# 4.2 Interpolation

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Introduction of interpolation and it's application.

## Interpolation

Interpolation is the process of upsampling and filtering a signal to increase its effective sampling rate. To be more specific, say that $x(m)$ is an (unaliased) $T$ -sampled version of ${x}_{c}(t)$ and $v(n)$ is an $L$ -upsampled version version of $x(m)$ . If we filter $v(n)$ with an ideal $\frac{\pi }{L}$ -bandwidth lowpass filter (with DC gain $L$ ) to obtain $y(n)$ , then $y(n)$ will be a $\frac{T}{L}$ -sampled version of ${x}_{c}(t)$ . This process is illustrated in .

We justify our claims about interpolation using frequency-domain arguments. From the sampling theorem, we know that $T$ - sampling ${x}_{c}(t)$ to create $x(n)$ yields

$X(e^{i\omega })=\frac{1}{T}\sum_{k} {X}_{c}(i\frac{\omega -2\pi k}{T})$
After upsampling by factor $L$ , implies $V(e^{i\omega })=\frac{1}{T}\sum_{k} {X}_{c}(i\frac{\omega L-2\pi k}{T})=\frac{1}{T}\sum_{k} {X}_{c}(i\frac{\omega -\frac{2\pi }{L}k}{\frac{T}{L}})$ Lowpass filtering with cutoff $\frac{\pi }{L}$ and gain $L$ yields $Y(e^{i\omega })=\frac{L}{T}\sum_{\frac{k}{L}\in \mathbb{Z}} {X}_{c}(i\frac{\omega -\frac{2\pi }{L}k}{\frac{T}{L}})=\frac{L}{T}\sum_{l} {X}_{c}(i\frac{\omega -2\pi l}{\frac{T}{L}})$ since the spectral copies with indices other than $k=lL$ (for $l\in \mathbb{Z}$ ) are removed. Clearly, this process yields a $\frac{T}{L}$ -shaped version of ${x}_{c}(t)$ . illustrates these frequency-domain arguments for $L=2$ .

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