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.
Questions & Answers
what is variations in raman spectra for nanomaterials
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
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Mahi
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How this robot is carried to required site of body cell.?
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Rafiq
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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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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