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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
-
=
Read More
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FAQs on PPT: Fourier Transform - Signals and Systems - Electrical Engineering (EE)

1. What is the Fourier Transform?
Ans. The Fourier Transform is a mathematical technique used to decompose a complex signal into its constituent frequencies. It converts a time-domain signal into a frequency-domain representation, providing information about the amplitude and phase of each frequency component present in the signal.
2. How does the Fourier Transform work?
Ans. The Fourier Transform works by representing a signal as a sum of sine and cosine waves with different frequencies. It decomposes the signal into its individual frequencies by analyzing the amplitude and phase of each frequency component. This transformation allows us to analyze and manipulate signals in the frequency domain.
3. What is the significance of the Fourier Transform in signal processing?
Ans. The Fourier Transform plays a crucial role in signal processing as it allows us to analyze and manipulate signals in the frequency domain. It helps in tasks such as filtering, noise removal, compression, and modulation/demodulation of signals. By understanding the frequency content of a signal, we can extract meaningful information and perform various operations on it.
4. What are some practical applications of the Fourier Transform?
Ans. The Fourier Transform finds applications in various fields, including telecommunications, image processing, audio processing, and medical imaging. It is used in signal analysis, spectrum analysis, image enhancement, equalization of audio signals, data compression, and many other areas where the frequency content of a signal is essential for analysis and manipulation.
5. Are there any limitations or constraints of the Fourier Transform?
Ans. The Fourier Transform assumes that the signal being analyzed is stationary, meaning that its properties do not change over time. It is also limited by the finite length of the signal being analyzed, which can introduce spectral leakage and aliasing effects. Additionally, the Fourier Transform may not be suitable for signals with rapidly changing frequency components or non-linear behavior.
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