Choosing Window Length to Resolve Sinusoids

Recall from §1.4 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.1.18 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.1.18 for the rectangular-window.

**Figure 1.18:** Two length--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.1.18, we must have $B_w \leq \Delta f$ Hz, where is the main lobe width in Hz, and $\Delta f$ 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 1.1. 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 1.2 lists minimum effective values of 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 the next section.

Table 1.2: Main-lobe width-in-bins

for various windows.

Window Type		$K^\ast$
Rectangular
Hamming
Hann
Generalized Hamming		--
Blackman
-term Blackman-Harris

Requiring $B_w \leq \Delta f$ Hz implies

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

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

(2.8)

Thus, to resolve the frequencies

and

, the window length must span at least periods of the difference frequency

, measured in samples, where

is the width of the main lobe, measured in side-lobe widths. 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 ``

-term'' Blackman-Harris window of length $M\ge KD=2LD$ samples may be said to resolve the sinusoidal frequencies

and

. Using Table 1.2, the minimum resolving window length can be determined using the sharper bound as $M\ge \lceil K^\ast \cdot D\rceil$ .

Subsections

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Previous: Other Definitions of Main Lobe Width
Next: Periodic Signals

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Spectral Audio Signal Processing
Spectrum Analysis of Sinusoids
Choosing Window Length to Resolve Sinusoids