- 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.
Sometimes, information in the time domain isn’t enough. So, we move to the frequency domain to get more details. This shift from one domain to another is called transformation. We use tools like the Fourier Series and Fourier Transform to change signals from time to frequency domain by breaking them into harmonically related sinusoids.
Most practical signals can be broken down into sinusoids. This breakdown of periodic signals is called a Fourier series.
Frequency Analysis
Just like white light splits into seven colors, a periodic signal can be broken into a weighted sum of harmonically related frequencies. This sum of sinusoids or complex exponentials is known as a Fourier series or transform. In general, breaking down any signal into its frequency parts is called frequency analysis. So, the Fourier series and Fourier transform are tools for frequency analysis.
This can be clearer from the following.
Suppose if we pass a light through a prism, it gets split into seven colors VIBGYOR. Each color has a particular frequency or a range of frequencies. In the same way, if we pass a periodic signal through a Fourier tool, which plays the role of prism, the signal is decomposed into a Fourier series.
Signals and Vectors Analogy
An N dimension vector needs N dimensions for its representation. Like an ant moving on a table needs two dimensions for the representation of its position on the table i.e. x and y. Also we are familiar with i, j, k coordinate system for a vector representation in three dimensions. This unit vector i, j and k are orthogonal to each other. In the same way if we treat a signal as a multidimensional vector we need many more dimensions which are orthogonal to each other. It was the genius of J. B. J. Fourier who invented multi-dimensions, which are orthogonal to each other. These are sinusoids with harmonically related sinusoids or complex exponential. Consider the dimensions (also called bases)
sinω0t sin2ω0t sin3ω0t sin4ω0t ……..sinnω0t
cosω0t cos2ω0t cos3ω0t cos4ω0t……..cosnω0t
Thus, all sinnω0t are orthogonal with Sinmω0t (n≠m) and we, therefore can use sinω0t, sin2ω0t… ∞ as the primary dimensions (also called bases) to express a periodic signal. Similarly, we can also use cosω0t, cos2ω0t, cos3ω0t… ∞ as the additional dimensions when sinω0t dimensions cannot be used. We will see for even signals only cosine terms will be suitable and for odd signal only sine terms will be suitable. For a periodic signal neither an odd nor even, we use both sine and cosine terms.
NOTE
Only periodic signals can be represented as Fourier series provided the signal follows the Dirichlet’s conditions. For non-periodic signals, we have Fourier transform tool which transform the signal from time domain to frequency domain.
Breaking a signal into its harmonically related frequencies is called Fourier Analysis. Recombining these parts is known as Fourier Synthesis.
Dirichlet’s Conditions
x (t) is absolutely integrable over any period, that is,
x (t) has a finite number of maxima and minima within any finite interval of t.
x (t) has a finite number of discontinuities within any finite interval of t, and each of these discontinuities are finite.
Note that the Dirichlet’s conditions are sufficient but not necessary conditions for the Fourier series representation.
