I. Previously on IET
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
2
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
3
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
4
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
5
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
6
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
7
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
8
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
9
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
10
Complex Exponential Function
e jωt = cos(ωt ) + j sin(ωt )
Im-Axis
ω
© Tallal Elshabrawy
Re-Axis
11
The Fourier Transform
Representing functions in terms of complex
exponentials with different frequencies
Gf  

 gt e

© Tallal Elshabrawy
 jωt

dt 
 gt e
 j2πft
dt

12
The Fourier Transform (Cosine Function)
e jωt  cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
+
-ω
Re-Axis
ejωt  e jωt  2cos  ωt 
+
© Tallal Elshabrawy
13
The Fourier Transform (Cosine Function)
e jωt  cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
+
-ω
Re-Axis
ejωt  e jωt  2cos  ωt 
+
© Tallal Elshabrawy
14
The Fourier Transform (Cosine Function)
e jωt  cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
+
-ω
Re-Axis
ejωt  e jωt  2cos  ωt 
+
© Tallal Elshabrawy
15
The Fourier Transform (Cosine Function)
e jωt  cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
+
-ω
Re-Axis
ejωt  e jωt  2cos  ωt 
+
© Tallal Elshabrawy
16
The Fourier Transform (Cosine Function)
e jωt  cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
+
-ω
Re-Axis
ejωt  e jωt  2cos  ωt 
+
© Tallal Elshabrawy
17
The Fourier Transform (Sine Function)
e jωt  cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
-
-ω
Re-Axis
ejωt  e jωt  2jsin  ωt 
© Tallal Elshabrawy
18
The Fourier Transform (Sine Function)
e jωt   cos  ωt   jsin ωt 
e jωt  cos  ωt   jsin ωt 
Im-Axis
Im-Axis
ω
Re-Axis
+
-ω
Re-Axis
ejωt  e jωt  2jsin  ωt 
+
© Tallal Elshabrawy
19
The Fourier Transform (Sine Function)
 je jωt   jcos  ωt   sin  ωt 
 jejωt   jcos  ωt   sin ωt 
Im-Axis
Im-Axis
Re-Axis


 j e jωt  e jωt  2sin  ωt 
ω
Re-Axis
-ω
© Tallal Elshabrawy
20
The Fourier Transform (Sine Function)
je jωt  jcos  ωt   sin ωt 
 jejωt   jcos  ωt   sin ωt 
-ω
Im-Axis
Re-Axis

+
Im-Axis
Re-Axis

 j e jωt  e jωt  2sin  ωt 
ω
+
© Tallal Elshabrawy
21
Fourier Transform of Sinusoids
e jωt  e jωt
cos  ωt  
2
1/2
sin  ωt 
1/2
-ω
0
ω
e

 j
jωt
 e  jωt

2
j(1/2)
-ω
0
ω
-j(1/2)
Notes


A real value for the coefficients in the frequency domain means that the starting
point for rotation is on the real axis
An Imaginary value for the coefficients in the frequency domain means that the
starting point for rotation is on the imaginary axis
© Tallal Elshabrawy
22
Fourier Transform of Real Valued Functions
Im-Axis
Im-Axis
ωn
ω1
Im-Axis
Re-Axis
Re-Axis
Re-Axis
ω2
Im-Axis
Im-Axis
Im-Axis
-ω2
Re-Axis
Re-Axis
Re-Axis
-ω1
-ωn
A real-valued function in time implies that
G(-f) = G*(f)
© Tallal Elshabrawy
23