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.

The fourier transform is used to transform one complex valued real variable to another function.    The resulting function, often called the frequency domain representation (or the spectral density) of the original function, describes which frequencies are present in the original function.

The notes on fourier series ref   Fourier Series deals primarily with functions of periods of T = 2π.     The notes also show that the theory can be applied to functions with periodic values other than 2π.     The fourier transform is the generalisation of the fourier series with periods aproaching infinity.    Effectively fourier series relate to periodic functions whilst fourier transforms are more generalised including non-periodic function and periodic function.    Fourier series are actually a subset of fourier transforms.

These notes do not provide detailed information of fourier transforms.    They provide a very basic introduction for mechanical engineers relating fourier transforms to the, much simpler, 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 )

Equation A can be expressed in exponential form as

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.
Using this feature the above equation can be re-arranged as ..

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 a₀ , a_n and b_n are as follows

This can be expressed in exponential form as

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

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.

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
Phase = φ = -πna/T
| c_n | = a/T .sinc (πna/T)

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
= 1/T = f₀ = ω₀ /2π
In the first example, which is an even function, the phase spectrum is zero for all values of n . This indicates that each harmonic is in phase with f₀ (the fundamental harmonic). The second example indicates that the nth harmonic is out of phase by -πnaf₀

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.

If T becomes very large ( tends to infinity ) then the function will tend to an isolated, non periodic, function.     This limiting process is used to develop the equations for the Fourier Transform from the Fourier Series.

Consider a function which has a width which is very very small relative to the period T. If T approaches infinity than the function is effectively non periodic.

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.

Notes:
f( t ) is also the Fourier transform of F( k ).
Only one of the of the integrals must have a minus sign in the exponent.    It is not important which one in terms of application of the transform. The rule selected at the onset of the analysis must be not be broken.

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.

Other methods of identifying the fourier transforms include

or to preserve symmetry