Some Math of the PAR Problem

Transcription

1 Some Math of the PAR Problem Peter Oswald, Jaobs University Bremen Summer Shool Math & Comm July Introdution A pilot tone design problem and PAR optimization 2. Unimodular ase Littlewood-Erdoes onjetures, ultraflat polynomials 3. L - L estimates p q Optimal PAR not possible for general oeffiient moduli but PAR penalty only logarithmi in number of frequenies

2 Problem Note: For math onveniene, we use sqrt of what power means to the engineer! Frequeny domain Time domain { ( } 2 ) 1/ 2 P(t) N-1 0 max P(t) exp(it) Average power Pea power (relevant for RF omponents) L norm of P(t) L norm of P(t) 2 Pea power at least PAR min redued! Average power

3 Dirihlet ernel: PAR 4 Example: 1

4 Pilot tones for time synhronization Suppose, we send a speifi signal (nown to the reeiver) P(t) N-1 exp(it) 2π N-1 0 P(t - τ) P(t -s) ds 0 0 and it arrives time-delayed (by an unnown delay ). To synhronize at the reeiver, we perform onvolution 2 exp(i(s - τ)) and an use this autoorrelation funtion to detet if it is peay. E.g., the Dirihlet ernel τ τ P(t) N 1 0 exp(it), r : 2 1 or the so-alled Dolph-Cheyshev design an be used.

5 Dirihlet ernel (r 1) For this d hoie: Side lobe maximum Pea value P(0) d2π/n d as N goes to infty. Can one do better?

6 Yes: Dolph-Chebyshev design For same d hoie: Side lobe maximum Pea value P(0) 1/osh(π) Optimal oeffiients ome from Chebyshev polynomials.! One problem remains: All peay designs have lose to worst PAR value ost a lot of powering of RF omponents!

7 Same proedure for arbitrary r! All pilot tones of the form N 1 iα Q(t) r e e 0 it have the same autoorrelation funtion, an freely hoose real phases α for PAR minimization! Tal by Werner Henel Be aware: Real-life PAR redution is not as easy, too many onstraints on what an be hanged in a design.

8 Littlewood-Erdoes onjetures (harmoni analysis, number theory, probability, optimization, ) Set L K N N : : {P(t) {P(t) N -1 N exp(it) exp(it) Flatness onjeture (Littlewood 195x): For some a,a and all large N : : ± 1} 1} P(t) L N : 0 < a max P(t) / N A < Maybe, a,a 1 as N goes to infinity (ultraflatness)? PAR onjeture (Erdoes 1957, 1995: 100$ reward) P(t) L N : max P(t) / N A > 1

9 Known results Salem/Zygmund (1954): Average PAR is higher by a logarithmi fator. Rudin/Shapiro (~1958): Upper PAR bound A < 5. Deterministi onstrution. Littlewood (1961): Asymptotially, A < 1.36 using Gauss sums. Beller (1971): Similarly, A < Kahane (1980): Ultraflat polynomials exist if arbitrary phases are allowed! Probabilisti arguments. Be (1991): Flat polynomials exist for phases from disrete set {2πm/400}. Methods less probabilisti, don t wor for 400 replaed by 2! Odlyzo (199x): Extensive experiments, generally supportive of the flatness onjeture. Reent wor by Borwein, Erdelyi, Saffari, and others (e.g. Approximation Theory X, 2001, pp. 1-40)

10 PAR optimal polynomial from L 16

11 The L - l PAR problem p q Denote N 1 it p, q : max N min N 0 r R C : r 0 e L / r p l par + This is the worst-ase PAR problem (for arbitrarily given oeffiient moduli) in more general norms. Important speial ase: par log(n), N,2 Worst ase ahieved for launary sequenes. Fait: PAR problem is always deently solvable if all phases are at the disposal. q

12 1/q O(1)? log(n)? N What is nown? 1 O(1)? O(N 1/2 1/q ) 0 1 par p,q 1/p