Although our primary interest in this chapter is with slow or fast FH/MFSK, it is perhaps more instructive to first discuss the most basic non-coherent FH acquisition configuration, which, although best suited to analog information transmitted as an amplitude modulation, or no information modulation at all, nevertheless represents the simplest illustration of the above-mentioned acquisition process dichotomy. As such, consider the single (fixed) dwell time serial search acquisition system illustrated in Figure 3.1. In this scheme, the received FH signal plus noise is correlated in a wideband mixer with the local hop sequence produced by an FH synthesizer driven by a PN generator
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960 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.1.A single dwell time serial search FH acquisition system.
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FH Acquisition Techniques 961
whose epoch is controlled in accordance with the decision to continue the search. The result of this correlation is passed through an IF filter followed by an energy detector implemented here as a baseband non-coherent demodulator.1(The bandwidths of the IF filter and demodulator low-pass filters are chosen based upon considerations to be discussed shortly.) Post- detection integration of the energy detector output produces a signal whose mean value is nominally zero when the two hop sequences are misaligned and non-zero when they are either partially or fully aligned. Thus, compar- ing this signal with a preset threshold allows a decision to be made as to whether or not FH acquisition has been achieved, or equivalently whether or not to step the PN code epoch and continue the search.
A similar serial search technique for acquiring PN sequences was dis- cussed in great detail in Chapter 1. Thus, our discussion here will be, in com- parison, brief and merely serve to highlight the essential differences between the two systems.
To begin, first suppose that the received FH signal and the locally gener- ated hop signal out of the mixer will appear as in Figure 3.2a.2Now, if the bandwidth of the IF filter is chosen to be less than twice the hop frequency spacing, then all of the frequency components of the mixer difference sig- nal will be outside this bandwidth, resulting in a zero correlation voltage at the demodulator input. Now suppose that the received sequence and syn- thesizer sequence are partially aligned, i.e., misaligned by less than a single hop interval. Then, the mixer difference signal, as illustrated in Figure 3.2b, will contain frequency components within the IF bandwidth (assuming a composite frequency error dfless than the hop spacing) which are effective for correlation detection. These “bursts” of sinusoids at a frequency fIFdf have random phases relative to one another resulting in an IF filter output x(t) which does not have a discrete spectral component at this frequency. In fact, for a given timing offset twith magnitude less than a hop interval Th, we can write x(t) as
(3.1) rectt1t2 à
0; 0tt 1t 7 02 Th 0t0 tTh 1t 6 02 1; ttTh 1t 7 02
0tTh 0t0 1t 6 02 x1t2 22S a
i
rectt1tiTh2cos32p1fIFdf2tci4
1Alternately, the baseband non-coherent demodulator could be replaced by a band-pass square- law envelope detector, as in our previous discussions of serial search acquisition of PN sequences in Chapter 1.
2For simplicity of this discussion, we shall for the time being ignore the information modula- tion and the additive noise.
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962 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.2a.Received FH signal and local hop signal misaligned by more than a single hop interval.
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FH Acquisition Techniques 963
Figure 3.2b.Received FH signal and local hop signal misaligned by less than a single hop interval.
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which has the power spectral density
(3.2) The non-coherent demodulator forms a filtered measure of the average power in x(t) by demodulating it with quadrature reference signals and and summing the squares of the resultant filtered low-pass signals. Thus, if 0H(j2pf)02denotes the squared magnitude of the low-pass filter transfer function, then the non-coherent demodulator out- put has an average value
(3.3) Assuming first that the bandwidth of the demodulator low-pass filters is much larger than the hop rate 1/Th, and, in addition, the frequency error is small (relative to this bandwidth), then (3.3) simplifies to
(3.4) i.e., a triangular correlation curve for width 2Th. In general, the non- coherent demodulator low-pass filter bandwidth must be chosen large enough to accommodate the information modulation bandwidth, the max- imum system frequency error, and the hop frequency modulating spectrum.
Thus, for slow frequency hopping (SFH), wherein the information symbol rate dominates this choice, the above assumption of a large low-pass filter bandwidth relative to the hop rate is valid and hence no significant filter- ing of the correlation curve occurs in accordance with (3.4). For fast fre- quency hopping (FFH) where the low-pass filter bandwidth is chosen on the order of the hop rate, significant filtering occurs and the correlation curve must be computed from (3.3). As an example of the filtering distor- tion of the triangular correlation function, consider the case of single-pole low-pass filters with single-sided noise bandwidth B. Then evaluation of
Sa1 0t0 Thb R1t2 S1Th 0t0 22
Th qqcsin pfpf¿1T¿1hTh0t00 2 dt0 2
2
df¿ S1Th 0t0 22
Th qq
csin5pfdf21Th 0t0 26 p1fdf21Th 0t0 2 d
20H1j2pf2 02df.
R1t2 qqSx1ffIF; t2 0H1j2pf2 02df
12 sin 2pfIFt 12 cos 2pfIFt
S1Th 0t0 22
Th csin5p3f1fIFdf2 4 1Th 0t0 26 p3f 1fIFdf2 4 1Th 0t0 2 d
2
. Sx1f; t2 1
Th 0f5x1t26 02
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(3.3) with df0 yields
(3.5)
which is plotted against t/Th in Figure 3.3 with BTh as a parameter. We observe from this figure that in the neighborhood of t0, the predominant effect of the filtering is a reduction of the correlation peak by an amount approximately given by 1/4BThand a corresponding broadening of the tri- angular shape.
When additive noise and possibly interference signals are present, the design of the demodulator low-pass filters will be governed by considera- tions additional to minimum correlation function degradation, which, as we noted above, requires their bandwidth to be large. In particular, minimiza- tion of the demodulator square-law noise output requires that these same bandwidths be chosen small. thus, as is characteristic of square-law demod- ulation systems, a tradeoff exists between signal signal and signal noise plus noise noise degradations.
In summary, then, for SFH of an information-bearing carrier, the low-pass filter bandwidths must be commensurate with the information modulation bandwidth and therefore the demodulator output signal-to-noise ratio will be set by this bandwidth. For FFH of the same information-modulated car- rier, the low-pass filter bandwidths must be large enough to accommodate
R1t2Sa1 0t0 Thb ≥1
1expe4BTha1 0t0 Thb f 4BTha1 0t0
Thb ¥
FH Acquisition Techniques 965
Figure 3.3. Filter effects on the FH correlation function.
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966 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.4.A serial search FH acquisition system with integrate-and-dump-type demodulator.
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the hop rate (which is larger than the modulation bandwidth) and can there- fore be set to maximize the demodulator output signal-to-noise ratio.
One way of avoiding the degradation of the correlation curve peak in the FFH cse is to replace the low-pass filters in the non-coherent demodulator with integrate-and-dump circuits as in Figure 3.4. Since for the rectangular bursts of signal correlation appearing at the demodulator input, the integrate-and-dumps act as matched filters, then the demodulator output samples have an average value
(3.6) or for df0,
(3.7) It is interesting at this point to note that this very same function of time and frequency offset as expressed by (3.6) will again be significant later on in the chapter when we study the effects of time and frequency errors on fine syn- chronization performance.
With the previous discussion as background, we now return our atten- tion to the case of primary interest in this chapter, namely, coarse acquisi- tion of fast or slow FH/MFSK. A basic serial search acquisition configuration for FFH/MFSK is illustrated in Figure 3.5. For the case when the received and local hop signals are misaligned by less than a single hop interval, the bottom line of Figure 3.6 is the sequence of frequencies char- acterizing the mixer difference signal. If, as before, the IF filter bandwidth is chosen narrow enough to eliminate the difference of two adjacent hop frequencies, but wide enough now to pass the entire MFSK signalling fre- quency band, then only the frequency components corresponding to the non-crosshatched areas in Figure 3.6 will pass through this filter and be available for correlation detection. Thus, we observe that in each symbol interval Ts, which is synchronous with the local FH synthesizer, the signal component of the IF filter output will consist of mTs/Thbursts of sinu- soid of duration Th2 0t0. These mbursts are all at the same frquency (cor- responding to the particular MFSK tone transmitted in that symbol interval) but have random phases which are independent of one another.
A measure of the lack of coarse time synchronization can therefore be obtained by separately combining the energies detected in each hop inter- val at each of the Mpossible MFSK frequencies {fsi;i1, 2, . . . ,M} and then choosing the largest of these m-fold diversity combinations. Since this selection is made only once per symbol interval, post-detection accumu- lation (over say NFFHsymbols) is required, the result of which is compared with a preset threshold to determine whether or not to continue the search.
R1t2Sa1 0t0 Thb2. R1t2S sin2pdf1Th 0t0 2
1pdfTh22
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968 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.5.A serial search acquisition system for FFH/MFSK.
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FH Acquisition Techniques 969
Figure 3.5(continued)
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970 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.6.Received FFH/MFSK signal and local hop signal misaligned by less than a single hop interval.
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In the absence of noise, for the symbol interval during which fiwas the frequency of the transmitted MFSK tone, the m-fold accumulated output samples of the Mnon-coherent demodulators are given by
(3.8) Thus if dfis small compared with the spacing between adjacent MFSK tones, then the largest Rk(t) will occur for the value of k for which fifsk. Since this occurrence will be true independent of which symbol we examine, the input to the post-detection accumulator will be given by R(t) of (3.6) mul- tiplied by m.
For ordinary (without diversity) SFH/MFSK, the appropriate serial search acquisition system analogous to Figure 3.5 is illustrated in Figure 3.7 with a corresponding time-frequency diagram in Figure 3.8. Since in each hop inter- val, the mixer difference signal contains sinusoidal bursts at different fre- quencies corresponding to the MFSK tones transmitted in that hop, then in contrast with Figure 3.5, no non-coherent combining ocfurs at the outputs of the Mnon-coherent demodulators. Rather, the largest of these Moutputs is selected each symbol interval. Post-detection accumulation (over say NSFH symbols) of these selections and comparison with a preset threshold again determines whether or not to continue the search.
Another characteristic of the acquisition system in Figure 3.7 is that the set of Mnon-coherent demodulator outputs does not necessarily remain unchanged as one passes from symbol to symbol within a given hop. For example, if the timing error t is less than a symbol interval Ts(as is the case illustrated in Figure 3.8), then for all symbols except the last in a given hop interval, the sets of Moutputs will be identical, although not neces- sarily ordered the same way within a given set. Since a maximum is sought in each symbol interval, the ordering is unimportant and thus in each case a decision is made among Menergy detections corresponding to an input of Tssec of a given MFSK tone and tsec of the adjacent transmitted tone.
For the last symbol of that hop, however, only Ts t sec of the corre- sponding MFSK tone is available for energy detection, the remaining t sec corresponding to noise only. As t increases beyond Tssec, fewer and fewer symbol decisions will be based upon a full Tssec of input signal.
Finally, when tTh, the entire mixer difference signal will contain noise only.
For SFH/MFSK with diversity m, the appropriate serial search acquisition system reverts back to one resembling Figure 3.5, where the outputs of the M non-coherent demodulators are now individually summed over the m chips (one per hop) corresponding to a given symbol. As such, the integrate- and-dumps in each non-coherent demodulator operate over a chip interval
k1, 2,p, M.
Rk1t2mS sin25p3df1fskfi2 4 1Th 0t0 26 5p3df1fskfi2 4Th62 ;
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972 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.7.A serial search acquisition system for SFH/MFSK.
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FH Acquisition Techniques 973
Figure 3.7.(continued)
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974 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.8.Received SFH/MFSK signal and local hop signal misaligned by less than a single hop interval.
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corresponding to Ts/m and the demodulator outputs must be delayed by intervals of Thsec before accumulation.
To evaluate the performance of the various serial search FH/MFSK acqui- sition systems in the presence of noise, we must determine their operating characteristic, i.e., the relation among false alarm probability PFA, detection probability PD, and dwell time td. To determine this relation, we shall assume that the post-detection accumulation size Nh(actually NFFHor NSFH) is large so that the statistics at the input to the threshold comparison in Figures 3.5 and 3.7 may be assumed Gaussian. A similar assumption was made in Chapter 1 in connection with our discussion of serial search PN code acqui- sition. Because of this simplifying assumption, it is sufficient to find only the mean and variance of the signal at the post-detection accumulator input in both the in-sync and out-of-sync conditions.
To illustrate the procedure, we consider first the case of no diversity and equal symbol and hop rates. Thus, with m1 and TsTh, Figures 3.5 and 3.7 are identical. Assuming first the out-of-sync condition (tTh), then the mixer output is noise only and the Mnon-coherent demodulator (normal- ized) outputs all have the same probability density function (pdf) given by (see (1.73) of Chapter 1)
(3.9) On the other hand, for the “ideal” in-sync condition (tdf0) and orthog- onal MFSK tone spacing (i.e., an integer multiple of the symbol rate), then M1 of the demodulator outputs will be characterized by (3.9), while the remaining output corresponding to the transmitted tone (say fsl) has the pdf (see (1.72) of Chapter 1)
(3.10) where gh STh/N0is the hop signal-to-noise ratio, or, in this case, also the symbol signal-to-noise ratio.
Letting denote the random variable corresponding to the largest of the Mnon-coherent demodulator normalized outputs at the i-th sampling (once per hop) instant, then the probability density function of is given by
(3.11) qN1Yi*2 d
dYi* 3PN1Yi*2 4MMpN1Yi*2 3PN1Yi*2 4M1 Yi* Yi*
^
pSN1yl*2 ee1yl*gh2I0122ghyl*2; yl*0
0; otherwise
k1, 2,p, M.
pN1yk*2 eeyk*; yk*0 0; otherwise
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for the out-of-sync condition and
(3.12) for the in-sync condition where PN(Y*) and PS1N(Y*) are, respectively, the probability distribution functions corresponding to the pdf’s pN(y*) and PSN(y*) of (3.9) and (3.10), i.e.,
(3.13) Substituting (3.9) and (3.10) into (3.13) and the results of these integral evaluations into (3.12) gives for the in-sync condition
(3.14)
The corresponding result for the out-of-sync condition is obtained by letting gh0 in (3.14), i.e.,
(3.15) As previously mentioned, we need to determine the first two central moments of in order to evaluate the operating characteristic of the acqui- sition system. From (3.14), we can determine the mean of for the per- fectly in-sync condition as
(3.16) 1gh1M12a
M2 k0
112k
1k221k122 aM2
k bexpcak1 k2bghd mSN^ qq
Yi*qSN1Yi*2dYi*
Yi* Yi*
qN1Yi*2 eM exp1Yi*2 31exp1Yi*2 4M1; Yi*0 0; otherwise.
qSN1Yi*2e
exp31Yi*gh2I0122ghYi*2 31exp1Yi*2M1 1M12exp1Yi*2 31exp1Yi*2 4M2 0Y*iexp31Ygh2 4I0122ghY2dY; Y0
0; otherwise.
PSN1Y*2 Y*q
pSN1y*2dy*.
PN1Y*2 Y*qpN1y*2dy*
1M12pN1Yi*2PSN1Yi*2 3PN1Yi*2 4M2 pSN1Yi*2 3PN1Yi*2 4M1
qSN1Yi*2 d
dYi*5PSN1Yi*2 3PN1Yi*2 4M16
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and for the out-of-sync condition,
(3.17) Note that the leading terms of (3.16) and (3.17) correspond to the means of y*as determined from pSN(y*) and pN(y*) in (3.10) and (3.9), respectively (see (1.75) and (1.76) of Chapter 1). Similarly, the mean-squared value of
under the two sync conditions is determined as
(3.18) and
(3.19) Thus, the in-sync and out-of-sync variances of are given by
(3.20) where once again the leading terms will correspond to the variances of y* (see (1.75) and (1.76) of Chapter 1).
Post-detection accumulation of produces the approximately Gaussian (Nhlarge) random variable
(3.21) which when compared with the normalized threshold gives rise to a false alarm probability (exceeding the threshold when in the out-of-sync condition)
(3.22) Qah*NhmN
2NhsN2 b ^ Q1b2 PFA h*q22pN1 hsN2 expc 1Z*NhmN22
2NhsN2 ddZ*
h*
Z* a
Nh
i1
Yi* Yi* sN2 1Yi*2N2 mN2
sS2N1Yi*2S2NmS2N
Yi* 1Yi*22N221M12a
M2 k0
112k
1k221k122 aM2
k b c 1
k2 1 k1d. c a 1
k2b a gh
k21b 1
k1dexpcak1 k2bghd
1Yi*2SN2 24ghgh221M12a
M2 k0
112k
1k221k122 aM2
k b Yi*
mN1 1M12 a
M2 k0
112k
1k221k122 aM2
k b.
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and a detection probability (exceeding the threshold when in the in-sync condition)
(3.23)
where, as in previous chapters,Q(x) is the Gaussian probability integral.
Eliminating bbetween (3.22) and (3.23) produces the desired system oper- ating characteristic
(3.24)
where SN,N, , and are determined from (3.16)—(3.20) and are all functions of the signalling alphabet size Mand hop signal-to-noise ratio gh. Alternately, since the dwell time tdof the system is related to the accu- mulation size Nhby
(3.25) then (3.24) can be expressed in terms of the dwell time-hop rate product, which produces a relation analogous to (1.81) of Chapter 1. Figures 3.9 and 3.10 are plots of false alarm probability PFAversus normalized dwell time td/Thwith detection probability PD as a parameter for gh 20 dB and either 2-ary or 8-ary FSK, respectively. Clearly, as the number of signalling levels Mincreases, the required post-detection accumulation increases pro- portionally.
When m-diversity is employed, then the appropriate pdf’s analogous to (3.9) and (3.10) become
(3.26) pN1yk*2 • 1yk*2m1
1m12! exp1yk*2; yk*0 0; otherwise
Nh td Th , sN2
sS2N PDQ°
Q11PFA2 1mSNmN2B Nh sN2 sSN>sN ¢ Q°
bB Nh
sN2 1mSNmN2 sSN>sN ¢ PDQah*NhmSN
2NhsS2N b Qah*NhmNNh1mSNmN2 2NhsN21sSN>sN2 b
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and
(3.27) Similarly the out-of-sync and in-sync pdf’s for the largest of the Mnonco- herently combined normalized demodulator outputs are given respectively by
(3.28) Yi*0
qN1Yi*2 •
M1Yi*2m1
1m12! exp1Yi*2 c1 a
m1 k0
1Yi*2k
k! exp1Yi*2 dM1; 0; otherwise
pSN1yk*2 • a yk*
mghb1m12>2exp31yk*mgh2 4Im1124mghyk*2; yk*0
0; otherwise.
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Figure 3.9. False alarm and detection probability performance of serial search FH/MFSK acquisition system;gh 20 dB.
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and
(3.29) c1 a
m1 k0
1Yi*2k
k! exp1Yi*2 dM20Yi*
a Y
mghb1m12>2
exp31Ymgh2 4Im1124mghY2dY; Yi*0 0; otherwise
a Yi*
mghb1m12>2exp31Yi*mgh2 4Im1124mghYi*2
c1 a
m1 k0
1Yi*2k
k! exp1Yi*2 dM1 1M12 1Yi*2m1
1m12! exp1Yi*2
980 Time and Frequency Synchronization of Frequency-Hopped Receivers
Figure 3.10. False alarm and detection probability performance of serial search FH/MFSK acquisition system;gh 20 dB.
à
qSN1Yi*2
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with first two moments
(3.30) where is the k-th generalized Laguerre polynomial of order m 1 [3] and Ckjare multinomial coefficients which satisfy the recursion rela- tionship
(3.31) Thus, recognizing that mSNand mNcorrespond to (3.30) with n1, and fur- ther obtaining sSNand sNfrom (3.20) using (3.30) evaluated for n2, one uses (3.24) to obtain the system operating characteristic for the m-diversity case.
We mention in passing that for acquisition in noise, but in the absence of information modulation (see Figure 3.4 for the appropriate system), one merely sets M1 in (3.16)—(3.20), which, when substituted into (3.24), gives the simple result
(3.32) Note that if the acquisition system of Figure 3.1 were used in place of that in Figure 3.4, then (1.32) would still be appropriate with ghreplaced by S/2N0Bwhere Bis again the single-sided low-pass filter noise bandwidth.
Under these conditions, (3.32) becomes analogous to (1.81) of Chapter 1. In fact, the same substitution is appropriate to all of the previous
PDQcQ11PFA2 2Nhgh 112gh
d. Ckj •
1; j0
1 j a
min1j, m12 l1
kllj
l! Ck, jl j1, 2,p, k1m12. Lk1m121x2
1Yi*2Nn 1Yi*2SnN0gh0; n1, 2 a
1k121m12
j0
Ck1, j 1jn2!
1k22jn Ljn1m12a mgh
k2b f a
mjn1
l0 ak1
k2blLl1m12a mgh
k2b 1
1k12n1k22m e a
k1m12 j0
Ckj 1mjn12! 1m12!1k12mj 1M12 a
M2
k0 aM2
k b112k expcak1 k2bmghd 1Yi*2SnN 3m11gh2 4n 1n12m112gh2
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