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Introduction The very high percentage of the science of physics involves vibrations and waves.
Mechanical engineering, electrical engineering, fluid mechanics , sound and vision engineering etc all involve
vibration and waves.
The detailed analysis of vibrations and waves generally involves the use of fourier transforms. A fourier
transform at is most basic level involves transforming a complex waveform into a form which is easily assessed.
A most convenient analogy is the transformation of a chord played by a musical instrument into
a formal discription of its component notes. Another useful analogy is when light passes through a prism
and is broken down into its component primary colours. Complex exponentials representation of Fourier Series The following identities show exponential  trigonometric relationship The fourier series shown below was developed in the notes on webpage Fourier Series. It is clear form from scrutiny of the above equations that
a _{n} = a _{n} ( even function ) and also that
that b _{n} =  b _{ n} ( odd function )..
( these feature are referenced below ) This can be rearranged as .. Now it is shown above that a _{n} = a _{n} and it can also be proved
that b _{n} =  b _{n}. The following solution also can be used. The Kronecker delta function δ_{kn} is often used for these equation .δ_{kn} = 1 if k = n and = 0 otherwise Therefore And finally A fourier series with a period T which is not 2π is represented it in terms of a function f( t )... [ t = T.x /2 π ] as follows The relevant formulas for determining a_{0} , a_{n} and b_{n} are as follows This can be expressed in exponential form as Example: Complex exponential based Fourier Series for function as shown below
Note: The expression sin(x) / x which occurs frequenty in Fourier Transforms is given the special name sinc(x). As limit sinc(x) for x > 0 = 1 then it is accepted that sinc (0) = 1. The above equation can therefore be rewritten as It is pointed out that the coeffiicient c_{n} for this even function is a real number. Note: The values shown for c_{n} are discrete values this is not a continuous function Example: Complex exponential based Fourier Series for function, as above offset by retarded by a/2 as shown below Note: As the expression sinc(0) = 1. The above equation can therefore be rewritten as It is pointed out that the coeffiicient c_{n} for this function is a complex number. Complex Spectra. The coefficient c_{n} is generally a complex number and is in the form c_{n} =  c_{n} e^{jφ} where,
for the example above Note: The values shown are discrete values the functions shown are not a continuous . The Phase spectra φ = πna/T indicates the phase of each harmonic
relative to the fundamental harmonic frequency Non periodic functions  fourier Transforms Introduction Fourier series are generally applicable only to periodic functions
but nonperiodic functions can also be transformed into fourier components  this
process is called a Fourier Transform. This expressed in exponential form results in If the product ( n / T ) in the exponent is replaced by a variable k_{n} then the the equation becomes It is clear that for large T then the the summation contains a large number of waves each with wavelength difference. The discrete summation tends to a continuous integration with k_{n} replaced by a variable k and c_{n} becoming a function of k . That is the equation evolves as shown below In these equations F( k ) is the Fourier transform of f( t ). The variables , for this
example of t and k are called conjugate variables. When conjugate variables are multiplied the product should be unitless. If the variables are t= time(say seconds) and k = frequency
say cycles per second then the equations shown are appropriate. Considering the example as shown above As T increases towards infinity and the distance between adjacent harmonics reduces towards zero i.e. the spectrum becomes a continuous function and n/T > k. Note: The above function is a continuous function. Alternative Representation of Fourier Transforms Other methods of identifying the fourier transforms include or to preserve symmetry 
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Last Updated 01/05/2010