Continuing our discussion of differential equations and transforms now I want to talk about Fourier series and transforms. So firstly, the Fourier transform and Fourier series are closely related topics. And in particular a Fourier series states that any periodic function can be written as the sum of an infinite number of sinusoidal terms which are sometimes called harmonic terms or harmonics. And the definition is given in this extract here and I have re-written the equation here. The Fourier transform f(t) is equal to, the Fourier series I'm sorry, f(t) is equal to a zero plus the summation from n equals one to infinity of a n cosine etc. Where t is equal to two pi over omega zero. Is the period of the function and the coefficients a n, et cetera are obtained from these expressions, a zero equals one over T, integral from zero to T, of the function of T. DT a n is two over T, F of T cosine. And omega zero T and BN is two over N, integral F of T sine, N omega T. And furthermore if the Fourier series representing a periodic function is truncated. After a term n=N the mean square value of the truncated series is given by Parseval's relation. Now, normally we won't be computing a Fourier series and there are several typical series which are given in the handbook. The first one is this rectangular wave form here of amplitude V0 and period T. The terms in the Fourier series are given by this expression here. Summation from n=1 to infinity minus 1, etc. Where N are just odd numbers in that summation. The next one is a somewhat similar looking function, rectangular wave form and the Fourier transform is given by this expression here. And finally the last example they give is the Fourier transform of a serious or train of impulses or delta functions separated by a period or a time interval capital T and the Fourier transform, or the Fourier series rather, is given by these expressions here. Let's do an example on that. The question is, what is the first term of the Fourier series of the rectangular voltage function with amplitude of plus minus two volts and a period T of one second? Which of these alternatives is the first term? So here is a situation. This is the rectangular wave train and the amplitude varies from +2 volts to -2 volts and the periodicity here is 1 second. So from the previous table on the previous slide we have the general term in the Fourier series is minus one race to the n minus one et cetera. So in this case we're only asked to evaluate the first term. So first we can compute the frequency, the angular frequency, omega 0, which is 2 pi over T. And in this case, the period T is one second, therefore, omega 0 is equal to 2 pi. So the first term then, putting N is equal to 1, we have minus 1 raised to the 1 minus 1 is 0. So minus one, any number raised to zero is one, times four zero of n pi and n is equal to one. A cosine and omega zero and n is equal to one and omega zero is equal to two pi. So combining those together, we find that the first term is equal to eight over pi, cosign two pi T. And the answer is A. Now, closely related to this is the idea of a Fourier transform which is also very important in a wide range of engineering and physical problems. Especially in terms of solving partial differential equations, spectroscopy, signal processing, image processing, and many more. The Fourier transform is defined as transforming a function of time, t, into a function of frequency, omega. And the definition, which is given in the extract here, is f omega of omega is equal to the integral for minus infinity of the function of time multiplied by e to the minus j omega t dt. Where j is the square root of minus 1. And this is called the Fourier transform. And the reverse of this lowercase function of t, in terms of the Fourier transform is given by this expression, and that is called the Inverse Fourier transform. So these two functions form a Fourier transform pair. And some examples of applications of this are given in the extract from the referenced handbook right here. So here again is the definition of the Fourier transform and its inverse from the handbook. And in the handbook they give you a number of transforms of various functions which can be used in various situations. One thing I'll mention about this, it uses this term, sinc. And the definition of sinc(x) is sin(x) over x. Although I don't think this is specifically stated in the handbook. So from this table you can evaluate the Fourier transforms of various functions and it seems unlikely that they would ask you to compute one from first principles. It also gives a number of transform theorems. For example, linearity, the first one here, the Fourier transform of the sum of two functions which are added together, is simply the sum of the Fourier transforms. And effects under scale change or time change etc. are all given here so they can be looked up in order to use any particular transform. So that completes the discussion of Fourier transforms and series.
