Discrete Fourier Analysis

4.6.2 Discrete Fourier Analysis

Although the data to be treated in Fourier analysis are often continuous in the time or space domain, for computational purposes these data must usually be represented in terms of a

158 Named after the American physicist J. William Gibbs, who in 1899 proved that this ringing effect will always occur when approximating a discontinuous function with a Fourier series.

488 Chapter 4. Interpolation and Approximation finite discrete sequence. For example, a function f (t) of time is recorded at evenly spaced

intervals 4t in time. Assume that the function f is known at equidistant arguments in the interval [0, 2π],

x k = 2πk/N, k = 0 : N − 1.

Such data can be analyzed by discrete Fourier analysis. Define the inner product

N −1

(f, g) =

f (x k ) ¯g(x k ), x k = 2πk/N. (4.6.16)

k =0

Then, with φ j (x) =e ij x we have

e j ij x k k = e −ikx k

e −k)hk .

k =0

k =0

From Lemma 3.2.2 it now follows that

(φ j ,φ k ) = N if (j − k)/N is an integer,

0 otherwise.

Theorem 4.6.4 (Trigonometric Interpolation). Every function, defined on the equidistant grid x k = 2πk/N, k = 0 : N − 1, can be interpolated by the trigonometric polynomial  k +θ

 2 + cos jx + b j sin jx) + θa k +1 cos(k + 1)x.

if N even, θ =

1 if N even,

0 if N odd,

= (N

− 1)/2 if N odd, and

1 N −1

(4.6.20) N k

f (x k )e −ijx k ,

=0 N

2 N −1 2 −1

f (x k ) sin jx k . (4.6.21) N k =0

f (x k ) cos jx k ,

N k =0

If the sums in (

4.6.18) are terminated when |j| < k + θ, then one obtains the trigono- metric polynomial which is the best least squares approximation, among all trigonometric

polynomials with the same number of terms, to f on the grid. Proof. The expression for c j is justified by (4.6.17). Further, by (4.6.20)–(4.6.21) it follows

that

a j =c j +c −j ,

b j = i(c j −c −j ),

c k +1 = a k

4.6. Fourier Methods 489 The two expressions for f (x) are equivalent, because

+ θc k +1 cos(k + 1)x

=c 0 +

(a j cos jx + b j sin jx) + θa k

cos(k + 1)x. The function f (x) coincides on the grid x 0 ,x 1 ,...,x N −1 with the function

e −i(N−j)x k

ij x =e k ,

c −j =c N −j .

However, the functions f and f ∗ are not identical between the grid points. If we set ω = e ix and ω k

=e ix k , then

where P (ω) is a polynomial of degree less than N. It becomes clear that trigonometric interpolation is equivalent to polynomial interpolation at the grid points ω k . The mapping

C N →C N (f 0 ,f 1 ,...,f N −1 ) 9→ (c 0 ,c 1 ,...,c N −1 ) is called the discrete Fourier transform (DFT).

The calculations required to compute the coefficients c j according to (4.6.20), Fourier analysis , are of essentially the same type as the calculations needed to compute f ∗ (x) at the grid points

x k = 2πk/N, k = 0 : N − 1,

when the expansion in (4.6.22) is known, so-called Fourier synthesis. Both calculations can be performed very efficiently using FFT algorithms; see Sec. 4.7.1.

Functions of several variables are treated analogously. Quite simply, one takes one variable at a time. In the discrete case with two variables we set

x k = 2πk/N,

y ℓ = 2πℓ/N,

and assume that f (x k ,y ℓ ) is known for k = 0 : N − 1, ℓ = 0 : N − 1. Set

1 N −1

c j (y ℓ ) =

f (x k ,y ℓ )e −ijx k ,

)e −iky = ℓ c j (y ℓ .

N ℓ =0

490 Chapter 4. Interpolation and Approximation From Theorem 4.6.4, then (with obvious changes in notations)

= (ij x e = k c j,k +iky ℓ ) .

j =0

j =0 k =0

The above expansion is of considerable importance in, e.g., crystallography. The Fourier coefficients

c ij x j (f ) =

f (x)e dx, j = 0, ±1, ±2, . . . , (4.6.23)

of a function f with period 2π are often difficult to compute. On the other hand the coefficients of the DFT

1 N −1 ˆc j (f )

2πk

, j = 0 : N − 1, (4.6.24) N k =0

f (x k )e −ijx , x k =

can be computed very efficiently by the FFT. Now, since f (x 0 ) = f (x N ) , the sum in (4.6.24) can be thought of as a trapezoidal sum

1 01 1 N 1 ˆc j (f ) =

f (x N ) N 2 2 approximating the integral (4.6.23). Therefore, one might think of using ˆc j as an approxima-

f (x 0 ) + f (x 1 )e −ijx 1 + · · · + f (x N −1 )e −ijx −1 +

tion to c j for all j = 0, ±1, ±2, . . . . But ˆc j (f ) are periodic in j with period N, whereas by Theorem 4.6.3 the true Fourier coefficients c j (f ) decay as some power j −(k+1) as j → ∞.

We now show a way to remove this deficiency. Let f k , k = 0, ±1, ±2, . . . , be an infinite N-periodic sequence with f k = f (x k ) . Let ϕ = Pf be a continuous function such that ϕ(x k ) =f k , k = 0, ±1, ±2, . . . , and approximate c j (f ) by c j (ϕ) . Then c j (ϕ) will decay to zero as j → ∞. It is a remarkable fact that if the approximation scheme P is linear and translation invariant, we have

(4.6.25) where the attenuation factors τ j depend only on the approximation scheme P and not

c j (ϕ) =τ j ˆc j (f ),

the function f . This implies that τ j can be determined from (4.6.25) by evaluating c j (ϕ) and ˆc j (f ) for some suitable sequence f k , k = 0, ±1, ±2, . . . . This allows the FFT to be

used. For a proof of the above result and a more detailed exposition of attenuation factors in Fourier analysis, we refer to Gautschi [141].

Example 4.6.2.

For a given N-periodic sequence f k , k = 0, ±1, ±2, . . . , take ϕ(x) = Pf to be defined as the piecewise linear function such that ϕ(x k ) =f k , k = 0, ±1, ±2, . . . . Clearly this approximation scheme is linear and translation invariant. Further, the function ϕ is continuous and has period 2π.

4.6. Fourier Methods 491 Consider in particular the sequence f k = 1, k = 0 mod N, and f k = 0 otherwise. We

f (x k )e −ijx =

N k =0

Further, setting h = 2π/N and using symmetry, we get

0 j 2 πh . 2 This gives the attenuation factors

Note that the coefficients c j (Pf ) decay as j −2 , reflecting the fact that the first derivative of Pf is discontinuous. If instead we use a cubic spline approximation, the first and second derivatives are continuous and the attenuation factors decay as j −4 .