Weighted Norms of Ambiguity Functions and Wigner Distributions

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1 Weighted Noms of Ambiguity Functions and Wigne Distibutions Pete Jung Faunhofe Geman-Sino Lab fo Mobile Communications (MCI) and the Heinich-Hetz Institute axiv:cs/06007v4 [cs.it] 7 Jul 006 Abstact In this aticle new bounds on weighted -noms of ambiguity functions and Wigne functions ae deived. Such noms occu fequently in seveal aeas of hysics and engineeing. In ulse otimization fo Weyl Heisenbeg signaling in wide-sense stationay uncoelated scatteing channels fo examle it is a key ste to find the otimal wavefoms fo a given scatteing statistics which is a oblem also well known in ada and sona wavefom otimizations. The same situation aises in quantum infomation ocessing and otical communication when otimizing ue quantum states fo communicating in bosonic quantum channels, i.e. find otimal channel inut states maximizing the ue state channel fidelity. Due to the non-convex natue of this oblem the otimum and the maximizes itself ae in geneal difficult find, numeically and analytically. Theefoe ue bounds on the achievable efomance ae imotant which will be ovided by this contibution. Based on a esult due to E. Lieb [], the main theoem states a new ue bound which is indeendent of the wavefoms and becomes tight only fo Gaussian weights and wavefoms. A discussion of this aticula imotant case, which tighten ecent esults on Gaussian quantum fidelity and coheent states, will be given. Anothe bound is esented fo the case whee scatteing is detemined only by some abitay egion in hase sace. I. INTRODUCTION Time-fequency eesentations ae an imotant tool in signal analysis, hysics and many othe scientific aeas. Among them ae the Woodwad coss ambiguity function Ãgγ(τ,ν), which can be ined as ( denotes comlex conjugate) Ã gγ (τ,ν) = g(t τ )γ(t+ τ )e iπνt dt () and the Wigne distibution W gγ (τ,ν) W gγ (τ,ν) = g(τ + t )γ(τ t )e iπνt dt () whee the functions g,γ : R C assumed to be in L (R). Both ae elated by W gγ (τ,ν) = Ãgγ (τ,ν) whee γ (t) = γ( t). Hence all esults which will esented late on aly on Wigne functions as well. Due to non commutativity of the shifts in τ and ν (in hase sace) thee exists many Which can be elaxed to othe saces by the Hölde inequality initions of these functions which diffe only by hase factos. In consideing noms only, the ambiguities due to these hase factos ae not imotant. To be consistent with the evious wok in [], [3] in this aticle the altenative inition A gγ (x) = g,s x γ = g(t)(s x γ)(t)dt (3) is used, whee S x is the time-fequency shift oeato given as (S x f)(t) = e iπxt f(t x ) (4) and x = (x,x ) R. Note that A gγ (x) = e iπxxã gγ( x, x ). These oeatos establish u to hase factos an unitay eesentation of the Weyl Heisenbeg gou on L (R) the so called Schödinge eesentation (see fo examle [4]). They equal (again u to hase factos) the so called Weyl oeatos (Glaube dislacement oeatos), i.e. efom hase sace dislacements in one dimension. It is an imotant and in geneal unsolved (non convex) oblem in many fields of hysics and engineeing to find nomalized function g and γ such that the following integal g,s x γ C(x)dx = A gγ (x) C(x)dx (5) is maximized whee C(x) could be some obability distibution and dx is the Lebesgue measue on R. Fo examle in ada and sona alication (5) is tyically elated to the coelation esonse with some filte g of a tansmitted ulse γ afte assing though a non-stationay scatteing envionment chaacteized by some C( ). This fomulation is obtained fo so called Weyl Heisenbeg signaling in wide-sense stationay uncoelated scatteing (WSSUS) channels [5], [], [6], [7] whee C( ) is called the scatteing function. If consideing γ as a obability wave function in quantum mechanics (5) can be consideed also as its ovela with some wave function g afte seveal hase sace inteactions.

2 In quantum infomation ocessing (5) is tyically witten as ue state fidelity (5) = T{Π g S x Π γ S xc(x)dx} = T{Π g A(Π γ )} (6) whee Π f = f f, f is the ank-one ojecto onto f and T( ) denotes the tace functional. The middle tem in (6) is the Kaus eesentation [8] of a bosonic quantum channel A( ) [9], [] which mas the inut stateπ γ (ank one density oeato) to the outut A(Π γ ). Minimizing the obability of eo P e = T{Π g A(Π γ )} (see fo examle [0]) fo ank one measuements is then the maximization of the ue state fidelity, i.e. the following otimization oblem: maximize g,γ T{Π g A(Π γ )} (7) Fo eachγ the oeatoa(π γ ) is a ositive semi inite tace class (thus comact) oeato, hence (7) likewise eads maximize λ max(a(x)) (8) TX=,X 0 whee the ank elaxation follows fom convexity of the maximal eigenvalueλ max ( ) and linea convexity of A( ) (see fo examle [], [3]). In a slightly moe geneal context this ae consides A gγ C which diectly gives the weighted noms of ambiguity functions in the fom of ( ) / A gγ,c = A gγ (x) C(x)dx = A gγ C / (9) whee C : R R + is now some abitay weight function. Fo = the esults match then the examles given so fa. Note futhemoe that this toic is also connected to Rényi entoies H() of time fequency eesentations H() = log A gγc (0) i.e. a measue of time fequency infomation content []. II. MAIN RESULTS The esults ae oganized in a main theoem esenting the geneal ue bound to A gγ C. Then, two secial cases ae investigated in moe detail. The fist is dedicated to the oveall equality case in the main theoem and imotant fo Gaussian bosonic quantum channels. The second case discusses anothe alication elevant situation motivated by WSSUS ulse shaing in wieless communications. But befoe stating, the following initions ae needed. Definition Let 0 < <. Fo functions f : R C and F : R C ( ) / ( ) / f = f(t) dt F = F(x) dx ae then the common notion of noms, whee dt and dx ae the Lebesgue measue on R and R. Futhemoe fo = is f = ess su f(t) F = ess su F(x) If f is finite f is said to be in L (R) (similaly if F is finite, F is said to be in L (R )). Fo discussion of the equality case fo the esented bound the fomulation to be Gaussian fo functions f : R C and F : R C is needed. Definition Functions f(t) and F(x) ae said to be Gaussian if fo a,b,c,c C, A C and B C f(t) = e at +bt+c and Re{a} > 0 and A A > 0. F(x) = e x,ax + B,x +C () Two Gaussians f(t) and g(t) ae called matched if they have the same aamete a. The main ingedient fo the esented analysis is the following theoem due to E. Lieb [] on (unweighted) noms of ambiguity functions. Theoem 3 (E. Lieb) Let A gγ (x) = g,s x γ be the coss ambiguity function between functions g L a (R) and γ L b (R) whee = a + b. If < < with q = a and q b, then holds A gγ H(,a,b) g a γ b () ( ) /q, whee H(,a,b) = c q ca/q c b/q c /q c = /() q /(q). Equality is achieved with g and γ being Gaussian if and only if both a and b > /( ). In aticula fo a = b = A gγ g γ (3) Actually Lieb oved also the evesed inequality fo <. Futhemoe, fo the case = it is well known that equality holds in (3) fo all g and γ. Then the otimal sloe (elated to entoy) A gγ (x) ln A gγ (x) dx (4) at = is achieved by matched Gaussians []. Fo simlifications it is assumed fom now that g = γ =. With the evious eaations the main theoem in this aticle is now:

3 Theoem 4 Let A gγ (x) = g,s x γ be the coss ambiguity function between functions g,γ with g = γ = and, R. Futhemoe let C( ) L q (R ) with q =. Then A gγ C holds fo each max{, }. ( ) C Poof: In the fist ste Hölde s inequality gives (5) A gγ C A gγ C q (6) fo conjugated indices and q, thus with = + q. Equality is achieved fo < < if and only if thee exists λ R such that C(x) = λ A gγ (x) ( ) (7) fo almost evey x. Simila conclusions fo = and = ae not consideed in this ae. Lieb s inequality in the fom of (3) fo A gγ gives fo hs of (6) A gγ C q = A gγ C q = ( ( ) A gγ ) C q C q (8) The latte holds fo evey, thus the case = is now included as aleady mentioned befoe. Equality in (8) is achieved if g and γ ae matched Gaussians. Futhemoe if stictly >, equality in (8) is only achieved if g and γ ae matched Gaussians. Relacing q = gives the desied esult. Note that aat fom the nomalization constaint the bound in Thm.4 does not deend anymoe on g and γ. Hence fo any given C( ) the otimal bound can be found by ( ( ) ) min C (9) R max{, } In the minimization has to be foced fo Hölde s inequality and fo Lieb s inequality. Two secial cases ae investigated now in moe detail which ae elevant fo alication. Fist the oveall equality case in Thm.4 is consideed. Coollay 5 Let C(x) = αe απ(x +x ) with R α > 0. Then fo each max{, } holds A gγ C ( ) α ( ) (0) The best bound is given as α A gγ α+ α C α ( /) / else Fo α equality is achieved at = α if g and γ ae matched Gaussian, i.e. then holds. A gγ C = α α+ () + if and only () Poof: The moments of L nomalized two dimensional Gaussians ae given as ( ) s s C s = α s (3) s Accoding to Thm.4 the ue bound ( ) ( f() α = C = ) ( ) (4) holds fo each max{, /}. The otimal (minimal) bound is attained as some oint min which can be obtained as min f() = f( min ) (5) R max{, } The fist deivative f of f at oint is f () = f() ln(( ) ) (6) α Thus f ( min ) = 0 gives only one stationay oint min ( min ) = min = α + > (7) α Due to f()/ > 0 and stict monotonicity of ln( ) follows easily that f ( min +ǫ) > 0 > f ( min ǫ) fo all ǫ > 0, hencef attains a minimum at min. The constaint min is stictly fulfilled fo evey allowed α and, hence the solution is feasible ( min ) if α. Then the otimal (minimal) bound is f( min ) = α α+ (8) Fo the infeasible case instead, i.e. fo 0 < α <, follows that minimal bound is attained at the bounday oint =. Thus f(/) = α ( /) /. Summaizing, min R max{, } f() = α α+ α α ( /) / else (9) is the best ossible ue bound. It emains to investigate the conditions fo equality. Lieb s inequality is fulfilled with equality if stictly min > and g,γ ae matched Gaussians. In this case follows A gγ (x) = e π (ax + α x )+ B,x +C (30)

4 fo some B C, a,c C and Re{a} > 0, thus A gγ ( ) is a two dimensional Gaussian. Next, to have equality in (6) > and equation (7), which is in this case A gγ (x) = e Re{ π (ax + α x )+ B,x +C} = λαe απ ( ) x = λ C(x) ( ) (3) have to be fulfilled fo almost evey x. Thus, it follows that Re{B} = (0,0), λαe Re{C} =, Re{a} = and most imotant again = α +. But, this is obviously also the minimum if α, hence in this and only this case equality is achieved. It is emakable that the sha if and only if conclusion fo Gaussians holds now fo α. Lieb s inequality alone needs α > but in conjuction with Hölde s inequality this is elaxed. The esults ae illustated in Fig.. Futhemoe note that fo = evey min is feasible. This esult is imotant fo so called bosonic Gaussian quantum channels [9], [], i.e. C( ) is a two dimensional Gaussian. In othe wods, accoding to (6) and C( ) as in Coollay 5, the solution of the Gaussian fidelity oblem [3], [3] is max T{Π ga(π γ )} = max λ max(a(x)) g,γ TX=,X>0 = α α+ (3) with Gaussian g and γ as aleady found in [3] using a diffeent aoach. But now this states the stong oosition that maximum fidelity is achieved only by coheent states = =.9 α/(α+) α / ( /) / combined bound /α Fig.. Nom bounds fo Gaussian weights: Both functions in () seaately and the combined vesion ae shown fo = and =.9. In ada and sona alications and also fo wieless communications the following ue bound is imotant. It is elated to the case whee scatteing occus with constant owe in some egion of hase sace (in this context also called time fequency lane). Fo examle in wieless communications tyically only the maximal disesions in time and fequency (maximum delay sead and maximum Dole sead) ae assumed and/o known fo some ulse shae otimization. Those situations ae coveed by the following esult: Coollay 6 Let U R a Boel set, U < and C(x) = U χ U(x) its L nomalized chaacteistic function. Then fo each max{, } holds A gγ C < ( ) U (33) It is not ossible to achieve equality. The shaest bound is e U A gγ e U e/ C < ( ) / (34) else whee = max{,}. U Poof: The oof is staightfowad by obseving that Accoding to Thm.4 follows C s = U χ U s = U s s (35) ( ) ( ) f() = C = = e U ln (36) U Equality is not ossible because Thm.4 equies C to be Gaussian fo equality. The otimal vesion is obtained by minimizing the function f() unde the constaint max{,/}. The fist deivative f of f at oint is f () = f() ( ln( ) ) (37) U Thus f ( min ) = 0 gives the only oint min = e U. The function f() is log-convex on (0, e3/ U ] = I. That is h() = lnf() = (ln U )/ is convex on I, because h () = U (ln ) h () = U 3( ln +3) (38) shows, that h () 0 fo all I. Hence f() is convex on I. Obviously min I, hence this oint is in the convexity inteval and theefoe must be the minimum of f. Futhe, this value is also feasible if still min max{,/} = / whee = max{,}, i.e. U < e (39)

5 has to be fulfilled. Then the desied esult isf( min ) = e U e. If min < /, i.e. is infeasible, the minimum is attained at the bounday, i.e. at = /. Thus ( ) / f( /) = (40) U The esults ae shown in Fig. fo =,, 3. Fo the inteesting case = the esult futhe simlifies to A gγ e U e U e C < (4) U else Examle: When using the WSSUS model [4] fo doubly undesead channels = =3 e U /e (/( * U )) (/* ) combined bound U Fig.. Nom bounds fo U χ U weights: Both functions in (34) seaately and the combined vesion ae shown fo =,,3. disesive mobile communication channels one tyically assumes time fequency scatteing with shae U = {(x,x ) 0 x τ d, x B d } (4) with B d τ d < e, whee B d denotes maximum Dole bandwidth B d and τ d is maximum delay sead. Then (34) edicts, that the best (mean) coelation esonse ( = ) in using filte g at the eceive and γ at the tansmitte is bounded above by A gγ C < e B d τ d e (43) = III. CONCLUSIONS In this contibution new bounds on weighted noms of ambiguity functions and Wigne distibutions ae esented which only deend on the shae of the weight function. Futhe the imotant equality case is discussed which is attained only by Gaussian weights and wave functions. The esults ae imotant in the field of wavefom otimization fo non stationay envionments as needed fo examle in WSSUS channels. This channel model is fequently used in ada and sona alications and of couse in wieless communications. Futhemoe these noms ae needed in quantum infomation ocessing fo bosonic quantum channels because they ovide insights on achievable fidelities in those quantum channels. In the secial case of the Gaussian quantum channel they ovide also the otimum inut states, i.e. only coheent states achieve this otimal fidelity as fequently conjectued. Hence, in the mentioned fields the esults establish limits on achievable efomance. IV. ACKNOWLEDGMENTS I would like to thank Igo Bjelakovic fo many useful discussions. REFERENCES [] E. Lieb, Integal bound fo ada ambiguity functions and Wigne distibutions, J.Math.Phys, vol. 3, no. 3, Ma 990. [] P. Jung and G. Wunde, The WSSUS Pulse Design Poblem in Multicaie Tansmission, submitted to IEEE Tans. on. Communications. [Online]. Available: htt://axiv.og/abs/cs.it/ [3], A Gou-Theoetic Aoach to the WSSUS Pulse Design Poblem, 005 IEEE Intenational Symosium on Infomation Theoy, 005. [Online]. Available: htt://axiv.og/abs/cs.it/05008 [4] G. B. Folland, Hamonic Analysis in Phase Sace. Pinceton Univesity Pess, 989. [5] W. Kozek, Matched Weyl-Heisenbeg exansions of nonstationay envionments, PhD thesis, Vienna Univesity of Technology, 996. [6] K. Liu, T. Kadous, and A. M. Sayeed, Othogonal Time-Fequency Signaling ove Doubly Disesive Channels, IEEE Tans. Info. Theoy, vol. 50, no., , Nov 004. [7] W. Kozek, Nonothogonal Pulseshaes fo Multicaie Communications in Doubly Disesive Channels, IEEE Jounal on Selected Aeas in Communications, vol. 6, no. 8, , Oct 998. [8] K. Kaus, States, effects and oeations, fundamental notions of quantum theoy. Belin: Singe, 983. [9] A. Holevo, Pobabilistic and statistical asects of quantum theoy. Noth Holland, 98. [0] C. Helstom, Quantum Detection and Estimation Theoy. Academic Pess, 976. [] R. Baaniuk, P. Flandin, A. Janssen, and O. Michel, Measuing timefequency infomation content using the Renyientoies, IEEE Tans. Info. Theoy, vol. 47, no. 4, , May 00. [] M. Hall, Gaussian Noise and Quantum-Otical Communication, Phys.Rev. A, vol. 50, no. 4, , Oct 994. [3] C. Caves and K. Wodkiewics, Fidelity of Gaussian Channels, Oen Systems and Infomation Dynamics, vol., , Se 004. [Online]. Available: htt://axiv.og/abs/quant-h/ [4] P. Bello, Chaacteization of andomly time vaiant linea channels, Tans. on Communications, vol., no. 4, , Dec 963.

c( 1) c(0) c(1) Note z 1 represents a unit interval delay Figure 85 3 Transmit equalizer functional model

c( 1) c(0) c(1) Note z 1 represents a unit interval delay Figure 85 3 Transmit equalizer functional model Relace 85.8.3.2 with the following: 85.8.3.2 Tansmitted outut wavefom The 40GBASE-CR4 and 100GBASE-CR10 tansmit function includes ogammable equalization to comensate fo the fequency-deendent loss of the