Since linear interpolation is a convolution of the samples with a
triangular pulse
(from Eq.(4.5)),
the frequency response of the interpolation is given by the Fourier
transform
, which yields a
sinc
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
frequency
response, as shown in Fig.4.9.

In slightly more detail, from , and sinc , we have

sinc

where we used the convolution theorem for Fourier transforms, and the fact that sinc .

The Fourier transform of
is the same function aliased on
a block of size
Hz. Both
and its alias are plotted
in Fig.4.9. The example in this figure pertains to an
output sampling rate which is
times that of the input signal.
In other words, the input signal is upsampled by a factor of
using linear interpolation. The ``main lobe'' of the interpolation
frequency response
contains the original signal bandwidth;
note how it is attenuated near half the original sampling rate (
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:

- The spectrum is ``rolled'' off near half the sampling rate. In fact, it is nowhere flat within the ``passband'' (-1 to 1 in Fig.4.9).
- Spectral imaging distortion is suppressed by only 26 dB (the level of the first sidelobe in Fig.4.9.

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