![]() What is the Fourier transform of this function:įind the Fourier transform of a square pulse: Π We have applied Cauchy’s integral theorem to obtained the Fourier transform: ![]() ![]() Recall that this was also how we could obtain the frequency response function: What is the Fourier transform of this function? For (5.1.11), we simply define the left hand-side by the right-hand-side operation. The reason we use (5.1.9) is that it is a continuation from our discussion of extension of Fourier’s theorem of periodic function to Fourier transform. In computer analogy, the equal sign is not = but only an assignment: =. Functions χ and x are not any specific functions and the equal sign doesn’t represent an equation, but a definition: the left hand side is always considered as a definition by the right hand side. There is absolutely nothing wrong with either way. One very important note: In the following, sometimes we will write: Not every function has a Fourier transform: for example function that diverges at infinity or does not vanish sufficiently fast at infinity: the integral value simplies does not exist. Where F denote the Fourier transform and denotes the inverse Fourier transform. We can avoid this factor if we integrate vs frequency f instead of ω. Notice that there is a 2π factor involved. Thus, the pair of functions x and χ are referred to as Fourier transform of each other. This means that we can express the input signal as:Īnd we refers to x as the Fourier transform of χ. So, indeed, as T→∞, the interval as would be in an integral. Where is the whole area of bar, we define, is the bar width, which will play the role of interval Δω in the integral. To be correct, we have to include the integration interval as well, right? Take a look at this:Īs illustrated with the bar chart above, to express the concept of (5.1.2) rigorously, we write: When we sum a function over a very dense sets of points, it approaches to an integral: What kind of curve is that? We’ll see that it is known as the Fourier transform. We observe that all the points of Fourier coefficients appear to morph into a curve. Increase the period and we note that becomes very dense, for the simple reason: the fundamental frequency becomes very small: Lect_3340_Fourier_background_review_part2Ĭonsider a periodic function of period T, if T→∞, what would become of the sumįirst, let’s take a look of a typical example, suppose we have a square wave as shown below:
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