One Sine and One Cosine ``Phase Quadrature'' Case

Figure 5.8 shows a similar spectrum analysis of two sinusoids

$$x(n) = \sin(\omega_1 n) + \cos(\omega_2 n), \quad n=0,1,\ldots,M-1,$$ (6.18)

using the same frequency separation and window lengths. However, now the sinusoids are 90 degrees out of phase (one sine and one cosine). Curiously, the top-left case ( $M=20=\hbox{1/2 difference-frequency cycle}$ ) now appears to be resolved! However, closer inspection (see Fig.5.9) reveals that the ``resolved'' spectral peaks are significantly far away from the sinusoidal frequencies. Another curious observation is that the lower-left case ( $M=40=\hbox{1 difference-frequency cycle}$ ) appears worse off than it did in Fig.5.7, and worse than the shorter-window analysis at the top right of Fig.5.8. Only the well resolved case at the lower right (spanning two full cycles of the difference frequency) appears unaffected by the relative phase of the two sinusoids under analysis.

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

Figure 5.9 shows the same plots as in Fig.5.8, but overlaid. From this we can see that the peak locations are biased in under-resolved cases, both in amplitude and frequency.

Figure 5.9: Overlay of the plots in Fig.5.8.

The preceding figures suggest that, for a rectangular window of length $M$ , two sinusoids are well resolved when they are separated in frequency by

$$\zbox {\Delta\omega\geq 2\Omega_M} \qquad \left(\Omega_M \isdef \frac{2\pi}{M}\right),$$ (6.19)

where the frequency-separation $\Delta \omega = \omega_2-\omega_1$ is in radians per sample. In cycles per sample, the inequality becomes

$$\zbox {\Delta {\tilde f}\geq \frac{2}{M}},$$ (6.20)

where the ${\tilde f}\isdef f/f_s = fT$ denotes normalized frequency in cycles per sample. In Hz, we have

$$\Delta f\geq 2\frac{f_s}{M}.$$ (6.21)

or

$$\zbox {M \geq 2\frac{f_s}{\Delta f}.}$$ (6.22)

Note that $f_s/f$ is the number of samples in one period of a sinusoid at frequency $f$ Hz, sampled at $f_s$ Hz. Therefore, we have derived a rule of thumb for frequency resolution that requires at least two full cycles of the difference-frequency under the rectangular window.

A more detailed study [1] reveals that $1.44$ cycles of the difference-frequency is sufficient to enable fully accurate peak-frequency measurement under the rectangular window by means of finding FFT peaks. In §5.5.2 below, additional minimum duration specifications for resolving closely spaced sinusoids are given for other window types as well.

In principle, we can resolve arbitrarily small frequency separations, provided

• there is no noise, and
• we are sure we are looking at the sum of two ideal sinusoids under the window.

One method for doing this is described in §5.7.2. However, in practice, there is almost always some noise and/or interference from other signals, so we normally prefer to require sinusoidal frequency separation by on the order of one main-lobe width or more.

The rectangular window provides an abrupt transition at its edge. While it remains the optimal window for sinusoidal peak estimation, it is by no means optimal in all spectrum analysis and/or signal processing applications involving spectral processing. As discussed in Chapter 3, windows with a more gradual transition to zero have lower side-lobe levels, and this is beneficial for spectral displays and various signal processing applications based on FFT methods. We will encounter such applications in later chapters.