AFS, More UNIX, and Printing
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Transcript AFS, More UNIX, and Printing
Fourier Transform
September 27, 2000
EE 64, Section 1
©Michael R. Gustafson II
Pratt School of Engineering
1
Recap of Chapter 3
• Fourier Series
– Frequency-space representation of periodic
signals
• Fourier Analysis
– Using the properties of the Fourier Series to
solve system and circuit equations for periodic
inputs
– Using the synthesis and analysis equations to
examine frequency content of periodic signals
2
Limitation of Chapter 3
• Fourier Series analysis only works for
periodic signals!
3
Fourier Transform
• Another tool for solving system
relationships in the frequency domain is the
Fourier Transform
• The Fourier Transform can almost be
thought of as the Fourier Series with an
infinite period
– Huh?
4
Fourier Transform (cont)
• For the Fourier Series, discrete frequencies
separated by the fundamental frequency are
considered
• For the Fourier Transform, a continuum of
frequencies separated by e are considered
• Because a continuum of frequencies are
used, the signal does not have to be periodic
5
Motivation
• Remember this?
S
st
e st
e
H s
S
s t
e st
h
e
d
s
H s h e d
S
st
st
s
e
e h e d
S
t
h (t )
6
Motivation (cont)
• Nothing in that derivation presumed a
periodic function, so the motivation is the
same as for Fourier Series - "If it can be
shown that a signal can be represented as a
combination of complex exponentials, then
the response of an LTI system to an input can
be represented as a combination of complex
exponentials times some function of the
impulse response and the exponential."
7
Fourier Transform Equations
• Synthesis
1
x t
2
• Analysis
X j e jt d
X j xt e jt dt
Note the factor is in front of the synthesis
equation this time!
8
But what does THAT mean?
• For a signal, X(j) gives the frequency
content of the signal
• For a signal, the magnitude squared of the
signal gives the power spectrum of the
signal
– BUT, non-impulse values of X(j) are only
truly meaningful when looked at as a spectrum
9
Probability Analogy
• This is a probability density function of
height (x-axis is difference from average)
10
Probability Analogy (cont)
• What is the probability of someone being
exactly average height?
• What is the probability of someone being 1"
taller than average?
• What is the probability of someone being 1"
shorter than average?
• Don't those add up to over 100%???
11
Probability Analogy (cont)
• The probability density is only useful for
determining ranges:
– What is the probability that someone is over a
foot taller then average?
– What is the probability that someone is average
height or shorter?
12
Back to X(j)
• The analogy works here - while you cannot
really say how much power is concentrated
in a particular frequency, you can say how
much is concentrated in a band of
frequencies
– Exception->if the signal has periodic
components, then the magnitudes squared
impulses at those frequencies do represent the
power there
13
But what about systems?
• If instead of x(t) you use h(t), H(j) gives an
indication of the input-to-output relationship of a
system:
– The magnitude gives the ratio of the output to the input
for a particular frequency
– The angle gives the phase difference between the output
and the input
– Ex: if H(j10)=1.4<Pi/4, then the component of the output
at 10 rad/s has 1.4 times the magnitude of the component
of the input at that frequency and is 45 degrees ahead in
phase
14
Convergence of the FT
• Same rules as FS except for periodicity:
Can' t blowup
Not infinitely squiggly
Not infinitely broken
xt
2
dt
15
Examples
• Find X(j) for x(t)=u(t+W)-u(t-W)
• Find X(j) for x(t)=exp(-at) u(t)
• Find x(t) if X(j)=1
• Find x(t) if X(j)=u(+W)-u(-W)
• Find x(t) if X(j)=2(- 0)
16
Periodic Parts
• The Fourier Transform will only return
finite values for energy signals.
• Power signals, if periodic, can be worked
with:
x t
jk 0t
a
e
k
k
X j 2
a
k
k
0
17
Examples (cont)
• Find X(j) for x(t)=cos(t)+u(t+1)-u(t-1)
• Find X(j) for x(t)=cos(3t) sin(5t)
18
Questions
19