Advanced Frequency Synthesis by Phase Lock

APPENDIX S

SPECTRUM ANALYSIS

Power spectrums of simulated waveforms are measured in spectrum analyzers using the fast Fourier transform (FFT) method of generating digital Fourier transforms (DFTs). An FFT is a DFT. Lyons [2004], especially pp. 120–122, is recommended as a reference.

S.1   SPECTRUMS

S.1.1   Periodicity

Since ΣΔ modulation is periodic, resulting waveforms are periodic. What we call quantization noise at the synthesizer output is actually a periodic waveform (assuming the synthesizer has settled). It can be characterized by a Fourier series consisting of harmonics of sinusoids at frequencies cycle/T_sequ, where T_sequ is the duration of the repeated sequence. If a Fourier series is obtained from a segment of a waveform of duration T_segm, it will represent a waveform with the form of that segment repeated every T_segm. This will be an accurate representation of the periodic waveform if T_segm = T_sequ (Fig. S.1) or if T_segm = nT_sequ (Fig. S.2), where n is an integer. A Fourier transform of a waveform segment of duration T_segm will have frequencies spaced by cycle/T_segm, but if n > 1, it follows that only one of each n of those will be nonzero, since a waveform that repeats at intervals of T_sequ contains only harmonics of cycle/T_sequ (Fig. S.2, right side).

FIGURE S.1   An accurate power spectrum is obtained when the analyzed segment T_segm equals the sequence duration