Two Cosines (``In-Phase'' Case)

Figure 5.7 shows a spectrum analysis of two cosines

$$x(n) = \cos(\omega_1 n) + \cos(\omega_2 n), \quad n=0,1,\ldots,M-1,$$ (6.17)

where $\omega_1 = \pi/2$ and $\omega_2 = \omega_1 + \Delta\omega$ , and the frequency separation $\Delta \omega = \omega_2-\omega_1$ is $2\pi/40$ radians per sample. The zero-padded Fourier analysis uses rectangular windows of lengths $M=20$ , $30$ , $40$ , and $80$ ( $\Delta\omega = \frac{1}{2}\Omega_M, \frac{3}{4}\Omega_M, \Omega_M, 2\Omega_M$ , where $\Omega_M\isdef 2\pi/M$ ). The length $N=1024$ FFT output is divided by $M$ so that the ideal height of each spectral peak is $\max_{\omega_k}\{\vert X(\omega_k)\vert\}=1/2$ .

Figure: DTFT of two closely spaced in-phase sinusoids, various rectangular-window lengths $M$ .

The longest window ($M=80$ ) resolves the sinusoids very well, while the shortest case ($M=20$ ) does not resolve them at all (only one ``lump'' appears in the spectrum analysis). In difference-frequency cycles, the analysis windows are two cycles and half a cycle in these cases, respectively. It can be debated whether or not the other two cases are resolved, and we will return to them shortly.