A method for improving the performance of spectral signal analysis using microcontrollers


Аuthors

Safaraleev V. A.*, Gusarov A. A.

Mlitary spaсe Aсademy named after A.F. Mozhaisky, Saint Petersburg, Russia

*e-mail: vka@mil.ru

Abstract

Various methods are used in signal processing. The most popular way to analyze signals is spectral analysis. However, the calculation of the spectral components of the signal requires high computational costs. The aim of the work is to develop a way to reduce the execution time of spectral analysis by optimizing the algorithm for calculating the discrete Fourier transform. The FFT algorithm is based on the application of an operator designated as WnkN In the literature, its application is often called the "butterfly algorithm". To convert a signal (window) with a duration of 8192 (213) samples, ≈2·106 clock cycles will be required.
At the same time, the following aspects should be noted:
1. Reducing the size of the window function (N) leads to an increase in distortion of the calculated spectral components of the signal
2. In the FFT algorithm, when the size of the window function decreases, the interval between the analyzed frequencies increases
It is known that the spectrum of those parts of the signal that are located close to the boundaries of the window function is significantly distorted. In order not to lose information about the signal under study, it is recommended to ensure that the window functions overlap by 50-75%. The proposed method is based on the DFT algorithm. The proposed method for optimizing the algorithm for calculating the signal spectrum makes it possible to reduce the frequency sampling step with the possibility of selecting the analyzed frequencies, remove the restriction on the size of the window function and maximize window overlap. The use of tabular function assignment is a widely used way to improve performance. In the proposed method, the number of values in the tabular representation of the harmonic function requires equality of the length of the table L = 2J, where J is an integer. The tabular representation of the harmonic function allows you to obtain values corresponding to multiples of frequencies without significantly increasing the amount of calculations. Another bottleneck in the calculation of spectral components is the summation of the results of processing a large number of samples of the original signal – the "window". To solve this problem, it is proposed to store in RAM the results of multiplying each input sample by the cos and sin values of each frequency under study. The sums corresponding to the following window function (shifted by one count) differ from the original one in that:
– the product for the count with the number n=N+1 is added;
– there is no product for counting with the number n=0.
Thus, the calculation time of each spectral component becomes independent of the window size. At the same time, the need for RAM increases significantly. Let's compare the performance indicators of the spectrum calculation. For example, for a window function equal to 128 counts, 16`576 clock cycles of the microcontroller are required to perform the FFT. According to the proposed method, only 5`248 (41 × 128) clock cycles will be required, which gives a gain of 3.16 times. Similarly, for a window equal to 1024 counts, the gain will be 4.51 times, and for a window equal to 8192 counts – 5.87 times. There are ways to further increase productivity. Conditionally, they can be called software (algorithmic) and hardware. Moreover, they can be used simultaneously.

Keywords:

spectral analysis, discrete Fourier transform, spectral analysis performance, digital signal processing

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