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Lecture

# SYDE 252 Fourier Transforms

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University of Waterloo

Systems Design Engineering

SYDE 252

John Zelek

Fall

Description

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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