Fourier series
Prof. Bill Lionheart
School of Mathematics
The University of Manchester
Joseph Fourier
Joseph Fourier
• Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830)
was a French mathematician and physicist
Joseph Fourier
• Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830)
was a French mathematician and physicist
• He was also a civil servant and Napoleon sent him to be governor
of Egypt...
Joseph Fourier
• Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830)
was a French mathematician and physicist
• He was also a civil servant and Napoleon sent him to be governor
of Egypt...
...where in his spare time he founded Egyptology!
Joseph Fourier
• He his best known for Fourier series A way of writing a function
as a sum of frequency components, that is sum of sine waves of
different frequencies
Joseph Fourier
• He his best known for Fourier series A way of writing a function
as a sum of frequency components, that is sum of sine waves of
different frequencies
• The mathematics of Fourier series under-pin much of digital
audio including
Joseph Fourier
• He his best known for Fourier series A way of writing a function
as a sum of frequency components, that is sum of sine waves of
different frequencies
• The mathematics of Fourier series under-pin much of digital
audio including
• Mobile phones
Joseph Fourier
• He his best known for Fourier series A way of writing a function
as a sum of frequency components, that is sum of sine waves of
different frequencies
• The mathematics of Fourier series under-pin much of digital
audio including
• Mobile phones
• MP3 players
Can we write a function as a series of sine and cosines?
Given a function f (x) defined for 0 ≤ x ≤ 2π we want to write
it as a sum of “frequency components”, that is cos x, cos 2x,cos 3x
etc and sin x, sin 2x,sin 3x.
For example consider
1
1
sin x + sin 3x + sin 5x + · · ·
3
5
Sums of frequency components
sin x
Sums of frequency components
sin 3x
Sums of frequency components
sin x + (1/3) sin 3x
Sums of frequency components
sin x + (1/3) sin 3x + (1/5) sin 5x
Sums of frequency components
sin x + (1/3) sin 3x + · · · + (1/21) sin 21x
Square wave
We see that
sin x + (1/3) sin 3x + (1/5) sin 5x + · · · x
tends to the “square wave” function.




f (x) = 
1 0≤x<π
−1 π ≤ x ≤ 2π
]
Sawtooth wave
sin x
]
Sawtooth wave
sin x − (1/9) sin 3x
]
Sawtooth wave
sin x − (1/9) sin 3x + (1/25) sin 5x − (1/49) sin 7x + (1/81) sin 9x
]
Sawtooth wave
sin x − (1/9) sin 3x + · · · − (1/192) sin 19x
]
Fourier Series Derivation
∞
a0
X
f (x) = +
ak cos kx + bk sin kx
2 k=1
Fourier Series Derivation
∞
a0
X
f (x) = +
ak cos kx + bk sin kx
2 k=1
2π
Z
0
f (x) cos mx dx =
Fourier Series Derivation
∞
a0
X
f (x) = +
ak cos kx + bk sin kx
2 k=1
2π
Z
f (x) cos mx dx =
0
=
2π
Z a
0
0
2
cos mx dx+
∞
X
2π
Z
k=1 0
2π
Z
ak cos kx cos mx dx+ bk sin kx cos mx dx
0
Fourier Series Derivation
∞
a0
X
f (x) = +
ak cos kx + bk sin kx
2 k=1
2π
Z
f (x) cos mx dx =
0
=
2π
Z a
0
0
2
cos mx dx+
=0+
∞
X
2π
Z
k=1 0
∞
X
k=1
ak
2π
Z
0
2π
Z
ak cos kx cos mx dx+ bk sin kx cos mx dx
cos kx cos mx dx + bk
0
2π
Z
0
sin kx cos mx dx
Remember...
What is cos A cos B?
Remember...
What is cos A cos B?
cos(A + B) = cos A cos B − sin A sin B
so
Remember...
What is cos A cos B?
cos(A + B) = cos A cos B − sin A sin B
so
cos(A + B) + cos(A − B) = 2 cos A cos B
Remember...
What is cos A cos B?
cos(A + B) = cos A cos B − sin A sin B
so
cos(A + B) + cos(A − B) = 2 cos A cos B
and we see that
2
2π
Z
0
cos mx cos nx dx =
2π
Z
0
cos(m + n)x dx +
2π
Z
0
cos(m − n)x dx
Remember...
What is cos A cos B?
cos(A + B) = cos A cos B − sin A sin B
so
cos(A + B) + cos(A − B) = 2 cos A cos B
and we see that
2
2π
Z
0
cos mx cos nx dx =
2π
Z
cos(m + n)x dx +
0
= 0, if m 6= n
2π
Z
0
cos(m − n)x dx
Remember...
What is cos A cos B?
cos(A + B) = cos A cos B − sin A sin B
so
cos(A + B) + cos(A − B) = 2 cos A cos B
and we see that
2
2π
Z
cos mx cos nx dx =
0
2π
Z
cos(m + n)x dx +
0
0
= 0, if m 6= n
but if m = n it is
2π
Z
2π
Z
0
1 dx = 2π
cos(m − n)x dx
Fourier Series Derivation
∞
a0
X
f (x) = +
ak cos kx + bk sin kx
2 k=1
2π
Z
f (x) cos mx dx =
0
=
2π
Z a
0
0
2
cos mx dx+
=0+
∞
X
2π
Z
k=1 0
∞
X
k=1
ak
2π
Z
0
2π
Z
ak cos kx cos mx dx+ bk sin kx cos mx dx
cos kx cos mx dx + bk
0
2π
Z
0
sin kx cos mx dx
Fourier Series Derivation
∞
a0
X
f (x) = +
ak cos kx + bk sin kx
2 k=1
2π
Z
f (x) cos mx dx =
0
=
2π
Z a
0
0
2
cos mx dx+
=0+
∞
X
2π
Z
k=1 0
∞
X
k=1
ak
2π
Z
2π
Z
ak cos kx cos mx dx+ bk sin kx cos mx dx
0
cos kx cos mx dx + bk
0
2π
Z
0
=
πam
sin kx cos mx dx