Fourier Transforms of
Generalised Functions
F    FT { f (t )} 



f (t )e it dt
Existence of FT
• The FT of a function exists provided the
function satisfies the Dirichlet conditions:

 f t  dt  ,
and any discontinuities are finite

• Generalised functions
- Do not satisfy the Dirichlet conditions
- Often used in signal processing
- We can still find their FTs
The Delta Function
• Properties
1.  t   0 for t  0
2. lim  t   
t 0

3.

Let
h0
1/h
 t dt  1

• Graph becomes a ‘spike’
h
• Models eg a very large current applied
for a very short time
The Delta Function
• Further properties...

 f t  t dt  f 0
 (t )
f (t )
f (t )
 (t  a)


 f t  t  a dt  f a 

b

a
 f  
f t  t   dt  
0
0
if a    b
otherwise
a
FT of the Delta Function

FT t    eit t dt

 e  i 0
1
In general…


  f t  t dt  f 0


 


FT t  a    eit t  a dt

 e  ia


  f t  t  a dt  f a 


 

Fourier Transform of 1
If
f t   δ t   F    1
Symmetry property:
FTF t   2πf  ω
 FT1  2    
 2  
Delta function
is even
FT of Cosine and Sine
If
f t    t  a   F    eia
Symmetry: FTF t   2πf  ω
 FTeiat  2    a 
Similarly
 2   a 
iat
FTe  2    a 
1
1
iat
iat


2   a   2   a 
 FT cosat   FT e  e 
2
2
     a    (  a)
1
1
iat
iat

 FT sin at  FT e  e 

2   a   2   a 
2i
2i
 i    a    (  a)
FT of Powers of t
Multiplication property:
dF
FTtf t   i
dω
d
FT
t  FTt 1  i FT1
d
d
2  
i
d
 2 i '  
Similarly
d
FTt
FTt  FTt t  i
d
d
2i '  
i
d
2
 2 ' '  
FT of sgn(t)
Definition
1
sgn t   
 1
t0
t0
Result from Homework 1…


2i
FT e sgn t    2
  a2
a t
Let a  0 : e sgn t   sgn t 
2i
a t
2
i



sgn t 
and FT e

2


2i
 FTsgn t   
a t



FT of the Heaviside Function
sgn t 
Hence
1 sgn t 
1
FTut   FT  1  sgn t 
2

1
 FT1 FTsgn t 
2
1
2i 
  2πδ ω   
2

i 

 π  δ ω 

 

1
ut   1  sgnt 
2
Now look at Tutorial 3