EE511 Day 4 Class Notes
Discussion of Fourier Transform
Laurence Hassebrook
Updated 9-5-03
Friday 9-5-03
Wednesday 8-31-05
Covered written notes for Fourier Transforms up to convolution theory (8-27-01).
GET VALUES TO BE USED IN DSSS VISUALIZATION
Projection Integral and the Fourier Transform
Recall the projection integral but let’s remove the normalization such that the projection has units of
energy rather than power s.t.
y ab   at bt  dt  a projected onto b
t2
t1
where we dropped the 1/T multiplier. Consider the case where
bt   e  j 2ft
so
y ab   a t e  j 2ft dt
t2
t1
The Fourier Transform (FT) is the limit of the above such that
lim
t2

Fa t   t1    at e  j 2ft dt   at e  j 2ft dt  A f 
t1

t2  
The FT is invertible s.t.

a t   F1 A f    A f e j 2ft df

PROPERTIES OF A FT
1. Duality
1
2. Time Shift
3. Frequency Shift
4. Convolution
5. Correlation
6. Multiplication
2