Swinburne Online Education Introductory Radio Astronomy SETI Module

Swinburne Online Education Introductory Radio Astronomy & SETI Module 6: Electronics and Mathematics FOURIER TRANSFORM Activity 1: Merci Monsieur Fourier © Swinburne University of Technology

Summary The Fourier Transform is one of the most important mathematical tools ever developed and is essential for radio astronomy. In later activities, we will see how it is used in creating images from multidish images and for identifying signals in searches for ETIs. In this Activity, we will examine: 1. the Fourier series (both finite and infinite); 2. the Fourier Transform; 3. time and frequency domains; and 4. the Fast Fourier Transform.

Introduction The Fourier Transform is, perhaps, the most useful mathematical tool ever developed with numerous applications in many different fields of science and engineering. A small selection of the fields where the Fourier Transform has proved invaluable are the studies of optics, acoustics, fluid mechanics, information theory and quantum mechanics. The Fourier Transform is also essential for radio astronomy, in both antenna theory and for the creation of images of astronomical data. In this Activity, we will introduce Fourier analysis and the Fourier Transform in a way which allows an appreciation of its nature and beauty, whilst avoiding mathematical nomenclature as far as possible.

Who was Fourier? Count Jean Baptiste Joseph Fourier was born in 1768 in Auxere, France. With a natural ability at mathematics, he quickly became a teacher of mathematics while only aged sixteen at the military school in Auxere! Fourier joined the faculty at the Ecole Normale at Paris in the year of its founding (1795) when he was 27. Later, Fourier’s success as a teacher led to the offer of the Chair of Analysis at the Ecole Polytechnique in 1807. The ideals of the French Revolution swept Fourier into politics. Accompanying Napoleon on his expedition to Egypt, Fourier undertook research on Egyptian antiquities, gave advice on engineering and diplomatic undertakings, and was secretary of the Institut d'Égypte, which Napoleon established in Cairo in 1798.

Following Napoleon's fall from power in 1815, Fourier was appointed director of the Statistical Bureau of the Seine. This allowed Fourier to further develop his mathematical work, leading a quiet academic life in Paris. He died aged 62 in 1830. Fourier’s work stimulated research in mathematical physics and today Fourier analysis remains in constant use in modern analysis, as well as being a mathematical tool used extensively in signal processes. Fourier's masterpiece was his mathematical Theory of Heat Conduction, commenced in 1807 and published in 1822 as “Theorie Analytique de la Chaleur”.

The Theory of Heat Conduction In this work, Fourier expounded upon his ideas of heat transfer and outlined his new method of mathematical analysis which we now call Fourier Analysis. y What is the temperature at this point? Plate Temperature on four edges known x Fourier was trying to determine the temperature at specific points of a thin, two-dimensional plate as a function of time, when the temperature along the edges of the plate was known.

The Heat Transfer Equation The equation Fourier needed to solve looked like: u t = 2 u x 2 + 2 u y 2 • u is the temperature; • x and y are the coordinates of the point we are interested in; • t is time; This symbol is the “partial derivative” (related to the gradient), but it is the method of solution we are interested in here, not the specifics of the heat transfer equation. . . Fourier expressed the solution of this problem (in the onedimensional case) as a sum of sine and cosine waves.

The Sine Wave 1. 0 The sine wave is an example of a periodic waveform. By wave, we mean a shape that repeats itself, or undergoes periodic oscillations. y 0. 5 0. 0 x -0. 5 -1. 0 One period of the “infinitely” repeating sine wave Other familiar examples of waveforms are: y Square Wave x

The Sine Wave 1. 0 The sine wave is an example of a periodic waveform. By wave, we mean a shape that repeats itself, or undergoes periodic oscillations. y 0. 5 0. 0 x -0. 5 -1. 0 One period of the “infinitely” repeating sine wave Other familiar examples of waveforms are: y Sawtooth Wave x

The Sine Wave 1. 0 The sine wave is an example of a periodic waveform. By wave, we mean a shape that repeats itself, or undergoes periodic oscillations. y 0. 5 0. 0 x -0. 5 -1. 0 One period of the “infinitely” repeating sine wave Other familiar examples of waveforms are: y Triangular Wave x

Fourier Series According to Fourier analysis, any waveform, no matter how complex, can be represented as the sum of a series of sine waves. This is known as a Fourier Series. We will show this result graphically first before delving into the mathematical description. Consider the graph: This is made up of three sine waves. . .

This sine wave. . . …is added to this sine wave, which has one half the period (or twice the frequency) of the first sine wave. . .

…to which is added a third sine wave, with one third the period (three times the frequency) of the first sine wave. . . …to give the resultant waveform we sought shown in red.

Infinite Series In many cases, in order to get a perfect representation of an arbitrary waveform, an infinite number of sine waves must be summed together. One example where this is the case is the sawtooth wave. To demonstrate this, we will add just the first four terms of the Fourier series. We do not recommend that you try the infinite case at home!

The original sine wave (two cycles are shown here). . .

…is added to a sine wave with twice the frequency and one half the amplitude of the original sine wave. . . Note that we are now only comparing the frequencies of the waves, not the periods.

…to give the waveform shown in red.

A sine wave (pink) with three times the frequency and one third the amplitude of the original sine wave. . .

…is added to give the waveform shown in red.

A sine wave (orange) of 4 times the frequency and one quarter the amplitude of the original sine wave. . .

…is added to give the waveform shown in red. And so on.

With just four terms we have an approximation to the sawtooth waveform. Even though the sawtooth wave requires an infinite series of sine waves to be represented exactly, an approximation is made by summing a finite Fourier Series. This process has been greatly aided by computers, which allow infinite an Fourier Series to be approximated to extreme precision by very a large finite series.

A Little Bit of Mathematics We have now demonstrated graphically how a Fourier Series works, but the time has come to add some mathematical precision. In conventional notation, we write any arbitrary waveform as a function* f(t). For the moment, we can think of the variable t as representing time. f(t) t * Click here to learn more about functional notation.

Sine Waves Again The basic sine wave is written f(t) = sin(t), which completes one cycle as t takes on values between t = 0 and t = 2. f(t) t t=0 A more general sine wave can be written: f(t) = A sin(Bt) where: Amplitude = A Period = T = 2 /B Frequency = f = 1 / T = B/2 t = 2

Fourier Series Notation We are now ready to write the Fourier Series for an arbitrary function as a summation of these “generalised” sine waves: f(t) = A 1 sin(B 1 t) + A 2 sin(B 2 t) + A 3 sin(B 3 t) +. . . Amplitude and “frequency” of first sine wave

Fourier Series Notation We are now ready to write the Fourier Series for an arbitrary function as a summation of these “generalised” sine waves: f(t) = A 1 sin(B 1 t) + A 2 sin(B 2 t) + A 3 sin(B 3 t) +. . . Amplitude and “frequency” of second sine wave

Fourier Series Notation We are now ready to write the Fourier Series for an arbitrary function as a summation of these “generalised” sine waves: f(t) = A 1 sin(B 1 t) + A 2 sin(B 2 t) + A 3 sin(B 3 t) +. . . Amplitude and “frequency” of third sine wave We can introduce some additional notation to simplify this expression. f(t) = n =1 An sin(Bn t) Note: We can write a The nth set of amplitude Finite Fourier Series and frequency values “Summation”: add together all terms where n takes on integer values from 1 to “Infinity” by replacing with N, the number of terms to sum.

Returning to our two previous examples, the first waveform may be represented by f(t) the finite Fourier Series: t 3 f(t) = sin(t) + sin(2 t) + sin(3 t) f(t) = sin(n t) n =1 The infinite Fourier Series for the sawtooth wave is: f(t) = 1 sin(t) + sin(2 t) + sin(3 t) + sin(4 t) +… Amplitude = f(t) = Frequency= 2/2 n =1 1 sin(n t) n

Sine and Cosine Everything we have said so far about sine waves works equally well with cosine waves - that’s because the waveforms have the same shape, only they are f(t) t=0 radians (= 90 degrees) out of phase. cosine t = 4 t = 2 t Maximum/minimum of cosine wave occurs before sine wave Thus we can also define the Fourier Series in terms of cosine terms: f(t) = Dn cos(En t) n =1

The Fourier Transform Sine and cosine waves have very elegant mathematical properties, and the analysis of an arbitrary function or wave is made easier when it is represented by a Fourier Series. A question which often arises when we are studying a particular waveform is this: If we start with a periodic waveform, how do we find out the amplitudes and the frequencies of the sine (or cosine) waves that make up the Fourier Series? The Fourier Transform is the mathematical tool which performs this task.

When we detect a wave, we are receiving information about how the waveform changes as time passes. This is referred to as information in the time domain. The Fourier Transform gives information in the frequency domain. We can represent information in the frequency domain pictorially by using lines on a graph. Each line represents one sine function in the Fourier series of which the waveform is composed. The height of each line is the amplitude of the sine wave it represents. The distance the line is along the horizontal axis is the frequency of the sine wave it represents.

Let’s look at the (finite) Fourier Series for the Sawtooth wave again: f(t) = 1 sin(t) + sin(2 t) + sin(3 t) + sin(4 t) Amplitude 1 1 st sine wave This is called a Fourier Spectrum 2 nd sine wave 4 th sine wave 1 2 3 2 x Frequency 4 The Fourier Transform of this function identifies the amplitudes and frequencies of the four components. (Note that we plot frequency multiplied by 2 ).

When we detect a wave, we are receiving information about how the waveform changes as time passes (the time domain). f(t) FOURIER TRANSFORM t The wave is a function of time The Fourier Transform gives information on the wave in the frequency domain. g(f) f The Fourier Transform is a function of frequency

When we detect a wave, we are receiving information about how the waveform changes as time passes (the time domain). f(t) INVERSE FOURIER TRANSFORM t The Inverse Fourier Transform is a function of time The Fourier Transform gives information on the wave in the frequency domain. We can use the Inverse Fourier Transform to get from the frequency domain to the time domain. g(f) f The Fourier Spectrum is a function of frequency

A Violin String The open D string of a violin has the following waveform in the time domain: A m p l i t u d e The amplitude and frequency of the twelve sine waves (the fundamental plus 11 overtones) which make up the vibration of the D-string are easily read from the graph. Frequency The Fourier Transform gives this frequency domain representation

Pulses and Aperiodic Functions What about when we receive waveforms which are not periodic? When working on the first correct theory of heat convection Fourier realised that it is not only periodic waveforms that can be represented by a series of sine waves: in fact all functions can be represented in such a manner. This is important as it means that we can use Fourier Transforms on wave pulses and aperiodic (non-periodic) waveforms. Actually we do consider aperiodic waveforms to have a period: only it is an infinite period, so the pattern never repeats. *

The representations in the frequency domain we have examined so far involved a summation of sine waves which have distinct frequencies - in the Fourier Spectrum, each sine wave component is a distinct line. This corresponds to our Fourier Series representation as a summation of integer values of the index, n. f(t) = n =1 An sin(Bn t) n = 1, 2, 3, 4, 5, … When we take the Fourier Transform of aperiodic waveforms we need to include an infinite number of sine waves, and the difference in the frequencies of these sine waves becomes infinitesimal.

Consequently, in the frequency domain, the spacing between the lines representing sine wave components move together. The is a very common wavepulse called a “tophat”. FOURIER TRANSFORM When we take the Fourier Transform, the frequency domain representation (Fourier Spectrum) is:

Information is read from this graph in the same way as before. Each of an infinite number of sine waves is infinitesimally close to the next. and all the sine waves in between this sine is added to this sine wave and this sine wave

Using a Computer If our waveform or pulse is very complicated, doing all the mathematics by hand for the Fourier Transform would be quite time-consuming. We have seen that an “infinite” number of sine waves are required in some cases. If we want to use a computer to perform Fourier Transforms, we need to make approximations: but we still need to achieve the greatest accuracy possible within the limitations of our computer’s processing abilities. The “Fast Fourier Transform” (FFT) is a very fast and efficient way of calculating Fourier Transforms on a computer whilst achieving great accuracy.

Digitising the Signal Consider the following waveform which is digitised into N discrete values, so that it can be processed on a computer. The more samples we have, the better the representation of the wave. Original waveform is “continuous” Digitised (sampled) waveform with N = 30 The number of computer operations required to compute the Fourier Transform is of order N 2. In this example, N=30, so 900 operations are required. This is because we use a different trial sine curve for up to N different frequency values on each of the N discrete points in the digitised wave.

Cutting Back on Calculations It was soon realised however, that many of these calculations were redundant, and that clever partial sums could be reused to give greater efficiency when computing the Fourier Transform. The FFT algorithm was a watershed, because it was much more efficient than “simple-minded” approaches to the Fourier Transform. The FFT requires the input data to be regularly spaced in the time domain so that some “tricks” can be used to speed up the processing. A candidate for the FFT as spacing between discrete points is regular!

How Fast is Fast? The resultant process takes the order of 5 N x log 2 N floating point operations to compute the transform instead of N 2 operations. For N=1000000, this is clearly an enormous saving! The FFT is most efficient for computing transforms of waveforms which have 2 N discrete (digitised) points. If many transforms are to be computed, additional speedups can be obtained by pre-computing the so called “twiddle factors” which involve calculations of sines and cosines that are computationally-intensive.

Are You Receiving Me? We have come to the end of our discussion of the basic theory behind the Fourier Transform. It is now time to see the Fourier Transform in action, which will happen in the next Activity: “Feeds and Backends” where we examine some of the electronic components required to convert radio signals into a form which we can analyse.

Image Credits AIP Emilio Segré Visual Archives http: //www. aip. org/history/esva AIP Emilio Segré Visual Archives: Jean Baptiste Joseph Fourier http: //www. aip. org/history/esva/photos/fourier_jean_a 1. jpg

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Functions f(x) y Suppose we have an equation which takes an x-value and returns the corresponding value on the y-axis: y = x 2 x It is sometimes convenient to use a shorthand notation for equations, particularly when the equation relating x and y is very complex (this could be because mathematicians are lazy!). We could write the equation above as: f(x) = x 2, and relabel the axes of the graph. . .

f(x) What does this mean? f(x) is called a function. You can think of it as a black box which takes a parameter, x, and returns a value, f(x). x The Function f(x) For example, for the case f(x) = x 2, we have: value

f(x) What does this mean? f(x) is called a function. You can think of it as a black box which takes a parameter, x, and returns a value, f(x). 2 The Function f(x) f(2) = 22=4 4

f(x) What does this mean? f(x) is called a function. You can think of it as a black box which takes a parameter, x, and returns a value, f(x). 7 The Function f(x) f(7) = 72=49 49

f(x) What does this mean? f(x) is called a function. You can think of it as a black box which takes a parameter, x, and returns a value, f(x). The Function f(x) = x 2 2

Functions 2 Of course, f and x are just labels, and have no intrinsic meaning. We could write our example equation as g(t) = t provided we label the axes g(t) of the graph as: t Our wave functions are generally functions of time so t would stand for time in this case.

Functions of more than one variable Suppose instead that our function depended on more than one variable. An example of this is the Gravitational Force: F = G M 1 M 2 R 2 We could write this equation in functional notation as: F(M 1, M 2, R) The notation tells us that there is some function, F, which requires 3 parameters as inputs, M 1, M 2 and R, in order to calculate a value. We don’t include G as a parameter as it is a constant.

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Period The period is the time taken for one cycle of the wave. one period

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Frequency The frequency is the number of cycles of the wave per second. The frequency is the inverse of the period. If the period is known the frequency is also known and vice-versa. frequency = 1/period = 1/frequency A wave with period has frequency 2. A wave with period 3 has frequency .

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Amplitude The amplitude is the height of the wave. amplitude

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