Page 397 - Analog and Digital Filter Design

394 Analog and Digital Filter Design

A similar process can be applied to other Window functions and other frequency
responses. That is, use the appropriate sinc(x) function and multiply this by a
Window function for each value of n. The Window will be a function of N, the
total number of coefficients (taps) required. Time-shifting must then be applied
to make the first tap coefficient become h[O] and the last tap coefficient become
h[N - 11.

A Data-Sampling Rate-Changer

Apart from filtering, it is also possible to use FIR filters to perform a change of
the data-sampling rate. Suppose a system is receiving signals from two sources
that have different sampling rates and that the system clock is operating at the
higher rate. Provided that one of these sampling rates is an integer multiple of
the other, it is possible to convert the signal that has the slower sampling rate
into one with a higher sampling rate.

An example of where differing sampling rates could occur is a system that is
receiving digitized telecommunication signals sampled at 8 kHz and 16 kHz. The
8 kHz sampled signals could be digitized speech in the analog band of 300 Hz
to 3.4kHz. The 16kHz sampled signals could be wider bandwidth speech (up
to 7kHz). The ratio of the higher sampling rate to the lower sampling rate is
two in this case.
Data from the speech channel sampled at 8 kHz is input to the system. Since the
system is processing data samples at twice the rate that they are being received,
intermediate samples are set to zero. Thus the data sequence is: D1, 0, D2, 0,
D3,0, D4,0, D5, and so on. This process is equivalent to mixing in radio systems
and results in aliases of the original spectrum being produced. Filtering is
needed to remove these aliases, which can be within the frequency range of the
wider bandwidth speech.

A suitable filter is known as an interpolator. The interpolator is a lowpass filter
that has a passband cutoff frequency equal to the highest frequency of the
sampled signals. The stopband of the filter must be below the alias frequency
produced by the system over-sampling. The zero-valued samples are replaced
by an average of the samples on either side. The exact value of the replacement
depends upon the frequency of the signals being processed.

References

1. Thede, Les. Analog and Digital Filter Design Using C. New Jersey:
Prentice-Hall, 1996.
2. Sanjit J. Mitra and James E Kaiser. Handbookfor Digital Signal Pro-
cessing. New York: John Wiley & Sons, 1993.