Previously, we considered periodic functions f(x) of period 1 and derived their Fourier series expansion f(x) = â. â k=ââ f(k) ei 27 k x. (1) where the function.
Lecture Notes on Fourier Transforms (IV) Christian Bauckhage B-IT, University of Bonn In this brief note, we have a closer look at the Fourier coefficients that appear in the Fourier series expansions of periodic functions. We will find that they are inner products between complex valued functions. This new insight will later on allow us to understand the Fourier transform as a linear basis transform in a Hilbert space.
Interpreting Fourier Coefficients and Fourier Series Previously, we considered periodic functions f ( x ) of period 1 and derived their Fourier series expansion ∞
∑
f (x) =
fˆ(k ) ei 2π k x
(1)
k =−∞
where the function fˆ(k) =
Z 1 0
f ( x ) e−i 2π k x dx
(2)
is called the kth Fourier coefficient. In this note, we will have a deeper look at the significance of equations (1) and (2). To begin with, we recall that complex valued functions over an interval, say [ a, b], form a vector space and that we can define the inner product of two such functions as
Z f ( x ), g ( x ) =
b a
f ( x ) g¯ ( x ) dx
(3) Recall that, if z ∈ C is given by
where g¯ ( x ) denotes the complex conjugate of g( x ).
z = a + i b,
Given these prerequisites, we next recall that we assumed the function f ( x ) that occurs in (2) to be real valued. However, real numbers are but a subset of complex numbers so that we may just as well think of f ( x ) as a complex valued function. Moreover, since the complex exponentials e−i 2π k x which feature prominently in (2) are the complex conjugates of ei 2π k x , we find fˆ(k) =
=
Z 1 0
Z 1 0
=
f ( x ) e−i 2π k x dx
then its complex conjugate amounts to z¯ = a − i b.
Recall that, since eix = cos x + i sin x,
(4)
its complex conjugate eix is given by cos x − i sin x = e−ix .
f ( x ) ei 2π k x dx
f ( x ), ei 2π k x
(5) (6)
This, however, is to say that the Fourier coefficient fˆ(k) in (2) is an inner product between two functions. Next, we recall that the complex exponentials which appear in equations (1) and (2) are periodic functions of x of period 1. Hence, given
lecture notes on fourier transforms (iv)
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any two such complex exponentials over the interval [0, 1], we may just as well compute their inner product and find that
hei 2π k x , ei 2π m x i = =
Z 1 0
Z 1 0
=
Z 1 0
ei 2π k x ei 2π m x dx ei 2π k x e−i 2π m x dx ei 2π x (k−m) dx.
(7)
At this point, we note that, a further evaluation of the expression on the right hand side of (7) requires us to distinguish two basic cases. First of all, for the case where k = m, it evaluates to Z 1 0
ei 2πx(k−m) dx =
Z 1 0
e0 dx =
Z 1 0
1 dx = 1.
(8)
Second of all, for k 6= m, we have 1 Z 1 ei 2π x (k−m) ei 2π (k−m) − 1 i 2π x (k −m) e dx = = = 0. i 2π (k − m) 0 i 2π (k − m) 0
(9)
Therefore, if we combine equations (7), (8), and (9) into a single expression, we obtain
i 2π k x i 2π m x 1 if k = m e ,e = (10) 0 if k 6= m. n o But this is to say that the family of complex exponentials ei 2π k x
k ∈Z
forms an orthonormal basis for (square integrable) functions over the interval [0, 1]. Consequently, equation (1), i.e. the Fourier series expansion ∞
f (x) =
∑
k =−∞
fˆ( x ) ei 2π k x =
∞
∑
f ( x ), ei 2π k x ei 2π k x
(11)
k =−∞
can now be understood as a way of expressing the f ( x ) in terms of the basis provided by the complex exponentials. Looking at (11), we note that it constitutes a Hilber space generalization of the familiar basis expansion of, say, Euclidean vectors. For instance, given a vector x = [ x1 , x2 ] T ∈ R2 and the standard basis e1 , e2 ⊂ R2 , we know that we can express x as 2
x = x1 e1 + x2 e2 = x, e1 e1 + x, e2 e2 =
∑
x, ei ei
(12)
i =1
and observe that the only “conceptual difference” between (11) and (12) is that the former involves infinitely many terms whereas the sum in the latter has finitely many terms.
Outlook The observation that Fourier series are basis expansions is a deep insight and, in a later note, we will generalize pur finding to the
Looking at the denominator in (9), we understand the need for having to distinguish two cases: for k − m = 0, (9) is undefined!
lecture notes on fourier transforms (iv)
more general case of the Fourier transform. This will allow us to see that the Fourier transform is nothing else but a basis transform in infinitely dimensional function spaces.
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