How to Calculate Fourier Transforms

Fourier transforms is easily can be grasped if certain steps followed in a carefully arranged paces. Fourier transforms is the foundation of many part of modern civilisation. Those include mobile communication and digital photography, lasers and optics. Fourier transform has been ramified to other tools such as the discrete Fourier transforms, wavelets (well known as being used in JPeg and MPeg), pattern recognition, finance, medical imaging and numerous other usages.

Steps

1. Learn what is a periodic function. A periodic function repeats its form in known interval of time. That is f ( t ) = f ( t + nT ), where n is any integer.
   • These intervals are called periods. In previous relation, T is the period.
     

     A Periodic Function
2. Learn the basic idea in Fourier his own language.
   • Any periodic function can be decomposed, can be written, in terms of certain number of basic sinusoid function with simple periods.
   • Each sinusoid function has frequency of integer multiple of the basic frequency.
     

     Fourier Expansion
     The previous equation says, that any periodic function can be written or expanded as the summation of
   • a constant value, ½ a₀, also called the DC value and a bunch of sinusoid functions. Depending of the original function part of the expansion could be zero.
   • ω₀ is the basic circular frequency that can be easily calculated from the basic period T.
   • It only remains to calculate a₀ and a formula that creates the whole bunch of a_n and whole bunch of b_n. You do it using orthogonality property of sinusoids.
3. Learn what is the meaning of orthogonal functions. Orthogonal functions are perpendicular to each other. It means if you take any two, say, f ( t ) and g ( t ) from a bunch of them, then
   • 

     Orthogonality
   • Sinusoid functions are such a bunch of orthogonal functions.
   • Compare this with the basic notion of perpendicular vectors where the dot product is equal to zero. Dot product is the summation of products of pairwise components of two vectors. Here instead of summation an integral should be calculated.
4. Know the difference between a vector and a phasor.
   • A vector carries a point on a straight line to some other point.
   • A phasor revolves a vector around a point with certain circular frequency ω. A phasor is a revolving vector.
5. Observe that when a fixed length vector is rotating around a point, its projection, its shadow on the imaginary axis changes from a maximum value gradually to zero and then to maximally negative number and again back to zero and again back to a maximum positive value.
   • 
     Length of projection of the rotating vector – shadowed on the imaginary axis – changes in a sinusoid manner.
6. Conclude that a sinusoid can be written as a phasor and in this fashion it is easier to deal with Fourier series. Compare this with sinusoid form. All the concerns about a₀ a_n b_n have been relaxed. There is only one factor a_k that should be calculated. Calculations are done by calculating a simple integral of f ( t ) that gives all the coefficients in one go. Now the before mentioned chef makes any kind of cake with just one ingredient.
   
7. Interpret the expansion for the f ( t ). What is unknown in that expansion?
   • You need to calculate an infinite number of a_k's.
   • All a_k's are easily can be calculated from integration of f ( t ) to result in the whole bunch of them.
     • Instead of the whole bunch wording, the notation { a_k } is used.
     • { a_k } is known as the spectrum of f ( t ).
   • f ( t ) is actually the synthesise of infinite number of phasors with different length rotating with frequencies harmonic to the basic frequency ω₀ of the f ( t ) in both directions, clockwise and counter-clockwise since k roams among the positive and negative integers, both.
8. Look at the pair of formulas as a transform, rather than as a series expansion. When you have f ( t ) then you have a_k. And inversely, when you have a_k then you'll get f ( t ). Values of a_k are transform of f ( t ). Value of f ( t ) is the inverse transform of a_k's. This is written as,
   
9. Note. It seems that there are two domains. f ( t ) is in the domain of time but a_k's are in domain of integer numbers. Hence, Fourier expansion transforms one domain to another domain and vice versa.
   
   • For this reason this is said to be a time continuous transform.
   • People who study waves, use an oscilloscope to watch the time continuous wave and use a spectrum analyser to watch the lines or spectrums of the concerned wave.
10. Look a the most frequent example. This is say, a rectangle shutter that regularly opens and closes. Or could be a clock time-stamping an event regularly. It is a train of fixed duration pulses.
    

    A Train of Pulses
    • This is the easiest example one can calculate using K-12 (British A-level Maths) or equivalent knowledge of calculus, since inside the integral f ( t ) is equal to one for part and equal to zero for other parts and you should calculate integral of an exponential which is equal to itself regardless of a coefficient. In that level you are familiar with converting a complex exponential to a sinusoid. Remains what is a Sinc function. Simply, Sinc ( x ) = Sin ( x ) / x. It scales a sinusoid to its angle, similar to percentage.
    • 

      Sinc Function as the Envelope
      Draw the envelope of the a_k .
    • Draw the envelope of the |a_k | to appreciate its dying hops.
    • Each lobe of the Sinc function is filled with certain number of spectrum lines.
    • Making Each pulse of the train narrower, makes the number of the lines in the spectrum to be increased and looks more dense and it seems as if the spectrum is actually a continuous Sinc function not a discrete any more.
11. Appreciate that now you are looking at the Fourier series expansion of a periodic function as a transform of two domains. Remains to see what is the transform of a non-periodic function.
12. Affirm your expectation that the expansion of a non-periodic function will be in the form of an integral instead of summation.
    
    • You are correct that this is the Fourier integral contrasted with the Fourier series.
13. Hence Fourier transform for continuous time" functions could be a Fourier series or a Fourier integral.
14. Consider a single rectangular pulse. You see that pulse if a rectangular shutter opens and be closed only once. Or a step motor becomes on and then off.

A Single Pulse