Constant, Sinusoid, Square, Triangle, and sawtooth waveforms, in depth and summarized.
Introduction
Once one has obtained a solid understanding of the fundamentals of
Fourier series
analysis and the
General Derivation of the Fourier Coefficients , it is useful to have an understanding of the common signals used in Fourier Series Signal Approximation.
Deriving the fourier coefficients
Consider a square wave f(x) of length 1. Over the range [0,1), this can be written as
Fourier series approximation of a square wave
Fourier series approximation to
.
The number of terms in the Fourier sum is indicated in eachplot, and the square wave is shown as a dashed line over two
periods.
Real even signals
Given that the square wave is a real and even signal,
EVEN
*
REAL
therefore,
EVEN
* REAL
Consider this mathematical question intuitively: Can a
discontinuous function, like the square wave, be expressed as asum, even an infinite one, of continuous signals? One should at
least be suspicious, and in fact, it can't be thusexpressed.
The extraneous peaks in the square wave's Fourier series
never disappear; they are termed
Gibb's phenomenon after the American physicist
Josiah Willard Gibbs. They occur whenever the signal isdiscontinuous, and will always be present whenever the signal
has jumps.
Deriving the fourier coefficients for other signals
The Square wave is the standard example, but other important signals are also useful to analyze, and these are included here.
Constant waveform
This signal is relatively self-explanatory: the time-varying portion of the Fourier Coefficient is taken out, and we are left simply with a constant function over all time.
Sinusoid waveform
With this signal, only a specific frequency of time-varying Coefficient is chosen (given that the Fourier Series equation includes a sine wave, this is intuitive), and all others are filtered out, and this single time-varying coefficient will exactly match the desired signal.
Triangle waveform
This is a more complex form of signal approximation to the square wave. Because of the
Symmetry Properties of the Fourier Series, the triangle wave is a real and odd signal, as opposed to the real and even square wave signal. This means that
ODD
*
REAL
therefore,
* IMAGINARY
Fourier series approximation of a triangle wave
Sawtooth waveform
Because of the
Symmetry Properties of the Fourier Series, the sawtooth wave can be defined as a real and odd signal, as opposed to the real and even square wave signal. This has important implications for the Fourier Coefficients.
Fourier series approximation of a sawtooth wave
Fourier series approximation vi
Interact (when online) with a Mathematica CDF demonstrating the common Fourier Series. To download, right click and save file as .cdf.
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