Two Multirate Identities
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Table of Contents
3.2.3 Two Multirate Identities
As we have seen so far, the basic multirate system2 consists of a sampling rate alteration device
and a digital filter. In many applications, these components are cascaded. Changes in
the position of these components will result in computationally efficient realization, keeping
the input/output (I/O) relation unaltered. For example, placing the filter HD(zD), the
impulse response of which is interpolated by D, in front of the downsampler is equivalent
to downsampling a signal by D first and then passing it through an LPF HD(z) created by
expanding the original filter impulse response by a factor of D. This concept is shown in
Figure 3.20.
A proof of the identity is given below. We know from Equation 3.9 that the subsampled
signal in Figure 3.20b can be written as
and the filtered signal in Figure 3.20b becomes
Likewise, Figure 3.20a can be represented as
This equation can be realized by Figure 3.20. For notational simplicity, we will express
Equation 3.28 as
where D ↓ corresponds to subsampling by a factor D. Equation 3.30 can be realized as
Figure 3.20b, and thus Figures 3.20a and 3.20b represent equivalent systems.
Figure 3.20: The Decimation Identity.
Figure 3.21: The Interpolation Identity.
Similar equivalence can be realized in the case of interpolation. There is a corresponding
interpolator identity that states that upsampling using a filter HI (zI ) is equivalent to
filtering with HI(z) and then zero-filling I − 1 zeros between samples of the filter output.
This identity is shown in Figure 3.21. The identities expressed by Figures 3.20 and 3.21
are called the Noble Identities.
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