 A warm welcome to the 29th session in the second module of the course Signals and Systems. We will now build on the example that we had in the previous session. Build as in derived properties of the Fourier transform beginning with that example. So, what we need by property of the Fourier transform is when we change the signal in some way or when we bring signals together what happens to the corresponding Fourier transform. Well, the first property that the Fourier transform has is what is called linearity, let us understand this. So, if H1t has the Fourier, now we are going to use a symbol to denote Fourier transformation. So, when we write script f like this we mean has a Fourier transform given by H1 of omega. Remember, you could write the Fourier transform either in terms of omega or in terms of f, in terms of the angular frequency or the cycles per second region and of course omega is equal to 2 pi f. Either way we would have the same properties, the properties do not change essentially. So, let H1t have the Fourier transform capital H1 omega and let H2t have the Fourier transform H2 omega. Then linearity says alpha times H1t plus beta times H2t has the Fourier transform alpha times H1 omega plus beta times H2 omega for all alpha beta belonging to a set of complex numbers. Remember, we must allow for complex numbers in general and all applicable H1 and H2. Let me first explain the meaning of this property to you. What this property means essentially is that when I take a linear combination of two functions each of which has a Fourier transform and the linear combination can be taken with respect to any complex coefficients. Then the Fourier transforms are also linearly combined in the same way. We now need to prove this property and let us get down to that job. So, approve, very simple. Of course H1 omega is equal to the integral from minus to plus infinity H1t, e raised to the power minus j omega t dt and that is also true of H2. Now we multiply the first equation by alpha and the second one by beta and add this and that gives us the property. Of course, when we add it, we will get alpha times H1 omega plus beta times H2 omega is integral from minus to plus infinity. Remember, you can take the constants inside and combine them and this essentially proves the property. Because as you see, the Fourier transform of alpha times H1t plus beta times H2t is indeed equal to the same linear combination of H1 omega and H2 omega. A very simple proof. In fact, while we are on the subject, let us prove also the same property for the inverse Fourier transform. So, the same property holds for the inverse. In fact, we will now write down a symbol for the inverse. So, if I write capital H1 omega or capital H1f, understand from the context. I must make this little remark here. You see, we often would talk of angular frequency and equally often talk of cycles per second frequency and from the context, it will be quite clear whether we are talking about angular frequency or cycles per second or you might call it Hertz frequency. And then we think of the Fourier transform either as a function of the angular frequency or as a function of the cycles per second frequency. And the relationships are very simple. Let us write down both these relationships here. So, we have H1 omega is of course equal. So, you know, we want to find the inverse Fourier transform. So, let us write down a symbol for that. We write the same things, script F but with an inverse sign. So, we interpret this to mean has an inverse Fourier transform equal to H1 of t. And we will use the same symbol here. From the context, it will be clear. The only difference is how we interpret that symbol. And the interpretation just has a small difference of a constant. So, when we are talking about angular frequency, the Fourier inverse should be calculated as follows. You need a factor of 1 by 2 pi. But then we can note that omega is equal to 2 pi F and therefore d omega is 2 pi dF whereupon when I take the inverse Fourier transform thinking of the Fourier transform as a function of the cycles per second frequency, I simply get the 2 pi cancels. Let me write down the 2 pi both outside and inside and show that it cancels. So, in a way dealing with cycles per second frequency has some conveniences. You do not need to remember that factor of 2 pi. There is a perfect symmetry between the Fourier transform and the inverse Fourier transform. Anyway, the exercise that I am going to leave for you to do is as follows. Show that the inverse Fourier transform also obeys linearity. In other words, alpha times h1 omega plus beta times h2 omega, all correspondingly with the cycles per second frequency would have the Fourier inverse alpha times h1 t plus beta times h2 t. Now, we shall take up a few more properties of the Fourier transform in the next session. Thank you.