Advanced DSP - FFT windows: FFT as series of filters

A productive way to visualise spectral leakage is to view the Fourier Transform as equivalent to a series of filters, one centred on each FFT bin. The filter's frequency response is the shape of the window function's kernel. Each FFT bin contains contributions from all other frequency components within the bandwidth of the filter, weighted by the filter's frequency response (the kernel).

The diagram shows the kernel of a rectangular window function. The frequency component we think we are measuring is the FFT bin at the middle of the kernel. But an interfering frequency component some distance away will also contribute to the measured value of this frequency component.

In fact, the measured value of the FFT bin will include contributions from all frequency component in the bandwidth of the kernel, weighted by the kernel's value at those frequencies. This will include contributions from broad band noise as well as from narrow band signals at other frequencies.

There are two common situations with which we have to deal:

• detection of a frequency component in the presence of broadband noise
• distinguishing between close narrow band frequency components

The choice of window function may be different in the two cases. To help in choosing a suitable window function we need some quantitative measures of the quality of a window function.