the frequencies in the inphase and quadrature components are always equal, .omega.i=.omega.j, providing a constant envelope and continuous phase 4N-ary signal of the type NFSK/4PSK.

8. A demodulation method for recovering the data in CP FPM of claim 7, in accordance with the description given above.