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
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.
Triangular Pulse as Convolution of Two Rectangular Pulses