 A warm welcome the 20th session of the second module in the core signals and systems. We are now well equipped to decompose a periodic waveform into either its sinusoidal components or its complex exponential rotating phasor components. What do we do with this decompose? Is what we are going to answer in this session. You will recall that quite a few sessions before, we had looked at what happens when a complex exponential goes into a linear shift invariant system. Now, we will take a periodic waveform being applied to a linear shift invariant system with a given impulse response and query what the output would be. So, in fact, for variety, let us not take a most general system, but let us take a particular case. Let us take the case of the symmetric square wave and a specific linear shift invariant system. Let us illustrate our concepts with that example and then generalize. So, what we are going to do is to consider the following signal system situation. You have this linear shift invariant system. Essentially, a resistance in a capacitance in series where the input voltage is applied across the series combination, let us call it V i t and we could also call it x t in our standard notation. And the output voltage V o t, it should be called y t in our standard notation is the voltage across the capacitor. The input x t is periodic with period t. So, of course, we can decompose it into its Fourier components. Now, we are going to start using this term. We are going to start using this term Fourier series. Fourier, I must mention very briefly, is the famous mathematician a few centuries ago who proposed this idea of decomposing a periodic waveform into its sinusoidal component. So, he suggested that we think of a periodic waveform as a linear combination of sinusoids all of which have the same period. Of course, the periods could be sub multiples of this period. That means the frequencies are all multiples of the fundamental frequency. And in fact, he thought of it in the context of solving certain differential equations which would lead to what are called boundary condition problems. I do not want to say too much about Fourier here. Maybe we will take it up some other time, but Fourier is a very well known name in all contexts in signals and systems, in system theory, in control engineering, a very famous mathematician, a very famous name with the whole concept of decomposition in terms of sinusoids named after him. This decomposition that we have been talking about, thinking of a periodic waveform as a linear combination of sinusoids or complex exponentials with frequencies equal to multiples of the fundamental is called a Fourier series decomposition. That is what we have written here. So, this term Fourier series is a term that we shall use again and again, essentially a decomposition into complex exponentials or sinusoidal components. In this case, we are talking about complex exponential components. Now, let us assume that xt is real. So, for example, xt could be this, a symmetric square wave. You know how to decompose a symmetric square wave into its Fourier series. We have done that in the last few sessions. We did it in the sinusoidal decomposition context and we know how to go from a sinusoidal decomposition to a complex exponential decomposition or if you like from a complex exponential decomposition to a sinusoidal decomposition. The ideas are very simple. However, we will use the complex exponential decomposition for a good reason. We have already seen that when a complex exponential goes through this linear shift invariant system, it comes out multiplied by a constant. That constant is essentially what is called a function of its impulse response operated upon by that particular sinusoid or that particular complex exponential. So, in fact, let us take this particular example of an RC circuit which is excited by one of these complex exponentials. So, let us apply Ck e raised to the power j 2 pi by t times kt to this RC circuit. So, we have Ck, the kth component e raised to the power j 2 pi by t times kt being applied to the RC circuit as the input. It is the voltage input given to the combination of r and c and we want to see the output. Now, we know how to analyze. We do not need to go through the whole gamut of generalized analysis. We could use what is called phasor analysis on this RC circuit. So, we know that each of these elements can be replaced by their phasor impedances. So, for example, the phasor impedance of this is equal to r independent of frequency. The phasor impedance of this is equal to 1 by JC times the angular frequency and the angular frequency is 2 pi by t times k. And therefore, the output phasor by the input phasor is equal to 1 by JC times 2 pi by t times k divided by r plus 1 by the same quantity. Let us simplify this. And therefore, the output is going to be Ck times this quantity. So, it is very interesting. You know, we have the output for any one of these complex exponents. And we know that if I give a complex exponential to this linear shift invariant system, it is going to give me a complex exponential of the same frequency. And I also have a very simple phasor analysis technique to find the output here. But here I did not need to worry about taking any impulse response and going through the impulse response. I could directly use my elementary knowledge of electric circuits. And in fact, you see the beauty is since the system is linear, I can take each of these components in turn, analyze what happens as a consequence of applying that component to the system and then superpose, the principle of superposition holds. So, now we can write down the output to the entire periodic input x t, that is very simple to do. So, we have the same r t circuit. And I applied the general periodic x t here, x t which is summation k going from minus to plus infinity Ck raised to the power j 2 pi by t k t. And I know what the output y t would be. The output y t would simply be summation k going, let me write it here. It is a little of longer expression. In fact, you see something very interesting. You gave a Fourier series as the input to this linear shift invariant system and you have ready made for you the Fourier series representation of the output. Let us mark that clearly for us. This is essentially the Fourier series representation of the output. And these are the Fourier coefficients here. Now, the problem is in the input we might have known what kind of a periodic waveform we have. For example, we could have taken the periodic waveform to be that symmetric square wave which I showed you. It could be a triangular wave, it could be what you will. Now, that is where the catch is. A very pertinent question at this point is can I have any and every periodic waveform being applied and can I analyze it this way? The answer is no. Yes and no. Yes, for most practical waveforms, practical periodic waveforms that you will encounter. No for some pathological cases of periodic waveforms which have some very peculiar properties. For example, if the waveform has an infinite number of maxima and minima in a finite interval in a period, suppose that waveform has an infinite number of maxima and minima. Let us take an example. Suppose you took this example of a function, sign 1 by t in the interval t between 0 and 1 and periodic and then let us call this equal to x t and x t is periodic with period 1. Unfortunately, this waveform has an infinite number of maxima and minima in this interval of 1 and this kind of a waveform is not amenable to Fourier analysis. There are certain conditions under which Fourier analysis can be done and they are called the Dirichlet conditions. So, this is said to violate what is called the Dirichlet condition. Dirichlet was also a mathematician. I cannot say too much about these conditions right now, but we should just be aware that these conditions are here. We will say more about Fourier analysis in the next session. Thank you.