- Fourier Series Definition: A Fourier series is defined as the decomposition of periodic signals into harmonically related sinusoids.
- Fourier Transform Definition: The Fourier transform is defined as a tool for converting non-periodic signals from the time domain to the frequency domain.
- Frequency Analysis: This process breaks down signals into their frequency components, similar to how a prism splits light into colors.
- Orthogonal Dimensions: Sinusoids are used as primary dimensions to express periodic signals, with cosine and sine functions providing additional dimensions.
- Fourier Series in Network Theory: Fourier series are essential in network theory for analyzing and understanding the frequency components of signals.
Time-domain and frequency-domain descriptions show different properties of the same signal. A Fourier Series and Fourier Transform are related but distinct representations. A Fourier series represents a periodic signal by discrete harmonics of its fundamental frequency. A Fourier transform represents an aperiodic continuous-time signal with a continuous frequency variable, subject to the relevant convergence framework.
A periodic waveform that meets suitable conditions can be reconstructed from a Fourier series of sines and cosines or complex exponentials.
Frequency Analysis
A periodic signal has a discrete line spectrum at integer multiples of its fundamental frequency. Its weighted sum of sinusoids or complex exponentials is a Fourier series or transform only in the series case; a Fourier transform generally uses an integral over a continuous frequency variable. Both tools support frequency analysis, but their domains, coefficients and convergence rules differ.
A prism gives a loose visual analogy for spectral separation.
White light contains a continuous distribution of wavelengths rather than only seven discrete colours. A prism separates those wavelengths spatially. Fourier analysis instead calculates coefficients or a spectrum for mathematical basis functions. For a periodic signal, those coefficients occur at discrete harmonic frequencies.
Signals and Vectors Analogy
A finite vector is expressed in a basis such as orthogonal Cartesian unit vectors. Signal space extends this idea to functions, using an inner product defined by integration over a period. Fourier did not invent geometric dimensions; his work established how trigonometric functions can represent broad classes of periodic functions. One real trigonometric basis contains:
sinω0t sin2ω0t sin3ω0t sin4ω0t ……..sinnω0t
cosω0t cos2ω0t cos3ω0t cos4ω0t……..cosnω0t
For integer n ≠ m, sine functions at nω0 and mω0 are orthogonal over a complete fundamental period. The cosine functions are similarly orthogonal, and sine-cosine cross terms integrate to zero. A complete real series also includes a constant term. With the time origin chosen for the stated symmetry, an even signal uses cosine terms and an odd signal uses sine terms. A general periodic signal usually needs both.
NOTE
A Fourier series is periodic by construction. The classic Dirichlet conditions are sufficient for pointwise convergence of many engineering waveforms, but they are not necessary. Broader mean-square and distributional frameworks cover additional signals. The Fourier transform is commonly used for aperiodic signals; periodic signals can also have transforms represented by impulse lines.
Calculating coefficients or a spectrum is Fourier Analysis. Reconstructing the signal from that representation is Fourier Synthesis.
Dirichlet’s Conditions
One common engineering statement of Dirichlet’s sufficient conditions requires x(t) to be absolutely integrable over one period:
Within a period, x(t) has a finite number of maxima and minima.
Within a period, x(t) has a finite number of finite discontinuities.
These conditions guarantee a useful form of Fourier-series convergence but are not necessary. At a jump, the series converges to the midpoint of the two one-sided limits rather than necessarily to the assigned point value.
