Note: Descriptions are shown in the official language in which they were submitted.
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~.
M QPPM CP QPPM CP PPM NPSK/2PSK QAM MPSK
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
T 3T
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)
11
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
B/2
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
M QFPM CP QFPM CP FPM QAM CP FSK NPSK/2PSK
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
M QFPM CP QFPM CP FPM QAM CP FSK NFSKI2PSK
2 o.PoS5K) ~ho-.83) (ho=.
4 0.1~~~ 0.83 0.83 0.1 0.79 U.I
8 0 95 0 15
16 0.2 1.-25 0 81 0.2 0.47 020
32 0 57 0 23
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
