# Overview of multirate signal processing

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Digital transformation of the sampling rate of signals, or signal processing with different sampling rates in the system.

## Applications

1. CD to DAT format change, for example.
2. oversampling converters; which reduce performance requirements onanti-aliasing or reconstruction filters
3. bandwidth of individual channels is much less than theoverall bandwidth
4. Eyes and ears are not as sensitive to errors in higher frequencybands, so many coding schemes split signals into different frequency bands and quantize higher-frequency bands withmuch less precision.

## General rate-changing procedure

This procedure is motivated by an analog-based method: one conceptually simple method to change the sampling rate is tosimply convert a digital signal to an analog signal and resample it! ( [link] )

${H}_{\mathrm{aa}}(\Omega )=\begin{cases}1 & \text{if \left|\Omega \right|< \frac{\pi }{{T}_{1}}}\\ 0 & \text{otherwise}\end{cases}$ ${h}_{\mathrm{aa}}(t)=\frac{\sin (\frac{\pi }{{T}_{1}}t)}{\frac{\pi }{{T}_{1}}t}$ Recall the ideal D/A:
${x}_{a}^{\prime }(t)=\sum_{n=()}$ x 0 n t n T 0 T 0 t n T 0 T 0
The problems with this scheme are:
1. A/D, D/A,filters cost money
2. imperfections in these devices introduce errors

Digital implementation of rate-changing according to this formula has three problems:

1. Infinite sum: The solution is to truncate. Consider $sinc(t)\approx 0$ for $t< {t}_{1}$ , $t> {t}_{2}$ : Then $m{T}_{1}-n{T}_{0}< {t}_{1}$ and $m{T}_{1}-n{T}_{0}> {t}_{2}$ which implies ${N}_{1}=\lceil \frac{m{T}_{1}-{t}_{2}}{{T}_{0}}\rceil$ ${N}_{2}=\lfloor \frac{m{T}_{1}-{t}_{1}}{{T}_{0}}\rfloor$ ${x}_{1}(m)=\sum_{n={N}_{1}}^{{N}_{2}} {x}_{0}(n){sinc}_{{T}^{\prime }}(m{T}_{1}-n{T}_{0})$
This is essentially lowpass filter design using a boxcar window: other finite-length filter design methods could beused for this.
2. Lack of causality : The solution is to delay by $\max\{\left|N\right|\}$ samples. The mathematics of the analog portions of this system can be implemented digitally.
${x}_{1}(m)=(t, , ({h}_{\mathrm{aa}}(t), {x}_{a}^{\prime }(t)))=\int_{()} \,d \tau$ n x 0 n m T 1 τ n T 0 T 0 m T 1 τ n T 0 T 0 τ T 1 τ T 1
${x}_{1}(m)=({T}^{\prime }, , \sum_{n=()} )$ x 0 n T m T 1 n T 0 T m T 1 n T 0 n x 0 n sinc T m T 1 n T 0
So we have an all-digital formula for exact digital-to-digital rate changing!
3. Cost of computing ${sinc}_{{T}^{\prime }}(m{T}_{1}-n{T}_{0})$ : The solution is to precompute the table of $\mathrm{sinc}(t)$ values. However, if $\frac{{T}_{1}}{{T}_{0}}$ is not a rational fraction, an infinite number of samples will be needed, so some approximation will have tobe tolerated.
Rate transformation of any rate to any other rate can be accomplished digitally with arbitrary precision (if somedelay is acceptable). This method is used in practice in many cases. We will examine a number of special cases andcomputational improvements, but in some sense everything that follows are details; the above idea is the centralidea in multirate signal processing.

Useful references for the traditional material (everything except PRFBs) are Crochiere and Rabiner, 1981 and Crochiere and Rabiner, 1983 . A more recent tutorial is Vaidyanathan ; see also Rioul and Vetterli . References to most of the original papers can be found in these tutorials.

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