Definition

Continuous Fourier transform for a piecewise continuous and integrable function, $f(t)$ by

\[F(s)=\int _{ -\infty  }^{ \infty  }{ f(t){ e }^{ -2\pi jst }dt } \]

where $F(s)$ represents Fourier transform and $j=\sqrt { -1 }$.

This is often referred to as "Forward Transform".

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Motivation

Consider a pure (Having a single frequency) tone with some high pitch and another with a low pitch. Now, what happens when we play both the sounds together? Yes, we get a new tone with some pitch. They obey the Principle of Superposition i.e. at some points, the two tones get resonated to produce a high pitch and at some points, they get canceled out with each. But in the end, we will be having the resultant pitch for a given time (Vector addition of both the pitches at that instant).

NOTE: The pitch of a sound is commonly referred for the sensation of its frequency.

Now consider we are given with some unpure tone i.e. a sound with different frequencies. Now can we obtain the pure tones present in it?

Mathematically, if we consider the unpure tone as some function of time, can we get all its constituent frequencies?

Will they apply only to periodic functions or can even be extended to non-periodic (rather functions with $\infty$ period) functions?

Now if the above set of approximations is possible, is it even possible to find the amount of each constituent frequency present in a given function? i.e. is it possible to convert the time domain function to a frequency domain function? (as in the case with Laplace transformations)

NOTE that in the entire lecture note, we will be using the terms, “sinusoids”, “sines and cosines” and “complex exponentials” interchangeably.

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Bird's eye view

“Fourier Series” is a tool to represent any periodic function in terms of sines and cosines or approximate any function with sines and cosines within a given interval!

That is, the Fourier series can't be used to approximate non-periodic functions over the entire region. But they can be approximated for a given interval of $x$, using Fourier Series.

Now to compute the non-periodic functions over entire $x$ (in terms of sinusoids), we use “Fourier Transform” which in-hand also converts it to a function with frequency domain (over the entire region). This helps to find the frequency distribution of the constituent frequencies of the function, Hence resulting in the sinusoids themselves indirectly!

Each of these sinusoids has a different frequency and amplitude which is a key to plot any function.

Since these transformations come with their unique properties and advantages such as making it easier to simply differential equations in the transformed forms than in original forms, make its identity and to have a wide range of applications, mainly in signal processing.

NOTE: If we use sinusoids as well as the exponentials for these transformations, we end up with a Laplace transform!

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Context of the Definition

What makes Fourier Transform different from a Fourier Series?

Whenever we try to distinguish between these two, we always get to whether we are considering a periodic or a non-periodic function.

Tp begin with, let's consider a periodic function. We can treat it as a function defined only for an interval (equal to its period) as it behaves the same over the rest of the intervals. Since it has a finite domain, we can get the exact amount of each constituent frequency present. These could be multiplied with normal periodic functions such as sines and cosines (whose value is bounded) to simply get the original function itself. Remember here we are building the function only over a specific interval with the width as its period. [We simply duplicate it over the other intervals even to get the original function with the same domain].

There is a set of formulas given to calculate the same.

For a periodic function, $f(x)$, its Fourier Series in the interval [-L,L] is given by

\[f(x)=\frac { { a }_{ 0 } }{ 2 } +\sum _{ n=1 }^{ \infty  }{ { a }_{ n }cos(\frac { n\pi x }{ L } )+\sum _{ n=1 }^{ \infty  }{ { b }_{ n }sin(\frac { n\pi x }{ L } ) }  } \]

where  ${ a }_{ 0 }=\frac { 1 }{ L } \int _{ -L }^{ L }{ f(x)dx } $ , $a_{ n }=\frac { 1 }{ L } \int _{ -L }^{ L }{ f(x)cos(\frac { n\pi x }{ L } )dx }$ and  $b_{ n }=\frac { 1 }{ L } \int _{ -L }^{ L }{ f(x)sin(\frac { n\pi x }{ L } )dx } $

Now it won't be the same case if we consider a non-periodic function. we need to consider the entire interval over which the function is defined rather than only a part of it. So we can't exactly get the constituent frequencies as we are dealing with infinite domains! So to draw something similar to Fourier Series, we simply try for alternative ways to represent the constituent frequencies present in it. Hence we try to establish its constituent frequencies distribution rather than calculating the amount of each frequency present (The case in Fourier Series). Hence by this, we can approximate the things defined in infinitely large intervals.
Consider some periodic function. Now we can consider some interval (equal to its period) by which we can explain the entire behaviour of the function itself. Now for the given periodic function, we can directly represent it in terms of sines and cosines using the Fourier Series.

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Simply Fourier Transform allows us to compute such a function that takes in the frequency and returns the amount of sinusoid present with that particular frequency in our function. and mathematically Fourier Transform is a generalization of the Fourier Series by making the intervals tend to infinity (i.e. $L\rightarrow \infty $).

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Diving further into the Fourier Transform

Now as mentioned in the motivation part, consider we are given with some frequency distribution curve. Can we form a function that has the same frequency distribution? i.e. can we draw out a function whose Fourier transform is already present?

For this purpose, we have Inverse Fourier Transform.

The Inverse Fourier Transforms for each Continuous Fourier Transform and Discrete Fourier Transform given by,

$$ f(x)=\int _{ -\infty  }^{ \infty  }{ F(s){ e }^{ 2\pi jsx }ds } $$

and

$$ { x }_{ p }=\frac { 1 }{ N } \sum _{ k=0 }^{ N-1 }{ { x }_{ k }{ e }^{ -2\pi jpk / N } } $$

respectively.

We often use the symbol $\Leftrightarrow$ to mean one as a Fourier transform as the other i.e.

$$ f\overset { \\F }{ \Longleftrightarrow  } F $$

Since we can represent any function in terms of sines and cosines with Fourier Series, let us analyze the Fourier Series of a Square Pulse. The equation of the square pulse (with period $1$) is given by

\[f(x)=1\quad 0\le x\le 1\] \[f(x)= -1\quad -1\le x<0\]

and its Fourier series is given by

\[f(x)=1\quad \quad 0\le x\le \frac { 1 }{ 2 }\] \[f(x)=-1\quad \frac { 1 }{ 2 } \le x<1\]

(Use the Fourier Series Equations)

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The overshoot we observe in the approximation of a function using the Fourier Series near the points of discontinuities is referred to as the Gibbs phenomenon. The phenomenon is illustrated for a square pulse in the above animation.

Here we are adding a set of sinusoids (of different frequencies and amplitudes) to approximate the function. Now, are they the only frequencies present in our function? Nope! We can even take any set of sinusoids together and simply add them to get another sinusoid of some different frequency!

Since the square pulse, we considered is a real-valued function, therefore we won't get any complex terms in our expansion!

Now we know Fourier Series to compute the amount of each sinusoid (i.e. with a particular frequency) present in our function.

NOTE: Given a sinusoid with amplitude $A$ and frequency $f$ in the Fourier series of some function, then we can say that the function contains A amount of sinusoid with frequency f!

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There are two types of Fourier Transform, Continuous-Time Fourier Transform (CTFT) and Discrete-Time Fourier Transform (DTFT). The former is used for functions that are aperiodic and continuous in time and frequency domain and the latter one is used for functions that are periodic and continuous in time and frequency domains.

Since in astronomical observations, we deal with discretely sampled signals and for a finite duration, for such signals, we use Discrete Fourier Transform. For $N$ uniformly sampled data points $x_{p}$ where $p=\{1,2,3..N\}$ DFT is given by

\[{ x }_{ k }=\sum _{ p=0 }^{ N-1 }{ { x }_{ p }{ e }^{ -2\pi jpk / N } } \]

which contains a finite number of sinusoids.

Here the continuous variable $s$ is replaced by the discrete variable $k$. (Just a convention)

Complex exponentials are periodic functions and are complete and orthogonal. This property enables them to represent any piecewise continuous function and also minimizes the least-square error between the original and transformed functions.

Consider the following coins analogy to get the basic intuition of the formula obtained for Fourier Transform.

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Here we can add (or rather integrate) over the entire region because of the Orthogonality property which the sinusoids hold. It takes care of the fact that every sine wave of a specific frequency is perpendicular to the sine wave with other frequencies and also to the cosine wave with any frequencies. Therefore the value of a sinusoid of a specific frequency doesn't affect the values of sinusoids with other frequencies. Hence enabling us to freely add (or integrate) the constituent frequencies to get a frequency domain function.

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Applications

• They have an immense application in control signals, where they are used to find the constituent frequencies present in a given signal. i.e. are used to analyze the spectrum of signals.
• They are also used for the filtering of sound signals. Consider some sound which has a high pitch. We can filter out the pure signal (due to which it is producing a very pitch sound) if we get to know its frequency. We can get that particular frequency by just taking down its Fourier Transform and search where there is a peak in the graph!
• They are used to find the response of the Linear time-invariant systems (LTIs)
• Discrete Fourier transforms are used for digital signal manipulation in digital signal processing and are used for making convolution as well.
• Fourier transforms are also used in image processing and image compression where they keep track of frequencies of pixels and hence use this data to keep aside the duplicate (or similar) pixels.
• Used for frequency analysis of linear circuits which include capacitors, inductors, etc as their components.
• These are also used in mobiles and other similar communication devices. Our mobiles use Fourier transform to manipulate the signals received and separate its components to base frequencies, hence distinguishing the signals received for the respective tasks.

History

It all started around 3^rd century BC, when ancient astronomers decomposed a periodic function into a sum of simple oscillating functions, though there wasn't any formalized method to perform the same. In the 17^th century, Jean-Basptiste Joseph Fourier used a special series to solve heat equation in metal plate, in his work “Memoire sur la propagation de la chaleur dans les corps solides”, in 1807. Till then, the equation was solvable only if the heat source was a sine or a cosine wave, and there wasn't any generalized method. Hence Fourier considered a complex heat source as a linear combination of sines and cosines to solve even these complex heat sources, which ended up getting a new series.

He termed the particular linear combination as Fourier Series.

Later around the 1820s, Fourier showed how the same idea can be extended to Non-periodic function (or functions with $\infty$ period) by simply writing them as an infinite sum of sinusoids.

The series didn't have any precise notion of function and integral until Peter Gustav Lejeune Dirichlet and Bernhard Riemann formalized the results in the early nineteenth century.

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References

• A. Sudhakar,Shyammohan , “Circuits and Networks –Analysis & synthesis” McGrawHill education.
• M.C.Anumaka , “Analysis and applications of Laplace &Fourier transformations in electric circuit’’.
• https://www.cv.nrao.edu/course/astr534/FourierTransforms.html

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Further Readings

• Fourier Transform by Ian N. Sneddon
• The Fourier Transform and its Applications by Ronald Newbold Bracewell.
• An Introduction to Laplace Transforms and Fourier Series by Phil Dyke.

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