Fourier Transform in Signals Processing: The Fourier Transform Spectrometer
Introduction:
This is a summary of the operating principles of the mFTS (Multichannel Fourier Transform Spectrometer), designed to deliver the material to an audience with a small to medium level of technical expertise. Here, one will find basic mathematical concepts of the Fourier Transform, its application to the art of signals processing, and the summary of the use of Fourier - Transform based instruments in Astronomy. The mFTS was designed and built by Dr. Arsen Hajian of the United States Naval Observatory, where it currently serves as an instrument for detecting Earth - sized planets in extra solar galactic stellar systems.
Joseph Fourier (1786 - 1830):
Fourier was not only a scientist; he had a turbulent political career. During his service in the French Army, he was assigned the position of a scientific expert to an expeditionary force to Egypt. During his tour in Cairo, Fourier helped to establish the Cairo Institute and was elected to the post of Institute Secretary, a position he has held for the entire time of French occupation.
Despite his political success, Fourier remained loyal to science, and has chosen to return to France and carry on his research in Mathematics and Mathematical Physics.
In 1817, Joseph Fourier was elected into the French Academy of Sciences.
In physics and engineering, problems involving periodic functions occur very frequently. The motions behaving like a sinusoid are reasonably simple to deal with; however, strictly simple harmonic motions do not themselves occur very often. In reality, most common functions are periodic but rarely sinusoidal.
Fourier has proposed an idea that ANY periodic function can be represented by an infinite sum of sinusoidal functions, sines and cosines in particular, amplitude modulated by some constant. Musicians will be very familiar with this concept - each string on a guitar has its own fundamental frequency. When a string is plucked at any fret, the resultant sound wave is essentially a pure wave, and the motion of the string itself is a standing wave. However, chords consist of a number of strings, plucked at different frets, and combining to give a note, whose function, though periodic, is not a sine wave. Thus, if one hears a guitar chord, he can break it up into components that are represented by individual strings. Each of these components represents a term in the Fourier Series expansion of the function representing the chord. One may ask about the infinite nature of the series. For the case of the guitar chord, the series are still infinite, yet some of the coefficients may be zero.
If we expand an arbitrary, periodic function of period λ into a series of sines and cosines, we get the following expression:
Where:
And by remembering Euler's Identity:
(Note that n = 1, 2, 3, 4, 5, ...)
It is often useful to map a function onto a different domain, where the problem becomes much simpler to deal with. In physics and engineering, we are often faced with functions that are not quite periodic, yet never the less have wave-like properties.
An integral transform is a linear operation that transforms one function in one domain into a different function in another. Fourier Integral Transforms deal specifically with time - domain representation of a function and the frequency - domain representation of that function. When a space-time based signal is Fourier Transformed, it is essentially analyzed for its frequency components. The concept is similar to that of the Fourier Series, with a difference that a function to be transformed does not necessarily have to be periodic, and the function depicting the frequency composition of a signal is continuous.
Imagine a wave-like disturbance in space: a monochromatic light bean that is infinite in time and space. The equation of this electromagnetic pulse can be written as E (x, t) = E0 Sin (kx + ωt), where k is the wave vector and ω is the angular frequency. Now the said disturbance is little more than a sine wave, with a single frequency component. If we are to analyze this function in wave - space, we would see a single value for the component frequency and with a height corresponding to the maximum amplitude. We have just showed qualitatively that the Fourier Transform of a sine wave is an amplitude-modulated Delta Function(a d. f. has a value of zero everywhere accept a definitive value). By the same principle, a beat - like interference of two waves would be two Delta Functions of approximately the same amplitude lying close together. )
One may ask how this differs from the Fourier Series. The difference is that the Fourier Transform is used to analyze a continuous spectrum that is not necessarily periodic.
The Fourier Transform is defined for functions of a single and double variable, respectively, as:
An important property of the Fourier Transform to mention is the Convolution Theorem, which states that the Fourier Transform of the convolution of two functions is a product of their Fourier Transforms:
Spectroscopy:
It is often useful to look at a source of light (or any other kind of EM radiation) and decompose the incident light into its frequency components. We have already seen that the Fourier Transform does just that. However, it is also well known that any material, if excited, will emit radiation of very specific colours, corresponding to the energy level transitions in the atom. The abundance of a given frequency depends on the probability of a given electron transition path. Thus, when looking at a light pulse of an unknown origin, by breaking the signal up into fundamental frequencies one can tell the chemical composition of the light source. Breaking up the signal into fundamental frequencies and analyzing the results for chemical composition is what is known as spectroscopy.
Very hot bodies, such as stars tend to radiate in all wavelengths, as the emitted radiation from a star does not come from the electron transitions in atoms, but rather from electron interaction within the plasma stars consist of. Similar effects can be observed when turning a radio on next to a metal welding apparatus - the result is white noise at all wavelengths. In fact, this is exactly how radar jammers work - they radiate intensively at the radio and radar wavelengths, thus making it impossible to detect any fainter signals. Spectroscopy has the answer for this problem as well. It turns out that there are absorption lines present inside such signals corresponding to the chemical composition. These result from the electrons in the chemical structure of the radiating body absorbing the energy and reemitting it in arbitrary directions. If one is to look at a visible spectrum emitted by our Sun, they are to find a rainbow pattern we are familiar with, but with a few lines missing (Frauenhofer Lines) predominantly at the hydrogen and helium wavelengths.
In summary, Spectroscopy is used to understand the chemical composition of a source. It has many applications that are not linked to Astronomy and Astrophysics. Spectrometry is used to find the chemical composition of materials in instances ranging from geology to engineering to matters of national security to environmental science.
Fourier Transform In Signal Processing and Spectroscopy:
· Interference:
Before I begin the description of how one can apply the concepts of the Fourier Transform, I must deviate once again to introduce a notion of Interference as a phenomenon associated with waves.
If one is to take two waves of the same amplitude and frequency and target them at a point on a screen, the result will be some kind of a combination of the two beams. They can add either completely constructively(where the total Electromagnetic Field will be a sum of Electromagnetic Fields associated with the two individual waves), completely destructively (where there will be no image on the screen), or in some kind of combination (where the resultant Electromagnetic Field is a linear combination of the two E-Fields associated with the waves). The resultant brightness of the spot is associated with the irradiance (which is in turn associated with the square of the Electromagnetic Field) of the two beams - and can be combined linearly. However, more importantly, the resultant signal will depend on the Difference in Path Length (in other words, how much longer did one wave have to travel than the other wave in terms of wavelength) if the two waves are coherent ("in - step"), or on the phase difference if the sources are incoherent. In fact, the path length difference introduces a phase difference between two coherent beams, so for two beams of identical frequency and amplitude, the deciding factor in the interference pattern on the screen will be the phase shift of one beam relative to another. One can refer to Young's Experiment for a more rigorous treatment of the phenomenon.
So, it can be shown that:
A special case arises when I1 = I2 :
· A Michelson Interferometer:
An interferometer is a device that produces an interference pattern resulting from two (or more) Electromagnetic waves. There are many kinds of interferometers that use almost the entire EM spectrum to measure desired quantities. For the purposes of this discussion, we will confine ourselves to a Michelson Interferometer operating in the visual bandwidth. This is a mirrored instrument that takes a source of light, splits it into two beams and recombines them. Because the two beams travel different distances, there is a phase shift introduced, which in turn makes an interference pattern possible.
A key feature of the Michelson Interferometer is a device called the Metrology System. This is essentially a mirror stationed on a plate that has a degree of freedom in a dimension perpendicular to the incident light beam. By varying the position of the Metrology Mirror, a varying phase shift between the beams is introduced.
As the Metrology Mirror (furthest to the right) moves in the x - direction, the interference pattern is changed. This effect is due to a different optical path length and hence, phase shift.
· The Applications to Spectroscopy:
We have already agreed that if we are to vary the position of the Delay Line, we will change the interference pattern the instrument outputs. Furthermore, if we are to have a sensitive detector at the screen and record the position of the delay line, we can create a graph of the interference pattern versus delay. This is what is called an Interferogram - a graph of intensity as a function of Delay Line position.
It can be shown (see "A Mathematical Treatment", below) that the Fourier Transform of the resultant interferogram is the spectrum of the incident beam.
· A Mathematical Treatment:
This is a reasonably rigorous and complete proof of the fact that the interference pattern as a function of delay line position is indeed the Fourier Transform of the source's spectrum:
i.) For a general non - monochromatic input function:
Let:
x = Path length Difference
At normal incidence, d = kx, the above expression can simplify to:
I
Therefore, for non - monochromatic light,
I (x) =
Now, re-write the above expression as:
W (x) =
To put things into a more physical sense, if we are to have a spectrum similar to those depicted in the spectroscopy graphic (coming from a distance star not unlike our Sun and bearing square - potential similarities), the interferogram will represent a function that is basically a sinusoid encased into a hyperbolic sine wave packet.
ii.) For a general, monochromatic source:
Essentially, the same treatment as above, accept the Fourier Transform shall consist of a delta function in the argument. I shall spare the reader of the derivation.
· Conclusion and Ideas to Consider:
In summary, a Fourier Transform Spectrometer is an instrument that calculates the spectrum of a source of light by creating and measuring an interference pattern resultant from splitting into two and recombining the original signal. The spectrometer consists of a Michelson Interferometer and analyzing software that approximates the interferogram to a function and makes the appropriate integral transformation into a wavelength domain. This is generally the most accurate way of measuring the component frequencies present in a signal. There are various engineering details that go along with the theory - such as keeping the temperature constant, making sure that there are no noise vibrations that will corrupt the signal. But these are details that are specific to the mission one is trying to accomplish.
An Application to Astrometric Measurements, in Particular to the Radial Expansion Method for Detecting Planets in Extrasolar Systems
One application of the technique described above is the detection of planetary companions to extrasolar galactic objects. In other words, the Fourier Transform Spectrometer is to be used to detect planets in other star systems, possibly as small as our Earth. Because of the small size of the planets compared to the bright stars, the planets cannot be detected directly. So, in order to see the planets, it is necessary to look for the dynamic effects that the planet has of the star. A lonely star is essentially a sphere of constant shape (disregarding matter ejection, etc). However, if a star has a companion, the companion will exert a gravitational force on the star, equal and opposite to the gravitational force exerted on the planet by the star (Newton's 3rd Law). This causes some of the star's material to be shifted toward the planet. Because the planet orbits, the "bulge" on the star shifts location. Essentially, the star "wobbles" as a result of an orbiting companion.
The phenomenon that is the resting pillar of the planetary detection via a Spectroscopic analysis is that of the Doppler Shift. Doppler Shift arises when a source of a wave (sound, E&M, mechanical, any wave) is moving in the direction perpendicular to the observer. The change in the displacement of the source relative to the emitted wave introduces a shift in the wavefront of the wave. The result is that the wavelength gets shorter if the source is moving toward an observer and longer if the source is moving away from the observer. This means that the light coming from the star gets either red or blue shifted if the source is moving away from or toward an observer (respectively). Doppler Shift is the fundamental principle of radar detection used by the police. And just like the moving car, the point on the surface of the star that is emitting light is blue - shifting its wavefront if the planet is inline with the observer and the star and is on the near side of the star, and red - shifting the light is on the far side of the star.
As the planet pulls on the star, a blue or a red shift is introduced. The shift is evident in the spectral analysis of the star - the absorption lines on the star serve as calibration markers
The spectral analysis of a star shows the red and the blue shifts. The top image is unshifted. Absorption lines serve as markers. The second image from the top denotes a red - shifted spectrum of the top graphic. As one can see, all of the absorption lines are shifted toward the red. The opposite is true in the bottom image. The absorption lines are shifted toward the blue end of the spectrum.
So, by analyzing the spectrum of the star obtained via the Fourier Transform Spectrometer, it is possible to detect planets in other solar systems. The Radial Expansion Method has been used to detect a number of extrasolar planets.
References:
· Techniques of Physics: Fourier Transforms and Their Physical Applications, D. C. Champeney, Academic Press, London and New York, 1973.
· The Fourier Transform and Its Applications, 3rd Edition, Ronald N. Bracewell, McGraw-Hill Higher Education, 2000.
· An Encyclopedia of Lasers and Optical Technology, Academic Press, 1991.
· Optics, 4th Edition, Eugene Hecht, Addison Wesley, 2002.
· Mathematical Methods in the Physical Sciences, 2nd Edition, Mary L. Boas, John Wiley & Sons, 1983.
· The U.S. Geological Survey website, www.usgs.gov/
· United States Naval Observatory Astrometric Department website, http://ad.usno.navy.mil/fts-public/FTSAbout.shtml
· Arizona Search for Planets website, http://www.psi.edu/~esquerdo/asp/asp.html
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