Fourier transformation allow us to convert functions with a space or time independent into a function with spatial frequency or temporal frequency as the independent variable
In other words, it decomposes the function
The Fourier series expansion is an intuitive and special version of the Fourier transformation
The Fourier series allows a periodic function to be represented as an infinite sum of sinusoidal functions at definite frequency equal to multiple of the fundamenta
The Fourier transform allows an aperiodic function to be expressed as an integral sum over a continuous range of frequency
A lot of the idea of Fourier transformation stems from abstract vector space, where the basis vectors are the sinusoids
The understanding of functions as vectors is therefore crucial
When working with functions, we need a basis to represent them
The usual basis set works well due to its orthonormality
However, it is possible to find other orthonormal basis
In Fourier analysis, we are interested in using a basis of sinusoidal functions to represents functions
In other words, we are interested in constructing periodic functions as a linear combination of sine and cosine functions of different angular frequency
f(x)=ω=0∑∞[Sωsin(ωx)+Cωcos(ωx)]
We choose our basis functions to be sine and cosine functions of different angular frequency
If we then express the angular frequency in terms of the period of the function (ωc+ωs) and (ωc−ωs) will become T(nc+ns)π and T(nc−ns)π respectively. If we then perform the integration with some simplication, we will obtain
If we then express the angular frequency in terms of the period of the function (ωα+ωβ) and (ωα−ωβ) will become T(nα+nβ)π and T(nα−nβ)π respectively. If we then perform the integration with some simplication, we will obtain
After the computation, our orthonormal basis will look like such
2T1Tsin(ωx)Tcos(ωx)
The implication of choosing sinusoids as the basis functions is that we can represent any periodic functions as their linear combination
f(x)=λ×2T1+ω=1∑∞[SωTsin(ωx)+CωTcos(ωx)]
To find the Fourier series expansion of a function is to find the component of each basis function
The main advantage of using an orthonormal basis is that the inner products between any two orthonormal basis functions must be 0 or 1, which means that the component of each basis can be obtained by determining the inner product between the periodic function and the corresponding basis function
⟨Basis i∣Basis j⟩=δij
The coefficient of the constant can be determined by finding the inner product between the function and 2T1
λ=⟨2T1∣f(x)⟩
The jth coefficient of the sine basis can be determined by finding the inner product between the function and the jth sine basis
Sj=⟨Tsin(jx)∣f(x)⟩
The jth coefficient of the cosine basis can be determined by finding the inner product between the function and the jth cosine basis
Cj=⟨Tcos(jx)∣f(x)⟩
Having determined the expression for each of the basis, we can rewrite the linear combination
We shall substitute the coefficients with the corresponding inner products
The consequence of choosing the sinusoids as our orthonormal basis is that we need to compute three different types of definite integrals in order to get the coefficients of the series
While it is very much doable, it is not efficient by any stretch of the imagination
The mathematics become easier if we use the exponentials instead of the sinusoids
We wish to keep the orthonormal basis we chose, but change how we represent the basis functions
Using Euler's Identity, we can rewrite the sine and cosine functions as exponentials
The Fourier series allows us to represent periodic functions in terms of sinusoids, which is often a lot easier to manipulate
Unfortunately, most functions are aperiodic and blindly using the method above will not work
However, we can modify the Fourier series expansion slightly such that the representation can be applied to aperiodic functions as well
Consider a simple aperiodic function
If we were to apply the Fourier series expansion in the same manner we did before, we will be able to replicate its shape with sinusoids, but we will also make an infinite number of unwanted replicates
One trick to prevent this is to treat the function as a periodic function, but with an infinite period such that it will never repeat again
We shall convert our aperiodic function into a periodic function of infinite period
The conversion is only possible by taking the limit, which strongly suggests us to define a new quantity, Δω
Δω=ωn+1−ωn=T(n+1)π−T(n)π=Tπ
We can redefine ω in terms of Δω
ω=Tnπ=nΔω
We can redefine T in terms of Δω
T=ωnπ=nΔωnπ=Δωπ
We are now in a position to generalize the Fourier series to aperiodic functions
For aperiodic functions, the Fourier series only works when the period approaches infinity
f(x)=T→∞limω=−∞∑∞Fωe−iωx
We can express Fω in terms of an integral
f(x)=T→∞limω=−∞∑∞e−iωx[2T1∫−T+Teiωxf(x)dx]
After making the appropriate substitution with our newly defined ω and T, we are left with
f(x)=T→∞limω=−∞∑∞e−iωx×2πΔω∫−T+Teiωxf(x)dx
Rearranging the expression will allow us to see things more clearly
f(x)=T→∞limω=−∞∑∞e−iωx(2π1∫−T+Teiωxf(x)dx)Δω
Since T is inversely proportional to Δω, applying the limit will convert the summation into an integral
f(x)=2π1∫−∞∞e−iωx(∫−∞+∞eiωxf(x)dx)dω
The aperiodic version of Fourier series expansion changes the nature of the coefficient
As the limit of the period approaches infinity, the change in angular frequency, Δω , will approach zero and there will be a continuous change in the angular frequency
Hence, the coefficient, Fω, is no longer discreet and will become a continuous function of ω
T→∞⟹Δω→dω⟹Fω→F(ω)
Any function, be it periodic or otherwise, can thus be represented as
The Fourier Transform is much more profound than just a mathematical trick to represents aperiodic functions using the sinusoids
The Fourier Transform "accidentally" gives us the tool to manipulate functions on a much higher level, which is why we apply the transformation to periodic functions as well
We have established the Fourier series expansion is nothing but expanding a vector in terms of its component in a basis of sinusoids
Expansion in vector space is unique, in the sense that, only one vector possess this set of components in that particular basis
In other words, knowing all the coefficient for each sinusoid is the same as knowing what the function is
We can therefore represent a function, not with an infinite series of sinusoids, but with the coefficient for each angular frequency
F(ω)=∫−∞∞eiωxf(x)dx
F(ω) is called the Fourier transform of f(x).It is a complex function, which can be represented as the sum of a real and imaginary function
The peaks corresponds to the angular frequency of sinusoids that contribute to the representing the function
The height of the peaks corresponds to contributions or coefficients of the sinusoids of that particular angular frequency
Since F(ω) contains equivalent information to that of f(x) as they are just two different ways of looking at a function, we can easily convert F(ω) back to f(x)
f(x)=2π1∫−∞∞F(ω)e−iωx
This is called the inverse Fourier transform
F(ω) and f(x) are describing the same function in the same vector space, but with different basis
We say that f(x) lives in the x-domain, whereas F(ω) lives in the x-frequency domain
