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Linear Interpolation Frequency Response

Since linear interpolation is a convolution of the samples with a triangular pulse $ h_l(t) = (p_T\ast p_T)(t)$ (from Eq.$ \,$ (4.5)), the frequency response of the interpolation is given by the Fourier transform $ H_l(f)$ , which yields a sinc$ ^2$ function. This frequency response applies to linear interpolation from discrete time to continuous time. If the output of the interpolator is also sampled, this can be modeled by sampling the continuous-time interpolation result in Eq.$ \,$ (4.5), thereby aliasing the sinc$ ^2$ frequency response, as shown in Fig.4.9.

Figure: (a) $ \protect$sinc$ ^2(fT) =
{\cal F}_f^2\{p_T/T\}$ .
(b) $ \protect\alias _{10}\{\protect$sinc$ ^2(fT)\} \propto
\protect\dtft \{\protect\sample _{T/10}\{p_T\ast p_T\}\}$ .
\includegraphics[width=3in]{eps/sincsquared}

In slightly more detail, from $ h_l(t) = (p_T\ast p_T)(t)$ , and $ {\cal F}_f\{p_T\}= T$sinc$ ( f T)$ , we have

$\displaystyle H_l(f) \isdef {\cal F}_f\{h_l\} = {\cal F}_f\{p_T\ast p_T\} = {\cal F}^2_f\{p_T\} = T^2$sinc$\displaystyle ^2(fT)
$

where we used the convolution theorem for Fourier transforms, and the fact that $ P_T(f) = T$sinc$ ( f T) = \sin(\pi f T) / (\pi f)$ .

The Fourier transform of $ h_l(nT/L)$ is the same function aliased on a block of size $ f_s=L/T$ Hz. Both $ H_l$ and its alias are plotted in Fig.4.9. The example in this figure pertains to an output sampling rate which is $ L=10$ times that of the input signal. In other words, the input signal is upsampled by a factor of $ L=10$ using linear interpolation. The ``main lobe'' of the interpolation frequency response $ H_l(f)$ contains the original signal bandwidth; note how it is attenuated near half the original sampling rate ($ f T=1/2$ in Fig.4.9). The ``sidelobes'' of the frequency response contain attenuated copies of the original signal bandwidth (see the DFT stretch theorem), and thus constitute spectral imaging distortion in the final output (sometimes also referred to as a kind of ``aliasing,'' but, for clarity, that term will not be used for imaging distortion in this book). We see that the frequency response of linear interpolation is less than ideal in two ways:

These qualitative remarks apply to all upsampling factors $ L\geq 2$ using linear interpolation. The case $ L=1$ is considered in the next section.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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