Resolving SinusoidsWe saw in §5.4.1 that our ability to resolve two closely spaced sinusoids is determined by the main-lobe width of the window transform we are using. We will now study this relationship in more detail.
For starters, let's define main-lobe bandwidth very simply (and somewhat crudely) as the distance between the first zero-crossings on either side of the main lobe, as shown in Fig.5.10 for a rectangular-window transform. Let denote this width in Hz. In normalized radian frequency units, as used in the frequency axis of Fig.5.10, Hz translates to radians per sample, where denotes the sampling rate in Hz. 3.1, the DTFT is given analytically by
where is frequency in Hz, and is the sampling interval in seconds ( ). The main lobe of the rectangular-window transform is thus ``two side lobes wide,'' or
as can be seen in Fig.5.10. Recall from §3.1.1 that the side-lobe width in a rectangular-window transform ( Hz) is given in radians per sample by
As Fig.5.10 illustrates, the rectangular-window transform main-lobe width is radians per sample (two side-lobe widths). Table 5.1 lists the main-lobe widths for a variety of window types (which are defined and discussed further in Chapter 3).
Other Definitions of Main Lobe WidthOur simple definition of main-lobe band-width (distance between zero-crossings) works pretty well for windows in the Blackman-Harris family, which includes the first six entries in Table 5.1. (See §3.3 for more about the Blackman-Harris window family.) However, some windows have smooth transforms, such as the Hann-Poisson (Fig.3.21), or infinite-duration Gaussian window (§3.11). In particular, the true Gaussian window has a Gaussian Fourier transform, and therefore no zero crossings at all in either the time or frequency domains. In such cases, the main-lobe width is often defined using the second central moment.6.5 A practical engineering definition of main-lobe width is the minimum distance about the center such that the window-transform magnitude does not exceed the specified side-lobe level anywhere outside this interval. Such a definition always gives a smaller main-lobe width than does a definition based on zero crossings.
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.
We make the main-lobe width smaller by increasing the window length . Specifically, requiring Hz implies
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 .
signals are periodic in nature, such as short segments of most tonal musical instruments and speech. The sinusoidal components in a periodic signal are constrained to be harmonic, that is, occurring at frequencies that are an integer multiple of the fundamental frequency .6.6 Physically, any ``driven oscillator,'' such as bowed-string instruments, brasses, woodwinds, flutes, etc., is usually quite periodic in normal steady-state operation, and therefore generates harmonic overtones in steady state. Freely vibrating resonators, on the other hand, such as plucked strings, gongs, and ``tonal percussion'' instruments, are not generally periodic.6.7 Consider a periodic signal with fundamental frequency Hz. Then the harmonic components occur at integer multiples of , and so they are spaced in frequency by . To resolve these harmonics in a spectrum analysis, we require, adapting (5.27),
Note that is the fundamental period of the signal in samples. Thus, another way of stating our simple, sufficient resolution requirement on window length , for periodic signals with period samples, is , where is the main-lobe width in bins (when critically sampled) given in Table 5.2. Chapter 3 discusses other window types and their characteristics. Specifically, resolving the harmonics of a periodic signal with period samples is assured if we have at least
- periods under the rectangular window,
- periods under the Hamming window,
- periods under the Blackman window,
- periods under the Blackman-Harris -term window,
1], is relatively advanced and may be skipped without loss of continuity.] Figures 5.14(a) through 5.14(d) show four possible definitions of main-lobe separation that could be considered for purposes of resolving closely spaced sinusoidal peaks.
sinusoidal peaks, it is important to know the minimum frequency separation of the peaks, and to choose an FFT window which is long enough to resolve the peaks accurately. Generally speaking, the window must ``see'' at least 1.5 cycles of the minimum difference frequency. The rectangular window ``sees'' its full length. Other windows, which are all tapered in some way (Chapter 3), see an effective duration less than the window length in samples. Further details regarding theoretical and empirical estimates are given in .
Sinusoidal Peak Interpolation
Effect of Windowing