The Fourier coefficientsare the coordinates of f in that basis.
The function isdisplayed in white, with the Fourier series approximation in red.
Fourier Series Applet  Paul Falstad
This brings us to the last member of the Fourier transform family: the . The time domain signal used in the Fourier series is and. Figure 1310 shows several examples of continuous waveformsthat repeat themselves from negative to positive infinity. Chapter 11 showedthat periodic signals have a frequency spectrum consisting of . Forinstance, if the time domain repeats at 1000 hertz, the frequency spectrum willcontain a first harmonic at 1000 hertz, a second harmonic at 2000 hertz, a thirdharmonic at 3000 hertz, and so forth. The first harmonic, i.e., the frequency thatthe time domain repeats itself, is also called the Thismeans that the frequency spectrum can be viewed in two ways: (1) thefrequency spectrum is , but zero at all frequencies except theharmonics, or (2) the frequency spectrum is , and only at theharmonic frequencies. In other words, the frequencies between the harmonicscan be thought of as having a value of zero, or simply not existing. Theimportant point is that they do not contribute to forming the time domain signal.
The recipe for calculating the Fourier transform of an image is quite simple:take the onedimensional FFT of each of the rows, followed by the onedimensional FFT of each of the columns. Specifically, start by taking the FFTof the pixel values in row 0 of the real array. The real part of the FFT's outputis placed back into row 0 of the real array, while the imaginary part of the FFT'soutput is placed into row 0 of the imaginary array. After repeating thisprocedure on rows 1 through 1, both the real and imaginary arrays containan intermediate image. Next, the procedure is repeated on each of the of the intermediate data. Take the pixel values from column 0 of the realarray, the pixel values from column 0 of the imaginary array, andcalculate the FFT. The real part of the FFT's output is placed back into column0 of the real array, while the imaginary part of the FFT's output is placed backinto column 0 of the imaginary array. After this is repeated on columns 1through 1, both arrays have been overwritten with the image's frequencyspectrum.
The Fourier Series  The Scientist and Engineer's Guide …
Remember that for realvalued inputs, the transformed data is Hermite symmetric. This means that the positive and negative real parts will be identical, and that the positive and negative imaginary parts will be the same, but of opposite sign. Equivalently, we say that that negative and positive values are complex conjugates of one another. When the Fourier transform completes, the data are in a somewhat odd order. Specifically, the values of positive frequencies are laid out from 0 to the maximum frequency. These are followed by the negative frequency values, which are laid out from the highest to the lowest frequency (1). The plots I put in the problem set are of the magnitude of the data: sqrt( Real^{2} + Imaginary^{2}). A complex number and its complex conjugate have the same magnitude. Thus I showed the plots only for positive frequency.
In spite of this, Fourier image analysis does have several useful properties. Forinstance, in the spatial domain corresponds to in thefrequency domain. This is important because multiplication is a simplermathematical operation than convolution. As with onedimensional signals, thisproperty enables FFT convolution and various deconvolution techniques. Another useful property of the frequency domain is the ,the relationship between an image and its projections (the image viewed fromits sides). This is the basis of , an xray imagingtechnique widely used medicine and industry.
Fourier Series Applet  Falstad
Figure 1311 shows an example of calculating a Fourier series using theseequations. The time domain signal being analyzed is a , a squarewave with unequal high and low durations. Over a single period from /2 to /2, the waveform is given by:
The corresponding for the Fourier series are usually writtenin terms of the of the waveform, denoted by , rather than thefundamental frequency, (where = 1/). Since the time domain signal isperiodic, the sine and cosine wave correlation only needs to be evaluated overa single period, i.e., /2 to /2, 0 to ,  to 0, etc. Selecting different limitsmakes the mathematics different, but the final answer is always the same. TheFourier series analysis equations are:
Differential Equations  Fourier Series

Fourier analysis of periodic signals  Miller Puckette
Fourier Series

Fourier analysis of periodic signals
Fourier Image Analysis  DSP

Fourier transform as additive synthesis.
Fourier analysis is used in image processing in much the same way as with onedimensional signals
A Pictorial Introduction to Fourier Analysis/Synthesis
The function f can be recovered as a real Fourier series
The functions , cos(nx), sin(n x) form an orthonormal set of functions on the space of periodic functions (*).
Fourier Analysis of a SquareWave Grating
The function f can be recovered as a complex Fourier series
The functions form an orthonormal setof functions on the space of complex valued periodic functions.
A Pictorial Introduction to Fourier Analysis
Fortunately, Excel has some built in functions that make it possible to perform Fourier transforms relatively easily. If you intend to use Excel for this purpose, I encourage you to look through their help files to understand it, but here are a few notes.
Harmonic Phasors and Fourier Series
Fourier analysis is used in image processing in much the same way as with onedimensional signals. However, images do not have their information encodedin the frequency domain, making the techniques much less useful. For example,when the Fourier transform is taken of an signal, the confusing timedomain waveform is converted into an easy to understand frequency spectrum. In comparison, taking the Fourier transform of an image converts thestraightforward information in the spatial domain into a scrambled form in thefrequency domain. In short, don't expect the Fourier transform to help youunderstand the information encoded in images.
Fourier Series Tutorial  YouTube
The Fourier series creates a continuous periodic signal witha fundamental frequency, , by adding scaled cosine and sine waves withfrequencies: , 2, 3, 4, etc. The amplitudes of the cosine waves are held in the variables: _{1}, _{2}, _{3}, _{3}, etc., while the amplitudes of the sine waves are held in: _{1}, _{2}, _{3}, _{4}, and so on. In other words, the and "" coefficients are the real andimaginary parts of the frequency spectrum, respectively. In addition, thecoefficient _{0} is used to hold the DC value of the time domain waveform. Thiscan be viewed as the amplitude of a cosine wave with zero frequency (aconstant value). Sometimes is grouped with the other "" coefficients, butit is often handled separately because it requires special calculations. There isno _{0} coefficient since a sine wave of zero frequency has a constant value ofzero, and would be quite useless. The synthesis equation is written:
Fourier Series  Harvard Department of Mathematics
In the late 1960's, suggested that the neurons in the visual cortex might process spatialfrequencies instead of particular features of the visual world. In English,this means that instead of piecing the visual world together like a puzzle, the brainperforms something akin to the mathematical technique of Fourier Analysis to detectthe form of objects. While this analogy between the brain and the mathematical procedureis at best a loose one (since the brain doesn't really "do" a Fourier Analysis),whatever the brain actually does when we see an object is easier to understand within thiscontext. Thus, a review of the basic concepts of Fourier Analysis will be very helpful.