Zero-Padding Techniques in Spectral Interpolation

Introduction

We start with the original function, $f(x)$ :

$\begin{equation} f(x) = sin(x) + 0.5cos(2x) + 0.25cos(3x) \end{equation}$

Then we take 16 ( $N$ ) evenly spaced samples labeled $f_n$ . The following formulation of the DFT equation was then used to get the frequencies $F_k$ :

$\begin{equation} F_k = \sum_{n=0}^{N-1} f_{n}e^{-2\pi ikn/N} \end{equation}$

The following image shows the original DFT spectra (real and imaginary):

Case 1: Zeros Interspersed

In Case 1 the zeros are interspersed in-between the original frequencies. The total number of frequencies are doubled to 32 ( $N'$ ) in the new spectra, $F_{k'}$ . The following image shows the resulting spectra:

After zero padding the inverse DFT is taken using the following formula:

$\begin{equation} f_{n'} = \frac{1}{N'} \sum_{k'=0}^{N'-1} F_{k'}e^{2\pi ik'n'/N'} \end{equation}$

Finally, $f_{n'}$ is scaled by a factor of two to account for the zero padding. Note that $f_{n' = 32}$ is discarded as it is an extrapolation, not an interpolation between the sampled values. The following image shows the original function and sampling as well as the interpolated values.

Notice that the result in this case is simply 2 of the original $f_n$ samples end to end. By manually computing the DFT it can be seen that this method is not an interpolation, it simply reproduces multiple sets of the original spectra.

Case 2: Zeros Centered

In Case 2 all 8 zeros are placed in the center of the spectrum. The following plots show the spectra with the zeros added in the center.

As before the inverse DFT is taken and the result is scaled. The following image shows the original function and sampling as well as the interpolated values.

From this result it is clear that the original values are reproduced and interpolated between by $f_{n'}$ when zeros are inserted into the center of $F_k$ .

References

R.G. Lyons Understanding Digital Signal Processing 3rd edl. 2010 Prentice Hall
B.L. Adams, S.R. Kalidindi, D.L. Fullwood Microstructure Sensitive Design for Performance Optimization 2012 Butterworth-Heinemann