# 6.1 Interpolation, decimation, and rate changing by integer fractions

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## Interpolation: by an integer factor l

Interpolation means increasing the sampling rate, or filling in in-between samples. Equivalent to sampling abandlimited analog signal $L$ times faster. For the ideal interpolator,

${X}_{1}(\omega )=\begin{cases}{X}_{0}(L\omega ) & \text{if \left|\omega \right|< \frac{\pi }{L}}\\ 0 & \text{if \frac{\pi }{L}\le \left|\omega \right|\le \pi }\end{cases}$
$y(m)=\begin{cases}{X}_{0}(\frac{m}{L}) & \text{if m=\{0, ±(L), ±(2L), \dots \}}\\ 0 & \text{otherwise}\end{cases}$
The DTFT of $y(m)$ is
$Y(\omega )=\sum_{m=-\omega }$ y m ω m n x 0 n ω L n n x n ω L n X 0 ω L
Since ${X}_{0}({\omega }^{\prime })$ is periodic with a period of $2\pi$ , ${X}_{0}(L\omega )=Y(\omega )$ is periodic with a period of $\frac{2\pi }{L}$ (see [link] ). By inserting zero samples between the samples of ${x}_{0}(n)$ , we obtain a signal with a scaled frequency response that simply replicates ${X}_{0}({\omega }^{\prime })$ $L$ times over a $2\pi$ interval!

Obviously, the desired ${x}_{1}(m)$ can be obtained simply by lowpass filtering $y(m)$ to remove the replicas.

${x}_{1}(m)=(y(m), {h}_{L}(m))$
Given ${H}_{L}(m)=\begin{cases}1 & \text{if \left|\omega \right|< \frac{\pi }{L}}\\ 0 & \text{if \frac{\pi }{L}\le \left|\omega \right|\le \pi }\end{cases}$ In practice, a finite-length lowpass filter is designed using any of the methods studied so far ( [link] ).

## Decimation: sampling rate reduction (by an integer factor m)

Let $y(m)={x}_{0}(Lm)$ ( [link] )

That is, keep only every $L$ th sample ( [link] ) In frequency (DTFT):
$Y(\omega )=\sum_{m=()}$ y m ω m m x 0 M m ω m n M m n x 0 n k δ n M k ω n M ω ω M n x 0 n k δ n M k ω n DTFT x 0 n DTFT δ n M k
Now $\mathrm{DTFT}(\sum \delta (n-Mk))=2\pi \sum_{k=0}^{M-1} X(k)\delta ({\omega }_{\prime }-\frac{2\pi k}{M})$ for $\left|\omega \right|< \pi$ as shown in homework #1, where $X(k)$ is the DFT of one period of the periodic sequence. In this case, $X(k)=1$ for $k\in \{0, 1, \dots , M-1\}$ and $\mathrm{DTFT}(\sum \delta (n-Mk))=2\pi \sum_{k=0}^{M-1} \delta ({\omega }_{\prime }-\frac{2\pi k}{M})$ .
$(\mathrm{DTFT}({x}_{0}(n)), \mathrm{DTFT}(\sum \delta (n-Mk)))=({X}_{0}({\omega }^{\prime }), 2\pi \sum_{k=0}^{M-1} \delta ({\omega }_{\prime }-\frac{2\pi k}{M}))=\frac{1}{2\pi }\int_{-\pi }^{\pi } {X}_{0}({\mu }^{\prime })2\pi \sum_{k=0}^{M-1} \delta ({\omega }_{\prime }-{\mu }_{\prime }-\frac{2\pi k}{M})\,d {\mu }^{\prime }=\sum_{k=0}^{M-1} {X}_{0}({\omega }_{\prime }-\frac{2\pi k}{M})$
so $Y(\omega )=\sum_{k=0}^{M-1} {X}_{0}(\frac{\omega }{M}-\frac{2\pi k}{M})$ i.e. , we get digital aliasing .( [link] ) Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequencycomponents above $\left|\omega \right|< \frac{\pi }{M}$ ( [link] ).

## Rate-changing by a rational fraction l/m

This is easily accomplished by interpolating by a factor of $L$ , then decimating by a factor of $M$ ( [link] ).

The two lowpass filters can be combined into one LP filterwith the lower cutoff, $H(\omega )=\begin{cases}1 & \text{if \left|\omega \right|< \frac{\pi }{\max\{L , M\}}}\\ 0 & \text{if \frac{\pi }{\max\{L , M\}}\le \left|\omega \right|\le \pi }\end{cases}$ Obviously, the computational complexity and simplicity of implementation will depend on $\frac{L}{M}$ : $2/3$ will be easier to implement than $1061/1060$ !

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