A nearly-optimum signal in communication theory

Signal Processing 3 (1981) 407-412 North-Holland Publishing Company

407

SHORT COMMUNICATION A NEARLY-OPTIMUM SIGNAL IN COMMUNICATION THEORY E. BORCHI Istituto di Energetica dell' Universitfi, via di S.Marta 3, 50139 Firenze, Italy G. PELOSI Istituto di Elettronica dell' Universitfi, via di S.Marta 3, 50139 Firenze, Italy M. POLI Istituto di Energetica dell' Universitd, via di S.Marta 3, 50139 Firenze, Italy Received 5 February 1981

Al~tratt. The method presented in this paper enables us to obtain, starting from a physical point of view, a nearly-optimum signal for communication theory. The product of pulse duration and spectral width, referred to the positive frequency spectrum only, is 1.1814... as compared with the minimum theoretical value 1.1802... obtained by Hilberg and Rothe for the Weber signal. With respect to the Weber signal our proposed function offers the not-negligible advantage of analytical simplicity. Some possible applications are outlined. Zusammenfauung. Die in dieser Arbeit vorgestellte Methode ermSglicht es uns - ausgehend von einem physikalischen Standpunkt - ein fast optimales Signal fiir die Nachrichtentheorie zu erhalten. Das Produkt aus Pulsdauer und Bandbreite - ausschlie~lich auf positive Frequenzen bezogen - betr~igt 1,1814 im Vergleieh mit dem kleinsten theoretischen Wert 1,1802, der von Hilbert und Rothe ffir das Weber Signal erhalten wurde. Was das Weber Signal betrifft, so bietet die yon uns vorgeschlagene Funktion den wichtigen Vorteil der analytischen Einfachheit. Einige m/Sgliche Anwendungen werden skizziert. 1 6 s m 6 . En partant du point de vue de la physique, la m6thode pr6sent6e dans cette communication permet d'obtenir un signal presque optimal pour ia th6orie des communications. L¢ produit de la dur6e de rimpulsion et de sa largeur de bande, en se rdf6rrant uniquement au spectre des fr6quences positives, est de 1,1814, compar6 ~ la valeur minimum th6orique de 1,1802 obtenue par Hilberg et Rothe pour la signal de Weber. Par rapport au signal de Weber, la fonction propos6e offre I'avantage non n6gligeable de la simplicit6 analytique. Quelques applications possibles sont mentionn6es. Keywords. Signal theory, communication theory.

1.

Introduction

and theory

It has b e e n r e c e n t l y s h o w n [1] t h a t t h e g e n e r a l u n c e r t a i n t y r e l a t i o n of H i l b e r g a n d R o t h e [2] f o r r e a l signals in c o m m u n i c a t i o n t h e o r y c a n b e d e r i v e d in t h e f r a m e w o r k o f q u a n t u m t h e o r y u s i n g t h e d o u b l e h a r m o n i c o s c i l l a t o r m o d e l [3]. T h e d o u b l e - h a r m o n i c o s c i•l l a t o r e l•g e n f u n c t t•o n 1 is, f o r to > 1 0 1 The double-harmonic oscillator eigenfunction is given by 0(x)= D~((2/h)l/2(mk)l/4(x-ao)), where m is the reduced mass, h = h/2~t is the reduced Planck constant, k is the elastic constant and ao is the distance at equilibrium of the oscillating particles. On the other hand, if we look for the applications in communication theory, it is convenient to consider 0(x) the frequency spectrum and x the angular frequency. 0165-1684/81/0000-0000/$02.50

© 1981 North-Holland

E. Borchi et al. / A nearly optimum signal

408

to --too

¢J(w) = D ~ ( ~ \ /20/

(1)

where tOo is the centroid of IO(to)l2 [1]:

°Olo

oO

oO

l ,o,l do

and/20 is a suitable parameter. The function D~ is known in the literature as the Weber function [4]. The Schr6dinger eigenvalue associate to the eigenfunction (1) is given by [3] E~ -- Eo(v +½)

(3)

where the constant Eo may be expressed in terms of the parameters of the problem and v, a function of X = Wo//2o, is obtained for even parity by solving the transcendental equation ~0'(0) ---D'~ ( - X ) = 0. The Hilberg and Rothe problem corresponds to find the ground state of the double-harmonic oscillator system. The minimum eigenvalue /"rain = -0.204947 is obtained for Xmi, --- 1.086375. Hilberg and Rothe assume, without any restriction too = 1 or equivalently/201 = 1.086375. Then the eigenfunction ¢J(to) = D ~ n [ 1 . 0 8 6 3 7 5 ( W - 1)],

w/>0

(4)

minimizes the uncertainty product if the spectral width is referred to the positive frequency spectrum only. The quantum-mechanical approach [1] enables us to propose a simple function which closely approximates the Hilberg and Rothe uncertainty product. Apart from the importance of this argument in the fielcl of communication theory, a nearly-optimum signal (according to the uncertainty principle) may be useful in many applications. A simple approximation to the Fourier transform of the Weber function (4), as a third-order polynomial in a restricted range of values, was found some years ago by Borchi et al. [5], and its application to digital filtering gave excellent results. For a review of the applications see also [6]. Our starting point is the application of the variational method to the double oscillator system using the trial function

¢J+(w) = N[~o(tO - wl) + ~o(w + o)1)]

(5)

where N is a normalization factor. This choice is motivated, on physical ground, from the fact that the even solution of the double oscillator can be reasonably approximated by an even combination of wavefunctions of two simple harmonic oscillators centred on to = ±to,. Since we are interested to the ground eigenstate (4), we confine ourselves to the ground eigenstates ~00of the simple harmonic oscillators ~Oo(to+wl) =

1 l" 1/to+to1\ 2] (vr-~/2,),/2 e x p [ - ~ ~ / ]

(6)

where/21 (fi/4~m) 1/2. To test the reliability of the trial function (5) we evaluate according to the variational method the energy of the system. A straightforward calculation gives =

E = El(v+ + 1)

(7)

where E , = fi~/k/m and 2z v+(z)=[z2(1-@(z))---~e-Z2]/[l

+e-~2];

here @(z) is the error-function and z = toff/21. Signal Processing

(8)

E. Borchi et al. / A nearly optimun 7 signal

409

Fig. 1 reports the behaviour of -v+(z~ versus z. The minimum of v+(z), given by (V+)ml, =--0.20383, is obtained for z = 0.772. The so-obt.med value (V+)mi, is in close agreement with the Vmi,-value: -0.20495 by Hilberg and Rothe.

0"2[--

-t2.0 ,

2

I

,'"

i

t OAF

l.Z

t". . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

0.5

~-1.2

""

0.77 0.86 z

=

1.0

~ 2.0

L5

OJllO ~

Fig. 1. Behaviour of -u+(z) versus z = oJl/D1 (continuous curve) and of A~oAt versus z (dashed curvel.

We can now evaluate the uncertainty product. The minimum value of the uncertainty product will allow us to determine the parameter t01//21 which appears in eq. (6). By using the normalization io I¢,+(,o)1=do~ = 1, eq. (5) becomes 1 0.) - - ( . D 1 2 + -z 2 --1/2 r 116o + o31"~2]] +expt-skT) J/ (9) and the Fourier transform of ¢J+(w) is 2 @+(t) = 212nlff-~]l/z [ ~ j cos(o~,t) e x p [ - ~xa x 2t z].

(10)

A simple calculation gives OD

a = Jo o,l~+(~,)l 2 dw $2x

zfl)( z ) +

=

~o

=TDE[1L

~= Io °'=1~°+('°)1~do, ~ = .I_T t2lq~+(t)12 dt i_i~ i~+(t)[2 dt : 2-~12[ 1

2 e-z2/x/'-~

1 + e -z2

2z 2 "1 + ~ J '

2 z 2 e_Z 2

~ +---~z] • +~

2 The Fourier transform of 0+(m) is defined, in our notations, by q~+(t)=LooO+(m)e (1/2,rr) [_oo q~+(t) e dt.

iwt

,

d~a, conversely: ~#+(m)= V o l . 3, N o . 4, O c t o b e r 1981

E. Borchi et al. / A nearly optimum signal

410

From the definitions [2] At = 2(~) 1/2, AoJ = 2[(to -----~] 1/2 we have at, a,,, = 4 [ ~ ( ~--'x- 032)],/2.

(11)

The behaviour of AtAw is reported in Fig. 1. For z -* 0 one has Atlloj -> 1.20562, the uncertainty product for a gaussian signal; for z ~ oo the uncertainty product goes to the asymptotic limit 2. The minimum of the uncertainty product, obtained for z = Zl = 0.859, is given by (AtAoJ)mi, = 1.1814.

(12)

This value has to be compared with the theoretical minimum 1.1802 . . . obtained by Hilberg and Rothe for the Weber signal. In Table 1 the most important properties are reported for the functions qx+(oJ) and 40+(t). Also the values are reported which result from the exact theory [1]. For the sake of completeness in the third column of Table 1 we report the properties of a gaussian signal. In Table 1 Ao~H means the Heisenberg variance 2[~--~]1/2. As stated by the Heisenberg uncertainty principle, the product AojHAt >-2; the value 2 is obtained only for a gaussian signal. For the Weber signal one has o5 = Wo (eq. (2)); moreover both the energy ratio Ev/Eo and the uncertainty product assume their minimum value for the same value Xmi, = 1.086375 (the X variable corresponds to q ~ z). For the nearly-optimum signal ¢+(oJ) the value 03, calculated for z = z l = 0 . 8 5 9 (the z-value corresponding to the minimum (12) of the uncertainty product) is 0.950~o~. The zx-value fixes the value of the parameter w~//21. We have in fact

¢ol/fll=zl =0.859.

(13)

At z = Zr, i, = 0.772 (the z-value corresponding to the minimum of the variational energy ratio E/Et, eq. (7)) the uncertainty product assumes the value 1.1841. In conclusion we have obtained a linear combination of two gaussian functions which closely approximates the Hilberg and Rothe uncertainty product. The Fourier transform is again a gaussian function multiplied by a cosine factor. Table 1 • 2 ~

Fourier transform

(a)

03 0

.1/2

-~

q~+(t)

eq. (10)

2(~/~/2)1/2 e-re'2/2

Aco

1.414214 ¢Oo

1.345 ¢01 (d)

((

A~oH

2.449490 ~Oo

2.327 col

(d)

~/2/2

2 1 -~

03

~oo

0.950 ~ l

(d)

/2/J~

At

0.834535/~Oo

0.878/¢oi

(d)

",/2//2

AcoAt

1.180212

1.1814

(d)

2(1 - ~2, /

2.043 -0.20383 0.29617 1.092 1.215

(d)

2 -----

A~oHAt Vmi n

(E~/Eo)min Xmi n

(b)

Xa

(c)

(a) (b) (c) (d) Signal

2.044185 -0.204947 0.295053 1.086375 1.086375

/2 = 0.852502/2

1/2

= 1.205621

the Fourier transform of D~ is not simply expressible. Xmi. is the X-value which corresponds to the minimum value of v (v+) and E_~/Eo (E/E1): Xml. = ~/2 zmin. Xx is the X-value which corresponds to the minimum value of A~oAt: Xx = ~/2 zl. evaluated for z = zl (or X = XI).

Processing

E. Borchi et al. / A nearly optimum signal

411

If one requires, as Hilberg and R o t h e do [2], for the centroid o3 the value 1, one has 0.950tol = 1, i.e. to1 = 1.053

(14)

and, from (13)/'1l = 1.226.

2.

Conclusions

It may be convenient to consider the signal @+(t) as an " e l e m e n t a r y signal" in the G a b o r sense and to apply to a signal F(t) the generalized G a b o r - H e l s t r o m integral transform [7] over @+-elementary signals. In this respect an application to the analysis of time-varying signals with small bandwidth-duration values, along the lines recently suggested by K o d e r a et al. [8], could be of some interest. W e only limit here to evaluate the G a b o r - H e l s t r o m transforms t ' +oo

g,(to, to) = |

Fi(t)w*(t, to, to) dt,

(i = 1, 2)

, 1 - oo

of a signal with a linearly increasing frequency Fx(t) = exp(i'rr/3t 2) and of a signal with a linearly varying frequency and a gaussian amplitude [8] F 2 ( t ) = exp[~rt:(-y+i/3)]; w is a (/'+-elementary signal centred at t = to

w(t, to, to) = @+(t-to) e x p [ i t o t - i ( a - ~)to/o] x and a,/3, 3' are arbitrary real constants. For this purpose we define the function

G(to, to; y, to1) = [A+(% to1) + i A - ( y , tox)] • exp[-B(to, to; y, tox)+iC(to, to; y, to1)] . {exp[H(to, to; y, to1) +iK(to, to; % to1)] + e x p [ - H ( t o , to; y, t o 1 ) - i K ( t o , to; y, to1)]} where

r 2.~/~n1[((2.~v + n~) ~ + (2xq3)2) 1/2 + (2"rr3~+ n 2 ) ] ] 1 / 2 [1 + e-('°Jn02][(2~y + .02) 2 + (2~/3) 2] J '

A±(~,, to~) = k

B(to, to; 3', t o 1 ) -

(2~r~, + n ])[(to - 2~/3to) 2 +,o2 + 2~r~,n ]to] + 8"rr~3Wo(to - ~Oto) 2[(2.tr.y +/2~)2 + (2~rO)2] , 4 2

2

2

2

C(to, to; % toi) = to(cz -~)to+ *r3(n~t°-to - t o t ) - t o t ° n l ( 2 " r r y + 0 2 ) (2~r3, + 02) 2 + (2~r/3) 2 H(to, to ; y, to1) = K (to, to; y, toa) -

to~Cto(2rr 3, + O21) - 2"tr/3to/'/~] (2.tr y +/22)2 + (2~r/3)2 ' 2~rtol[/3 (to - 2"rr3to) - yto(2~ry +/22)] (2~r~, + n~) 2 + (2,~0) 2

T h e n the G a b o r - H e l s t r o m transforms of F1 and F2 are given by

gl(to, to)=G(to, to;O, tol)

and

g2(to, to)=G(to, to; %tol). Vol. 3, No. 4, October 1981

412

E. Borchi et al. / A nearly optimum signal

F o r c o m p l e t e n e s s w e also r e p o r t t h e G a b o r - H e l s t r o m t r a n s f o r m s of F1 a n d F2 e v a l u a t e d with a gaussian e l e m e n t a r y signal w (t, to, oJ) = 2(/'21x/~) 1/2 e -(t-t°)2n~/2 exp[imt - i ( a - ½)oJto]. We obtain

gl(to, O~)=G(to, oJ;O,O),

g2(to, oJ)=G(to, oJ; y,O).

F i n a l l y we m e n t i o n a p o s s i b l e a p p l i c a t i o n of o u r a p p r o x i m a t e d W e b e r f u n c t i o n to n u m e r i c a l digital filtering. A t h o r o u g h o n e - a n d t w o - d i m e n s i o n a l s t u d y of this p r o b l e m is a c t u a l l y in p r o g r e s s .

References [1] E. Borchi and G. Pelosi, "A quantum mechanical derivation of the general uncertainty relation for real signals in communication theory", Signal Processing, Vol. 2, 1980, pp. 289-292. [2] W. Hilberg and P.G. Rothe, "The general uncertainty relation for real signals in communication theory", Information and Control, Vol. 18, 1971, pp. 103-125. [3] E. Merzbacher, Quantum Mechanics, Wiley, New York, 1970, pp. 21-26. [4] P.M. Morse and H. Feshbach, Methods of TheoreticalPhysics, McGraw-Hill, New York, 1953. [5] E. Borchi, V. Cappellini and P.L. Emiliani, "A new class of FIR digital filters using a Weber-type weighting function", Alta Frequenza, Vol. 44, 1975, pp. 469--470. [6] V. Cappellini, A.G. Costantinides and P.L. Emiliani, Digital Filters and their Applications, Academic Press, New York, 1978. [7] L.K. Montgomery and I.S. Reed, "A generalization of the Gabor-Helstrom transform", IEEE Trans. Information Theory, Vol. IT-13, 1967, pp. 344-345. [8] K. Kodera, R. Gendrin and C. de Villedary, "Analysis of time-varying signals with small BT values", IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-26, 1978, pp. 64-76.

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