Linear Systems 101 - Part 7 - Sinusoidal response (0)

 Ok, we have studied until now the impulse and step response. Our next subject is the sinusoidal response and we are going to study it in 3 posts. 3 posts?! Oh yes, it is that important (and also a little bit more complicated to do the calculations). Here in this post, we are just motivating and justifying, anyway, presenting the reasons why it is important to study how sinusoidals are affected by dynamic (LTI) systems. We will also make some detours from Linear and Control Systems and invade the realm of Signal Processing, but just a bit, I swear!

First, let us recall what a sinusoid is: \begin{equation}s(t) = s \sin(\omega t + \phi), \end{equation} where s is the amplitude, ω is the (angular) frequency, and ϕ is the phase.  A typical sinusoidal wave is shown here. Note that, a priori, the signal extends from -∞ to ∞ but, as we are doing here throughout this series, we consider that s(t) = 0 for t < 0. 

Right, so what? Well for starters,  it turns out that all periodic signals can be written as the sum of sinusoids via the Fourier series. That is, the square wave, the triangular wave, the sawtooth wave, and so on and so forth, have a representation via this series. Every one of them has an application in Engineering, for instance, the square wave arises as the clock in digital systems. Also, the three waves appear in older audio synthesizers, see also this very interesting video showing how musicians could use those waves to emulate the sounds of different instruments. It is a must-see for music and Engineering aficionados! With that in mind, let us have a brief look at this definition of the Fourier Series: \begin{equation}c(t) =A_0 + \sum^{\infty}_{n=1}A_n\cos(2\pi n t/T)+ \sum^{\infty}_{n=1}B_n\sin(2\pi n t/T), \end{equation} where An and Bn are coefficients calculated through integral formulae and T is the signal period. This is an exciting formula indeed:

  • A0 is the average of c(t) over t, \begin{equation}\bar c= \int_{0}^{\infty}c(t) dt = A_0\end{equation}It is also known as the DC level of the signal in Electrical Engineering (since it is not associated with oscillations)
  • The values of n define multiples of the natural frequency f = 1/T. They are called harmonics.

Thus, all the information that characterizes the signal is concentrated on those harmonics. We can say that all the signal's energy is also concentrated there. You can visualize this by having a look at these interactive plots. As you can see there, if we plot the values of An and Bn against n, we have this visualization of the signal's frequency content. 

Ok, cool André. Every periodic signal can be expressed as a Fourier series. However, as you said before, we are considering signals that are 0 for t < 0, but a periodic signal must satisfy s(t+T) = t for all real t's. Touché! Buuuut, we can apply a similar concept to some continuous-time signasl, even if they are not periodic! This is done through the Fourier transform,\begin{equation}\hat s(f) = \int_{-\infty}^{\infty}f(t)e^{-i2\pi f t}dt\end{equation}

Ok, wait a minute, what's that? Well, a transform is an operation that transforms (duh!) the original domain of a function to another one. In this case, we are changing the domain of this signal defined in t to its equivalent form in the frequency domain f. There are a lot of discussions regarding the Fourier transform. The main point here is that we can also obtain the frequency content of any continuous-time signal that has a Fourier transform. I'm not going to give much detail here, but there is also a discrete-time Fourier transform which gives rise to the celebrated Fast Fourier Transform (FFT). And we finally got to the point: 

Since "any" signal can be written by an equivalent representation on the frequency domain, it is possible to study how the system affects, shapes, and modifies this signal's frequency content. By studying how the system affects a sinusoidal signal, we can consequently describe how the system also affects a more general class of signals.

Well, is it just that? It turns out that this is very very useful. Applications can be found in communication channels, audio applications such as equalizers and effects, image processing, data compressing, and much, much more. It is one of the pillars of signal processing, man! 

If you are already hooked, bear with me! Be not afraid and let us go!



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