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
Figure 3.1.A single dwell time serial search FH acquisition system.
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 fIF⫺df 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; 0ⱕtⱕt 1t 7 02 Th⫺ 0t0 ⱕtⱕTh 1t 6 02 1; tⱕtⱕTh 1t 7 02
0ⱕtⱕTh⫺ 0t0 1t 6 02 x1t2⫽ 22S a
i
rectt1t⫺iTh2cos32p1fIF⫺df2t⫹ci4
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.
Figure 3.2a.Received FH signal and local hop signal misaligned by more than a single hop interval.
Figure 3.2b.Received FH signal and local hop signal misaligned by less than a single hop interval.
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 Th
b R1t2⫽ S1Th⫺ 0t0 22
Th 冮⫺qqcsinpf¿1Tpf¿1Th⫺h⫺0t00 2t0 2d2df¿
⫽S1Th⫺ 0t0 22
Th 冮⫺qqcsinp1f5pf⫹⫹dfdf21T21Thh⫺⫺0t0t0 20 26d20H1j2pf2 02df. R1t2⫽ 冮⫺qqSx1f⫹fIF;t2 0H1j2pf2 02df
12 sin 2pfIFt 12 cos 2pfIFt
⫽S1Th⫺ 0t0 22 Th
csin5p3f⫺ 1fIF⫺df2 4 1Th⫺ 0t0 26 p3f⫺1fIF⫺df2 4 1Th⫺ 0t0 2 d2. Sx1f;t2⫽ 1
Th
0f5x1t26 02
(3.3) with df⫽0 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 t⫽0, 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
R1t2⫽Sa1⫺ 0t0 Th
b ≥1⫺
1⫺expe⫺4BTha1⫺ 0t0 Th
b f 4BTha1⫺ 0t0
Th
b ¥
Figure 3.3. Filter effects on the FH correlation function.
Figure 3.4.A serial search FH acquisition system with integrate-and-dump-type demodulator.
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 df⫽0,
(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 intervalTs, which is synchronous with the local FH synthesizer, the signal component of the IF filter output will consist of m⫽Ts/Thbursts of sinu- soid of duration Th20t0. 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;i⫽1, 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.
R1t2⫽Sa1⫺ 0t0 Th
b2. R1t2⫽Ssin2pdf1Th⫺ 0t0 2
1pdfTh22
Figure 3.5.A serial search acquisition system for FFH/MFSK.
Figure 3.5(continued)
Figure 3.6.Received FFH/MFSK signal and local hop signal misaligned by less than a single hop interval.
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 kfor which fi⫽fsk. 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 M outputs 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 ofTssec 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 tincreases beyond Tssec, fewer and fewer symbol decisions will be based upon a full Tssec of input signal.
Finally, when t⫽Th, 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
k⫽1, 2,p,M.
Rk1t2⫽mSsin25p3df⫺ 1fsk⫺fi2 4 1Th⫺ 0t0 26 5p3df⫺ 1fsk⫺fi2 4Th62 ;
Figure 3.7.A serial search acquisition system for SFH/MFSK.
Figure 3.7.(continued)
Figure 3.8.Received SFH/MFSK signal and local hop signal misaligned by less than a single hop interval.
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 probabilityPD, and dwell time td. To determine this relation, we shall assume that the post-detection accumulation size Nh(actuallyNFFHorNSFH) 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 m⫽1 and Ts⫽Th, Figures 3.5 and 3.7 are identical. Assuming first the out-of-sync condition (tⱖTh), 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 (t⫽df⫽0) and orthog- onal MFSK tone spacing (i.e., an integer multiple of the symbol rate), then M⫺1 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 theMnon-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 4M⫽MpN1Yi*2 3PN1Yi*2 4M⫺1 Yi* Yi*
⫽^
pS⫹N1yl*2⫽ ee⫺1yl*⫹gh2I0122ghyl*2; yl*ⱖ0
0; otherwise
k⫽1, 2,p,M. pN1yk*2⫽ ee⫺yk*; yk*ⱖ0
0; otherwise
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 PS⫹N(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 gh⫽0 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)
⫽1⫹gh⫹1M⫺12a
M⫺2 k⫽0
1⫺12k
1k⫹221k⫹122 aM⫺2
k bexpc⫺ak⫹1 k⫹2bghd mS⫹N⫽^ 冮⫺qqYi*qS⫹N1Yi*2dYi*
Yi* Yi*
qN1Yi*2⫽ eM exp1⫺Yi*2 31⫺exp1⫺Yi*2 4M⫺1;Yi*ⱖ0 0; otherwise.
qS⫹N1Yi*2⫽e
exp3⫺1Yi*⫹gh2I0122ghYi*2 31⫺exp1⫺Yi*2M⫺1
⫹1M⫺12exp1⫺Yi*2 31⫺exp1⫺Yi*2 4M⫺2
⫻冮0Y*iexp3⫺1Y⫹gh2 4I0122ghY2dY;Yⱖ0
0; otherwise.
PS⫹N1Y*2⫽ 冮⫺qY*pS⫹N1y*2dy*.
PN1Y*2⫽ 冮⫺qY*pN1y*2dy*
⫹ 1M⫺12pN1Yi*2PS⫹N1Yi*2 3PN1Yi*2 4M⫺2
⫽pS⫹N1Yi*2 3PN1Yi*2 4M⫺1 qS⫹N1Yi*2⫽ d
dYi* 5PS⫹N1Yi*2 3PN1Yi*2 4M⫺16
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 pS⫹N(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⫽ 冮h*q22pN1 hsN2 expc⫺ 1Z*2⫺NhNshN2mN22ddZ* h* Z*⫽ a
Nh
i⫽1
Yi* Yi* sN2 ⫽ 1Yi*2N2 ⫺mN2
sS⫹N2 ⫽ 1Yi*2S⫹N2 ⫺mS⫹N2
Yi* 1Yi*22N⫽2⫹21M⫺12a
M⫺2 k⫽0
1⫺12k
1k⫹221k⫹122 aM⫺2
k b c 1
k⫹2⫹ 1 k⫹1d.
⫻ c a 1
k⫹2b a gh
k⫹2⫹1b ⫹ 1
k⫹1dexpc⫺ak⫹1 k⫹2bghd 1Yi*2S⫹N2 ⫽2⫹4gh⫹gh2⫹21M⫺12a
M⫺2 k⫽0
1⫺12k
1k⫹221k⫹122 aM⫺2
k b Yi*
mN⫽1⫹ 1M⫺12 a
M⫺2 k⫽0
1⫺12k
1k⫹221k⫹122 aM⫺2
k b.
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.
Eliminatingbbetween (3.22) and (3.23) produces the desired system oper- ating characteristic
(3.24)
whereS⫹N,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 levelsMincreases, 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*2m⫺1
1m⫺12! exp1⫺yk*2;yk*ⱖ0 0; otherwise
Nh⫽ td
Th
, sN2
sS⫹N2 PD⫽Q°
Q⫺11PFA2⫺ 1mS⫹N⫺mN2 B
Nh
sN2
sS⫹N>sN
¢
⫽Q° b⫺B
Nh
sN2 1mS⫹N⫺mN2 sS⫹N>sN
¢ PD⫽Qah*⫺NhmS⫹N
2NhsS⫹N2 b ⫽Qah*⫺NhmN⫺Nh1mS⫹N⫺mN2 2NhsN21sS⫹N>sN2 b
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*ⱖ0
qN1Yi*2⫽ •M1Yi*2m⫺1
1m⫺12! exp1⫺Yi*2 c1⫺ a
m⫺1 k⫽0
1Yi*2k
k! exp1⫺Yi*2 dM⫺1; 0; otherwise
pS⫹N1yk*2⫽ • a yk* mgh
b1m⫺12>2exp3⫺1yk*⫹mgh2 4Im⫺1124mghyk*2;yk*ⱖ0 0; otherwise.
Figure 3.9. False alarm and detection probability performance of serial search FH/MFSK acquisition system;gh⫽ ⫺20 dB.
and
(3.29)
⫻ c1⫺ a
m⫺1 k⫽0
1Yi*2k
k! exp1⫺Yi*2 dM⫺2冮0Y*i
a Y
mgh
b1m⫺12>2
⫻exp3⫺1Y⫹mgh2 4Im⫺1124mghY2dY;Yi*ⱖ0 0; otherwise
a Yi* mgh
b1m⫺12>2exp3⫺1Yi*⫹mgh2 4Im⫺1124mghYi*2
⫻ c1⫺ a
m⫺1 k⫽0
1Yi*2k
k! exp1⫺Yi*2 dM⫺1
⫹ 1M⫺12 1Yi*2m⫺1
1m⫺12! exp1⫺Yi*2
Figure 3.10. False alarm and detection probability performance of serial search FH/MFSK acquisition system;gh⫽ ⫺20 dB.
à
qS⫹N1Yi*2⫽
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 mS⫹NandmNcorrespond to (3.30) with n⫽1, and fur- ther obtaining sS⫹NandsNfrom (3.20) using (3.30) evaluated for n⫽2, 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 M⫽1 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/2N0BwhereBis 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
PD⫽QcQ⫺11PFA2⫺ 2Nhgh
11⫹2gh
d. Ckj⫽ •
1; j⫽0
1 j a
min1j,m⫺12 l⫽1
kl⫹l⫺j
l! Ck,j⫺l j⫽1, 2,p,k1m⫺12. Lk1m⫺121x2
1Yi*2Nn ⫽ 1Yi*2S⫹Nn 0gh⫽0;n⫽1, 2
⫺ a
1k⫹121m⫺12
j⫽0
Ck⫹1,j
1j⫹n2!
1k⫹22j⫹nLj⫹n1m⫺12a⫺ mgh
k⫹2b f
⫻ a
m⫹j⫹n⫺1 l⫽0
ak⫹1
k⫹2blLl1m⫺12a⫺ mgh
k⫹2b
⫻ 1
1k⫹12n1k⫹22m e a
k1m⫺12 j⫽0
Ckj
1m⫹j⫹n⫺12!
1m⫺12!1k⫹12m⫹j
⫹ 1M⫺12 a
M⫺2 k⫽0
aM⫺2
k b1⫺12k expc⫺ak⫹1 k⫹2bmghd 1Yi*2S⫹Nn ⫽ 3m11⫹gh2 4n⫹ 1n⫺12m11⫹2gh2
results for acquisition with MFSK modulation if the integrate-and-dumps in Figures 3.5 and 3.7 are replaced by low-pass filters with noise band- widthB.
To modify the previous results to allow for a non-ideal in-sync condi- tion, we proceed as follows. Assuming that the MFSK tones are orthogo- nally spaced, e.g., for FFH we could have fsj⫺Fs,j⫺1⫽k/Thsince 1/This the minimum tone separation for orthogonality, then in the absence of fre- quency error (df ⫽0), the m-fold accumulated output samples of the M non-coherent demodulators would in the absence of noise be given by (see (3.8))
(3.33) wherefiis the transmitted tone. Thus, multiplying ghby the loss factor L (1⫺0 0)2in (3.30) is sufficient to account for this degrading effect. A simi- lar result was obtained in connection with our previous discussion of PN code acquisition (see (1.86) of Chapter 1).
When, in addition, frequency error is present, then, even for the above orthogonal tone spacing, the m-fold accumulated output samples of the M non-coherent demodulations will in the absence of noise allcontain non-zero signal components. Further, this set of output signal components will depend on which MFSK tone was indeed transmitted; i.e.,
(3.34) As a result, all of the normalized output samples have pdf’s of the form in (3.27) with mghreplaced by mRl(t)Th/N0;l⫽1, 2, . . . ,M. Although for- mally the procedure is straightforward, the computation of acquisition per- formance for this case is tedious and is not presented here. However, a similar problem will be considered in detail later on in the chapter relative to the computation of error probability performance in the presence of resid- ual time and frequency tracking errors.
It should be obvious by now that once the system operating characteris- tic is determined in (3.24), the specification of acquisition performance in
yk* l⫽1, 2,# # #,M;l⫽i. Rl1t2⫽mS sin25 3pdfTh⫺ 1l⫺i2k4 11⫺ 0e0 6
3pdfTh⫺ 1l⫺i2k42 ;
Ri1t2⫽mSsin23pdf1Th⫺ 0t0 2 4
1pdfTh22 ⫽mSsin23pdfTh11⫺ 0e0 4
1pdfTh22 e
⫽^ Rl1t2⫽0;l⫽1, 2,p,M;l⫽i
Ri1t2⫽mSa1⫺ 0t0 Th
b2⫽^ mS11⫺ 0e0 22