1

I need to interpolate a complex valued bandlimited periodic function using local interpolation.

I can have the signal sampled at any frequency I want over at equispace intervals.

I am aware that for bandlimited periodic functions the best interpolation method is global interpolation (FFT).

However, for this project I need to use local interpolation. I managed to perform the interpolation using Lagrange polynomials of order 10 but I require an oversampling of 2 to achieve the desired results. My goal is to reduce oversampling to any extend even if I need a longer interpolator.

I recently read as much as I could about FIR filters but somethings are not clear to me.

Since the interpolation factor is of the order of 2, I am planning to use a halfband pass interpolator (many books say this is the best since one gets a cheap interpolator due to the zeros in the impulse response).

My questions are:

  • How do I design a halfband pass filter? In Matlab there are nice functions in the dsp package that allow directly to construct such filters, unfortunately I have to do this implementation in C. I searched and I found that the remez() function from Octave to compute the coefficients is available in C and is the one I am planning to use.
    Designing a low pass filter with the Parks-Mclellan algorithm is it by default a halfband pass filter (provided I set up correctly the frequency bands) or do I have to enforce the time domain constraints?

  • Regarding the linear phase of the FIR filter, what I understand is that my signal will be shifted by some samples, how do I compensate to remove the shifting? What comes to my mind is to apply the corresponding shifting on the opposite direction, is this necessary?

  • If you could provide an working example of an interpolation of a periodic function (e.g. sum of sines) using a FIR I would be very grateful.
Marcus Müller
  • 30,525
  • 4
  • 34
  • 58
user1084135
  • 19
  • 2
  • Reading more about this topic, I found out that for my project I require zero-phase filters since this is an "offline" application. – user1084135 May 23 '18 at 19:00
  • It's an advantage of offline signal processing that non-causal filters can be used, and hence, zero-phase filtering is possible. It's not a necessity. – Marcus Müller Apr 27 '19 at 17:08
  • and no, "global interpolation (FFT)" (what's that?) is not inherently the "best" option; I think you might be thinking of padding the FFT of the original signal with zeros and then transforming it back. That has very serious problems! – Marcus Müller Apr 27 '19 at 17:11
  • For remez half-band filters that has almost half of the coefficients zero-valued, see my answer to FIR Filter design: Window vs Parks-McClellan and Least-Squares. – Olli Niemitalo Apr 27 '19 at 17:57
  • @MarcusMüller, when the signal is periodic, bandlimited, and you have a complete period, what kind of interpolation is better than global interpolation using trigonometric polynomials and implemented with the FFT? What are the problems you refer to? – user1084135 Apr 29 '19 at 06:29
  • edge effects, the circular convolution property of the DFT. But if you know all your signal components fit exactly multiple period times into the original DFT length, there's no problem. But why would you need to interpolate a signal that you know? – Marcus Müller Apr 29 '19 at 08:02
  • Efficiency. In my application, I required to multiply 2 bandlimited periodic functions S1 and S2, with bandwidth B1 and B2, respectively, with B1<B2. I could compute S1 and S2 to any resolution since they are known in closed form, however, the computations are expensive (CPU and memory usage). For this reason, I computed S1 and S2 to the minimal number of samples using the Nyquist criteria, then interpolate S1 to have the same number of points as S2 and perform multiplication. Later, I did some modifications in my algorithm to make it faster and I required to use local interpolation on S1. – user1084135 Apr 29 '19 at 18:38

1 Answers1

-1

The impulse response and frequency response of a length N zero-phase FIR lowpass filter can be designed using the Remez exchange algorithm.

N = 11; % filter length - must be odd

b = [0 0.1 0.2 0.5]*2; % band edges

M = [1 1 0 0 ]; % desired band values

h = remez(N-1,b,M); % Remez multiple exchange design

The impulse response h is returned in linear-phase form, so it must be left-shifted (N-1)/2 = 5 samples to make it zero-phase. Answer taken from https://www.researchgate.net/post/How_can_I_design_a_null_phase_FIR_filter.

I implemented this solution some time ago and still works very well in my code.

user1084135
  • 19
  • 2