4.1 Fundamentals of multirate signal processing

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,
  $$\begin{array}{|c|}\hline y\left(n\right)=x\left(n\right)e^{j{\omega }_{o}n}\\ \hline\end{array}.$$
  In the z -domain,
  $$\begin{array}{c}\hfill Y\left(z\right)=\sum _{n}y\left(n\right)z^{-n}=\sum _n\left(x,\left(n\right),e^{j{\omega }_{o}n}\right)z^{-n}=\sum _{n}x\left(n\right){\left(e^{-j{\omega }_o},z\right)}^{-n}=\begin{array}{|c|}\hline X\left(e^{-j{\omega }_o},z\right)\\ \hline\end{array}.\end{array}$$
  We can evaluate the result of modulation in the frequency domain by substituting $z=e^{j\omega }$ . This yields
  $$\begin{array}{c}\hfill Y\left(\omega \right)=\sum _{n}y\left(n\right)e^{-j\omega n}=\begin{array}{|c|}\hline X(\omega -{\omega }_o)\\ \hline\end{array}.\end{array}$$
  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 $N-1$ zeros between every sample of the input process.Formally, upsampling can be expressed in the time domain as
  $$\begin{array}{c}\hfill \begin{array}{|c|}\hline y\left(n\right)=\left\{\begin{array}{cc}x(n/N)\hfill & \text{when}n=mN\text{for}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else.}\hfill \end{array}\right)\\ \hline\end{array}\end{array}$$
  In the z -domain, upsampling causes
  $$\begin{array}{c}\hfill Y\left(z\right)=\sum _{n}y\left(n\right)z^{-n}=\sum _{m}x\left(m\right)z^{-mN}=\begin{array}{|c|}\hline X\left(z^N\right)\\ \hline\end{array},\end{array}$$
  and in the frequency domain,
  $$\begin{array}{c}\hfill Y\left(\omega \right)=\sum _{n}y\left(n\right)e^{-j\omega n}=\begin{array}{|c|}\hline X\left(N,\omega \right)\\ \hline\end{array}.\end{array}$$
  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
  $$\begin{array}{|c|}\hline y\left(m\right)=x\left(mN\right).\\ \hline\end{array}$$
  In the z domain,
  $$\begin{array}{c}\hfill Y\left(z\right)=\sum _{m}y\left(m\right)z^{-m}=\sum _{m}x\left(mN\right)z^{-m}=\sum _n\tilde{x}\left(n\right)z^{-n/N},\end{array}$$
  where
  $$\tilde{x}\left(n\right)=\left\{\begin{array}{cc}x\left(n\right)\hfill & \text{when}n=mN\text{for}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else}.\hfill \end{array}\right)$$
  The neat trick
  $$\begin{array}{c}\hfill \frac{1}{N}\sum _{p=0}^{N-1}{e}^{j\frac{2\pi }{N}np}=\left\{\begin{array}{cc}1\hfill & \text{when}n=mN\text{for}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else}\hfill \end{array}\right)\end{array}$$
  (which is not difficult to prove) allows us to rewrite $\tilde{x}\left(n\right)$ 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,
  $$\begin{array}{c}\hfill \begin{array}{|c|}\hline Y\left(\omega \right)=\frac{1}{N}\sum _{p=0}^{N-1}X\left(\frac{\omega -2\pi p}{N}\right)\\ \hline\end{array}.\end{array}$$
  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.,
  $$\begin{array}{|c|}\hline y\left(n\right)=\left\{\begin{array}{cc}x\left(n\right)\hfill & \text{when}n=mN\text{for}m\in \mathbb{Z}\hfill \\ 0\hfill & \text{else}.\hfill \end{array}\right)\\ \hline\end{array}$$
  Using trick [link] ,
  $$\begin{array}{ccc}\hfill Y\left(z\right)& =& \sum _{n}y\left(n\right)z^{-n}\hfill \\ & =& \sum _{n}x\left(n\right)\left(\frac{1}{N},\sum _{p=0}^{N-1},e^{j\frac{2\pi }{N}np}\right)z^{-n}\hfill \\ & =& \frac{1}{N}\sum _{p=0}^{N-1}\sum _{n}x\left(n\right){\left(e^{-j\frac{2\pi }{N}p},z\right)}^{-n}\hfill \\ & =& \begin{array}{|c|}\hline \frac{1}{N}\sum _{p=0}^{N-1}X\left(e^{-j\frac{2\pi }{N}p},z\right)\\ \hline\end{array},\hfill \end{array}$$
  which implies
  $$\begin{array}{c}\hfill \begin{array}{|c|}\hline Y\left(\omega \right)=\frac{1}{N}\sum _{p=0}^{N-1}X\left(\omega ,-,\frac{2\pi p}{N}\right)\\ \hline\end{array}.\end{array}$$
  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.