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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 $ \Delta f$ 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 $ \Delta f$ 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.

Figure: Two length-$ M$ -rectangular-window transforms displaced by $ 2\pi \Delta f T=2\Omega _M=4\pi /M$ rad/sample.
\includegraphics[width=\twidth]{eps/sinesAnn}

To obtain the separation shown in Fig.5.12, we must have $ B_w
\leq \Delta f$ Hz, where $ B_w$ is the main-lobe width in Hz, and $ \Delta f$ is the minimum sinusoidal frequency separation in Hz.

For members of the $ L$ -term Blackman-Harris window family, $ B_w$ can be expressed as $ B_w = 2L f_s/M$ , as indicated by Table 5.1. In normalized radian frequency units, i.e., radians per sample, we have $ 2\pi B_w T = 2L\Omega_M\isdeftext
K\Omega_M$ . For comparison, Table 5.2 lists minimum effective values of $ K$ for each window (denoted $ K^\ast$ ) given by an empirically verified sharper lower bound on the value needed for accurate peak-frequency measurement [1], as discussed further in §5.5.4 below.


Table: Main-lobe width-in-bins $ K$ for various windows.
Window Type $ K$ $ K^\ast$
Rectangular $ 2$ $ 1.44$
Hamming $ 4$ $ 2.22$
Hann $ 4$ $ 2.36$
Generalized Hamming $ 4$ --
Blackman $ 6$ $ 2.02$
$ L$ -term Blackman-Harris $ 2L$  


We make the main-lobe width $ B_w$ smaller by increasing the window length $ M$ . Specifically, requiring $ B_w
\leq \Delta f$ Hz implies

$\displaystyle B_w = K \frac{f_s}{M} \leq \Delta f \quad \Rightarrow \quad M \ge K \frac{f_s}{\Delta f}$ (6.26)

or

$\displaystyle \zbox {M \ge K \frac{f_s}{\left\vert f_2-f_1\right\vert}.} \protect$ (6.27)

Thus, to resolve the frequencies $ f_1$ and $ f_2$ , the window length $ M$ must span at least $ K$ periods of the difference frequency $ f_2-f_1$ , measured in samples, where $ K$ is the effective width of the main lobe in side-lobe widths $ f_M\isdeftext f_s/M$ . Let $ D\isdeftext \lceil f_s/\vert f_2-f_1\vert\rceil$ denote the difference-frequency period in samples, rounded up to the nearest integer. Then an ``$ L$ -term'' Blackman-Harris window of length $ M\ge KD=2LD$ samples may be said to resolve the sinusoidal frequencies $ f_1$ and $ f_2$ . Using Table 5.2, the minimum resolving window length can be determined using the sharper bound as $ M\ge \lceil K^\ast \cdot D\rceil$ .


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