Fourier Analysis II
Today we continue our investigation of the theory of Fourier Analysis. Previously we looked at Fourier Series and how they can be used to represent periodic functions as sums of sines and cosines. Now however, we shall take a look at the new concept: the Fourier Transform.
Fourier Transform transforms the given function f(t) in the time domain into F(k) in the frequency domain (we will learn more about why it is called the frequency domain further down). In order to understand the Fourier transform, we first introduce the time domain and frequency domain.
The time domain is the domain in which the actual signal is present in, in other words, the x axis is time. In the frequency domain, the x axis is frequency. Now it is quite non-intuitive at the beginning to understand the frequency domain. Consider the following:
In my previous Fourier Analysis I post, we proved that any periodic function can be broken down into sums of cosines and sines of different amplitude and frequency. This frequency is exactly what the frequency domain consists of. What the frequency domain does is it plots the amplitude of the sinusoidal wave with a particular frequency k. Take a look at the following frequency plot:
We have one sine wave with amplitude 1 and at frequency 1 (it is Hertz in this case, but it could also be angular frequency depending on the exponent in the transform), and one sine wave with amplitude 2 at frequency 2. The signal can then be written as the following sum:
However, in the case that there are several of these sinusoidal decompositions in the frequency plot, we can write it as a summation:
Now, we can consider the case where the frequency plot is a continuous function and hence the summation will now become an integral:
This looks very similar to the inverse Fourier Transform and for good reason. What we have done here is essentially transform a function in the frequency domain into a function in the time domain. However, the actual inverse Fourier Transform has a clever trick to incorporate the phase shift of the various sinusoidal waves. This is very necessary in applications such as signal processing as the signals are rarely all in phase. Therefore, in order to account for phase shift, we introduce the concept of representing phase and amplitude using complex numbers.
In the diagram above, r is the amplitude of the sinusoidal and phi is the phase. Now, since we can use complex numbers to account for amplitude and phase, we can also use deMoivre’s theorem to make the sin(kx) exponential, simplifying the integral. Putting it all together, we get to the following result:
It does not takes much work to prove that to transform f(t) into F(k), we can use the following:
Convolution
Now that we have worked our way to defining the Fourier transform and its inverse, we need to find a way to solve for the inverse transform of a product of Fourier transforms. For example, given F(k) and G(k), suppose we want to find a function, which when Fourier transformed gives us F(k)G(k). The first guess would obviously be f(t)g(t), but as we know from elementary calculus, the integral of products is not the product of the constituent integrals (in other words, the integral operator is not distributive over multiplication). In order to be able to find such a function, there exists a method called convolution. The convolution of two functions is defined as:
Being able to interchange the last two expressions often helps in simplifying the integral as we can choose the function that is more suited to have the x-t term. Deriving this expression is not simple and requires a leap of ingenuity, but we shall go ahead and prove it.
Using Fubini’s Theorem, we can interchange the integrals (mathematicians forgive me for not proving it is possible).
Now using the substitution y=t-x,
Hence, the equivalent of multiplying two functions in the frequency domain is convoluting the two function in the time domain. This will prove useful in the future when we have multiple functions and their Fourier transforms to deal with.
Conclusion
Fourier Transforms become very useful when doing signal analysis as it decomposes the signals into the different sinusoidal inputs and the amplitude and phase of those inputs. For example, if there is a signal incoming from an satellite, usually an radio frequency (RF) signal, then the input will not necessarily be clean, due to the interference with other radio waves and distortion due to the Doppler effect and other factors. In order to accurately filter out the satellite signal, we would use Fourier Transform to figure out which part of the signal is noise and which part is from the satellite itself. Furthermore, an interesting application is in radar detection technologies where a radio signal is sent out and the reflected signals from planes and ships is analysed to figure out the locations of the planes and make sure enemy planes are not invading airspaces.
Interestingly, stealth planes manage to evade this RF signals by not reflecting the waves back and sometimes even adding noise to the reflected waves to make it harder to filter. Recently, researchers at University of Waterloo have developed quantum radar that uses quantum entanglement of photons to make such stealth technologies obsolete.
