2.3 PERFORMANCE OF DQASK IN THE PRESENCE OF
2.3.2 Receiver Characterization and Performance
Figure 2.8 depicts a receiver used to perform differentially coherent detec- tion of differentially phase-encoded QASK. The structure combines the ele- ments of a differentially coherent receiver for a constant envelope modulation such as MPSK with a non-coherent envelope detector. The out- put from these two receiver components, namely, detected envelope and dif-
f⫽tan⫺1am
nb; m,n⫽;1,;3,p ;1K⫺12, u1i2⫽u1i⫺12⫺f1i2.
s1i21t2⫽ 12A1i2 cos10t⫹u1i22, s1i2⫽d3bn1i2 ⫺jam1i24 ⫽A1i2e⫺jf1i2,
3When we discuss the differentially coherent detection process in the next section, we shall switch back to the rectangular representation to accommodate the rectangular-shaped decision regions.
Figure 2.8.A receiver for differentially coherent detection of differentially phase-encoded QASK.
4Without any loss in generality, we shall set u⫽0 for simplicy of notation.
ferential phase are then converted to equivalent in-phase and quadrature signals upon which multilevel decisions are made, as is done in the more con- ventional coherent QASK receiver.
Appearing at the receiver input in the i-th signalling interval is the trans- mitted signal of (2.58) to which the channel has added a random phase shift uand an AWGN which has the usual narrowband representation (repeated here for convenience)
(2.61) whereNc(t) and Ns(t) are statistically independent low-pass Gaussian noise processes with single-sided power spectral density N0 w/Hz. Thus, the received signal in the i-th signalling interval is of the form
(2.62) The receiver first performs in-phase and quadrature carrier demodulation with a pair of quadrature reference signals of known frequency v0 but unknown phase fa. The results of these demodulations are then passed through integrate-and-dump (I & D) filters whose outputs are given by4
(2.63) where
(2.64) The receiver next generates the equivalent envelope and phase of the I
& D outputs, namely,
(2.65) h1i2⫽tan⫺1aVs1i2
Vc1i2b. V1i2⫽ 21Vs1i222⫹1Vc1i222
ns⫽^ 冮1i⫺Ts12T
s
Ns1t2dt. nc⫽^ 冮1i⫺iTs12T
s
#Nc1t2dt
⫽A1i2Ts cos1u1i2⫹fa2⫹nc cos fa⫺ns sin fa, Vc1i2⫽ 冮1i⫺iTs12T
s
y1i21t2 312 cos1v0t⫺fa2dt
⫽A1i2Ts sin1u1i2⫹fa2⫹nc sin fa⫹ns cos fa
Vs1i2⫽ 冮1i⫺iTs12T
s
y1i21t2 3⫺12 sin1v0t⫺fa2dt
y1i21t2⫽s3t,u4 ⫹n1t2⫽ 12A1i2cos10t⫹u1i2⫹u2 ⫹n1t2. n1t2⫽ 123Nc1t2cos10t⫹u2 ⫺Ns1t2sin10t⫹u2 4,
Finally, the differential phase h(i)⫺h(i⫺1)is formed and used to produce the in-phase and quadrature decision variables V(i)cos (h(i)⫺h(i⫺1)and⫺V(i)sin
(h(i)⫺h(i⫺1)upon which K-level decisions are made.
At this point, it is convenient to redraw Figure 2.8 in its equivalent form illustrated in Figure 2.9 by recognizing that
. (2.66) Figure 2.9 has the advantage of resembling a conventional coherentQASK receiver [8] with a noisy carrier demodulation reference and thus its error probability performance can be obtained almost by inspection. In particu- lar, from (2.63) and (2.66), we obtain the decision variables
(2.67) Letting
(2.68) and using (2.59), we can rewrite (2.67) as
where
(2.70) Finally, recognizing that (2.69) resembles the decision variables for a coher- ent QASK receiver whose carrier demodulation reference signals are in error by ha1i⫺12 radians, we can immediately write down an expression for
N2⫽ns sin u1i⫺12⫹nc cos u1i⫺12. N1⫽ns cos u1i⫺12⫺nc sin u1i⫺12
⫽A1i2Ts cos1f1i2⫹ha1i⫺122⫹N1 sin ha1i⫺12⫹N2 cos ha1i⫺12
⫹ns sin1u1i⫺12⫹ha1i⫺122
Uc1i2⫽A1i2Ts cos1f1i2⫹ha1i⫺122⫹nc cos1u1i⫺12⫹ha1i⫺122
⫽A1i2Ts sin1f1i2⫹ha1i⫺122⫺N1 cos ha1i⫺12⫹N2 sin ha1i⫺12
⫺ns cos1u1i⫺12⫹ha1i⫺122
Us1i2⫽A1i2Ts sin1f1i2⫹ha1i⫺122⫹nc sin1u1i⫺12⫹ha1i⫺122 ha1i⫺12⫽h1i⫺12⫺u1i⫺12⫺fa
⫺ns sin1fa⫺h1i⫺122.
Uc1i2⫽ ⫺A1i2Ts cos1u1i2⫹fa⫺h1i⫺122⫹nc cos1fa⫺h1i⫺122
⫺ns cos1fa⫺h1i⫺122
Us1i2⫽ ⫺A1i2Ts sin1u1i2⫹fa⫺h1i⫺122⫺nc sin1fa⫺h1i⫺122 Vs1i2 Vc1i2
⫺V1i2 sin1h1i2⫺h1i⫺122⫽ ⫺V1i2 sin h1i2 cos h1i⫺12⫹V1i2 cos h1i2 sin h1i⫺12
Vc1i2 Vs1i2
V1i2 cos1h1i2⫺h1i⫺122 ⫽V1i2 cos h1i2 cos h1i⫺12⫹V1i2 sin h1i2 sin h1i⫺12
1aˆm and bˆ
n2
à à
à à
Figure 2.9.An alternate and equivalent implementation of the receiver shown in Figure 2.8.
the symbol error probability of differentially coherent detected QASK (conditioned on the (i⫺1)-st symbol SNR), namely,
(2.71) where , the probability density function (pdf) of the nor- malized phase in the (i⫺1)-st signalling interval, is given by [2]
(2.72) with the (i⫺1)-sttransmission interval symbol SNR defined by
(2.73) andQ(x) denoting, as in previous chapters, the Gaussian probability inte- gral. Also from (51) of [8], with freplaced by
(2.74) In (2.74), the sum over jis for values ⫾1,⫾3, . . . ⫾(K⫺1) while the sums overkandlare for values 0,⫾2,⫾4, . . . ,⫾(K⫺2). Also,
(2.75) where
(2.76) is the average symbol SNR of the QASK-K2signal set with average power Sdefined in (2.56).
Finally, the average symbol error probability,Ps, is obtained by averag- ing (2.71) over the pdf of . To obtain this pdf, we note that, for a given K, the (i⫺1)-st symbol signal power (A(i⫺1))2ranges over K(K⫹2)/8 dif- ferent values.K/2 of these values correspond to signal points on the diago- nal of any quadrant and occur with probability 4/K2. The remaining
gs1i⫺12
gs⫽^ STs
N0
¢⫽^ B
3gs
K2⫺1
⫻Q5¢3l⫹ 11⫺l2cosha1i⫺12⫺ 1k⫺12sinha1i⫺124 6.
⫺ 4 K2 a
k,l
Q5¢3k⫹ 11⫺k2cosha1i⫺12⫹1l⫺12sinha1i⫺124 6
Ps1ha1i⫺122⫽ 4
K2 a
j,l
Q5¢3l⫹ 1l⫺12cosha1i⫺12⫺j sin ha1i⫺124 6
ha1i⫺12,
gs1i⫺12⫽ 1A1i⫺1222T N0
,
gs1i⫺12
⫽ à 1
2p exp1⫺gs1i⫺122 51⫹22pgs1i⫺12 cos ha1i⫺12 exp1gs1i⫺12cos2ha1i⫺122
⫻Q3⫺22gs1i⫺12 cos ha1i⫺124 6; 0ha1i⫺120 ⱕp
0; elsewhere
p11ha1i⫺12;gs1i⫺122
ha1i⫺12
p11ha1i⫺12;gs1i⫺122
Ps1gs1i⫺122⫽ 冮⫺ppPs1ha1i⫺122p11ha1i⫺12;gs1i⫺122dha1i⫺12
K(K ⫺ 2)/8 values correspond to off-diagonal signal points either above or below the diagonal of any quadrant and occur with probability 8/K2. Thus, takes on the discrete set,r, of values ⌬2(m2⫹n2)/2;m,n⫽1, 3, . . . , (K⫺1);mⱕnand the corresponding pdf is then
(2.77) where⌬is defined in (2.75). Averaging (2.71) over the pdf of (2.77) gives the desired result
(2.78) Equivalently, letting
(2.79) represent the “effective” pdf of the (i ⫺ 1)-st symbol phase, then (2.78) becomes the simple result
(2.80) Figure 2.10 is a plot of Psversusgsin decibels as evaluated from (2.80) forK⫽4 (DQASK-16). Also shown is the corresponding result for coher- ent detection of QASK which, for K⫽4, is given by [8]:
(2.81) For comparison, the performance of coherent and differentially coherent detection of MPSK with M⫽16 (i.e., PSK-16 and DPSK-16) is presented in Figure 2.10 [5]. For small gs, the coherent PSK-16 and the DQASK-16 per- form almost identically, but for large gs, the DQASK-16 approaches the per- formance of DPSK-16. Also, for large gs, coherent QASK-16 is about 4 dB better than coherent PSK-16 showing the more favorable exchange of aver- age power for bandwidth with the QASK-16 than with the PSK-16. While
Ps⫽3Q1¢2 31⫺34Q1¢2 4.
Ps⫽ 冮⫺ppPs1ha1i⫺122p1ha1i⫺122dha1i⫺12.
⫹ 8 K2 a
K⫺1
m,n⫽1, 3p m6n
p1aha1i⫺12; am2⫹n2
2 b¢2b
⫽ 4 K2 a
K⫺1
m⫽1, 3,pp11ha1i⫺12;m2¢22
p1ha1i⫺122⫽ a
r
p11ha1i⫺12;gs1i⫺122p1gs1i⫺122
⫽ 冮⫺ppPs1ha1i⫺122 c ar
p11ha1i⫺12;gs1i⫺122p1gs1i⫺122 ddha1i⫺12
Ps⫽ a
r
Ps1gs1i⫺122p1gs1i⫺122
p1gs1i⫺122⫽e
4
K2 ; gs1i⫺12⫽m2¢2; m⫽1, 3,p,1K⫺12
8
K2 ; gs1i⫺12⫽ am2⫹n2
2 b¢2;
m,n⫽1, 3,p 1K⫺12; m 6 n
gs1i⫺12
it is true that DQASK-16 suffers a significant performance deegradation with respect to coherent QASK-16 at large gs, we must recall our initial moti- vation, namely to use DQASK-16 along with frequency hopping (i.e., FH/DQASK) to protect a conventional QASK communication system against jamming. In the next two sections, we present the FH/DQASK per- formance in the presence of partial-band jamming and noise jamming.