Development of Fourier Series Theory using Exponential Representation
Non Period Functions
Fourier Transforms options
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 re-arranged 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 δkn is often used for these equation .δkn = 1 if k = n and = 0 otherwise
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 a0 , an and bn 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 cn for this even function is a real number.
Note: The values shown for cn 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 cn for this function is a complex number.
The coefficient cn is generally a complex number and is in the form
cn = | cn |ejφ 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 non-periodic 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 kn 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 kn replaced by a variable k and cn 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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