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Conservation of the passband signal amplitude using a filter based on the Fast Fourier Transform algorithm

Flavio Dalossa Freire, Isabel Gebauer Soares

TL;DR

This work tackles amplitude loss in passband filtering by introducing an FFT-based filter design that operates in the frequency domain and returns to time domain via IFFT. The method leverages shifted-symmetric frequency masking to achieve sharp, rectangular passbands and minimizes unintended alterations to the signal. Compared with a traditional FIR equiripple approach, the FFT-based method shows lower RMSE against a theoretical signal and better preservation of passband amplitude in tested bands. The authors provide practical code in Python, R, and MATLAB and outline directions for more rigorous evaluation and computational cost comparisons in future work.

Abstract

In this work, we propose an algorithm for a filter based on the Fast Fourier Transform (FFT), which, due to its characteristics, allows for an efficient computational implementation, ease of use, and minimizes amplitude variation in the filtered signal. The algorithm was implemented using the programming languages Python, R, and MATLAB. Initial results led to the conclusion that there was less amplitude loss in the filtered signal compared to the FIR filter. Future work may address a more rigorous methodology and comparative assessment of computational cost.

Conservation of the passband signal amplitude using a filter based on the Fast Fourier Transform algorithm

TL;DR

This work tackles amplitude loss in passband filtering by introducing an FFT-based filter design that operates in the frequency domain and returns to time domain via IFFT. The method leverages shifted-symmetric frequency masking to achieve sharp, rectangular passbands and minimizes unintended alterations to the signal. Compared with a traditional FIR equiripple approach, the FFT-based method shows lower RMSE against a theoretical signal and better preservation of passband amplitude in tested bands. The authors provide practical code in Python, R, and MATLAB and outline directions for more rigorous evaluation and computational cost comparisons in future work.

Abstract

In this work, we propose an algorithm for a filter based on the Fast Fourier Transform (FFT), which, due to its characteristics, allows for an efficient computational implementation, ease of use, and minimizes amplitude variation in the filtered signal. The algorithm was implemented using the programming languages Python, R, and MATLAB. Initial results led to the conclusion that there was less amplitude loss in the filtered signal compared to the FIR filter. Future work may address a more rigorous methodology and comparative assessment of computational cost.
Paper Structure (12 sections, 12 figures, 1 table)

This paper contains 12 sections, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Types of Filters according to frequency interval
  3. The Algorithm
  4. Pseudo-Code suggestion
  5. Code Examples
  6. Python Example
  7. R Example
  8. MATLAB Example
  9. Discussion and Results
  10. Signal Amplitude: Filtered versus Theoretical
  11. Signal Amplitude: FFT Filter versus FIR Equiripple Filter
  12. Conclusion

Figures (12)

  • Figure 1: FFT Result structure for N even
  • Figure 2: FFT Result structure for N odd
  • Figure 3: Signal with Noise Visualization
  • Figure 4: Comparison between band-pass filter $2$ to $4$ Hz and single-pass $3$ Hz
  • Figure 5: Comparison between band-pass filter $9$ to $11$ Hz and single-pass $10$ Hz
  • ...and 7 more figures