The introduction

Recently, encountered when conducting issues need to be trapezoidal wave Fourier series expansion of the problem, some of the information (ashamed, not from the start thinking of yourself points), and then could not find the results they want (there are similar, but not at any period, and there was no change), finally began to calculate, Here to make a little note, is a review of the previous knowledge, face.

Because the waveform like triangle wave, rectangular wave, trapezoidal wave is not continuous, it is easy to appear in the simulation software calculation convergence. Therefore, in this case, using a series of harmonic superposition form is equivalent to the original waveform, which can optimize the model well.

Preliminary knowledge

Formula 1.

Given a period of zeroThe function of, then it can be expressed as an infinite series:


Where, the Fourier coefficient is:


Nature of 2.

  • The convergence

The Fourier series of functions that satisfy the Dirichlet condition on a closed interval converge. Dirichlet conditions are as follows:

  • In defining the interval,Must be absolutely integrable;
  • In any finite interval,You can only take a finite number of extreme points;
  • On any finite interval,There can only be a limited numberThe first kind of discontinuity.

Satisfying the above conditionsThe Fourier series converge, and:

  • whenisWhen, the series converges to
  • whenisAt the discontinuity point of, the series converges to
  • orthogonality

Orthogonal means that the inner product of two different vectors is zero, which means that there is no correlation between them. For example, in three-dimensional Euclidean space, vectors that are perpendicular to each other are orthogonal. The orthogonality of the family of trigonometric functions can be expressed by the formula:


  • parity

Odd functionIt can be expressed as a sine series, and an even functionCan be expressed as cosine series:


Fourier series expansion of several common waveforms

1. Trapezoidal wave (odd function)

Trapezoidal wave

As shown in the figure above, the trapezoidal wave is an odd function with period T and its amplitude is, rising along time isIn the rangeIs expressed as:


According to the parity, the waveform is in the intervalThe Fourier series expansion of is:


Where, the Fourier coefficient is:


willSubstituting the function into the Fourier coefficient expression, it can be obtained:


by


Available:


To sum up, the trapezoidal wave can be obtained in the intervalThe Fourier series expansion of is:


Among them:

2. Pulse wave (even function)

Pulse wave

As shown in the figure above, the impulse wave is an even function of period T and its amplitude is, the pulse width isIn the rangeIs expressed as:


According to the parity, the waveform is in the intervalThe Fourier series expansion of is:


Where, the Fourier coefficient is:


willSubstituting the function into the Fourier coefficient expression, it can be obtained:


Therefore, the trapezoidal wave can be obtained in the intervalThe Fourier series expansion of is:


Among them:

3. Square wave (odd function)

Square wave

Similarly, the square wave is in the intervalThe Fourier series expansion of is:


Among them:

4. Triangle wave (odd function)

Triangle wave

Similarly, the triangle wave is in the intervalThe Fourier series expansion of is:


5. Sawtooth wave (non-odd and non-even function)

Sawtooth wave

The sawtooth wave is shown above in the intervalIs expressed as:


Since the function is non-odd and non-even, the waveform is in the intervalThe Fourier series expansion of is:


Where, the Fourier coefficient is:


willSubstituting the function into the Fourier coefficient expression, it can be obtained:


Therefore, the sawtooth wave can be obtained in the intervalThe Fourier series expansion of is:


conclusion

Here only a small part of the waveform of the Fourier series expansion is listed, for other waveforms, similar to the calculation can be substituted, after giving the formula, more is the test of mathematical integration calculation.

reference

[1] Wikipedia editor. Fourier series

[2] Editor of Baidu Encyclopedia. Fourier series

[3] Fourier Series Examples