Trigonometric Interpolation

Interpolation

2.3 Trigonometric Interpolation

EXAMPLE [taken from Bulirsch and Rutishauser (1968)]. For the function f(x) = cot x the values cot 1 0, cot 2°, ... have been tabulated. The problem is to determine an approximate value for cot 2°30'.

Polynomial interpolation of order 4, using the formulas (2.1.2.4), yields the tableau

Xi ^{/; = cot} (Xi) 1° 57.28996163

14.30939911

2° 28.63625328 21.47137102

23.85869499 22.36661762

3° 19.08113669 23.26186421 22.63519158

21.47137190 23.08281486

4° 14.30066626 22.18756808

18.60658719 5° 11.43005230

Rational interpolation with (j.l, v) = (2, 2) using the formulas (2.2.3.8) in contrast gives

1° 57.28996163

22.90760673

2° 28.63625328 22.90341624

22.90201805 22.90369573

3° 19.08113669 22.90411487 22.9037655J.

22.91041916 22.90384141

4° 14.30066626 22.90201975

22.94418151 5° 11.43005230

The exact value is cot 2°30' = 22.903 765 5484 ... ; incorrect digits are underlined.

A similar situation is encountered in extrapolation methods (see Sections 3.4,3.5, 7.2.3, 7.2.14). Here a function T{h) of the step length h is interpolated at small positive values of h.

(2.3.1.1b) ^A M-l A

\}I(x)

= T ⁺

^{h~l (Ah cos hx}

⁺

^{Bh sin hx)}

⁺

2M cos Mx of, respectively, N = 2M

+

1 or N = 2M support points (Xk'

he),

k = 0, ... , N - 1. Interpolation by such expressions is suitable for data which are periodic of known period. Indeed, the expressions \}I(x) in (2.3.1.1) represent periodic functions of x with the period 2n.l

Considerable conceptual and algebraic simplifications are achieved by using complex numbers and invoking De Moivre's formula

e^kix

=

cos kx

+

ⁱ sin kx.

Here and in what follows, i denotes the imaginary unit. If c

=

a

+

ib, a, b real, then

c

⁼ a - ib is its co~lex con' u ate, a is the real part of c, bits imaginary part, and

I

^c

I ⁼

^{.j cc}

⁼

^a2

+

b² its absolute value.

Particularly important are uniform partitions of the interval [0, 2n]

Xk

=

2nkjN, k

=

0, ... , N-1

to which we now restrict our attention. For such partitions, the trigonomet- ric interpolation problem can be transformed into the problem of finding a phase polynomial

(2.3.1.2)

with complex coefficients

P

j such that

k

=

0, ... , N - 1.

Indeed

and therefore (2.3.1.3)

Making these substitutions in expressions (2.3.1.1) for \}I(x) and then collect- ing the powers of e^ixt produces a phase polynomial p(x), (2.3.1.2), with

1 If sin u and cos u have to be both evaluated for the same argument u, then it may be advantageous to evaluate t = tan(u/2) and to express sin u and cos u in terms of t:

. 2t

sm u = 1 + ^{t2 ' cosu=1+t1 - t2 2 '}

This procedure is numerically stable for 0 ,,;; u ,,;; 1[/4, and the problem can always be trans- formed so that the argument falls into that range.

coefficients

P

j ,j

=

0, ... , N - 1 which are related to the coefficients Ah , Bh of 'P(x) as follows:

(2.3.1.4)

(a) If N is odd, then N = 2M

+

1 and

PO=2'

Ao Ao

=

2Po,

(b) If N is even, then N = 2M and

Po =

Ao

2'

Pj

=

t(Aj - iBj), PN-j

=

t(Aj

+

iBj ), j

=

1, ... , M - 1,

PM=T;

AM

Ao

=

2Po, Ah

=

Ph

+

PN-h, Bh

=

i(Ph - PN-h), h

=

1, ... , M - 1, AM

=

2PM.

The trigonometric expression 'P(x) and its corresponding phase polyno- mial p(x) agree for all support arguments Xk = 2nk/N of an equidistant partition of the interval [0, 2n]:

k = 0, 1, ... , N - 1.

However, 'P(x) = p(x) need not hold at intermediate points x+-^Xk. The two interpolation problems are equivalent only insofar as a solution to one problem will produce a solution to the other via the coefficient relations (2.3.1.4).

The phase polynomials p(x) in (2.3.1.2) are structurally simpler than the trigonometric expressions 'P(x) in (2.3.1.1). Upon abbreviating

and since W j +-Wk for j +- k,

°

^{~ j, k ~ N} - 1, it becomes clear that we are faced with just a standard polynomial interpolation problem in disguise:

find the (complex) algebraic polynomial P of degree less than N with k

=

0, ... , N - 1.

The uniqueness of polynomial interpolation immediately gives the following

(2.3.1.5) Theorem. For any support points (Xk,

h),

k = 0, ... , N - 1, with

h

complex and Xk

=

2nk/ N, there exists a unique phase polynomial p(x)

=

^Po

+

^{Pl e ix}

+ ... +

PN-l e(N-l)ix

with

for k = 0, 1, ... , N - 1.

The coefficients

Pi

of the interpolating phase polynomial can be expressed in closed form. To this end, we note that, for 0 ::;;; j, h ::;;; N - 1

(2.3.1.6)

More importantly, however, we have for 0 ::;;; j, h ::;;; N - 1 (2.3.1.7) ^{N - l .}

L

^{w{.W;h = {N for j = h,}

k = 0 0 for j =F h.

PROOF. Wj-h is a root of the polynomial

N-l w N - 1 = (w -1)

L

^{w k,}

k=O

from which either ^Wj-h = 1, and therefore j = h, or

N-l N-l N-l

"" ,.,; W- h -_{L. UlJc: k -} "" ,.,;- h -_{L. UlJc: -} "" _L. w^k _{j-h -}- 0 _.

o

k=O k=O k=O

Introducing the N-vectors

W(h) = (1, w~, ... , W~_dT, h = 0, ... , N - 1,

we see that the sums in (2.3.1.7) are the complex scalar products of the vectors ^w(j) and ^W(h):

N-l

(2.3.1.8)

L

^evi,w;h

=

^WU)TW(h)

=

[w(j), ^W(hl].

k=O

This definition of the scalar product of two complex vectors is standard;

it implies that [w. w] =

2;;J IWh

12 ~ 0 for each complex vector w.

Thus the vectors ^W(h) are seen to form an ortho onal basis of the complex space

eN.

Note that the vectors are of length ^{[W(h), w( )]} =

jN

instead of length 1, however.

From the orthogonality of the vectors ^W(h) follows:

(2.3.1.9) Theorem. The phase polynomial p(x) = L7~Ol Pjei^ix satisfies P(Xk) =

h ,

k = 0, 1, ... , N - 1,

for j,. complex and Xk

=

2nk/N,

if

^{and only}

if

I N - 1 I ^{N - 1}

f3j=-

L

^j,.w;j=-

L

j,.e-21tijk/N,

N k=O N k=O ^j

=

0, 1, ... , N - 1.

PROOF. With the vector notation f

=

(fo, fl' ... ,fN _ 1 )T,

1 N-l . 1 . 1 .

N k~O j,.w;J

=

N[f, w(})]

=

N[f3o w(O)

+ ... +

^f3N_1W(N-l), w(})]

=

f3j. 0 For phase polynomials q(x) of degree at most s, s ~ N - 1 given, it is in general not possible to make all residuals

k

=

0, ... , N - 1,

vanish~ as they would for the interpolating phase polynomial. In this context, the s-segments

Ps{x)

= 130 +

^f31eiX

+ ... +

f3se'iX

of the interpolating polynomial p(x) have an interesting best-approximation property:

(2.3.1.10) Theorem. The s-segment Ps(x),

°

^{~ s <} N, of the interpolating phase polynomial p(x) minimizes the sum

N-l

S(q) =

L

¹

^{he -}

^{q(Xk) 12}

k=O

[note that S(p) = 0] of the squared absolute values of the residuals over all phase polynomials

q(x)

=

Yo

+

^{1'1 eix}

+ ... +

^yse'iX.

The phase polynomial Ps(x) is uniquely determined by this minimum property.

PROOF. We introduce the vectors

P. = (P.(xo), ... , Ps(xN-tlf, q

=

(q(xo), ... , q(XN-1W and use the scalar product (2.3.1.8) to write

S(q)

=

[J - ^q,f - q].

By Theorem (2.3.1.9), f3j == (I/N)[f, w(j)] for j

=

0, ... , N - 1. For j ~ s,

~

^{[J - Ps' w(j)]}

⁼ ~

^{[f -}

htlhW(h),

^w(j)]

⁼

^{f3j - f3j}

⁼

^0,

and

[J - p., Ps - q] =

L •

[J - p., (f3j - Yj)w(j)] =

o.

j=O

But then we have

S{q) = [f - q,f - q]

=

[f - Ps

+

Ps - q,

f -

Ps

+

Ps - q]

=

[f - Ps,

f -

Ps]

+

[Ps - q, Ps - q]

~ [f - Ps,f - Ps]

=

S{Ps).

Equality holds only if [Ps - q, Ps - q]

=

0, i.e., if the vectors Ps and q are equal. Then the phase polynomials Ps(x) and q(x) are identical by the

uniqueness theorem (2.3.1.5). 0

Returning to the original trigonometric expressions (2.3.1.1), we note that Theorems (2.3.1.5) and (2.3.1.9) translate into the following:

(2.3.1.11) Theorem. The trigonometric expressions 'I'{x)

= ~O + h~O

(Ah cos hx

+

Bh sin hx)

A M-l A

'I'{x) =

i ⁺

^h~O (Ah cos hx

+

Bh sin hx)

+

2M cos Mx, where N = 2M

+

1 and N = 2M, respectively, satisfy

'I'{Xk)

=

fb k

=

0,1, ... , N - 1,

for Xk = 2nk/N

if

and only

if

the coefficients of'l'(x) are given by

2 ^{N - 1} 2 ^{N - 1} 2nhk

Ah

=

N

L

fk cos hXk

=

N

L he

cos -N '

k=O k=O

2 N - 1 2 N - 1 2nhk

Bh = N

L he

sin hXk = N

L he

sin -N .

k=O k=O

PROOF. Only the expressions for Ah, Bh remain to be verified. For by (2.3.1.4)

. N-l

Bh

=

i{Ph - PN-h)

=

~

L

he{e- hixk - e-(N-h)ixk), N k=O

and the substitutions (2.3.1.3) yield the desired expressions.

o

Note that if the support ordinates

he

are real, then so are the coefficients Ah, Bh in (2.3.1.11).

2.3.2 Fast Fourier Transforms

The interpolation of equidistant support points (Xk' f,,), Xk = 2nk/ N, k = 0, ... , N - 1, by a phase polynomial p(x) = L7~Ol

P

j

ei

ix leads to expressions of the form [Theorem (2.3.1.9)]

1 N-l

P

j ⁼ N

L

f"e-21tijk/N,

k=O

(2.3.2.1 ) j = 0, ... , N - 1.

The evaluation of such expressions is of prime importance in Fourier analysis. The expressions occur also as discrete approximations-for N equidistant arguments s-to the Fourier transform

H(s):=

f

+00 f(t)e-21tist dt,

-00

which pervades many areas of applied mathematics. However, the numerical evaluation of expressions (2.3.2.1) had long appeared to require on the order of N² multiplications, putting it out of reach for even high-speed electronic computers for those large values of N necessary for a sufficiently accurate discrete representation of the above integrals. The discovery [Cooley and Tukey (1965)] of a method for rapidly evaluating (on the order of N log N multiplications) all expressions (2.3.2.1) for large special values of N has therefore opened up vast new areas of applications. This method and its variations are called fast Fourier transforms. For a detailed treatment see Brigham (1974) and Bloomfield (1976).

There arc two main approaches, the original Coo ley- Tukey method and one described by Gentleman and Sande (1966), commonly called the Sande- Tukey method. Both approaches rely on an integer factorization of Nand decompose the problem accordingly into subproblems of lower degree.

These decompositions are then carried out recursively. This works best when

N= 2^n, n >

°

^integer.

We restrict our presentation to this most important and most straightfor- ward case, although analogous techniques will clearly work for the more general case N = N 1 N 2 ... N n' N m integer.

The Cooley-Tukey approach is best understood in terms of the interpola- tion problem described in the previous section (2.3.1). Suppose N = 2M and consider the two interpolating phase polynomials q(x) and r(x) with

q(X2h) = f2h' r(x2h) = f2h+ b h = 0, ... , M - 1.

The phase polynomial q(x) interpolates all support points of even index, whereas the phase polynomial r(x) = r(x - 2n/N) = r(x - n/M) interpo- lates all those of odd index. Since

eMixk _ e21tiMk/N _ e1tik _ _{- - - I} 1+1,

\ -1,

keven, k odd,

the complete interpolating phase polynomial p(x) is now readily expressed in terms of the two lower-degree phase polynomials q(x) and r(x):

(2.3.2.2) _{p(x) = q(x)} ( 1

+

₂ eMiX)

₊

_{r(x - 1t/M)}

(1 _

₂ ^eMiX) _.

This suggests the following n step recursive scheme. For m ~ n, let M = 2^{m -1} and R = 2^{n - m•}

Step m then consists of determining R phase polynomials P^{(m) = a(m)}

+

^a(m)eix

+ ... +

^a(m) e(2M - l)ix

r PrO Prl Pr,2M-l , r = 0, ... , R - 1,

from 2R phase polynomials p~m-l)(x), r = 0, ... , 2R - 1, using the recursion (2.3.2.2):

This relation gives rise to the following recursive relationship between the coefficients of the above phase polynomials:

(2.3.2.3) ⁼

j3(~-I) _rj

+

j3(m-1)B j _{R+r,j m} I _r

2j3(m) _r,M+j . = j3(~-_rj 1) _ j3(m- 1)'B j _R+r,j I ~ m /

r = 0, ... , R - 1, j = 0, ... , M - 1, where

m=O, ... , n.

The recursion is initiated by putting

a(0) := {' k

°

^{N 1}

PkO Jb =, ... , - ,

and terminates with

j = 0, ... , N - 1.

This recursion typifies the Cooley-Tukey method.

The Sande-Tukey approach chooses a clever sequence of additions in the sums L~~l he e-jiXk. Again with M = N/2, we assign to each termhe e-jiXk an opposite term fk+Me-jiXk+M. Summing respective opposite terms in (2.3.2.1) produces N sums of M = N/2 terms each. Splitting those N sums into two sets, one for even indices j = 2h and one for odd indices j = 2h

+

1, will lead to two problems of evaluating expressions of the form (2.3.2.1), each prob- lem being of reduced degree M = N/2.

Using the abbreviation

again, we can write the expressions (2.3.2.1) in the form 1 N-l

j3j = N

L

heB~k,

k;O j = 0, ... , N - 1.

Here n is such that N = 2n. Distinguishing between even and odd values ofj and combining opposite terms gives

1 ^{N -} 1 1 M- 1 1 M- 1

P2h

=

N

L

^f,.c;hk

⁼

^N

L

^(f,.

+

f,.+M)C="--1

=

N

L

fleC="--1

k=O k=O k=O

1 ^{N -} 1 1 M- 1 1 M- 1

P 2h+ 1 -_ ~ N k~O " Jkcn # (2h+ 1)k _ _ "(( - N k~O Jk Jk+M c# _ # ) k) hk ⁿ cⁿ-1 -- _ " f" hk N k~O kCn-1 for h =

° ... ,

M - 1 and M:= N/2, since c; = Cn-1' c';; = -1. Here

fie

=

f,.

+

f,.+M

I

f~ ⁼

^{(f,. - f,.+M)C:}

~

^k

⁼

0, ... , M - 1.

In order to iterate this process for m

=

n, n - 1, ... , 0, we let M ^:= 2m -1,

R ^:= 2n- m and introduce the notation

f~k)' r

=

0, ... , R - 1, k

=

0, ... , 2M - 1,

with f~~ = f,., k = 0, ... , N - 1. f~k-¹⁾ and 11k- 1) represent the quantities fk and f~

,

respectively, which were introduced above. In general we have, with M = 2m -¹ and R = 2n - m,

1 2M-1 (2.3.2.4) PjR+r = -

L

^f~k)cf::,

N k=O ^r = 0, ... , R - 1, j = 0, ... , 2M - 1, with the quantities f~k) satisfying the recursions:

I

m= n, ... ,1,

f~k-1) = f~k)

+

f~~k+M

I

I ; r = 0, ... , R - 1, f~~R~l = (f~k) - f~~~+M)C~ ~

I I

k

=

0, ... , M - 1.

(2.3.2.5)

PROOF. Suppose (2.3.2.4) is correct for some m:::; n, and let M':= M/2

=

2^{m - 2 ,} R' :=2R

=

2^{n - m}+1• For j

=

2h and j

=

2h

+

1, respec- tively, we find by combining opposite terms

1M-1 1 2M'-1

PhR'+r

=

PjR+r

=

N

L

(f~~~

+

f~~~+M)cf::

=

N

L

f~~k-1)c::;"_1'

k=O k=O

1 M-1

PhR'+r+R

=

PjR+r

=

N

L

^{(f~~~ -} f~~~+M)ci,:

k=O

1 M-1 1 2M'-1

= N

L

^{(f~~~ -} f~~~+M)C~C::;"-1 = N

L

f~~I/lc::;"-1'

k=O k=O

where r

=

0, ... , R - 1, j

=

0, ... , 2M - 1.

o

The recursion (2.3.2.5) typefies the Sande-Tukey method. It is initiated by putting

f

^(n):=# _{Ok Jk,} k

=

0, ... , N - 1,

and terminates with

P

, N ^:=_/(0) 1

,0,

^{r =} 0, ... , N - 1.

Returning to the Cooley-Tukey method for a more detailed algorithmic formulation, we are faced with the problem of arranging the quantities f3';.j) in an array:

I(

=

0, ... , N - 1.

Among suitable maps ^I( = I((m, r, j), the following is the most straightforward:

m

=

0, ... , n, r

=

0, ... , 2^{n - m -} 1, j

=

0, ... , 2m - 1.

It has the advantage, that the final results are automatically in the correct order. However, two arrays

PT ], P[ ]

are necessary to accommodate the left- and right-hand sides of the recursion (2.3.2.3).

We can make do with only one array

P[ ]

if we execute the transforma- tions "in place," that

,is,

if we let each pair of quantities f3';.j), f3';.~k + i occupy the same positions in P[ ] as the pair of quantities

f3';.j-l), P<;;/},

from w~ich the former are computed. In this case, however, the entries in the array P[ ]

are being permuted, and the maps which assign the positions in

P[ ]

^{as a}

function of the integers m, r, j become more complicated. Let r = r(m, r,j)

be a map with the above mentioned replacement properties, namely

p[r]

= f3';.j) with

r(m, r, j)

=

r{m - 1, r, j)

(2.3.2.6) .,

r(m, r,j

+

2m-1 ) = r(m - 1, r

+

2n-m,j)~

~m=l, ... ,n,

i r

=

0, ... , 2^{n - m -} 1,

I

^j ='0, ... , 2^{m -1 -} 1, and

(2.3.2.7) r(n, 0, j) = j, j

=

0, ... , N - 1.

The last condition means that the final result

Pi

will be found in positionj in the array

P[ ]: Pi

=

PU].

The conditions (2.3.2.6) and (2.3.2.7) define the map r recursively. It remains to determine it explicitly. To this end, let

rt.p = 0, 1 for p = 0, ... , n - 1, be a binary representation of an integer t, 0 :::; t < 2n. Then putting (2.3.2.8) p(t) ^:=rt.n^-1

+

^rt.n^-2 ·2

+ ... +

rt.o ·2n-¹

defines a permutation of the integers t

=

0, ... , 2n - 1 called bit reversal. The bit-reversal permutation is symmetric, i.e. p(p(t»

=

t.

With the help of the bit-reversal permutation p, we can express r(m, r,j) explicitly:

(2.3.2.9) r(m, r, j) = p(r)

+

j,

m = 0, ... , n, r = 0, ... , 2n - m - 1, j = 0, ... , 2m - 1.

PROOF. If again

t

=

^(Xo

+

^{(Xl •} 2

+ ... +

^{(Xn-l •} 2ⁿ

-1,

^(Xp

=

0, 1 for p

=

0, ... , n - 1, then by (2.3.2.6) and (2.3.2.7)

_ ( ° ) _

^{{r(n - 1, 0, t) if (Xn-l}

⁼ °

t - r n, ,t - _r ( n - , 1 1 ,t - 2n - 1)·f ^{1 (Xn-l}

= .

1 Thus

t

=

r(n, 0, t)

=

r(n - 1, (Xn- b (Xo

+ ... +

^(Xn-2 ·2n - 2 ),

and, more generally,

t = r(n, 0, t) = r(m, ^(Xn-l

+ ... +

^(Xm· 2n - m - 1, (Xo

+ ... +

^(Xm-l ·2^{m -}1 )

for m

=

0, ... , n - 1. For r

=

^(Xn-l

+ ... +

(Xm· 2n - m - 1, we find p(r) = ^{(Xm •} 2^m

+ ... +

(Xn-l • 2n - 1

and t = p(r)

+

j.

By the symmetry of bit reversal,

r(m, p(r), j) = r

+

j,

D

where

r

is a multiple of 2m,

°

^~

^r

^{< 2n, and}

°

^{~ j < 2m.} Observe that if

°

^{~ j < 2m - 1, then}

t = r(m, p(r), j) = r(m - 1, p(r), j) = r

+

^j,

t

= r(m, p(r), j

+

2^{m -} 1 ) = r(m - 1, p(r)

+

2^{n - m,} j) = r

+

j

+

2^{m - 1} mark a pair of positions in

P[ ]

which contain quantities connected by the Cooley-Tukey recursions (2.3.2.3).

In the following pseUdO-ALGOL formulation of the classical Cooley- Tukey method, we assume that the array

P[ ]

is initialized by putting

P[P(k)] :=fb k = 0, ... , N - 1,

where p is the bit-reversal permutation (2.3.2.8). This" scrambling" of the initial values can also be carried out "in place," because the bit-reversal permutation is symmetric and consists, therefore, of a sequence of pairwise interchanges or "transpositions." In addition, we have deleted the factor 2 which is carried along in the formulas (2.3.2.3), so that finally

P

j ^:= N 1 -

PU],

j = 0, ... , N - 1.

The algorithm then takes the form for m := 1 step! until n do

begin for j ^:=

°

step 1 until 2m - 1 - 1 do begin e ^:=

e!..;

end

end;

for

r

^:=

°

^{step 2m} until 2" - 1 do

begin u ^:=

p[r +

j]; V :=

p[r +

j

+

2m -1] X e;

p[r +

j] ^{:= U}

+

^v;

p[r +

j

+

^{2m -} 1] ^{:= U - V}

end

If the Sande-Tukey recursions (2.3.2.5) are used, there is again no prob- lem if two arrays of length N are available for new and old values, respec- tively. However, if the recursions are to be carried out" in place" in a single array

J[ ],

then we must again map index triples m, r, j into single indices r.

This index map has to satisfy the relations r(m - 1, r, k)

=

r(m, r, k), r(m - 1, r

+

2"-m, k) = r(m, r, k

+

2m-^{1 )}

for m

=

n, n - 1, ... ,1, r

=

0,1, ... , 2"-m - 1, k

=

0,1, ... , 2m-^{1 -} 1. If we assume

r(n, 0, k) = k for k = 0, ... , N - 1,

that is, if we start out with the natural order, then these conditions are precisely the conditions (2.3.2.6) and (2.3.2.7) written in reverse. Thus r

=

r(m, r, k) is identical to the index map r considered for the Cooley- Tukey method.

In the following pseUdO-ALGOL formulation of the Sande-Tukey method, we assume that the array ][ ] has been initialized directly with the values fk :

J[k] ^:=

h,

k = 0, 1, ... , N - 2.

However, the final results have to be "unscrambled" using bit reversal, 1 _

Pj ^:= N [J[pU)], j

=

0, ... , N - 1:

for m := n step - 1 until 1 do

begin for k ^:=

°

step 1 until 2m - 1 - 1 do begin e:= t:~;

for

r

^:=

°

^{step 2m} until 2" - 1 do

begin u ^:=

IF +

k]; ^{V :=}

J[r +

k

+

^{2m -} I];

J[r +

k] :=u

+

V;

J[r

+k

+

2^{m -} I ] :=(u - V) X e end

end

end;

If all values

h,

^k = 0, ... , N - 1 are real and N = 2M is even, then the problem of evaluating the expressions (2.3.2.1) can be reduced in size by putting

gh = i2h

+

^{if2h+ b}

and evaluating the expressions 1 M-l ..

Y . = _ ^~ g e- 2'tlJh/M

J M L... h ,

h=O

h = 0, ... , M - 1,

j = 0, ... , M - 1.

The desired values

p

j , j = 0, ... , N - 1, can be expressed in terms of the values ^Yj,j = 0, ... , M - 1. Indeed, one has with ^YM :=yo

(2.3.2.10)

P _

^{1 ( _ )}

+

~

( _ - ) -

^27tij/N

j -

4.

^Yj

+

^{YM - j 4i Yj YM - j e ,}

j=o, ...

,M,

PN -

^{j =}

P

j , j = 1, ... , M - 1.

PROOF. It is readily verified that

1 1 M-l

-4(Yj

+

^1M-j) = N

I

i2he-27tij2h/N, h=O

1 1 M-l

_(y._y- .)=_ ~ r e- 27tij(2h+l)/N+27tij/N

4 · ^{J M-J} N L... J2h+l •

I h=O D

In many cases, particularly if all values f,. are real, one is actually in- terested in the expressions

2 ^N-l 2njk A^j

=

N I i k cos -N '

k=O

2 ^N-l . 2njk Bj=N _k=O

Ih

^{sm -N '}

which occur, for instance, in Theorem (2.3.1.11). The values A j ' Bj are con- nected with the corresponding values for

P

j via the relations (2.3.1.4).

2.3.3 The Algorithms of Goertzel and Reinsch

The problem of evaluating phase polynomials p(x) from (2.3.1.2) or trigo- nometricexpressions 'P(x) from (2.3.1.1) for some arbitrary argument x = ~ is called Fourier synthesis. For phase polynomials, there are Horner-type eval- uation schemes, as there are for expressions (2.3.l.1a) when written in the form 'P(x) =

If= _

^{M pje jix.} The numerical behavior of such evaluation schemes, however, should be examined carefully.

For example, Goertzel (1958) proposed an algorithm for a problem closely related to Fourier synthesis, namely, for simultaneously evaluating the two sums

N-l N-l

I

^{Yk cos k~,}

I

^{Yk sin k~}

k=O k=l

for a given argument ~ and given values Yk, k = 0, ... , N - 1. This algorithm is not numerically stable unless it is suitably modified. The algorithm is based on the following:

(2.3.3.1) Theorem. For ~ =1= nt, r = 0, ± 1, ±2, ... , define the quantities 1 N-1

Uj:=~ LYksin(k-j+1g, j=0,1, ... ,N-1,

SIll .. k=j UN:= UN+1 :=0.

These quantities satisfy the recursions

(2.3.3.1 a) Uj

=

Y + 2Uj+ 1 cos ~ - Uj+ 2, j

=

^{N -} 1, N - 2, ... , O.

I n particular

N-1

(2.3.3.1b)

L

Yk sin k~

=

U 1 sin ~, k=1

N-1

(2.3.3.1c) L Yk cos k~

=

Yo + U ¹ cos ~ - U ^{2 .}

k=O

PROOF. For 0 ~j ~ N - 1, let

A:=Yj+2Uj+1COS~- Uj+2.

By the definition of Uj+ b Uj+2 ,

1 { N - 1 N-1 }

A = Yj + -. -" .2(cos

0

^{L Yk sin(k - j)~ ~ L Yk} sin(k - j - 1)~

SIll .. k=j+1 k=j+2

1 N-1

= Yj + ~.

I

^{Yk[2 cos ~} sin(k - jg - sin(k - j - 19].

SIll.. k=j+1 Now

2 cos ~ sin(k - jg

=

sin(k - j + 1)~ + sin(k - j - 19.

Hence

1 [ N - 1 ]

A=~ Yjsin~+

L

y"sin(k-j+1g =Uj •

SIll .. k=j+1

This proves (2.3.3.1 a ). (2.3.3.1 b) restates the definition of U l' To verify (2.3.3.1c), note that

1 N-1 1 N-1

U 2

=

~.

L

y" sin(k - 1 g

=

~.

L

y" sin(k - 1 )~,

. SIll..

,,=

^{2 SIll.. k}

=

¹

and

sin(k -

19 =

^cos ~ sin k~ - sin ~ cos k~.

o

Goertzel's algorithm applies the recursions (2.3.3.1) directly:

U[N]:= U[N

+

1] :=0; C :=cos(~); cc :=2 x c;

for j ^:= N - 1 step - 1 until 1 do

UU]:= yU]

+

^cc x UU

+

1] - UU

+

2];

s1:= y[O]

+

^C x U[1] - U[2];

s2:= U[1] x sin(~);

to find the desired results s1

= L;:J

Yk cos k~, s2

= L;:l

Yk sin k~.

This algorithm is unfortunately not numerically stable for small absolute values of~,

I

^~

I

^~ 1. Indeed, having calculated c

=

cos ~, the quantity s1 =

L;:J

Yk cos k~ will depend solely on c and the values Yk' We can write s1

=

<p(c, Yo, ... , YN-1)' where

N-1

<p(c, Yo, ... , YN-

d = L

^Yk cos k(arccos c).

k=O

As in Section 1.2, we denote by eps the machine precision. The roundoff error Ac = eeC' leel ~ eps, which occurs during the calculation of c, causes an absolute error A.s1 in s1, which in first-order approximation amounts to

A 1 . O<p ^{A _} Ee cos ~ N~1k . kJ:

OeS = ; ) oC - . J: L. Yk sm ..

uC sm.. k=O

= Ee(Cot~) N-1

L

^{kYk sin} k~.

k=O

An error A~

=

E~~,

I

^E~

I

^{~ eps in ~,} on the other hand, causes only the error

a

^{{N -1 }}

A~sl == o~ k~/k cos k~ . A~

= -E~~ N-1

L

^{kYk sin k~}

k=O

in s1. Now cot ~ ::::; 1/~ for small

I

^~

I.

The influence ofthe roundoff error in c is consequently an order of magnitude more serious than that of a corre- sponding error in ~. In other words, the algorithm is not numerically stable.

In order to overcome these numerical difficulties, Reinsch has modified Goertzel's algorithm [see Bulirsch and Stoer (1968)]. He distinguishes the two cases cos ~ > 0 and cos ~ ~ O.

Case (a): cos ~ > O. The recursion (2.3.3.la) yields for the difference bUj := Uj - Uj+1

the relation

bUj

=

Uj - Uj+1

=

Yj

+

(2 cos ~ - 2)Uj+1

+

Uj+1 - Uj+2

= Yj

+

;"Uj+ 1

+

bUj+ 1,

where

A. :=2(cos ~ - 1) = -4 sin2(~/2).

This suggests the algorithm A.:= -4 sin2(~/2);

U[N

+

1] :=bU[N] :=0;

for j:= N - 1 step -1 until 0 do

begin U[j

+

1] :=bUU

+

1]

+

UU

+

2];

bUU] :=A. x UU

+

1]

+

bU[j

+

1]

+

yU]

end;

sl :=bU[O] - A.12 x U[1];

s2:= U[I] x sin(~);

This algorithm is well behaved as far as the propagation of the error L\A.

=

c).A., I c).1 ~ eps in A. is concerned. The latter causes only the following error L\).sl in sl:

osl osl lOA.

L\).sl ==

8I

L\A. = c).A. .

a[

o~

sin2(~/2)

N-l .

= - c).

sin(~/2) cos(~/2)

.

k~O

kYk sm

k~

(

^~)N-l

= -c). tan:2 k~O kYk sin k~.

and tan(~/2) is small for small

I

~

I.

^Besides,

I

tan(~/2)

I

< 1 for cos ~ > O.

Case (b): cos ~ ~ O. Here we put

bUj:= Uj

+

Uj+ 1

and find

bUj = Uj

+

Uj+1 = Yj

+

(2 cos ~

+

2)Uj+1 - Uj+1 - Uj+2

=Yj+A.Uj+1 -bUj+b

where now

A.:= 2(cos ~

+

1)

=

4 cos2(~/2).

This leads to the following algorithm:

A. ^:= 4 cos2(~/2);

U[N

+

1] :=bU[N] :=0;

for j:= N - 1 step - 1 until 0 do

begin UU

+

1] :=bUU

+

1] - UU

+

2];

bUU] :=A. x UU

+

1] - bUU

+

1]

+

yU]

end;

sl :=bU[O] - U[I] x A.12;

s2:= U[I] x sin(~);

It is readily confirmed that a roundoff error AA =

8;.

A,

18;.1 ::::;

^{eps, in} A causes an error of at most

(

^~)N-l

A;.s1 == 8;. cot

2

k~O kYk sin k~

in s1, and

I

^cot(~/2)

I ::::;

1 for cos ~ ::::; O. The algorithm is therefore well behaved as far as the propagation of the error AA is concerned.

2.3.4 The Calculation of Fourier Coefficients.

Attenuation Factors

Let ~ be the set of all absolutely continuous² real functionsJ: ~ --+ ~ which are periodic with period 2n. It is well known [see for instance Achieser (1956)] that every function

J

^{E ~} can be expanded into a Fourier series (2.3.4.1 ) J(x)

= " c·ei

_L..J ⁰⁰ _J ^ix _'

j= ^-00

which converges towardsJ(x) for every x E ~. The coefficients Cj

=

cj(f) of this series are given by

1 f21t ..

(2.3.4.2) Cj

=

Cj(f):= 2n ⁰ J(x)e-^JIX dx, ^{j =} 0,

±

1,

±

2, ....

In practice, frequently all one knows of a functionJare its values./k := J(Xk) at equidistant arguments xk^:= 2nk/N, where N is a given fixed positive integer. The problem then is to find, under these circumstances, reasonable approximate values for the Fourier coefficients cAf). We will show how the methods of trigonometric interpolation can be applied to this problem.

By Theorem (2.3.1.9), the coefficients

P

j of the interpolating phase polynomial

with

2 A real function f: [a, b] -+ IR is absolutely continuous on the interval [a, b] if for every e > 0 there exists b > 0 such that

L; I

^f(b;} - f(a;}

I

< e for every finite set of intervals [a;, bJ with a ,,; a I < b I < ... < ^an < bn ,,; band

L; I

b; - a;

I

< b. If the function f is differentiable every- where on the closed interval [a, b] or, more generally, if it satisfies a .. Lipschitz condition"

I f(xl } - f(x2 } I ,,;

e

I ^{Xl -} x21 on [a, b], then f is absolutely continuous, but not conversely:

there are absolutely continuous functions with unbounded derivatives. If the function is abso- lutely continuous, then it is continuous and its derivative f' exists almost everywhere.

Moreover,f(x} = f(a} + J= f'(t} dt for X E [a, b]. The absolute continuity of the functionsf, g in an integral of the form J~ f(t}g'(t} dt also ensures that integration by parts can be carried out safely.

for k

=

0, ± 1, ±2, ... , are given by 1 N-l

Pj

=

N

L

fke- jixk,

k=O

j=0,1, ... ,N-1.

Since

fo = fN,

the quantities

P

j can be thought of as a "trapezoidal sum"

[compare (3.1.7)]

P

.=![fo

+!e-jiXl+ ... + ^I" e-jiXN-l+fNe-iiXN]

J N 2 ¹ IN-l 2

approximating the integral (2.3.4.2), so that one might think of using the sums

(2.3.4.3 )

for all integers j

=

0,

±

1,

±

2, ... as approximate values to the desired Fourier coefficients cj(f). This approach appears attractive, since fast Four- ier transforms can be utilized to calculate the quantities pj(f) efficiently.

However, for large indices j the value

PAf)

is a very poor approximation to cj(f). Indeed, Pj+kN

=

Pj holds for all integers k, j, while on the other hand

limlil~oo Cj = 0. [This follows immediately from the convergence of the Four- ier series (2.3.4.1) for the argument x

=

0.] A closer look also reveals that the asymptotic behavior of the Fourier coefficients

cAf)

depends on the degree of differentiability off:

(2.3.4.4) Theorem. If the 2n-periodic function f has an absolutely continuous

r th derivative prl, then

PROOF. Successive integration by parts yields 1 2" ..

c· _J

= -f

_{2n 0} ^{f(x)e-J1X dx}

1 2" ..

= - ..

_2nJI

f

^{f'(x)e-JIX dx}

0

1 2" ..

= -.-.

_2n(JI

_Y f

^{Prl(X)e-JIX dx}

0

1 2"

= _2n(jiY+

f

^{e- jix dprl(x).}

1 0

in view of the periodicity off This proves the proposition.

o