Linear Interpolation Frequency Response

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

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.

These qualitative remarks apply to all upsampling factors using linear interpolation. The case is considered in the next section.

---

Previous: Triangular Pulse as Convolution of Two Rectangular Pulses
Next: Special Cases

Order a Hardcopy of Physical Audio Signal Processing

---

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.