The two-dimensional minimum-bias window function

Signal Processing 4 (1982) 65-71 North-Holland Publishing Company

65

SHORT COMMUNICATION

THE TWO-DIMENSIONAL

MINIMUM-BIAS

WINDOW

FUNCTION

E. BORCHI and M. POLI Istituto di Energetica dell' Uni~'ersitd, ria di S. Marta 3. 50139 Firenze, Italy Received 8 March 1981 Revised 29 September 1981

Abstract. Using the criterion of minimization of the bias integral, by means of a quantum-mechanical analogy, the optimum two-dimensional window function for spectral estimates is determined. The optimum window and its two-dimensional Fourier transform are numerically evaluated. A comparison is carried out with some other standard windows. A three-dimensional generalization is briefly outlined. Zusammenfassung. Anhand des Kriteriums der Minimisierung des asymmetrischen Fehlerintegrals (mit Hilfe einer Analogie zur Quantenmechanik) wird die optimale Fensterfunktion f/Jr die Absch~itzung zweidimensionaler Spektralfunktionen hergeleitet. Das optimale Fenster und seine Fouriertransformierte werden numerisch bestimmt und mit einigen bekannten Fensterfunktionen vergleichen. Die Erweiterung auf den driedimensionalen Fall wird kurz umrissen. Rrsumr. Une fonction fen~tre optimale, bidimensionnelle pour l'estimation spectrale est d&ermin6e en utilisant le critrre de minimisation de l'intrgrale du biais au moyen de l'analogie de la mdcanique quantique. La fen~tre optimale et sa transformde de Fourier bidimensionnelle soar e~valures numrriquement. Une comparaison est effecture avec certaines fonctions fen&res standard. Une gdndralisation ~ trois dimensions est bri~vement esquissde. Keywords. Signal theory, spectral estimation, quantum mechanics.

1. Introduction In recent years much work has been devoted to the study of optimum windows for spectral estimation. At present many theoretical and empirical criteria are available [ 1-3] which take into account different, and often conflicting, requirements on the estimation procedure. Among others, the remarkable method by Papoulis [2] for highresolution spectral estimates is based on the +c~ 2 minimization of the bias integral ~-o~ co W(w) dw, where W(o~) is the Fourier transform of the timelimited window function w(t). In [3] it has been pointed out the existing analogy between the problem of minimum-bias for spectral estimates, as stated by Papoulis, and the quantum-mechani-

cal problem of the ground-state of a particle in the one-dimensional potential box [4]. Using this formal analogy a new class of window functions was proposed, which includes, as a limiting case, the Papoulis minimum-bias function. On the other hand, it is of increasing interest in different areas of digital processing and communications the determination of two-dimensional window functions. In a simplified procedure one may use the "rotated" one-dimensional window function w(t)=w(~/t~+t~). Then the frequency-dependent window function, which has also rotational symmetry, is related tO w(t) through the two-dimensional Fourier transform. However, from a conceptual point of view, this approach is quite unsatisfactory. A better

0165-1684/82/0000-0000/$02.75 © 1982 North-Holland

66

E. Borchi and [~I. Poli / 2 - D minimum-bias window function

procedure is the following. Consider the twodimensional window function

w(t)=w(~/t~+t~),

w(t) = 0

for It[ >'r, (1)

such that

w(O) = ~

wW(w) dw = 1.

(2)

Here

W(w) = 2w

f/

tw(t)Jo(wt) dt

(3)

is the two-dimensional Fourier transform of w(t), and ~o = ~/w~ +w,2. The bias term can be written in the form

B=

CX3

dxdyS(wl-x, w2-y)W(x,y)

- S(~oi, o~2)

=

io2-dO

,

S ( ~ / w - + p ' - 2 w p cos 0) (4)

where 0 is the angle between the p and w-directions, p = ~/x---~y 2, and the unknown two-dimensional spectrum S(w) has been supposed having circular symmetry. If W(p) takes significant values only in a small p-interval near p = 0, we can write (extending the Papoulis arguments [2] to twodimensional functions) that the bias is proportional to the integral ~

f)

0)3 W(w) dw.

(5)

Hence the problem of the minimum-bias twodimensional window functions is to find a signal w(t) satisfying (1) and (2) and such that the moment I3 of its Fourier transform is minimum. This problem is meaningful only if we assume W(o)) ~ 0 for every co. In order to solve this problem in the next section we extend our previously proposed quantummechanical analogy [3, 5] to the two-dimensional case. Our purpose is to determine a new twoSignal Processing

2. A n e w t w o - d i m e n s i o n a l w i n d o w

We review briefly the quantum-mechanical analogy introduced in previous works [3, 5]. We consider a particle of mass m in a two-dimensional potential box in the region 0 ~< r<½a, where r = i x 2 + y2. As the potential is infinite for r/> ½a, the correct wave-function of the particle in the box vanishes for r >>½a. The total energy of the particle in the quantum-mechanical state described by the wave-function t0(r) is proportional to the integral

I=

x W(p)p d o - S ( w )

13

dimensional window. The so-determined minimum-bias window is compared with some standard two-dimensional windows. Finally in Section 3 we will give a three-dimensional generalization of the work.

fo p3[tI'(p)i2 dp

(6)

where ~ ( p ) = 2 ~ r S o / 2 r o ( r ) J o ( p r ) d r is the twodimensional Fourier transform of ¢(r) and [ ~ ( p ) r is the probability density (normalized so that 2 ~ I o p l q ' ( p ) l ' - d p = l ) of the particle in the momentum r e p r e s e n t a t i o n - p [4]. We can note here the analogy between the quantum-mechanical problem, eq. (5), and the spectral estimation problem, eq. (5). On this line, by using the correspondences r ~ t, p -* w, a --, r and by putting Iq~(~o)l-'= W(o~)

(7)

the probability density of the particle in the twodimensional box (in the momentum representation) gives the window function W(w). From the relation (7) it follows that the Fourier transform of W(w) is given by

w(t) = 2,-r

WJo(~ot)dw

1. -r/2

x[

tlJo(wtOtO(tO dtl

d0

Xf a0

r/2

t,Jo(wt2)tO*(t2) dt2.

(8)

E. Borchi and M. Poll / 2-D minimum-bias window function

Since the wave-function t0(t) of the particle in the box vanishes for t/> ½r, a consequence of (8) is that w(t) vanishes for t/> 7. This property follows from the result [6]

o'=Jo(ax)Jo(bx)Jo(cx)x dx = 1/(2,-rA)

2J,,(2A,(rnl t/r) e im,b

tom..(t, 6)=

r4-~ L,±,(aT~)

(9)

where q5 is the azimuthal variable and a{ff ~ is the nth root of the mth Bessel function of the first kind (i.e. J~(Z~"~)=0). The + or - signs obviously make no difference since only the absolute value is concerned. For the ground state eq. (9) gives (Am' ~ A o = 2.40482 • • .)

2Jo(2Aot/r) to°(t)

From (7) we have

/ aoTJo(½,oT)~2 Wo(w) = WtA2o_ (½~o7)2}

(10)

r',/w J,(Ao) "

The two-dimensional Fourier transform of too(t)

"i

~

ax.

t

(13)

Since we have been not able to carry out the analytical integration in eq. (13), a numerical calculation of the window wo(t) is reported in Table 1 together with that of the two-dimensional Fourier transform Wo(w). For our optimum window the asymptotic behaviour is Wo(w)/r2 = 32Ao(i +sin wr) (for large o~), ~o7((2Ao)2 - (09,;-)2) 2 and the moment I3 =5o o~3W0(°~) dto takes the value ~ /3 = 1 4 5 . 3 5 / r 2. Fig. 1 reports the behaviour of the optimum window WOO).In order to allow a comparison, also the behaviour is reported of some standard onedimensional time-limited windows which through rotation are used for two-dimensional spectral estimation and digital processing. The corresponding two-dimensional Fourier transforms are shown in Fig. 2. The considered windows are:

Hamming [10]: WH(t) = 0 . 5 4 + 0 . 4 6 COS (Trt/7);

wK(t) =

Io(c~41-t-Z~r )/Io(a)

I It may be observed that the function Oo(t) = Jo(Aot/r), for t ~< r. and zero for t ~> r, represents the two-dimensional minim u m variance window. In fact the Fourier transform of 4~o(t):

~o(w) = 2"rrfo r/2 ttoo(t)Jo(~Ot)dt

2

Jo(wr)

q~o(W) = 2wA0r Jz(Ao) Ao _ (w"-"~_)z

D

a o~ - ( ½ , , , 7 ) 2

xJg (x)Jo(2x /r) • ~

Kaiser [11]:

is

Aor'Jrr Jo(½W7)

(12)

and from (8)

Wo(t) = 2A~, f o

where A is the area of a triangle whose sides are a, b, and c. In the case in which a, b, and c cannot form a triangle the value of the integral is zero. The above result represents the two-dimensional version of the generalized Fejer-Riesz theorem [2]. From the preceeding, it follows that the determination of the optimum time-limited window WoO) is reduced to the determination of the ground state (i.e. the state of lowest energy) eigenfunction ~o(t) for the two-dimensional potential box. It is known [7] that the wave function, with the previously specified normalization, of the twodimensional potential box has the form

67

(11)

where we have used the Lommel integrationformula [8].

minimizes the variance integral 2"~Jo oa34po(W)d~o. T h e n the time-limited function ~bo(t) may be considered the t w o - d i m e n sional generalization of the one-dimensional m i n i m u m variance G a b o r window [9] woO) = cos(-rr t,/2 r), for It] ~< r, and zero for It! ~ r. Vol. 4, No. 1. January 1982

E. Borchi and M. Poll / 2-D minimum-bias window function

68 Table I

Table 2

t/r

w(t)

oJT

0.00 0.05 0.I0 0,15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

1.00000 0.98600 0.94621 0.88465 0.80603 0.71547 0,61814 0.51903 0.42259 0.33261 0.25199 0,18271 0.12577 0.08126 0.04845 0.02597 0.01197 0.00437 0.00105 0.00009 0.00000

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

Wto~)/r 2 0.54323 0.53802 0,52265 0.49789 0.46498 0,42549 0.38127 0.33426 0,28642 0,23953 0.19518 0.15461 0.11874 0.08807 0.06278 0.04271 0.02745 0.01642 0.00892 0.00420 0.00155

Window

ET/r 2

Es/r

wo we Ww w+ WK W~

0.287 0.341 0.494 0.502 0.527 0.616

0.537 0.587 ~ 0.710 0.716 0.733 0.795

a There is an error in Table II of [3] for that concerns the numerical value of Papoulis sectional energy. The correct value is Es = (~+ 5/2w2)r.

energy E T = 2rr'Z2

fo

x[w(x)[ 2 dx,

x = t/7,

and the sectional energy

Zs = 2rfo' Iw(x)l: dx, that are of interest in spectral estimation.

where Io is the modified Bessel function of the first kind and zero order, and a is a p a r a m e t e r (the value used in our calculation is a = 6); polynomial approximation to the Weber function [12, 133: Ww(t) = a + b(t/r) + c(t/r) 2 + d ( t / r ) 3 with a = 1, b = 0.05112, c = -3.61459, d = 2.68342, for 0 ~< t / r <~0.5, and a = 1.62768, b=-3.07008, c=1.50653, d=-0.06193, for 0 . 5 < t / r < ~ l;

Papoulis [2]: wp(t) = (1/rr) s i n ( v t / r ) + (1 - t/r) cos(rrt/r). M o r e o v e r it has been considered a new approximation to the W e b e r function [14]:

w+/,) = cos(

{(7]

q exp -/3

\2rl

7

with/3 = -~(w/0.859) 2, In Table 2 we list, for the o p t i m u m window and the other previously considered windows, the total Signal Processing

3. Three-dimensional generalization We consider briefly the generalization of our results to a three-dimensional space. For a window with sphe'rical s y m m e t r y

w(t) = w(,/t~ + t~_ + t2),

w(t) = 0 for It[ > r,

the bias integral will b e p r o p o r t i o n a l to the integral

I4 =

I[

w 4 W ( w ) do)

where W(o)) is the three-dimensional transform

W(o)) =--47r

Fourier

tw(t) sin(wt) dt.

o)

Using the same p r o c e d u r e as in Section 2 one obtains that the ground-state wave function of a particle in a spherical box t < ½r is given by [7] ¢,(t) =

sin(2wt/r) ",/7r T t

E. Borchi and M. Poll / 2-D minimum-bias window function

69

1,0

0.9 \

\

0.8 \

,kx\ 0.7 X,

\

0.6

\

0.5

0.4

0.3

o2l

0.1

0 0.1

0

0.2

013

0.4

05

0.6

0.7

0.8

0.9

i.O

Fig. 1. Behaviour of time-limited two-dimensional window functions versus t/r. The considered windows are: minimum-bias window w0; . . . . Papoulis window wp; ......... polynomial approximation to Weber function ww; . . . . . new approximation to Weber function w+; . . . . Kaiser window wK; . . . . Hamming window wH. with the usual normalization 4"rr

Io"/2 IO(t)ft = dt=

The three-dimensional

2 This last result follows from the three-dimensional version of generalized Fejer-Riesz theorem. In fact, writing the function w(t) in a form analogous to eq. (8), one obtains:

1.

F o u r i e r t r a n s f o r m o f t0(t) is

w(t)=~Io~/2t,~(t,)dtJIo'Zt20(t2)dt2

sin z 1/)'((,.O) = (Ti'T) 3/2 Z(,172__ Z 2 )

w h e r e z = ½wr.

x

~1

--sin~otsinwqsinwt2dw.

a o o)

Then

W(w) results

w(,o)

The integral

3 3[ sin z \2 = I~(,o)12 = ~,, ,. --r-- 2-, \z(w -z

I( ~1s i n cot sin wt~ sin wt: d~o

)1

vanishes for t > q + t 2 [6] and therefore is certainly zero for and the three-dimensional

minimum-bias

window

t>r. Vol. 4, No, 1, January 1982

70

E. Borchi and M. Poll / 2-D minimum-bias window [unction

1.1 ~ .

\ \ N

1.0

x \ \

o.9 7..~ '~...x.\

\

"~.~.~,~ ..

\

.

\

.. \

0,7,

0.6'

\

~h', •

~\

o.2 I o.1 ]

- o.1 o

i

2

3

a

5

6

7

8

~

Fig. 2. Behaviour of two-dimensional Fourier transforms of window functions versus o~r. The considered windows are: - minimum-bias window Wo; . . . . Papoulis window Wp; ......... polynomial approximation to Weber function Ww; . . . . . new approximation to Weber function W+; . . . . Kaiser window WK; . . . . Hamming window W~.

w(t)

is given by

w~,/_-{(1-~) ,'~'~,,.~ . -:- j

w~,~= 2~, r~( ~,o. / ~ 1 ~,n(~),x 7

J0

\'IT2--X2/

X

for t~ r . 2 The analytical integration, for t ~< ~-, gives: Signal

Processing

.o,~T;j)/(~). The bias integral takes the value 14 = 8w4/'r 2.

E. Borchi and M. Poli / 2-D minimum-bias window function

4. Conclusions In this p a p e r the a n a l o g y b e t w e e n the q u a n t u m m e c h a n i c a l expression for the energy of a particle in a t w o - d i m e n s i o n a l box a n d the expression for the bias of a window f u n c t i o n has b e e n used in order to o b t a i n the final i m p o r t a n t result of the m i n i m u m - b i a s t w o - d i m e n s i o n a l w i n d o w function. W e r e m a r k that with a p r o p e r approach to the p r o b l e m , along the lines of the Papoulis work [2], the m i n i m u m - b i a s t w o - d i m e n s i o n a l w i n d o w could have b e e n o b t a i n e d w i t h o u t any reference to the particle in the box idea. O n the o t h e r h a n d we observe that our m e t h o d is easier since the particle lowest energy e i g e n f u n c t i o n is really k n o w n from q u a n t u m - m e c h a n i c a l books.

References [1] D.F. Palmer, "Bias criteria for the selection of spectral windows", IEEE Trans. Information Theory, Sept. 1969, pp. 613-615. [2] A. Papoulis, "Minimum-bias Windows for high-resolution spectral estimates". IEEE Trans. Information Theory, Vol. IT-19, No. 1, Jan. 1973, pp. 9-12.

71

[3] E. Borchi, V. Cappellini and E. Del Re. "'Some minimum bias window functions", Alta Frequenza. Vol. XLVI, No. 8, 1977, pp. 364-369. [4] V. Rojansky. Introductory Quantum Mechanics, Prentice Hall, Englewood Cliffs, NJ, 1959, pp. 157-162. [5] E. Borchi, V. Cappellini and E. Del Re. "'Minimum-bias windows defined through a quantum-mechanical analogy", Internal Report. Istituto di Elettronica, Facolth di Ingegneria, Universit~ di Firenze, R7-76, 1976. [6] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products. Academic Press, New York, 1965, p. 696 and p. 422. [7] V.I. Kogan and V.M. Galitskiy, Problems in Quantum Mechanics, Prentice-Hall, Engle~vood Cliffs, NJ, 1963, pp. 54-57. [8] N.W. McLachlan. Bessel Functions for Engineers, Oxford University Press. London, 1934, pp. 94-100. [9] D. Gabor, "Theory of communication". J. Inst. Electr. Eng.. No. 93, 1946, pp. 429-441. [10] R.B. Blackman and J.W. Tukey, The Measurement of Power Spectra, Dover, New York, 1958. Appendix B5, pp. 95-100. [11] J.F. Kaiser, "Digital filters", in: J.F. Kaiser and F. Kuo eds.. System Analysis by Digital Computer, Wiley, New York, 1966, pp. 218-277. [12] E. Borchi, V. Cappellini and P.L. Emiliani, "A new class of FIR digital filters using a Weber-type weighting function", Aria Frequenza. No. 44, 1975. pp. 469-470. [13] S. Bartoloni, Thesis (1980) and to be published. [14] E. Borchi, G. Pelosi and M. Poli, "A nearly-optimum signal in communication theory", submitted for publication.