Here we present some background material on multirate signal processing that is necessary to understand the filterbank processing used in sub-band coding. In particular, we describe modulation, upsampling, and downsampling in several domains: the time-domain, z-domain, and DTFT domain. In addition, we describe the aliasing phenomenon.
The presence of upsamplers and downsamplers in the diagram of
Figure 2 from "Introduction and Motivation" implies that a basic knowledge of multirate signal processing is
indispensible to an understanding of sub-band analysis/synthesis.This section provides the required background.
Modulation:[link] illustrates modulation using a complex exponential
of frequency
ω
_{o} .
In the time domain,
Note that
$X(\omega -{\omega}_{o})$ represents a shift of
$X\left(\omega \right)$up by
ω
_{o} radians, as in
[link] .
Upsampling:[link] illustrates upsampling by factor
N .
In words, upsampling means the insertion of
$N\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1$ zeros between
every sample of the input process.Formally, upsampling can be expressed in the time domain as
As shown in
[link] , upsampling shrinks
$X\left(\omega \right)$ by a factor of
N along the
ω axis.
Downsampling:[link] illustrates downsampling by factor
N .
In words, the process of downsampling keeps every
${N}^{th}$ sample
and discards the rest.Formally, downsampling can be written as
As shown in
[link] , downsampling expands
each
$2\pi $ -periodic repetition of
$X\left(\omega \right)$ by a factor of
N along
the
ω axis.
Note the spectral overlap due to downsampling, called “aliasing.”
Downsample-Upsample Cascade:
Downsampling followed by upsampling (of equal factor
N ) is
illustrated by
[link] .
This structure is useful in understanding analysis/synthesis filterbanksthat lie at the heart of sub-band coding schemes.
This operation is equivalent to zeroing all but the
$m{N}^{th}$ samples
in the input sequence, i.e.,
The downsampler-upsampler cascade causes the appearance
of
$2\pi /N$ -periodic copies of the baseband spectrum of
$X\left(\omega \right)$ .
As illustrated in
[link] , aliasing may result.
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Source:
OpenStax, An introduction to source-coding: quantization, dpcm, transform coding, and sub-band coding. OpenStax CNX. Sep 25, 2009 Download for free at http://cnx.org/content/col11121/1.2
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