Processing for compression - The MPEG - page 63

The filter makes one accumulation for every output sample, but that accumulation is the result of multiplying all
relevant input samples in the filter window by an appropriate coefficient. The number of points in the filter is
determined by the number of input samples in the period of the filter window, but the number of multiplications per
second is obtained by multiplying that figure by the output rate. If the filter is not integrated with the decimator, the
number of points has to be multiplied by the input rate. The larger the rate-reduction factor, the more advantageous
the decimating filter ought to be, but this is not quite the case, as the greater the reduction in rate, the longer the
filter window will need to be to accommodate the broader impulse response.
Figure 3.17: The spectrum of a typical digital sample stream at (a) will be subject to aliasing as in (b) if the
baseband width is not reduced by an LPF. At (c) an FIR low-pass filter prevents aliasing. Samples are clocked
transversely across the filter at the input rate, but the filter only computes at the output sample rate. Clearly this will
only work if the two are related by an integer factor.
When the sampling rate is to be increased by an integer factor, additional samples must be created at even
spacing between the existing ones. There is no need for the bandwidth of the input samples to be reduced since, if
the original sampling rate was adequate, a higher one must also be adequate.
Figure 3.18 shows that the process of sampling-rate increase can be thought of in two stages. First, the correct rate
is achieved by inserting samples of zero value at the correct instant, and then the additional samples are given
meaningful values by passing the sample stream through a low-pass filter which cuts off at the Nyquist frequency
of the original sampling rate. This filter is known as an interpolator, and one of its tasks is to prevent images of the
lower input-sampling spectrum from appearing in the extended baseband of the higher-rate output spectrum.
Figure 3.18: In integer-ratio sampling, rate increase can be obtained in two stages. First, zero-value samples are
inserted to increase the rate, and then filtering is used to give the extra samples real values. The filter necessary
will be an LPF with a response which cuts off at the Nyquist frequency of the input samples.
All sampled systems have finite bandwidth and need a reconstruction filter to remove the frequencies above the
baseband due to sampling. After reconstruction, one infinitely short digital sample ideally represents a sin x / x pulse