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Finite Impulse Response filters

A Finite Impulse Response (FIR) filter produces an output, y(n), that is the weighted sum of the current and past inputs, x(n).


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This is shown in figure 4 with tex2html_wrap_inline2837 representing a unit delay.

  figure92
Figure 4: A FIR filter

Consider supplying this filter with a sine wave, tex2html_wrap_inline2839:


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Using the identity tex2html_wrap_inline2841:


eqnarray103

The terms in parantheses are independent of time and hence the output is a sinusoid with amplitude:


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and phase:
displaymath117

This method may be used to provide the amplitude and phase response for any FIR filter. The transform is called the Fourier transform (defined in section 4), and it has a simple inverse. Conversely the filter coefficents may be obtained from the desired filter response using the same technique.

As a simple example, consider a low pass filter where the desired response, tex2html_wrap_inline2843 is:


eqnarray125


eqnarray131

But this means we need an infinite number of filter coefficients! True enough - real ``brick wall'' filters are impossible and sharp filters are hard to design. Some solutions:

Matlab implements all these, for example ``fir1(14, 0.5)'' is a 15 tap low pass filter that has a cutoff at half the maximum frequency. The filter coefficents are a windowed sinc funtion, plotted in figure 5 and the amplitude response is plotted in figure 6.

  figure151
Figure 5: Example filter coefficiants: plot(fir1(14, 0.5), '+')

  figure158
Figure 6: Example filter response: freqz(fir1(14, 0.5))

FIR filters are computationally expensive to implement but need not introduce phase distortions - useful in processing high quality speech.

IIR filters are often much more efficient, but can not be designed to have exact linear phase.


next up previous contents
Next: Infinite Impulse Response filters Up: Linear filters Previous: Linear filters

Speech Vision Robotics group/Tony Robinson