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whose central peak width is determined by the response of the reconstruction filter, and whose amplitude is
proportional to the sample value. This implies that, in reality, one sample value has meaning over a considerable
timespan, rather than just at the sample instant. Were this not true, it would be impossible to build an interpolator.
Performing interpolation steps separately is inefficient. The bandwidth of the information is unchanged when the
sampling rate is increased; therefore the original input samples will pass through the filter unchanged, and it is
superfluous to compute them. The combination of the two processes into an interpolating filter minimizes the
amount of computation.
As the purpose of the system is purely to increase the sampling rate, the filter must be as transparent as possible,
and this implies that a linear- phase configuration is mandatory, suggesting the use of an FIR structure.
Figure 3.19
shows that the theoretical impulse response of such a filter is a sin
x
/
x
curve which has zero value at the position of
adjacent input samples. In practice this impulse cannot be implemented because it is infinite. The impulse
response used will be truncated and windowed as described earlier.
Figure 3.19:
A single sample results in a sin
x
/
x
waveform after filtering in the analog domain. At a new, higher,
sampling rate, the same waveform after filtering will be obtained if the numerous samples of differing size shown
here are used. It follows that the values of these new samples can be calculated from the input samples in the
digital domain in an FIR filter.
To simplify this discussion, assume that a sin
x
/
x
impulse is to be used. There is a strong parallel with the operation
of a DAC where the analog voltage is returned to the time-continuous state by summing the analog impulses due to
each sample. In a digital interpolating filter, this process is duplicated.
[
5
]
If the sampling rate is to be doubled, new samples must be interpolated exactly halfway between existing samples.
The necessary impulse response is shown in
Figure 3.20
;
it can be sampled at the
output
sample period and
quantized to form coefficients. If a single input sample is multiplied by each of these coefficients in turn, the impulse
response of that sample at the new sampling rate will be obtained. Note that every other coefficient is zero, which
confirms that no computation is necessary on the existing samples; they are just transferred to the output. The
intermediate sample is computed by adding together the impulse responses of every input sample in the window.
The figure shows how this mechanism operates. If the sampling rate is to be increased by a factor of four, three
sample values must be interpolated between existing input samples.
Figure 3.21
shows that it is only necessary to
sample the impulse response at one-quarter the period of input samples to obtain three sets of coefficients which
will be used in turn. In hardware- implemented filters, the input sample which is passed straight to the output is
transferred by using a fourth filter phase where all coefficients are zero except the central one, which is unity.
