PPT: Fourier Transform | Signals and Systems - Electrical Engineering (EE) PDF Download

 Page 1


Fourier Transform
Page 2


Fourier Transform
How to Represent Signals?
 Option 1: Taylor series represents any function using
polynomials.
 Polynomials are not the best - unstable and not very
physically meaningful.
 Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Page 3


Fourier Transform
How to Represent Signals?
 Option 1: Taylor series represents any function using
polynomials.
 Polynomials are not the best - unstable and not very
physically meaningful.
 Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Jean Baptiste Joseph Fourier (1768-1830)
 Had crazy idea (1807):
 Any periodic function
can be rewritten as a
weighted sum of Sines and
Cosines of different
frequencies.
 Don’t believe it?
– Neither did Lagrange,
Laplace, Poisson and
other big wigs
– Not translated into
English until 1878!
 But it’s true!
– called Fourier Series
– Possibly the greatest tool
used in Engineering
Page 4


Fourier Transform
How to Represent Signals?
 Option 1: Taylor series represents any function using
polynomials.
 Polynomials are not the best - unstable and not very
physically meaningful.
 Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Jean Baptiste Joseph Fourier (1768-1830)
 Had crazy idea (1807):
 Any periodic function
can be rewritten as a
weighted sum of Sines and
Cosines of different
frequencies.
 Don’t believe it?
– Neither did Lagrange,
Laplace, Poisson and
other big wigs
– Not translated into
English until 1878!
 But it’s true!
– called Fourier Series
– Possibly the greatest tool
used in Engineering
A Sum of Sinusoids
 Our building block:
 Add enough of them to
get any signal f(x) you
want!
 How many degrees of
freedom?
 What does each control?
 Which one encodes the
coarse vs. fine structure of
the signal?
) + f wx A sin(
Page 5


Fourier Transform
How to Represent Signals?
 Option 1: Taylor series represents any function using
polynomials.
 Polynomials are not the best - unstable and not very
physically meaningful.
 Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Jean Baptiste Joseph Fourier (1768-1830)
 Had crazy idea (1807):
 Any periodic function
can be rewritten as a
weighted sum of Sines and
Cosines of different
frequencies.
 Don’t believe it?
– Neither did Lagrange,
Laplace, Poisson and
other big wigs
– Not translated into
English until 1878!
 But it’s true!
– called Fourier Series
– Possibly the greatest tool
used in Engineering
A Sum of Sinusoids
 Our building block:
 Add enough of them to
get any signal f(x) you
want!
 How many degrees of
freedom?
 What does each control?
 Which one encodes the
coarse vs. fine structure of
the signal?
) + f wx A sin(
Fourier Transform
 We want to understand the frequency w of our signal.  So, let’s
reparametrize the signal by w instead of x:
) + f wx A sin(
f(x) F(w)
Fourier
Transform
F(w) f(x)
Inverse Fourier
Transform
 For every w from 0 to inf, F(w) holds the amplitude A and phase
f of the corresponding sine
– How can F hold both?  Complex number trick!
) ( ) ( ) ( w w w iI R F + =
2 2
) ( ) ( w w I R A + ± =
) (
) (
tan
1
w
w
f
R
I
-
=