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SYDE 252 Fourier Transforms

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University of Waterloo
Systems Design Engineering
SYDE 252
John Zelek

Lecture 10 Fourier Transform (Lathi 7.1-7.3) Definition of Fourier Transform  The forward and inverse Fourier Transform are defined for aperiodic signal as:   Fourier series is used for periodic signals. L7.1 p678 Tuesday, 30 October, 12 3 The Fourier Transform Fourier’s Song Integrate your function times a complex exponential, It’s really not so hard you can do it with your pencil, And when you’re done with this calculation You’ve got a brand new function—the Fourier Transformation! What a prism does to sunlight, what the ear does to sound, Fourier does to signals, it’s the coolest trick around. Now filtering is easy, you don’t need to convolve; All you do is multiply in order to solve. From time into frequency—from frequency to time. Every operation in the time domain Has a Fourier analog—that’s what I claim. Think of a delay, a simple shift in time It becomes a phase rotation—now that’s truly sublime! And to differentiate, here’s a simple trick, Just multiply by j omega, ain’t that slick? Integration is the inverse, what you gonna do? Divide instead of multiply—you can do it too. From time into frequency—from frequency to time. Let’s do some examples...consider a sine. It’s mapped to a delta, in frequency—not time. Now take that same delta as a function of time Mapped into frequency—of course—it’s a sine! Sine x on x is handy, let’s call it a sinc. Its Fourier Transform is simpler than you think. You get a pulse that’s shaped just like a top hat... Squeeze the pulse thin, and the sinc grows fat. Or make the pulse wide, and the sinc grows dense, The uncertainty principle is just common sense. Dr. Time and Brother Frequency (Bob Williamson and Bill Sethares) 3.1 Introduction The FS is a valuable tool for analysing periodic functions, but many signals of interest are aperiodic. However, we can use our knowledge of FS to generate a related tool for aperiodic functions called the Fourier Transform or the Fourier Integral. 3.2 Analysis of an Aperiodic Function Suppose we have an aperiodic pulse signal such as that shown on the left in Fig.6. We can generate a periodic signalT0(t) from it by repeating it after a per0od T as shown above on the right. The FS for this periodic function is ∞ ▯ gT0(t) = Gnexp(jnω t0 n=−∞ where ω0= 2π/T 0 19 Connection between Fourier Transform and Laplace Transform  Compare Fourier Transform:  With Laplace Transform:  Setting s = jω in this equation yield:  Is it true that: ?  Yes only if x(t) is absolutely integrable, i.e. has finite energy: L7.2-1 p697 Define three useful functions  A unit rectangular window (also called a unit gate) function rect(x):  A unit triang
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