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### Simple Sufficient Condition for Peak Resolution

Recall from §5.4 that the frequency-domain image of a sinusoid through a window'' is the window transform scaled by the sinusoid's amplitude and shifted so that the main lobe is centered about the sinusoid's frequency. A spectrum analysis of two sinusoids summed together is therefore, by linearity of the Fourier transform, the sum of two overlapping window transforms, as shown in Fig.5.12 for the rectangular window. A simple sufficient requirement for resolving two sinusoidal peaks spaced Hz apart is to choose a window length long enough so that the main lobes are clearly separated when the sinusoidal frequencies are separated by Hz. For example, we may require that the main lobes of any Blackman-Harris window meet at the first zero crossings in the worst case (narrowest frequency separation); this is shown in Fig.5.12 for the rectangular-window. To obtain the separation shown in Fig.5.12, we must have Hz, where is the main-lobe width in Hz, and is the minimum sinusoidal frequency separation in Hz.

For members of the -term Blackman-Harris window family, can be expressed as , as indicated by Table 5.1. In normalized radian frequency units, i.e., radians per sample, we have . For comparison, Table 5.2 lists minimum effective values of for each window (denoted ) given by an empirically verified sharper lower bound on the value needed for accurate peak-frequency measurement , as discussed further in §5.5.4 below.

Table: Main-lobe width-in-bins for various windows.
 Window Type  Rectangular  Hamming  Hann  Generalized Hamming -- Blackman   -term Blackman-Harris We make the main-lobe width smaller by increasing the window length . Specifically, requiring Hz implies (6.26)

or (6.27)

Thus, to resolve the frequencies and , the window length must span at least periods of the difference frequency , measured in samples, where is the effective width of the main lobe in side-lobe widths . Let denote the difference-frequency period in samples, rounded up to the nearest integer. Then an -term'' Blackman-Harris window of length samples may be said to resolve the sinusoidal frequencies and . Using Table 5.2, the minimum resolving window length can be determined using the sharper bound as .

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