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For wideband baseband data containing a single
burst in noise, the model can be expressed as
x(n)=a*s(n-t0)*exp(j*2pi*fc*n)+(СКО)*v(n), n=0...N-1; (1)
N number of samples of wideband data Given that the noise is white Gaussian in nature, the start time and center frequency can be estimated by minimizing the least squares error given by e^2=from n=0 to N-1 sum(|x(n)-a's(n-t0')exp(j*2pi*fc'*n)|^2) where a',t0' and fc' are the estimated quantities of interest. There is no known closed S(t,f)= from n=0 to N-1 sum(x(n)*s"(n-t)*exp(-j*2pi*f*n)) (3) where the optimum values of (t, f) = (to', fc') are those which maximize |S(t,f)|. ... To begin the analysis, the data model for x(n) given in (3) is expanded using the signal S(t,f)=a*(from n=0 to N-1 sum(s(n-t0)*s"(n-t))*exp(j*2pi(fc-f)*n))+ The first summation term goes to zero if there is no overlap between the actual burst from (n=0) to (N-1) sum(s(n-t0)*s"(n-t)*exp(j*2pi*(fc-f)*n))= where v1 is complex zero-mean Gaussian variable with a variance of one. Hence (5) becomes S(t,f)=a*sqrt(M-|t-t0|)*v1+CKO*sqrt(M)*v2
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M length of hop in terms of the number of data samples
t0 burst start time index (i.e. the burst starts at n = t0)
fc burst center frequency normalized with respect to the sampling frequency fs
a complex-valued signal amplitude
s(m) complex baseband signal envelope for 0 ?m
form solution for this expression, however by minimizing e^2 with respect to a', and
assuming that the data contains the entire burst (i.e. M+t0<=N), then a' may be eliminated and the following simplified expression obtained:
plus noise definition given in (1), and the case where (t, f) не равно (to, fc) is considered.
Expanding (3) as discussed, then
СКО*(from n=t to t+M-1 sum(v(n)*s"(n-t)*exp(-j*2pi*f*n))) (5)
s(n-t0) and the predicted burst s(n - t). If there is some overlap, then
s(n-t0)s"(n-t) can be represented by a process with constant amplitude but random
phase. Using the central limit theorem, the result of the summation will be
sqrt(M-|t-t0|)*v1
The second
term can be replaced by CKO*sqrt(M)*v2 (where v2 has the same statistical properties as v1).
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