1. Fourier Analysis

2. Periodic Signals For all t, x(t + T) = x(t)
x(t) is a periodic signal
Smallest value of T is the fundamental period
Fundamental frequency 1/T
Periodic signals have a Fourier series representation
Fourier series coefficient Cm quantifies the strength of the component of x(t) at frequency m/T
Fourier transforms (defined next) are for both periodic and aperiodic signals

3. Fourier Integral Conditions for Fourier transform of x(t) to exist
x(t) is single-valued with finite maxima and minima in any finite time interval
x(t) is piecewise continuous; i.e., it has a finite number of discontinuities in any finite time interval
x(t) is absolutely integrable
Conditions not obeyed for cos(t), sin(t) and u(t)
We’ll find ways to define Fourier transforms for them

4. Laplace Transform Generalized frequency variable s = s + j w
Laplace transform consists of an algebraic expression and a region of convergence (ROC)
For substitution s = j w or s = j 2 p f to be valid, ROC must contain the imaginary axis
Laplace transform of u(t) is 1/s with ROC of Re{s} > 0
This ROC does not include the imaginary axis

5. Fourier Transform What system properties does it possess?
Memoryless (in fact requires infinite memory)
Causal
Linear
Time-invariant (doesn’t apply)
What does it tell you about a signal?
Answer: Measures frequency content
What doesn’t it tell you about a signal?
Answer: When those frequencies occurred in time

6. Fourier Transform Pairst w
F(w)
-6p
-4p
-2p 2p 4p 6p 1
t/2
-t/2 t
f(t) F

7. Fourier Transform Pairs
From the sifting property of the Dirac delta,
Consider a Dirac delta in the Fourier domain
Using linearity property, F{ 1 } = 2p d(w) 1 t
x(t) = 1
X(w) = 2 p d(w) F
(2p) w
(2p) means that the area under the Dirac delta is (2p)

8. Fourier Transform Pairs
F(w)
f(t)
(p)
(p) F t w
-w0 w0