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-
ftan1am
nb; m, n;1, ;3,p ;1K12, u1i2 u1i12 f1i2.
s1i21t2 12A1i2 cos10tu1i22, s1i2d3bn1i2jam1i24 A1i2ejf1i2,
732 Differentially Coherent Modulation Techniques
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.
12_c02 8/16/01 5:28 PM Page 732
http://jntu.blog.com
Performance of DQASK in the Presence of Additive White Gaussian Noise 733
Figure 2.8.A receiver for differentially coherent detection of differentially phase-encoded QASK.
http://jntu.blog.com
734 Differentially Coherent Modulation Techniques
4Without any loss in generality, we shall set u0 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) where Nc(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) h1i2tan1aVs1i2
Vc1i2b. V1i2 21Vs1i2221Vc1i222
ns^ 1i12TTs
s
Ns1t2dt.
nc^ 1iiTs12T
s
#N
c1t2dt
A1i2Ts cos1u1i2fa2nc cos fans sin fa,
Vc1i2 1i12TiTs
s
y1i21t2 312 cos1v0tfa2dt
A1i2Ts sin1u1i2 fa2nc sin fans cos fa Vs1i2 1iiTs12T
s
y1i21t2 312 sin1v0tfa2dt
y1i21t2 s3t, u4 n1t2 12A1i2cos10tu1i2u2n1t2. n1t2 123Nc1t2cos10tu2 Ns1t2sin10tu2 4,
12_c02 8/16/01 5:28 PM Page 734
http://jntu.blog.com
Finally, the differential phase h(i)h(i1)is formed and used to produce the in-phase and quadrature decision variables V(i)cos (h(i)h(i1)and V(i)sin (h(i)h(i1)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 ha1i12 radians, we can immediately write down an expression for
N2ns sin u1i12nc cos u1i12. N1ns cos u1i12nc sin u1i12
A1i2Ts cos1f1i2ha1i122N1 sin ha1i12N2 cos ha1i12
ns sin1u1i12ha1i122
Uc1i2A1i2Ts cos1f1i2ha1i122nc cos1u1i12ha1i122 A1i2Ts sin1f1i2ha1i122N1 cos ha1i12N2 sin ha1i12
ns cos1u1i12ha1i122
Us1i2A1i2Ts sin1f1i2 ha1i122nc sin1u1i12ha1i122 ha1i12 h1i12u1i12fa
ns sin1fah1i122.
Uc1i2 A1i2Ts cos1u1i2fah1i122nc cos1fah1i122 ns cos1fah1i122
Us1i2 A1i2Ts sin1u1i2fah1i122nc sin1fah1i122 Vs1i2 Vc1i2
V1i2 sin1h1i2h1i122 V1i2 sin h1i2 cos h1i12V1i2 cos h1i2 sin h1i12 Vc1i2 Vs1i2
V1i2 cos1h1i2h1i122V1i2 cos h1i2 cos h1i12 V1i2 sin h1i2 sin h1i12 1aˆm and bˆn2
Performance of DQASK in the Presence of Additive White Gaussian Noise 735
à à
à à
http://jntu.blog.com
736 Differentially Coherent Modulation Techniques
Figure 2.9.An alternate and equivalent implementation of the receiver shown in Figure 2.8.
12_c02 8/16/01 5:28 PM Page 736
http://jntu.blog.com
the symbol error probability of differentially coherent detected QASK (conditioned on the (i1)-st symbol SNR), namely,
(2.71) where , the probability density function (pdf) of the nor- malized phase in the (i1)-st signalling interval, is given by [2]
(2.72) with the (i1)-st transmission interval symbol SNR defined by
(2.73) and Q(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, . . . (K1) while the sums over kand lare for values 0,2,4, . . . ,(K2). 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 (i1)-st symbol signal power (A(i1))2ranges over K(K2)/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
gs1i12
gs^ STs N0
¢ ^ B 3gs K21
Q5¢3l 11l2cos ha1i12 1k12sin ha1i124 6. 4
K2 a
k, l
Q5¢3k 11k2cos ha1i12 1l12sin ha1i124 6 Ps1ha1i122 4
K2 a
j, l
Q5¢3l 1l12cos ha1i12j sin ha1i124 6 ha1i12,
gs1i12 1A1i1222T N0 ,
gs1i12 à
1
2p exp1gs1i122 5122pgs1i12 cos ha1i12 exp1gs1i12cos2ha1i122 Q322gs1i12 cos ha1i124 6; 0ha1i120 p 0; elsewhere
p11ha1i12; gs1i122 ha1i12 p11ha1i12; gs1i122
Ps1gs1i122 ppPs1ha1i122p11ha1i12; gs1i122dha1i12
Performance of DQASK in the Presence of Additive White Gaussian Noisehttp://jntu.blog.com 737
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(m2n2)/2;m,n1, 3, . . . , (K1);mnand 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 Psversus gsin decibels as evaluated from (2.80) for K4 (DQASK-16). Also shown is the corresponding result for coher- ent detection of QASK which, for K4, 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
Ps3Q1¢2 3134Q1¢2 4. Ps ppPs1ha1i122p1ha1i122dha1i12.
8 K2 a
K1 m, n1, 3p
m6n
p1aha1i12; am2n2 2 b¢2b 4
K2 a
K1 m1, 3,p
p11ha1i12; m2¢22 p1ha1i122 a
r
p11ha1i12; gs1i122p1gs1i122
ppPs1ha1i122 c ar p11ha1i12; gs1i122p1gs1i122 ddha1i12
Ps a
r
Ps1gs1i122p1gs1i122 p1gs1i122e
4
K2 ; gs1i12m2¢2; m1, 3,p, 1K12 8
K2 ; gs1i12 am2n2 2 b¢2;
m, n1, 3,p 1K12; m 6 n gs1i12
738 Differentially Coherent Modulation Techniques
12_c02 8/16/01 5:28 PM Page 738
http://jntu.blog.com
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.