FINAL REPORT OF FOURIER ANALYSIS- THE ORTHOGONALITY OF THE FOURIER BASIS -

9313098, Chi-Pin Cheng, Department of Communication Engineering, NCTU.

Abstract In this report, I will show that the elements of the Fourier basis aremutually orthogonal. Then, change the basis into another form, by the propertiesof even and odd functions.

I. The Fourier BasisBy definition, a periodic function f(x) with period 2π can be written as

f(x) =a0

2+∞∑

n=1

an cos nx + bn sinnx

where am = 1π

∫ π

−πf(x) cos mxdx, and bn = 1

π

∫ π

−πf(x) sinnxdx,

for m = 0, 1, 2, ... and n = 1, 2, ...

Then, we call that {1, cos mx, sinnx|∀m,n ∈ N} is the Fourier basis.

II. The Orthogonality of the Fourier Basis(a) find the norms of each element of the Fourier Basis.(sol)

‖ 1 ‖2=∫ π

−π

1 · 1dx = 2π

‖ cos mx ‖2=∫ π

−π

cos mx · cos mxdx = π,∀m ∈ N

‖ sinnx ‖2=∫ π

−π

sinnx · sinnxdx = π,∀n ∈ N

(b) claim: the elements of Fourier Basis are mutually orthogonal.(proof)< 1, cos mx >=

∫ π

−πcos mxdx = 0,∀m ∈ N

< 1, sinnx >=∫ π

−πsinnxdx = 0,∀n ∈ N

< cos mx, sinnx >=∫ π

−π12 [sin(m + n)x− sin(m− n)x]dx = 0,∀m,n ∈ N

< cos mx, cos nx >=∫ π

−π12 [cos(m + n)x + cos(m− n)x]dx = 0,m, n ∈ N,m 6= n

< sinmx, sinnx >=∫ π

−π−12 [cos(m + n)x− cos(m− n)x]dx = 0,m, n ∈ N,m 6= n

⇒ the elements of Fourier Basis are mutually orthogonal. �

1

2 FINAL REPORT OF FOURIER ANALYSIS - THE ORTHOGONALITY OF THE FOURIER BASIS -

III. Change of the Fourier BasisFirst, review the properties of even and odd functions. For any function f(x), wecan write it into f(x) = fe(x) + fo(x), where fe(x) is even, fo(x) is odd, by letting{

fe(x) = f(x)+f(−x)2

fo(x) = f(x)−f(−x)2

claim: the Fourier Basis can be changed into {eikx|k ∈ Z}.(proof)Since eikx = cos kx + i sin kx, for cos kx is even, and i sin kx is odd,

⇒

{cos kx = eikx+e−ikx

2

i sin kx = eikx−e−ikx

2

Then, we get

f(x) =a0

2+∞∑

k=1

ak cos kx + bk sin kx

=a0

2ei0x +

∞∑k=1

akeikx + e−ikx

2− ibk

eikx − e−ikx

2

let c0 = a02 , ck = an−ibn

2 , c−k = an+ibn

2 ,∀k ∈ N, and rewrite the equation:

f(x) =∞∑

k=−∞

ckeikx

Finally we successfully change the basis into {eikx|k ∈ Z}. �

IV. Coefficients of the Fourier BasisSince that we know the norms and the orthogonality of the Fourier Basis, we canget the coefficients by the properties learned from linear algebra.(a) for a0:

a0 = 2 ·∫ π

−πf(x) · 1dx

‖ 1 ‖2=

1π

∫ π

−π

f(x)dx

(b) for am,m 6= 0:

am =

∫ π

−πf(x) · cos mxdx

‖ cos mx ‖2=

1π

∫ π

−π

f(x) cos mxdx

(c) for bn, n 6= 0:

bn =

∫ π

−πf(x) · sinnxdx

‖ sinnx ‖2=

1π

∫ π

−π

f(x) sinnxdx

Note that we also verify the definition of Fourier Basis in I.

FINAL REPORT OF FOURIER ANALYSIS - THE ORTHOGONALITY OF THE FOURIER BASIS - 3

V. ConclusionsWe verified the orthogonality of Fourier Basis, and also changed it into the complexform. In fact, The Basis can be applied on all periodic functions, say, with period2L, by letting the Fourier Basis become {1, cos π

Lmx, sin πLnx|∀m,n ∈ N}, and

f(x) =a0

2+∞∑

n=1

an cosπ

Lnx + bn sin

π

Lnx

where am = 1L

∫ L

−Lf(x) cos π

Lmxdx, and bn = 1L

∫ L

−Lf(x) sin π

Lnxdx,for m = 0, 1, 2, ... and n = 1, 2, ...

VI. References1. ”An Introduction and Experiments of Communication Engineering,” Chen-YiChang, 2002.2. ”A First Course in Fourier Analysis,” Kammler, David W., Prentice Hall, 1999.