# 4.1 Fundamentals of multirate signal processing

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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,
$\begin{array}{|c|}\hline y\left(n\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}x\left(n\right){e}^{j{\omega }_{o}n}\\ \hline\end{array}.$
In the z -domain,
$\begin{array}{c}\hfill Y\left(z\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}y\left(n\right){z}^{-n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}\left(x,\left(n\right),{e}^{j{\omega }_{o}n}\right){z}^{-n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}x\left(n\right){\left({e}^{-j{\omega }_{o}},z\right)}^{-n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\begin{array}{|c|}\hline X\left({e}^{-j{\omega }_{o}},z\right)\\ \hline\end{array}.\end{array}$
We can evaluate the result of modulation in the frequency domain by substituting $z={e}^{j\omega }$ . This yields
$\begin{array}{c}\hfill Y\left(\omega \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}y\left(n\right){e}^{-j\omega n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\begin{array}{|c|}\hline X\left(\omega -{\omega }_{o}\right)\\ \hline\end{array}.\end{array}$
Note that $X\left(\omega -{\omega }_{o}\right)$ 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
$\begin{array}{c}\hfill \begin{array}{|c|}\hline y\left(n\right)=\left\{\begin{array}{cc}x\left(n/N\right)\hfill & \phantom{\rule{4pt}{0ex}}\text{when}n=mN\phantom{\rule{4pt}{0ex}}\text{for}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else.}\hfill \end{array}\right)\\ \hline\end{array}\end{array}$
In the z -domain, upsampling causes
$\begin{array}{c}\hfill Y\left(z\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}y\left(n\right){z}^{-n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{m}x\left(m\right){z}^{-mN}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\begin{array}{|c|}\hline X\left({z}^{N}\right)\\ \hline\end{array},\end{array}$
and in the frequency domain,
$\begin{array}{c}\hfill Y\left(\omega \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}y\left(n\right){e}^{-j\omega n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\begin{array}{|c|}\hline X\left(N,\omega \right)\\ \hline\end{array}.\end{array}$
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
$\begin{array}{|c|}\hline y\left(m\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}x\left(mN\right).\\ \hline\end{array}$
In the z domain,
$\begin{array}{c}\hfill Y\left(z\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{m}y\left(m\right){z}^{-m}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{m}x\left(mN\right){z}^{-m}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{n}\stackrel{˜}{x}\left(n\right){z}^{-n/N},\end{array}$
where
$\stackrel{˜}{x}\left(n\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{cc}x\left(n\right)\hfill & \text{when}\phantom{\rule{4.pt}{0ex}}n=mN\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else}.\hfill \end{array}\right)$
The neat trick
$\begin{array}{c}\hfill \frac{1}{N}\sum _{p=0}^{N-1}{e}^{j\frac{2\pi }{N}np}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{cc}1\hfill & \text{when}\phantom{\rule{4.pt}{0ex}}n=mN\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else}\hfill \end{array}\right)\end{array}$
(which is not difficult to prove) allows us to rewrite $\stackrel{˜}{x}\left(n\right)$ in terms of $x\left(n\right)$ :
$\begin{array}{ccc}\hfill Y\left(z\right)& =& \sum _{n}x\left(n\right)\left(\frac{1}{N},\sum _{p=0}^{N-1},{e}^{j\frac{2\pi }{N}np}\right){z}^{-n/N}\hfill \\ & =& \frac{1}{N}\sum _{p=0}^{N-1}\sum _{n}x\left(n\right){\left({e}^{-j\frac{2\pi }{N}p},{z}^{1/N}\right)}^{-n}\hfill \\ & =& \begin{array}{|c|}\hline \frac{1}{N}\sum _{p=0}^{N-1}X\left({e}^{-j\frac{2\pi }{N}p},{z}^{1/N}\right)\\ \hline\end{array}.\hfill \end{array}$
Translating to the frequency domain,
$\begin{array}{c}\hfill \begin{array}{|c|}\hline Y\left(\omega \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{1}{N}\sum _{p=0}^{N-1}X\left(\frac{\omega -2\pi p}{N}\right)\\ \hline\end{array}.\end{array}$
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.,
$\begin{array}{|c|}\hline y\left(n\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\left\{\begin{array}{cc}x\left(n\right)\hfill & \text{when}n=mN\phantom{\rule{4.pt}{0ex}}\text{for}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else}.\hfill \end{array}\right)\\ \hline\end{array}$
$\begin{array}{ccc}\hfill Y\left(z\right)& =& \sum _{n}y\left(n\right){z}^{-n}\hfill \\ & =& \sum _{n}x\left(n\right)\left(\frac{1}{N},\sum _{p=0}^{N-1},{e}^{j\frac{2\pi }{N}np}\right){z}^{-n}\hfill \\ & =& \frac{1}{N}\sum _{p=0}^{N-1}\sum _{n}x\left(n\right){\left({e}^{-j\frac{2\pi }{N}p},z\right)}^{-n}\hfill \\ & =& \begin{array}{|c|}\hline \frac{1}{N}\sum _{p=0}^{N-1}X\left({e}^{-j\frac{2\pi }{N}p},z\right)\\ \hline\end{array},\hfill \end{array}$
which implies
$\begin{array}{c}\hfill \begin{array}{|c|}\hline Y\left(\omega \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{1}{N}\sum _{p=0}^{N-1}X\left(\omega ,-,\frac{2\pi p}{N}\right)\\ \hline\end{array}.\end{array}$
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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