Fourier theory and delta functions
 (t  to )



 (t  to )du 
Fourier theory and delta functions



f (t ) (t  to )du 
FT  (t )  
FT  (t  to )  
Fourier theory and delta functions
  t  to   FT
-1
FT  t  t  
1
2
o
Or, for any two variables u,v:
1
2

 exp i t  t d
o


 exp iu  v  v  du     v  v 
o

Explain conceptually!
FT e
e
 io t
 
 i t
2    o 
o
So
appears at a negative frequency in the FT (because of our
i ( kz t ) as a “typical” wave at frequency .
convention of
e
i ( kz o t )

 
FT e
o
Convolutions
A convolution of two functions is the area under the
product of the two functions, which changes as the one of
the functions is shifted.

 g  h  (t )   g  t ' h  t  t ' dt '


 g  h  ( )   g  ' h    ' d '

more correct:
“…as one of the functions (inverted about the origin) is
shifted.”
Sketch the convolution of the two functions
(vertically displaced for display).
a) I got it mostly right
b) I tried, but got it mostly wrong
Sketch the convolution of the two functions
(vertically displaced for display).
a) I got it mostly right
b) I tried, but got it mostly wrong
Convolution Theorems
1
FT  g (t )h(t )  
 g  h  ( )
2
1
1
FT  g   h   
 g  h  (t )
2
Take transform inverses of above equations!
FT  g  h  (t )  2 g   h  
FT
1
 g  h  ( ) 
2 g (t )h(t )
products in one “space” become convolutions in the other, and
vice versa.
2 belongs on the product side
Prove carrier-frequency envelope principle
If a pulse is a steady (“carrier”) wave multiplied by an
envelope function
f (t )  g (t )e
i t
f (t )  g (t ) sin  t
…the FT f() is the FT of the envelope, g(, but
centered at ± . the width D is the width of g(.
Find
FT  cos  t sin  t 
You should be able to approach this two ways
without a traditional FT integral!
A gaussian pulse is g(t) with FT given by g()
g (t )  e
 t 2 /2 2
,
g ( ) 
e


2

 t 2 /2 2
 2 2
Find the complex FT of
e
cos
First write the cos as a sum of exponentials
a) I got it mostly right
b) I tried, but got it mostly wrong
ot
Find the FT of a double square pulse, each of width ,
centered at –t1, and t1
The FT of a single square pulse g(t) centered at t = 0
is
g ( )  A
sin  / 2

From today’s theory, find the FT of the double square pulse
a) I got it mostly right
b) I tried, but got it mostly wrong
Sketch FT(g(t)h(t)).
g()
h()