Page 359 - Analog and Digital Filter Design

356 Analog and Digital Filter Design

Interpolation
Interpolation is the opposite of decimation. Suppose our process described
above must produce an output at 48 kHz sample rate, to be compatible with the
rate of the input. We have reduced the internal data rate to 24kHz, so addi-
tional samples must be inserted between the data samples. These samples are
usually the average value of the previous sample and the following sample. In
some cases the same data may be in pairs of samples, but this does not add value
to the signal. By inserting average data values the signal output from the digital-
to-analog converter is smoother. It also contains a lower signal power at the
sampling frequency.

Decimation and interpolation are usually arranged to reduce or multiply the
data rate by a power of two. So we may have decimation rates of ‘h? ‘I4, and
so on. This means that there is an equal spacing between samples, which is
important for reconstructing the signal.

Decimation takes place ”naturally” in sigma-delta analog-to-digital converters.
These devices sample the signal by typically 64 times the output data rate. A
converter having a 40 kHz output data rate may sample a signal with a 10 kHz
bandwidth, so the signal is sampled at 2.56MHz in this case. The digitized signal
is decimated in the conversion process because each sample is used to produce
one bit of data. The output is logic 1 if the signal is higher than the previous
sample. The output is logic 0 if the signal voltage is lower than before. The
binary word is thus built up by adding or subtracting bits until the data word
represents the signal level.

Digital Filtering

This book has described analog filtering in some depth, both in terms of the
frequency response and in terms of the pole and zero locations. It is somehow
easy to imagine (for me, at least) signals flowing in a circuit. I can imagine the
potential divider action as the impedance of an inductor increases with fre-
quency while the impedance of a capacitor reduces. In a filter diagram the signal
path is usually through a single wire.

Digital filtering is a completely different concept from analog filtering. Digital
filtering processes signals in the time domain. Therefore, if a certain frequency
domain response is required, it is necessary to convert this response into the
equivalent time domain. It is not always intuitively obvious what this time
domain signal looks like. What is more, the signals are usually in parallel digital
form, in other words, they are a binary-coded version of an analog signal on,
perhaps, a 16-bit-wide bus. In some digital filter diagrams it is not obvious
that the signal path is a data bus (unless you are familiar with microprocessor