Fourier
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MIT OpenCourseWare http://ocw.mit.edu
MAS.160 / MAS.510 / MAS.511 Signals, Systems and Information for Media Technology Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Period of a Discrete Sinusoid
y [n ] = sin(2 501 n)
y [ n ] = sin(2 503 n)
T=50 samples
y [ n ] = y[n + 50]
y [ n ] = y[n + T]
T=?? samples [integer]
50/3 � integer
sin(0) = sin(2 )
irrational frequency
1 0.8 0.6
y ( t ) = sin(2 2 t) T = 12 sec
y=sin(2*pi*sqrt(2)/25*n)
0.4 0.2 0
continuous function periodic
-0.2 TextEnd -0.4
-0.6
-0.8
-1
0
y [ n ] = sin(2 503 n) y [ n ] = y[n + T] sin(0) = sin(2k) k=1,2…
T=?? samples
2 503 n = 2k n 50 samples = k 3 cycle
periodic T=n=50 samples, k=3 cycles
0.5
1
1.5
2 time (sec)
2.5
3
3.5
4
Ts=1/25 sec Ratio of integers
rational number
y [ n ] = sin(2
2 25
n)
y [ n ] = y[n + T]
sin(0) = sin(2k)
T=?? samples k=1,2…
Equiv. discrete sinusoid not periodic
2 252 n = 2k n 25 2 = k 2 irrational number
Period of Sum of Sinusoids
1
0.5
0
-0.5
-1
0
1
2
3
4
5
6
5
6
T1=0.2 seconds, T2=.75 seconds 2
1
0
-1
-2
0
1
2
3 time (sec)
y ( t ) = y (t + T )
Tsum=3 seconds
Least common multiple seconds to complete cycles T1=1/5 seconds
15 cycles
Complex Conversions
seconds to complete cycles
cartesian s=a+jb
polar
s= a +b e 2
T2=3/4 seconds 3/4s, 6/4s, …
1/5s, 2/5s, 3/5s … 4/20s, 8/20s, 12/20s, 16/20s, 20/20s, 24/20s, 28/20s, 32/20s, 36/20s, 40/20s, 44/20s, 48/20s, 52/20s, 56/20s, 60/20s
15/20s. 30/20s, 45/20s, 60/20s 4 cycles 1/5*k=3/4*l k/l=15/4
Tsum=15*T1=15/5=3 seconds
4
rational number
Tsum=4*T2=3/4*4=3 seconds
Tsum=3 seconds
2
j� a tan ( b a )
polar s=rej�
cartesian
s = r cos� + jr sin �
Complex Arithmetic Addition
cartesian
(a1 + jb1 ) + (a2 + jb2 ) = ( a1 + a2 ) + j (b1 + b2 )
Subtraction
cartesian
(a1 + jb1 ) � (a2 + jb2 ) = ( a1 � a2 ) + j (b1 � b2 )
Multiplication
polar
r1e j� 1 r2e j� 2 = r1r2e j (� 1 +� 2 )
Division
polar
r1e j� 1 r1 j (� 1 �� 2 ) = e r2e j� 2 r2
Powers
polar
(re )
Roots
polar
j� n
= r n e jn�
z n = s = re j� z = s1/ n = r1/ n e j (� / n + 2 k / n )
k = 1,2Kn �1
Representations of Sinusoids
Complex Conversions cartesian
polar
3 + j4 = 32 + 4 2 e
j� a tan (
polar 4
3
)
= 5e j� 0.927
2e
j 3
Re{ Ae j 2 ft +� }
Acos(2kf 0 t + � k )
cartesian
= Re{ Ae j� e j 2 ft }
= 2cos 3 + j2sin 3 = 1+ j 3
Subtraction
cartesian cartesian
(1+ j2) + ( 3 + j4 ) = (4 + j6)
j 3
6e
polar
5e
Division
polar
10e
Powers
polar
(3e )
Roots
polar
j� 2
j 4
j 4
÷ 5e 3
k=1
j 3 j 3 = 33 e ( 4 ) = 27e ( 4 )
z 3 = 64 = 64e j 0
4
e j 2 ft +e � j 2 ft 2 e j 2 ft + e � j 2 ft 2
4e j ( 2 / 3) 4e j ( 4 / 3)
n
n
k=1
k=1
cos(2�ft + � k ) = � Re{ Ak e 2 �ft +� k } = � Re{ Ak e� k e 2 �ft } �n = � Re{ Ak e� k } e 2 �ft � k=1
Ex.
3cos(2 40t + 2 ) �1cos(2 40t � 6 ) + 2 cos(2 40t + 3 ) � � j �j Re� 3e 2 e j2 40t �1e 6 e j2 40t + 2e 3 e j2 40t � � � �� j � �
�j Re� 3e 2 �1e 6 + 2e 3 e j2 40t
� � �
Re{5.234 e j1.545e2 40t }
5.234 cos(2 40t + 1.545)
Composite signals (waveform synthesis)
multiply cosines of different frequency
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
A1 cos(�1t ) � A2 cos(� 2 t + � ) � e j�1 t + e� j�1 t � e j (� 2 t +� ) + e� j (� 2 t +� ) A1 A2 2 2 �
�
decompose a periodic signal x(t) into a sum of a series of sinusoids - the Fourier series.
(
A1 A2 j�1 t j (� 2 t +� ) e e + e j�1 t e� j (� 2 t +� ) + e� j�1 t e j (� 2 t +� ) + e� j�1 t e� j (� 2 t +� ) 4
(
A1 A2 j (�1 t +� 2 t +� ) � j (� 2 t��1 t +� ) e +e + e j (� 2 t��1 t +� ) + e� j (�1 t +� 2 t +� ) 4 A1 A2 cos((�1 + � 2 ) t + � ) + cos((� 2 � �1 ) t + � ) 2
k
j� � j� = ( 105 )e ( 2 4 ) = 2e 4
z = 641/ 3 e j ( 0 / 3+2 k / 3) = 4e j ( 2 k / 3)
(
n
�A
j + j 7 = 5 6e ( 3 4 ) = 30e 12
j� 4
(
(
Sum multiple cosines same frequency
(1+ j2) � ( 3 + j4 ) = (�2 � j2)
Multiplication
=X�
= Re{ Xe j 2 ft }
Complex Arithmetic Addition
Ae j� �
)
)
)
Note: The sum of periodic functions is periodic. ex.
�� �8 X k = � 2k 2 �� 0
k odd k even
f 0 = 25Hz
)
)
Composite signals (waveform synthesis)
Composite signals (waveform synthesis)
� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 �� �8 �� 8 j k odd e k odd X k = � 2k 2 X k = � 2k 2 �� 0 �� 0 k even k even �
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
�
�� 8 e j X k = � 2k 2 �� 0
x(t) = 0.8105cos(2 25t + )
x(t) = 0.8105cos(2 25t + ) + 0.0901cos(2 75t + )
1
1
0.8
0.8
0.6
0.6
0.4
0.4
1
=
0
-1
0.02
0.04
0.06
0.08
0.1
0.12
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
0.4
=
0.2
0
-0.6
-0.6
0.6
0.4
-0.4
-0.4
0.8
0.6
-0.2
-0.2
1
0.8
0.2
0.2
0
k even
k=3
k=1
-1
k odd
-0.8
-0.8 0
0.02
0.04
0.06
0.08
0.1
0.12
-1
-1
0
0.02
0.04
0.06
0.08
0.1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.12
1
Composite signals (waveform synthesis)
0.8 0.6
�� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 �� 8 e j k odd X k = � 2k 2 �� 0 k even �
spectrum
0.4 0.2 0
0.45
-0.2
0.4053e� j
0.4
0.4053e j
-0.4
-0.6 -0.8
0.35
-1
k=5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.3
x(t) = 0.8105cos(2 25t + ) + 0.0901cos(2 75t + ) + 0.0324 cos(2 125t + ) 1
1
0.8
0.8
0.6
0.6
0.4
0.4
=
0.2
X 0.25 0.2
0.15
0.2
0.1
0.0450e� j
0
0
-0.2
-0.2
0.0450e j
0.0162e� j
0.05
0.0162e j
-0.4
-0.4
0 -300
-0.6
-0.6 -0.8
-200
-100
-125
-75
-25
0
25
75
100
125
200
f 300
-0.8 -1
-1
0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
x(t) = 0.8105cos(2 25t + ) + 0.0901cos(2 75t + ) + 0.0324 cos(2 125t + ) + ...
Fourier Series
Fourier Series For a given signal, how do we find X k = Ak e for each k ?
x(t) = t 0 � t < T0
j� k
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
Fourier Analysis
1 X 0 =
T0
�� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 �
where
1 X0 = T0
Xk =
2 T0
=
T0
� x(t)dt
f 0:fundamental frequency T0 = 1/ f 0
0
T0
� x(t)e
� j 2 kt
T0
dt
0
1 T0
T0
� x(t)dt
0.04
t
0
T0
� tdt = 0
2 T0
1 t T0 2
0
=
1 T0 2 T0 = T0 2 2
Mathematica: athena%add math athena%math In[1]:=1/T*Integrate[t,{t,0,T}] Out[1]:=T/2
Fourier Series
0.04
0.02
0
0 0 0
0.01
Xk =
2 T0
0.02 t
0.03
T0
� x(t)e
0.04 0.04
� j 2 kt
T0
dt
0
Fourier Series
x(t) = t 0 � t < T0
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
X0 =
Xk =
Mathematica:
In[2]:= 2/T*Integrate[t*Exp[-I*2*Pi*k*t/T],{t,0,T}]
T0 2
2 T0
T0
� te
� j 2 kt
T0
Xk =
2 T0
dt
0
T0 ( j2k + 1) �2 jk T e � 20 2 2 2 2 k 2 k T0 ( j2k + 1) T0 Xk = � 2 2 2 2k 2 2 k Xk =
T 2k T T X k = j 0 2 2 + 20 2 � 20 2 2 k 2 k 2 k T T j Xk = j 0 = 0 e 2 k k
e�2 jk = (e� j 2 ) = 1k =1
k
(2 I) k Pi - (2 I) k Pi) T) -((-1 + E Out[2]= ----------------------------------- (2 I) k Pi 2 2
k Pi 2E In[3]:= Simplify[%,Element[k,Integers]] IT Out[3]= ---k Pi
Xk = j
T0 T0 j 2 = e k k
e�2 jk = 1 e� jk = �1k
T0
� te 0
� j 2 kt
T0
dt
Fourier Series
Fourier Series
x(t) = t 0 � t < T0
x(t) = t 0 � t < T0
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
X0 =
T0 2
Xk =
T0 j 2 e k
x(t) =
0.04
f 0 :fundamental frequency T0 = 1/ f 0
T0 � T0 + � cos(2�kf 0 t + 2 k=1 �k
x(t) =
T0 T0 + cos(2f 0 t + 2
2
� 2
) T
) + 20 cos(2 2 f 0 t + 2 ) + K
0.04
0.02 t
T0 j 2 e k
0.03
T0 � T0 + � cos(2�kf 0 t + 2 k=1 �k
T0 T0 + cos(2f 0 t + 2
z t
1.1
Defined between 0
0.02
0
0 0.04 0.04
0.02
2 Xk = T0
0
1
X0 =
1 T0
T0
2
1
0.01
0.02 t
0.03
T0
� 1dt + T � �1dt 0
0
T0
0 t
0.02
2
0.04
0.04
T0
2
� 1e
� j 2 kt
T
T0
dt +
0
� j 2 kt 2 0 �1e T0 dt � T0 T0 2
In[2]:=2/T*Integrate[Exp[-I*2*Pi*k*t/T],{t,0,T/2}]+ 2/T*Integrate[- Exp[-I*2*Pi*k*t/T],{t,T/2,T}] I k Pi -I k Pi -I (1 - E ) I (-1 + E ) Out[2]= ----------------- + --------------- k Pi (2 I) k Pi k Pi E In[3]:= Simplify[%,Element[k,Integers]]
In[1]:=1/T*Integrate[1,{t,0,T/2}]+ 1/T*Integrate[-1,{t,T/2,T}] Out[1]:=0 X0 = 0
0.04
0.04
Fourier Series:Square Wave
1
0 0
T
) + 20 cos(2 2 f 0 t + 2 ) + K
t
0.04 0.04
1.1
)
0.04
Fourier Series:Square Wave
� 1 0 � t < T0 2 x(t) = � ��1 T0 2 � t < T0
2
� 2
y t
7 terms 0.02 t
Xk =
0.04
0 0.01
T0
2
x(t) =
f 0 = 25Hz T0 = 1/ f 0 = 0.04
0 0
X0 =
x(t) =
y t
0
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 �
k=1
k2 -I (-1 + (-1) ) Out[7]= ---------------k Pi
Xk =
(
� j �1+ (�1)
�� 4 �j X k = � k �� 0
k
k
)
� j (�1�1) j (�2) =� k k j4
=� k
2
2
k odd k even
Xk =
Xk =
� j (�1+ 1) k
2
2
� 1 0 � t < T0 2 x(t) = � ��1 T0 2 � t < T0
X0 = 0 �� 4 �j X k = � k �� 0
�� 4 � j 2 e X k = � k �� 0
k odd k even
k odd k even
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 1.2 1
x(t) = cos(2f 0 t � 4
2
)+
4 3
cos(2 3 f 0 t �
2
) +K
0.5
y t 0 z t
0.5
1 1.2 0 0
0.005
0.01
0.015
0.02 t
0.025
0.03
0.035
0.04 0.04
MAS.160 / MAS.510 / MAS.511 Signals, Systems and Information for Media Technology Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Period of a Discrete Sinusoid
y [n ] = sin(2 501 n)
y [ n ] = sin(2 503 n)
T=50 samples
y [ n ] = y[n + 50]
y [ n ] = y[n + T]
T=?? samples [integer]
50/3 � integer
sin(0) = sin(2 )
irrational frequency
1 0.8 0.6
y ( t ) = sin(2 2 t) T = 12 sec
y=sin(2*pi*sqrt(2)/25*n)
0.4 0.2 0
continuous function periodic
-0.2 TextEnd -0.4
-0.6
-0.8
-1
0
y [ n ] = sin(2 503 n) y [ n ] = y[n + T] sin(0) = sin(2k) k=1,2…
T=?? samples
2 503 n = 2k n 50 samples = k 3 cycle
periodic T=n=50 samples, k=3 cycles
0.5
1
1.5
2 time (sec)
2.5
3
3.5
4
Ts=1/25 sec Ratio of integers
rational number
y [ n ] = sin(2
2 25
n)
y [ n ] = y[n + T]
sin(0) = sin(2k)
T=?? samples k=1,2…
Equiv. discrete sinusoid not periodic
2 252 n = 2k n 25 2 = k 2 irrational number
Period of Sum of Sinusoids
1
0.5
0
-0.5
-1
0
1
2
3
4
5
6
5
6
T1=0.2 seconds, T2=.75 seconds 2
1
0
-1
-2
0
1
2
3 time (sec)
y ( t ) = y (t + T )
Tsum=3 seconds
Least common multiple seconds to complete cycles T1=1/5 seconds
15 cycles
Complex Conversions
seconds to complete cycles
cartesian s=a+jb
polar
s= a +b e 2
T2=3/4 seconds 3/4s, 6/4s, …
1/5s, 2/5s, 3/5s … 4/20s, 8/20s, 12/20s, 16/20s, 20/20s, 24/20s, 28/20s, 32/20s, 36/20s, 40/20s, 44/20s, 48/20s, 52/20s, 56/20s, 60/20s
15/20s. 30/20s, 45/20s, 60/20s 4 cycles 1/5*k=3/4*l k/l=15/4
Tsum=15*T1=15/5=3 seconds
4
rational number
Tsum=4*T2=3/4*4=3 seconds
Tsum=3 seconds
2
j� a tan ( b a )
polar s=rej�
cartesian
s = r cos� + jr sin �
Complex Arithmetic Addition
cartesian
(a1 + jb1 ) + (a2 + jb2 ) = ( a1 + a2 ) + j (b1 + b2 )
Subtraction
cartesian
(a1 + jb1 ) � (a2 + jb2 ) = ( a1 � a2 ) + j (b1 � b2 )
Multiplication
polar
r1e j� 1 r2e j� 2 = r1r2e j (� 1 +� 2 )
Division
polar
r1e j� 1 r1 j (� 1 �� 2 ) = e r2e j� 2 r2
Powers
polar
(re )
Roots
polar
j� n
= r n e jn�
z n = s = re j� z = s1/ n = r1/ n e j (� / n + 2 k / n )
k = 1,2Kn �1
Representations of Sinusoids
Complex Conversions cartesian
polar
3 + j4 = 32 + 4 2 e
j� a tan (
polar 4
3
)
= 5e j� 0.927
2e
j 3
Re{ Ae j 2 ft +� }
Acos(2kf 0 t + � k )
cartesian
= Re{ Ae j� e j 2 ft }
= 2cos 3 + j2sin 3 = 1+ j 3
Subtraction
cartesian cartesian
(1+ j2) + ( 3 + j4 ) = (4 + j6)
j 3
6e
polar
5e
Division
polar
10e
Powers
polar
(3e )
Roots
polar
j� 2
j 4
j 4
÷ 5e 3
k=1
j 3 j 3 = 33 e ( 4 ) = 27e ( 4 )
z 3 = 64 = 64e j 0
4
e j 2 ft +e � j 2 ft 2 e j 2 ft + e � j 2 ft 2
4e j ( 2 / 3) 4e j ( 4 / 3)
n
n
k=1
k=1
cos(2�ft + � k ) = � Re{ Ak e 2 �ft +� k } = � Re{ Ak e� k e 2 �ft } �n = � Re{ Ak e� k } e 2 �ft � k=1
Ex.
3cos(2 40t + 2 ) �1cos(2 40t � 6 ) + 2 cos(2 40t + 3 ) � � j �j Re� 3e 2 e j2 40t �1e 6 e j2 40t + 2e 3 e j2 40t � � � �� j � �
�j Re� 3e 2 �1e 6 + 2e 3 e j2 40t
� � �
Re{5.234 e j1.545e2 40t }
5.234 cos(2 40t + 1.545)
Composite signals (waveform synthesis)
multiply cosines of different frequency
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
A1 cos(�1t ) � A2 cos(� 2 t + � ) � e j�1 t + e� j�1 t � e j (� 2 t +� ) + e� j (� 2 t +� ) A1 A2 2 2 �
�
decompose a periodic signal x(t) into a sum of a series of sinusoids - the Fourier series.
(
A1 A2 j�1 t j (� 2 t +� ) e e + e j�1 t e� j (� 2 t +� ) + e� j�1 t e j (� 2 t +� ) + e� j�1 t e� j (� 2 t +� ) 4
(
A1 A2 j (�1 t +� 2 t +� ) � j (� 2 t��1 t +� ) e +e + e j (� 2 t��1 t +� ) + e� j (�1 t +� 2 t +� ) 4 A1 A2 cos((�1 + � 2 ) t + � ) + cos((� 2 � �1 ) t + � ) 2
k
j� � j� = ( 105 )e ( 2 4 ) = 2e 4
z = 641/ 3 e j ( 0 / 3+2 k / 3) = 4e j ( 2 k / 3)
(
n
�A
j + j 7 = 5 6e ( 3 4 ) = 30e 12
j� 4
(
(
Sum multiple cosines same frequency
(1+ j2) � ( 3 + j4 ) = (�2 � j2)
Multiplication
=X�
= Re{ Xe j 2 ft }
Complex Arithmetic Addition
Ae j� �
)
)
)
Note: The sum of periodic functions is periodic. ex.
�� �8 X k = � 2k 2 �� 0
k odd k even
f 0 = 25Hz
)
)
Composite signals (waveform synthesis)
Composite signals (waveform synthesis)
� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 �� �8 �� 8 j k odd e k odd X k = � 2k 2 X k = � 2k 2 �� 0 �� 0 k even k even �
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
�
�� 8 e j X k = � 2k 2 �� 0
x(t) = 0.8105cos(2 25t + )
x(t) = 0.8105cos(2 25t + ) + 0.0901cos(2 75t + )
1
1
0.8
0.8
0.6
0.6
0.4
0.4
1
=
0
-1
0.02
0.04
0.06
0.08
0.1
0.12
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
0.4
=
0.2
0
-0.6
-0.6
0.6
0.4
-0.4
-0.4
0.8
0.6
-0.2
-0.2
1
0.8
0.2
0.2
0
k even
k=3
k=1
-1
k odd
-0.8
-0.8 0
0.02
0.04
0.06
0.08
0.1
0.12
-1
-1
0
0.02
0.04
0.06
0.08
0.1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.12
1
Composite signals (waveform synthesis)
0.8 0.6
�� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 �� 8 e j k odd X k = � 2k 2 �� 0 k even �
spectrum
0.4 0.2 0
0.45
-0.2
0.4053e� j
0.4
0.4053e j
-0.4
-0.6 -0.8
0.35
-1
k=5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.3
x(t) = 0.8105cos(2 25t + ) + 0.0901cos(2 75t + ) + 0.0324 cos(2 125t + ) 1
1
0.8
0.8
0.6
0.6
0.4
0.4
=
0.2
X 0.25 0.2
0.15
0.2
0.1
0.0450e� j
0
0
-0.2
-0.2
0.0450e j
0.0162e� j
0.05
0.0162e j
-0.4
-0.4
0 -300
-0.6
-0.6 -0.8
-200
-100
-125
-75
-25
0
25
75
100
125
200
f 300
-0.8 -1
-1
0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
0.06
0.08
0.1
0.12
x(t) = 0.8105cos(2 25t + ) + 0.0901cos(2 75t + ) + 0.0324 cos(2 125t + ) + ...
Fourier Series
Fourier Series For a given signal, how do we find X k = Ak e for each k ?
x(t) = t 0 � t < T0
j� k
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
Fourier Analysis
1 X 0 =
T0
�� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 �
where
1 X0 = T0
Xk =
2 T0
=
T0
� x(t)dt
f 0:fundamental frequency T0 = 1/ f 0
0
T0
� x(t)e
� j 2 kt
T0
dt
0
1 T0
T0
� x(t)dt
0.04
t
0
T0
� tdt = 0
2 T0
1 t T0 2
0
=
1 T0 2 T0 = T0 2 2
Mathematica: athena%add math athena%math In[1]:=1/T*Integrate[t,{t,0,T}] Out[1]:=T/2
Fourier Series
0.04
0.02
0
0 0 0
0.01
Xk =
2 T0
0.02 t
0.03
T0
� x(t)e
0.04 0.04
� j 2 kt
T0
dt
0
Fourier Series
x(t) = t 0 � t < T0
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
X0 =
Xk =
Mathematica:
In[2]:= 2/T*Integrate[t*Exp[-I*2*Pi*k*t/T],{t,0,T}]
T0 2
2 T0
T0
� te
� j 2 kt
T0
Xk =
2 T0
dt
0
T0 ( j2k + 1) �2 jk T e � 20 2 2 2 2 k 2 k T0 ( j2k + 1) T0 Xk = � 2 2 2 2k 2 2 k Xk =
T 2k T T X k = j 0 2 2 + 20 2 � 20 2 2 k 2 k 2 k T T j Xk = j 0 = 0 e 2 k k
e�2 jk = (e� j 2 ) = 1k =1
k
(2 I) k Pi - (2 I) k Pi) T) -((-1 + E Out[2]= ----------------------------------- (2 I) k Pi 2 2
k Pi 2E In[3]:= Simplify[%,Element[k,Integers]] IT Out[3]= ---k Pi
Xk = j
T0 T0 j 2 = e k k
e�2 jk = 1 e� jk = �1k
T0
� te 0
� j 2 kt
T0
dt
Fourier Series
Fourier Series
x(t) = t 0 � t < T0
x(t) = t 0 � t < T0
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1
X0 =
T0 2
Xk =
T0 j 2 e k
x(t) =
0.04
f 0 :fundamental frequency T0 = 1/ f 0
T0 � T0 + � cos(2�kf 0 t + 2 k=1 �k
x(t) =
T0 T0 + cos(2f 0 t + 2
2
� 2
) T
) + 20 cos(2 2 f 0 t + 2 ) + K
0.04
0.02 t
T0 j 2 e k
0.03
T0 � T0 + � cos(2�kf 0 t + 2 k=1 �k
T0 T0 + cos(2f 0 t + 2
z t
1.1
Defined between 0
0.02
0
0 0.04 0.04
0.02
2 Xk = T0
0
1
X0 =
1 T0
T0
2
1
0.01
0.02 t
0.03
T0
� 1dt + T � �1dt 0
0
T0
0 t
0.02
2
0.04
0.04
T0
2
� 1e
� j 2 kt
T
T0
dt +
0
� j 2 kt 2 0 �1e T0 dt � T0 T0 2
In[2]:=2/T*Integrate[Exp[-I*2*Pi*k*t/T],{t,0,T/2}]+ 2/T*Integrate[- Exp[-I*2*Pi*k*t/T],{t,T/2,T}] I k Pi -I k Pi -I (1 - E ) I (-1 + E ) Out[2]= ----------------- + --------------- k Pi (2 I) k Pi k Pi E In[3]:= Simplify[%,Element[k,Integers]]
In[1]:=1/T*Integrate[1,{t,0,T/2}]+ 1/T*Integrate[-1,{t,T/2,T}] Out[1]:=0 X0 = 0
0.04
0.04
Fourier Series:Square Wave
1
0 0
T
) + 20 cos(2 2 f 0 t + 2 ) + K
t
0.04 0.04
1.1
)
0.04
Fourier Series:Square Wave
� 1 0 � t < T0 2 x(t) = � ��1 T0 2 � t < T0
2
� 2
y t
7 terms 0.02 t
Xk =
0.04
0 0.01
T0
2
x(t) =
f 0 = 25Hz T0 = 1/ f 0 = 0.04
0 0
X0 =
x(t) =
y t
0
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 �
k=1
k2 -I (-1 + (-1) ) Out[7]= ---------------k Pi
Xk =
(
� j �1+ (�1)
�� 4 �j X k = � k �� 0
k
k
)
� j (�1�1) j (�2) =� k k j4
=� k
2
2
k odd k even
Xk =
Xk =
� j (�1+ 1) k
2
2
� 1 0 � t < T0 2 x(t) = � ��1 T0 2 � t < T0
X0 = 0 �� 4 �j X k = � k �� 0
�� 4 � j 2 e X k = � k �� 0
k odd k even
k odd k even
� �� � x(t) = A0 + � Ak cos(2�kf 0 t + � k ) = X 0 + Re�� X k e j 2 �kf 0 t � � k=1 � k=1 1.2 1
x(t) = cos(2f 0 t � 4
2
)+
4 3
cos(2 3 f 0 t �
2
) +K
0.5
y t 0 z t
0.5
1 1.2 0 0
0.005
0.01
0.015
0.02 t
0.025
0.03
0.035
0.04 0.04
