n, 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.2 Now, 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 of less than the hop spacing) which are effective for correlation detection. These "bursts" of sinusoids at a frequency fIF of 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 T with magnitude less than a hop interval Th, we can write x(t) as 05157(7>0) Th-StTh(<0) (3.1) 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. For simplicity of this discussion, we shall for the time being ignore the information modula- tion and the additive noise. RECEIVED f1+8f f2+8f f3+8f f4+8f SEQUENCE SYNTHESIZER f4++IF f5+fIF 6+TIF SEQUENCE f4-f1 f4-12+f1f-8f f5-t2 16-f3 MIXER +1/F-8f +1/F-8f f5-f3+f1f-8f f6-f4+f1f-8f DIFFERENCE +1F-87 Figure 3.2a. Received FH signal and local hop signal misaligned by more than a single hop interval. RECEIVED f++ff f2+8f f3+8f f4+8t SEQUENCE SYNTHESIZER f2+ +F 13+TF SEQUENCE MIXER DIFFERENCE IF-8f 1-12-1F-81 fIF-Sf f2-f3+fi--8f fIF-8f f3-14+4F-87 COMPONENTS EFFECTIVE FOR CORRELATION DETECTION Figure 3.2b. Received FH signal and local hop signal misaligned by less than a single hop interval. Time and Frequency Synchronizati