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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
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