Filtering and Windowing

The Fourier transform of a signal $s(t)$ is a decomposition into frequency or spectral components. The DFT also performs a spectral decomposition but with a finite spectral resolution. The spectrum of a signal $s(t)$ obtained using a DFT operation is the convolution of the true spectrum of the signal $S(f)$ convolved by the FT $W(f)$ of the window function, and sampled at discrete frequencies. Thus a DFT is equivalent to a filter bank with filters spaced at $\Delta \omega$ in frequency. The response of each filter is the Fourier transform of the window function used to restrict the number of samples to $N$. For example, in the above analysis (see Section 8.3) the response of each `filter' is the sinc function, (which is the FT of the rectangular window $\Pi(t)$). The spectral resolution (defined as the full width at half maximum (FWHM) of the filter response) of the sinc function is $\frac{1.21\Delta\omega}{2\pi}$. Different window functions $w(n)$ give different `filter' responses, i.e. for

$$S(k) = \sum_{n=0}^{N-1} w(n) s(n) e^{-j2\pi nk/N}$$ (8.3.5)

the Hanning window

$$w(n) = 0.5(1 + \cos(2\pi n/N))\;\; \mbox{for } -N/2 \le n \le N/2-1$$ (8.3.6)

$$= 0\;\; \mbox{elsewhere}$$

has a spectral resolution $\frac{2\Delta\omega}{2\pi}$. Side lobe reduction and resolution are the two principal considerations in choosing a given window function (or equivalently a given filter response). The rectangular window (i.e. sinc response function) has high resolution but a peak sidelobe of 22% while the Hanning window has poorer resolution but peak sidelobe level of only 2.6%.