Processing for compression - The MPEG - page 60

the number of samples in the window is established, the continuous functions shown here are sampled at the
appropriate spacing to obtain window coefficients. These are multiplied by the truncated impulse response
coefficients to obtain the actual coefficients used by the filter. The amplitude responses I-V correspond to the
window functions illustrated. (Responses courtesy Philips Technical Review )
The Dolph window shown [ 4 ] in Figure 3.12 results in an equiripple filter which has the advantage that the
attenuation in the stopband never falls below a certain level.
Figure 3.12: The Dolph window shape is shown at (a). The frequency response is at (b). Note the constant height
of the response ripples.
Filter coefficients can be optimized by computer simulation. One of the best-known techniques used is the Remez
exchange algorithm, which converges on the optimum coefficients after a number of iterations.
In the example of Figure 3.13 , a low-pass FIR filter is shown which is intended to allow downsampling by a factor of
two. The key feature is that the stopband must have begun before one half of the output sampling rate. This is most
readily achieved using a Hamming window because it was designed empirically to have a flat stopband so that
good aliasing attenuation is possible. The width of the transition band determines the number of significant sample
periods embraced by the impulse. The Hamming window doubles the width of the transition band. This determines
in turn both the number of points in the filter, and the filter delay. For the purposes of illustration, the number of
points is much smaller than would normally be the case in an audio application.
Figure 3.13: A downsampling filter using the Hamming window
As the impulse is symmetrical, the delay will be half the impulse period. The impulse response is a sin x / x function,
and this has been calculated in the figure. The equation for the Hamming window function is shown with the
window values which result. The sin x / x response is next multiplied by the Hamming window function to give the
windowed impulse response shown. If the coefficients are not quantized finely enough, it will be as if they had been
calculated inaccurately, and the performance of the filter will be less than expected. Figure 3.14 shows an example
of quantizing coefficients. Conversely, raising the wordlength of the coefficients increases cost.