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Decimation by a Factor D 477
as h(n, m). Hence the input x(n) and the output y(m) are related by the
superposition summation for time-variant systems.
The sampling rate conversion process can also be understood from the
point of view of digital resampling of the same analog signal. Let x a (t)
be the analog signal that is sampled at the first rate F x to generate x(n).
The goal of rate conversion is to obtain another sequence y(m) directly
from x(n), which is equal to the sampled values of x a (t) atasecond rate
F y .Asisdepicted in Figure 9.2b, y(m) isatime-shifted version of x(n).
Such a time shift can be realized by using a linear filter that has a flat
magnitude response and a linear phase response (i.e., it has a frequency
response of e −jωτi , where τ i is the time delay generated by the filter). If
the two sampling rates are not equal, the required amount of time shifting
will vary from sample to sample, as shown in Figure 9.2b. Thus, the rate
converter can be implemented using a set of linear filters that have the
same flat magnitude response but generate different time delays.
Before considering the general case of sampling rate conversion, we
shall consider two special cases. One is the case of sampling rate reduction
by an integer factor D, and the second is the case of a sampling rate
increase by an integer factor I. The process of reducing the sampling rate
by a factor D (downsampling by D) iscalled decimation. The process of
increasing the sampling rate by an integer factor I (upsampling by I) is
called interpolation.
9.2 DECIMATION BY A FACTOR D
The basic operation required in decimation is the downsampling of the
high-rate signal x(n) intoalow-rate signal y(m). We will develop the
time- and frequency-domain relationships between these two signals to
understand the frequency-domain aliasing in y(m). We will then study
the condition needed for error-free decimation and the system structure
required for its implementation.
9.2.1 THE DOWNSAMPLER
Note that the downsampled signal y(m) isobtained by selecting one out
of D samples of x(n) and throwing away the other (D − 1) samples out
of every D samples—i.e.,
y(m) =x(n)| n=mD
= x(mD); n, m, D ∈{integers} (9.6)
The block diagram representation of (9.6) is shown in Figure 9.3. This
downsampling element changes the rate of processing and thus is fundamentally
different from other block diagram elements that we have used
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