Derivation of Gabor transform relations using Bessel's equality

Signal Processing 30 (1993) 257-262 Elsevier

257

Short communication

Derivation of Gabor transform relations using Bessel's equality* Richard S. Orr Atlantic Aerospace Electronics Corporation, 6404 Ivy Lane, Suite 300. Greenbelt, MD 20770, USA Received 26 August 1991 Revised 9 April 1992 and 25 June 1992

Abstract. Derivation of the key formulas of Gabor representation is usually facilitated by relationships taken from Fourier and Zak transform theory. We show that several of these fundamental formulas can be obtained directly from Bessel's equality for biorthogonal systems, aided only by the shift theorem for inner products of Gabor basis functions. The method is brought to the attention because it simplifies derivations in some cases and may be useful in finding new results in time frequency representation theory.

Zusammenfassung. Die Herleitung der Grundgleichungen f/Jr die Gabor Darstellung erfolgt im allgemeinen mit Hilfe der Fourier- und Zak-Transformationstheorie. Wir zeigen, dab mehrere dieser Grundgleichungen direkt von Bessels Beziehungen f/Jr biorthogonale Systerne abgeleitet werden k6nnen, wobei nur das Verschiebungstheorem f/ir innere Produkte yon Gabor Basisfunktionen hinzukommt. Auf diese Methode wird aufmerksam gemacht, da die Ableitung in einigen Fallen vereinfacht wird und sie mSglicherweise niitzlich bei der Herleitung neuer Resultate in der Theorie der Zeit Frequenz-Darstellung sein mag.

R6sum6. La d~rivation des formules cl6s de la repr6sentation de Gabor est habituellement facilit6e par les relations prises de la thborie des transform6es de Fourier et de Zak. Nous montrons que plusieurs de ces formules fondamentales peuvent &re obtenues directement ~ partir de l'6galit6 de Bessel pour des syst6mes bi-orthogonaux, compl&6e seulement par le th6or6me de d~calage pour les produits scalaires de fonctions de Gabor de base. Cette m6thode est mise en lumi6re parce qu'elle simplifie les d6rivations dans certains cas et peut 6tre utile pour d+couvrir de nouveaux rbsultats en th6orie de la repr+sentation temps fr6quence.

Keywords. Gabor transform; biorthogonal; Zak transform; Bessel's equality; ambiguity function.

Correspondence to: Dr. Richard S. Orr, Atlantic Aerospace Electronics Corporation, 6404 Ivy Lane, Suite 300, Greenbelt, MD 20770, USA. Tel.: (301) 982-5215. * This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under contract No. F49620-90-C-0016. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon. 0165-1684/93/$05.00 © 1993 Elsevier Science Publishers B.V. All rights reserved

R.S. Orr / Gabor transform relations using Bessel's equality

258

I. Introduction In analytic geometry we learn that the dot product of vectors a and b in three-dimensional space is found by summing the pairwise products of their coordinates :

a . b = (a~u~ + ayUy+ a:u:) • (b~u:,.+ byuy + b:u:) = a~bx + ayby + a:b-.

( 1)

If we survive through linear algebra to functional analysis, we learn that this neat little rule is just a special case of something called Bessel's equality, which speaks about inner products in spaces with an L2 norm:

( f i g ) = ~ ( f [ qS.)(c~. [g) n

= Z (f]dP.)(g]q3.) *.

(2)

n

In (2), the functions {~b.} are any complete, orthonormal basis for the space. If the {~b.} are incomplete, there is the Bessel's inequality for g =f:

< f l f ) = I)fl12~ [ [e,

In a finite dimensional space the { ~'n} exist as long as the {0n} are linearly independent. Use of (5) puts the representation o f f into the form

f = ~ (f[~n)O~.

The roles of { 0n} and { ~'n} are perfectly symmetric, and can be interchanged in (6):

f=~ (f[O,)~,.

We are now just steps away from Bessel's equality for complete biorthogonal systems [5]"

( f i g ) = Z ( f [ 0,)(u/, ]g).

This equation facilitates derivation of some of the basic relationships in the theory of Gabor expansions, although it appears not to have been much used in this way. In this note we show that several key results of Gabor theory can be obtained from (8) without recourse to the Zak or Fourier transform relations that usually enable the derivations. This is prefaced by a short primer on the Gabor transform.

A Gabor representation of a time function f(t) is a series expansion of the following form [1]"

f(t)=

~ m=--oo

g=Z(glc~m)¢m,

( 0 . ) ~,m) = 6.,,.. Processing

am,~W(t-nT)exp(j2~mt/T), n=--oo

(9)

(4)

m

we can readily put ( f i g ) into the form (2). Not all complete bases are orthogonal. If the set {0~} spans the space of interest but is not orthogonal, we can still use it as a basis for expansions. One option is to become a closet Gram-Schmidt orthogonalist, forcing things to be more familiar. Alternatively one sticks with the {0~} and finds expansion coefficients using a biorthogonal set { ~,}, which has the distinguishing property

Signal

(8)

n

2. Summary of the Gabor transform

which says that the total energy in the components cannot exceed the signal energy. We will not be concerned with incomplete bases here. If we express f and g in the complete basis {~b.},

n

(7)

n

(3)

n

f = Z ( f [ ~b.)~b.,

(6)

n

(5)

where the {am,.}, m, n~Y, are the Gabor coefficients, and w(t) the window. The translates of w(t) form the Gabor basis {Wm,.(t)}, as follows'

wm,.(t) = w ( t - n T ) exp(j2rtmt/T),

m, n~27.

(10) The grid of time-frequency points, (t.,f.,) = (nT, re~T), over which w(t) is displaced defines a unit area cell of dimension (T× l / T ) in the time frequency plane. In general, the {w,.,~(t)} are nonorthogonal, and (9) is a nonorthogonal expansion. One way to

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R.S. Orr / Gabor transform relations using Bessers equality

invert the expansion uses a related set of biorthogonal functions {b,,,n(t)},

bm,n(t)=b(t-nT) exp(j2nmt/T),

m, n(EY_, (11)

generated from a function b(t) having the property

(w .... t b,,,~>

=

w(t-nT)b*(t-qT) x exp[j2rc(m-p)t/T] dt=~m,pS.,q.

L

f(t)b*m,,(t)dt.

1

(17)

TZw*(v, r) "

When the {Win,,} constitute a frame, that is, when the following is true for any feL2(N) and 0
m,n

(12)

(13)

3. Existence of the biorthogonal function

IoT [,/v Zf(v, r) J0

(18)

it can be shown that [3]

A

IZw(v, r) I2
(19)

The 'tighter' the frame - equivalently, the smaller the ratio B/A - the closer to orthogonal are the basis vectors, and the less ill-conditioned is the expansion. Using (17) we can also express the frame condition through the biorthogonal function 1

Some comments on the existence of the biorthogonal function are in order. One can alternatively write an expression for the Gabor coefficients in terms of the Zak transforms of the signal and window as follows [ 1] :

"m'°=

Zb(v, r ) -

AIITII2~<~ I(f[Wm,,)12<~nllfll 2,

When such b(t) exists, (12) permits anaysis o f f into its coefficients by inner products o f f ( t ) and the {bm..(t)} according to the formula

am,n= (f[bm.n> =

One can easily show that the Zak transform of the biorthogonal function has the formal expression

1

--<~ IZb(v, v) 12,~< . BT AT

(20)

If w is an L2(R) function, we can see from the inner product relationship (16) that b is one also, since its Zak transform is bounded over [0, T) x [0, 1/ T). A good example of such a case has been presented by Friedlander and Porat [4], where the window is the one-sided decaying exponential,

w(t) = x / ~ exp(-at)u(t),

(21)

and the corresponding biorthogonal function is where the Zak transform of a functionf(t) is given by

Zf(v, r) = •

k =-oc

f(kT+ r) exp(-j2xkvT).

(15) The Zak transformation is a unitary operator, preserving inner products according to

fOT Zf(v, rlZ*g(v, r) dr dv (ZfIZg> = -~I/T 0 =IT(fig>.

(16)

b(t)=

I 2~lalTexp(at), -,

-T~
~ 1 q~aa Texp(at),

~

0
k 0,

elsewhere.

(22)

If, on the other hand, Zw has a zero, good behavior of the biorthogonal function is far from assured. The integrand Zf(v, r)/Zw(v, r) must have a Fourier series that converges to the periodic extension of Zf/Zw on [0, 1/T)× [0, T), which requires that Zf/Zw and its first partials be piecewise continuous over the integration region. If Zw has a zero at some interior point, Zf/Zw Vol. 30. No 2. January 1993

R.S. Orr / Gabor transform relations using Bessel's equality

260

approaches + ~ at the point (unless by happenstance f has a zero at the same point) and cannot be continuous in any interval containing the point. It is also true that if w is an L2(~) function for which Zw has a zero, the corresponding b will not be L2(~). The Gaussian function employed by Gabor fits the latter category. Its Zak transform has a single zero; the biorthogonal function, first calculated by Bastiaans [2], belongs to LI([~) but not to L2(~), and provides a very poor basis for numerical calculations. This close association of the Gaussian window with the Gabor expansion is possibly a reason that these expansions have not previously been prominent in the signal processing literature. If one selects windows whose Zak transform magnitudes are bounded above and below by positive numbers, the biorthogonal function is well behaved and can be calculated accurately from the window function.

4. Derivations using the Bessel equality Let {Wm,~}, {bm,n} be a biorthogonal pair, and write Bessel's equality in the form ( f i g ) = )-', (fibp,q)(Wp,qlg),

(23)

P,q

We can solve (26) for the Gabor coefficients using the alternative form of Bessel's equality:

(f[g)= Y" ( f ]

(27)

Wp,q)(bp,q ] g ) .

P,q

Set g = bin.,, in (27), yielding

a,,,,,,= Z (f[ Wp,q)(b Ibm_p.,,_q).

(28)

p,q

Equation (28) which says that the Gabor coefficients can be recovered by convolving the sampled short-time Fourier transform (STFT) o f f with the sampled auto-ambiguity function of the biorthogonal function - was obtained in one step from (27), but is rarely presented in this way. In a more surprising development, we find that we can obtain a special case of a result in the WeylHeisenberg frame theory via just the Bessel equality. In frame theory, the quantity ] ( f [ 0,)12, where { 0n} is a frame, plays a key role. Its behavior governs the extent to which the frame mapping { ( f ] 0 n ) } preserves energy, as described by the frame constants A and B in (18). In a WeylHeisenberg frame, one can exploit a Fourier transform property of the cross-ambiguity function A (x, y)(v, r),

A(x,y)(v, r) =

x(t)y*(t+ v) exp(-j2gvt) dt, oo

(29)

and let g = Wm.n:

(f[Wm,n)=~, (flbp,q)(Wp,q[Wm,n).

(24)

as follows [6]:

P,q

The inner product in (24) representing the sampled ambiguity function of w is invariant to a common shift in the time or frequency indices of its two components:

(Wp.qlWin,n)= (W l Wm-p,n-q).

(25)

Using this, (24) becomes (f]

Win,n)= Z ap,q(W]Wm-p,n-q). p,q

(26)

Without passing through any Zak transforms, we have obtained the convolution formula for the Gabor coefficients. Signal Processing

f f l A ( x , y)(v, r)[ 2 exp[jZTz(vt +fr)] dr dv = A(x)( t, - f )A*( y)( t, - f ).

(30)

In the above, the shorthand A(x) is used for the auto-ambiguity function A(x, x). Coupled with the Poisson summation formula in two dimensions, one has [7, 8]

~m~ A(x'Y)(~T'nT) 2 =EZA(x ) m

n

n,mK

A (y)

,inK

.

(31)

R.S. Orr / Gabor transform relations using Bessers equality In (31), K denotes the density of basis functions per unit area in the time-frequency plane, which must satisfy K>~ 1 for the basis to be complete. This formula can be used to prove theorems about the structure of frames and to develop bounds for the frame constants. The lattices over which the summations occur differ on the two sides of (31). On the left side the lattice is ~ = (m/KT, nT), whereas on the right it is KZ-fold more dense: ~q'=(n/T, mKT). In the Gabor case, where K = 1 and ~ = 5 °*, we can prove, without recourse to the remarkable theorem (30), the following version of (31):

Z [(flgm,.)12=~(flL,~)(glg~,o)*. (32)

m,n

u,v

g and the interior indices match:

(gm.n [bp,q) = (g lbp_,.,q_n), (wr,~[g,...) = (Wp-m,q-n [gp-r,q-s).

(36)

This transforms (34) to

E [(f[gm.n)[ 2

m,n

= E E Z (flwp,q)(bp,qlfp-r,q-~) m,n p,q r,s

× (glbp ,.,q_,,) * (Wp_,.,q_. ]gp_r.q_D *. (37) Now define new index variables,

u=p-r,

v=q-s,

x=p-m,

y=q-n,

(38)

and eliminate m, n, r and s, on the right-hand side of (37), to get

PROOF. Starting with Z [(flgm,n)[2=~ ( f , gm..)(flgm,.)*,

m,n

261

m,n

I ( f l g m , . ) I2 (33)

express the two inner products via the two symmetric forms of the Bessel equality:

re/,n

= ~, E E (flwp,q)(bp,qlfu,v) u,v p,q x,y

× (glbx,y) (wx.ylgu,~)

Y" [(flgm..)l 2

m,n

= ~ I Z (flWp,q) (bp,qlf.,v) 1 u,u p,q

=2Z m,nli,q~flwp,q)~bp,qlgm,n)

1

X[x,~,(glbx,y)(Wx,Ylgu,v)]*

[r,,(f [br,s)(Wr,slgm,n)]*

= E (flf~,v) (g [gu,o) *,

× 2

= E E E (flWp.q)(br,s[f)

which reproduces (30).

m,n p,q r,s

× (gm,n Ibp,q) *
*.

(34)

Now apply the index shift theorem to the second factor in (34) so that the associated functions in the biorthogonal system will have the same index,

(br,s[f) = (bp,qlfp r,q-s),

(39)

u,v

(35)

and apply the same formula to the third and fourth factors so that the left bracket of the third becomes

It is not clear that the work involved in getting to (39) is any less than that required to prove both the Fourier transform property of the magnitudesquared cross-ambiguity function and the Poisson sum formula, and it is not obvious that the method extends to the arbitrary Weyl-Heisenberg case. But it is somewhat remarkable that we can get there at all knowing only Bessel's inequality and the shift theorem for inner products of Gabor functions. Vol. 30, No. 2, January 1993

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R.S. Orr / Gabor transform relations using Bessers equality

5. Conclusions T h e e x a m p l e s p r e s e n t e d here illustrate the utility o f Bessel's e x p a n s i o n in o b t a i n i n g G a b o r - t h e o r e t i c r e l a t i o n s h i p s w i t h o u t the usual i n t e r m e d i a t i o n o f Z a k a n d F o u r i e r t r a n s f o r m s . P e r h a p s this tool will be revisited m o r e often in t i m e - f r e q u e n c y theory, either to m o r e c o n v e n i e n t l y r e p r o d u c e k n o w n results or to discover new ones.

[2] M.J. Bastiaans, "A sampling theorem for the complex spec-

[3] [4]

[5] [6]

Acknowledgment [7] I w o u l d like to t h a n k the a n o n y m o u s reviewers for their c o m m e n t s , especially those c o n c e r n i n g the existence o f the b i o r t h o g o n a l function, which p r o m p t e d the inclusion o f Section 3.

References [1] L. Auslander, I. Gertner and R. Tolimieri, "The discrete Zak transform application to time frequency analysis and synthesis of non-stationary signals", IEEE Trans. Signal Process., Vol. 39, No. 4, April 1991, pp. 825-835.

Signal Processing

[8]

trogram, and Gabor's expansion of a signal in Gaussian elementary signals", Optical Engrg., Vol. 20, No. 4, July/ August 1981, pp. 594 598. I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis", IEEE Trans. Inform. Theory, Vol. 36, No. 5, September 1990, pp. 961 1005. B. Friedlander and B. Porat, "Detection of transient signals by the Gabor representation", IEEE Trans. Acoust. Speech Signal Process., Vol. 37, No. 2, February 1989, pp. 169 180. W. Schmeidler, Linear Operators in Hilbert Space, Academic Press, New York, 1965, Chapter 5, pp. 36-42. S.M. Sussmann, "Least-square synthesis of radar ambiguity function", IRE Trans. Inform. Theory, Vol. IT-8, April 1962, pp. 246 254. R. Tolimieri and R.S. Orr, "Characterization of Weyl Heisenberg frames via Poisson summation relationships", Proc. Internat. Conf. Acoust. Speech Signal Process. 92, Vol. 4, Digital Signal Processing 1, pp. 277 280. R. Tolimieri and R.S. Orr, "Poisson summation, the ambiguity function and the theory of Weyl Heisenberg frames", IEEE Trans. Inform. Theory, to appear.