Next: Digital Delay Up: Discrete Fourier Transform Previous: Discrete Fourier Transform   Contents

## Filtering and Windowing

The Fourier transform of a signal 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 obtained using a DFT operation is the convolution of the true spectrum of the signal convolved by the FT of the window function, and sampled at discrete frequencies. Thus a DFT is equivalent to a filter bank with filters spaced at in frequency. The response of each filter is the Fourier transform of the window function used to restrict the number of samples to . 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 ). The spectral resolution (defined as the full width at half maximum (FWHM) of the filter response) of the sinc function is . Different window functions give different filter' responses, i.e. for

 (8.3.5)

the Hanning window
 (8.3.6)

has a spectral resolution . 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%.

Next: Digital Delay Up: Discrete Fourier Transform Previous: Discrete Fourier Transform   Contents
NCRA-TIFR