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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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Last Updated 01/05/2010