fourier series - school of mathematics | the university of ... fourier he his best known for fourier...

Fourier series

Prof. Bill LionheartSchool of MathematicsThe University of Manchester

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Joseph Fourier

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Joseph Fourier

• Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830)was a French mathematician and physicist

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Joseph Fourier


• He was also a civil servant and Napoleon sent him to be governorof Egypt...

2

Joseph Fourier


• He was also a civil servant and Napoleon sent him to be governorof Egypt...

...where in his spare time he founded Egyptology!

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Joseph Fourier

• He his best known for Fourier series A way of writing a functionas a sum of frequency components, that is sum of sine waves ofdifferent frequencies

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Joseph Fourier


• The mathematics of Fourier series under-pin much of digitalaudio including

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Joseph Fourier



• Mobile phones

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Joseph Fourier



• Mobile phones

• MP3 players

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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 cosx, cos 2x,cos 3xetc and sinx, sin 2x,sin 3x.

For example consider

sinx +1

3sin 3x +

1

5sin 5x + · · ·

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Sums of frequency componentssinx

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Sums of frequency componentssin 3x

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Sums of frequency componentssinx + (1/3) sin 3x

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Sums of frequency componentssinx + (1/3) sin 3x + (1/5) sin 5x

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Sums of frequency componentssinx + (1/3) sin 3x + · · · + (1/21) sin 21x

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Square waveWe see that

sinx + (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

]

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Sawtooth wave

sinx

]

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Sawtooth wave

sinx− (1/9) sin 3x

]

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Sawtooth wave

sinx− (1/9) sin 3x+ (1/25) sin 5x− (1/49) sin 7x+ (1/81) sin 9x

]

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Sawtooth wave

sinx− (1/9) sin 3x + · · · − (1/192) sin 19x

]

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Fourier Series Derivation

f (x) =a0

2+∞∑k=1

ak cos kx + bk sin kx

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f (x) =a0

2+∞∑k=1


2π∫0f (x) cosmxdx =

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f (x) =a0

2+∞∑k=1



=2π∫0

a0

2cosmxdx+

∞∑k=1

2π∫0ak cos kx cosmxdx+

2π∫0bk sin kx cosmxdx

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f (x) =a0

2+∞∑k=1



=2π∫0

a0

2cosmxdx+

∞∑k=1



= 0 +∞∑k=1

ak2π∫0

cos kx cosmxdx + bk2π∫0

sin kx cosmxdx

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Remember...What is cosA cosB?

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cos(A +B) = cosA cosB − sinA sinB

so

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so

cos(A +B) + cos(A−B) = 2 cosA cosB

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so


and we see that

22π∫0

cosmx cosnx dx =2π∫0

cos(m + n)x dx +2π∫0

cos(m− n)x dx

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so


and we see that

22π∫0



cos(m− n)x dx

= 0, if m 6= n

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so


and we see that

22π∫0



cos(m− n)x dx

= 0, if m 6= n

but if m = n it is2π∫0

1 dx = 2π

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f (x) =a0

2+∞∑k=1



=2π∫0

a0

2cosmxdx+

∞∑k=1



= 0 +∞∑k=1

ak2π∫0


sin kx cosmxdx

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f (x) =a0

2+∞∑k=1



=2π∫0

a0

2cosmxdx+

∞∑k=1



= 0 +∞∑k=1

ak2π∫0


sin kx cosmxdx

= πam

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fourier series - school of mathematics | the university of ... fourier he his best known for fourier...

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