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Sinusoids: continuous time
Amplitude A
Frequency F0  1 / T0 Hz
x(t )  A cos2 F0t     A cos2 F0 (t  t0 )
Phase   2F0t0 radians
delay
t0 sec
amplitude
A
period
T0 sec
Example of a Sinusoid
delay
period
amplitude
t0 3 9m sec
T0  10m sec
A  2 mVolts
mVolts
2
1
0
-1
-2
-3
0
20
40
60
80
100
msec
Then we can compute:
F0  1 / T0  100 Hz
  2F0t0  2 100  9 103  1.8 rad

x(t )  2 cos( 200t  1.8 )
Sampled Sinusoids
x[n]  x(nTs )
x(t )
1
Fs 
Ts
n
t
Ts sampling interval

x(t )  A cos2F0t   
x[n]  A cos 2
t  nTs  n
1
Fs
F0
Fs
n 

Analog and Digital Frequencies
x[n]  x(nTs )
x(t )
t
x(t )  A cos2F0t   
F0
Analog Frequency
in Hz (1/sec)
1
Fs 
Ts
n
x[n]  A cos0 n   
F0
0  2
Fs
Digital Frequency
in radians (no
dimensions)
Example
Given:
Analog Frequency:
F0  1.5kHz
Sampling Frequency: Fs  10kHz
compute:
Digital Frequency:
1.5 103
0  2
 0.3 rad .
3
10 10
Complex Numbers
A complex number is defined as
Real Part
x  a  jb
Imaginary Part
where
j  1
It can be expressed as a vector in the Complex Plane:
Im
b  Im{ x}
x
Re
a  Re{x}
Complex Numbers: Magnitude and Phase
You can represent the same complex number in terms of
magnitude and phase:
x
b  Im{ x}

a  Re{x}
a | x | cos( )
b | x | sin(  )
Then:
x | x | cos( )  j sin(  )
Complex Numbers
Recall:
e j  cos( )  j sin(  )
where
Then:
j  1
j
x | x | e  a  jb
a  Re{x} | x | cos  
b  Im{x} | x | sin  
| x | magnitude{x}  a 2  b 2
if a  0
arctan  b / a 
  phase{x}  
arctan  b / a    if a  0
Example
Let:
then
x  1 2 j
Re{ x}  1
Im{ x}  2
| x | 4  1  5
phase{x}  arctan{ 2}  1.1071rad
x  1  2 j  5e  j1.1071
Example
Let:
then
x  1 2 j
Re{ x}  1
Im{ x}  2
| x | 4  1  5
phase{x}  arctan{ 2}    1.1071    2.0344rad
x  1  2 j  5e j 2.0344
Euler Formulas
Since:
e j  cos( )  j sin(  )
e  j  cos( )  j sin(  )  cos( )  j sin(  )
Then:


1 j
cos( )  e  e  j
2
1 j
sin(  ) 
e  e  j
2j


Complex Exponentials
Using Euler’s Formulas we can express a sinusoidal
signal in terms of complex exponentials:

A j 2F0t    j 2F0t  
x(t )  A cos2F0t     e
e
2

This can be written as:
 A j  j 2F0t  A  j   j 2F0t
x(t )  A cos2F0t      e e
  e e
2

2

Complex Exponentials
Same for Discrete Time:

A j 0 n    j 0 n  
x[n]  A cos0 n     e
e
2

which can be written as:
 A j  j0 n  A  j   j0 n
x[n]  A cos0 n      e e
  e e
2

2

Why this is Important?
It is much easier to deal with complex exponentials than with sinusoids.
In fact:
d j 2F0t
e
  j 2F0  e j 2F0t
dt
j 2F0t
j 2F0t


e
dt

1
/
j
2

F

e
0



e j 2F0 (t  a )  e  j 2F0 a  e j 2F0t
differentiation, integration, time delay are just multiplications and
division, for complex exponentials only.