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 ( ) 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 ( ) 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.
The preceding figures suggest that, for a rectangular window of length , two sinusoids are well resolved when they are separated in frequency by
where the frequency-separation is in radians per sample. In cycles per sample, the inequality becomes
where the denotes normalized frequency in cycles per sample. In Hz, we have
Note that is the number of samples in one period of a sinusoid at frequency Hz, sampled at 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  reveals that 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.
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.
Phase Interpolation at a Peak
Two Cosines (``In-Phase'' Case)