3.3 FREQUENCY SYNCHRONIZATION OF NON-COHERENT FH / MFSK SYSTEMS
3.3.3 The Effects of Frequency Synchronization Error on FH/MFSK Error Probability Performance
The presence of a frequency synchronization error in an FH/MFSK receiver causes a degradation in system error probability performance due to factors
s; 2
s22 x1, x2, s12
p1bˆ0b2 s;
21b2^ E51z;m;220b6 12s222
rNh 312rg;4 m;1b2^ E5z;0b62s231g;4
z;R; 2
R;B 1 Nh a
rNh
j1
rj;
2 11r2j21sinc3p11>2 < b2, 11 0e0 2 4 22. mbˆ0b
Bˆ
sbˆ0b
e
mbˆ0b
sbˆ0b
mbˆ0b
sT2 s12
s22
s12s22 . s12 s2
32s2112bˆ2242 ; s22 s2 32s2112bˆ2242 ; x1 m
112bˆ222s2 ; x2 m 112bˆ222s2
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not unlike those produced by a timing synchronization error. In fact, both signal attenuation and loss of orthogonality degradations once again exist;
however, the functional dependence of these degradation components on the frequency error is quite different from the corresponding relationships for a timing error. Nevertheless, once these differences are identified, the remaining task of relating the above degradations to their impact on system error probability performance is straightforward in view of our previous detailed discussion for the effects of time synchronization errors. Thus, our discussion here will be brief and, wherever possible, draw heavily upon the results given in Section 3.2.3.
Starting first with the simple case of FH/MFSK with no diversity and orthogonal tones spaced by k/Th, then if, for example, the transmitted MFSK symbol in the j-th hop interval is given by
(3.114) then following the development in (3.97)—(3.103), and taking the sine and cosine Fourier transforms at ffIF1/2Th, the spectral estimate rjcorre- sponding to the transmitted MFSK symbol has the conditional pdf
(3.115) where
(3.116) Similarly, the spectral estimate rjnfor an incorrect correlator spaced in fre- quency by nk/Thfrom the correct one has the pdf of (3.115) with Dj( ,b) replaced by
(3.117) Note that for b 0,Dj( ,b) of (3.116) and Djn( ,b) of (3.117) reduce, respectively, to Dj( ) of (3.69) and Djn( ) of (3.71).
Comparing (3.115) with (3.73), it is clear that the bit error probability per- formance conditioned on fixed frequency and time synchronization errors is given by (3.78) together with (3.75) and (3.77) where Dj( ,b) and Djn(e, b) are used in place of Dj( ) and De jn( ) in (3.75).e
e e
e
e e
Djn1e, b2sinc23p1nkb2, 1 0e0 4.
e Dj1e, b2sinc23pb, 1 0e0 4.
p1rj0e, b2 à rj
s2 expec rj2
2s2ghDj1e, b2 d fI0c22ghDj1e, b2rj sd; 0rjq 0; otherwise
s1j21t2 22S sinc2pafofj 1
2Thbtd; 1j12ThtjTh
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Figures 3.27, 3.28, and 3.29 are the analogous results for performance in the presence of frequency error alone to those in Figures 3.18, 3.19, and 3.20, which depict performance in the presence of timing error alone.
Comparing Figure 3.28 with Figure 3.19, for example, we observe that in the case of frequency error, the loss of orthogonality is the dominant degra- dation component for all b, whereas, as previously mentioned, for time syn- chronization error, loss of orthogonality dominates only for large error values.
Returning to Figure 3.21, we observe that increasing the MFSK tone spac- ing to twice its minimum orthogonal value has a more pronounced effect on the performance improvement achieved in the presence of frequency error alone than that when timing error alone is present, particularly when the syn- chronization errors are large. When both time and frequency errors exist simultaneously, Figure 3.30 illustrates a set of curves that represent contours of constant Eb/NJdegradation for a fixed bit error probability. Both 4-ary and 8-ary FSK results are provided.
Frequency Synchronization of Non-Coherent FH / MFSK Systems 1019
Figure 3.27. 4-ary FSK with frequency error (minimum orthogonal tone spacing) (reprinted from [13]).
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1020 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.28. Performance degradation due to frequency error (reprinted from [13]).
Figure 3.29. 8-ary FSK performance degradation due to frequency error (reprinted from [13]).
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Frequency Synchronization of Non-Coherent FH / MFSK Systems 1021
Figure 3.30. Performance degradation due to time and frequency synchronization errors (reprinted from [13]).
Figure 3.31. Performance degradation due to frequency error with chip combining (reprinted from [13]).
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Finally, when m-diversity and non-coherent combining are superimposed on the basic FH/MFSK system of above, then again all of the results of Section 3.2.3.2 apply after making the same replacements of Dj( ,b) and Djn( ,b) for Dj( ) and Djn( ). In this regard, Figure 3.31 provides, for fre- quency error alone, the analogous results to Figure 3.22. Again we observe that increasing mimproves performance; however, a comparison of the two figures reveals that the reduction of loss of orthogonality by non-coherent combining plays a more prominent role with frequency errors than with tim- ing errors.When both time and frequency errors are present and fixed, Figure 3.32 illustrates the performance degradation as a function of m, the number of chips combined.
3.3.3.1 Average Error Probability Performance in the Presence of Frequency Synchronization Error Estimation
If the estimator of (3.106) is used for FH frequency synchronization and and is the fixed normalized time synchronization error, then a residual fre- quency offset arises, which, as in our previous discussions, affects system performance by reducing the signal energy available for the
n1b2^ bbˆ e
bˆ
e e
e
e
1022 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.32. Effect of chip combining on degradation due to synchronization errors (reprinted from [13]).
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non-coherent detection of the transmitted MFSK symbol and spreading sig- nal energy into adjacent MFSK frequency detectors. In particular, it is clear that in the presence of the residual offset, a signal attentuation degradation occurs that is given by
(3.118) and a loss of orthogonality also occurs that is given by
(3.119) Thus, if pn0b(n/b) denotes the b-conditional pdf of the residual error n(b), then analogoous to (3.93)
(3.120) It then follows that conditioned on , the average bit error probability Pb(e) is
(3.121)
where Pb( ,n) is obtained from (3.78) together with (3.77) and (3.75) with Dj( ,n) and Djn( ,n) used in place of Dj( ) and Djn( ).
Since, as previously noted, is comparatively small, we may make a sim- plifying assumption analogous to (3.95), namely,
(3.122) in which case (3.121) simplifies to
(3.123) Analogous to Figure 3.23, Figure 3.33 is an illustration of the average bit error probability performance of (3.123) for 4-ary and 8-ary FSK with no timing error ( 0) and Nh10. Comparing these two figures, we observe that the average bit error probability is considerably less degraded by a resid- ual frequency error than by a residual timing error.
e
Pb1e2201>2Pb1e, bmbˆ0b2db.
pbˆ0b1bn0b2d1n 1bmbˆ0b22 sbˆ0b
e e
e e
e
Pb1e2201>2dbbb11>2>2Pb1e, n2pbˆ0b1bn0b2dn
e
pn0b1n0b2pbˆ0b1bn0b2. Djn1e, n2sinc23p1nkn1b22, 1 0e0 4.
sinc23pn1b2, 1 0e0 4 Dj1e, n2sinc23p1bbˆ2, 1 0e0 4
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Finally, if both the fine time and fine frequency estimators of (3.53) and (3.106), respectively, are employed, then the overall error probability per- formance in the presence of the combination of the two residual synchro- nization errors can be computed from
(3.124) peˆ0e1eh0e2dndh
Pb401>2de01>2dbee11>2>2bb1>12>2Pb1e, n2pbˆ0b1bn0b2
1024 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.33. Average bit error probability performance in the presence of frequency error estimation;e0,Nh10.
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or from the approximate simplified expression
(3.125)