Copyright license. Exchanging Information with the Stars. The goal. Some challenges
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1 Copyright license Exchanging Information with the Stars David G Messerschmitt Department of Electrical Engineering and Computer Sciences University of California at Berkeley messer@eecs.berkeley.edu Talk at: SETI Institute 30 March 0 c Copyright David G. Messerschmitt, 0. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this License, visit or write to Creative Commons, 559 Nathan Abbott Way, Stanford, California 9305, USA. The goal Some challenges Exchanging information with other solar systems would be an exciting voyage The capabilities and limitations of our Universe to support such exchanges is little understood This work is a first step toward such understanding No experimentation Relevant astronomical observations No coordination
2 Implicit coordination Some distinctions Design guidance based on: Simplicity: Occam s razor Fundamental limits and resulting optimization Where physical impairments are least controlling Assumptions about capabilities and resources Awareness of motivations and incentives Attractor beacon vs. information-bearing signal Discovery vs. ongoing communication This talk focuses on: Radio frequencies Design of an information-bearing signal Receiver design for discovery of that signal Impairments addressed in this talk Complex-valued baseband equivalent signal White noise Radio-frequency interference Dispersion in the ionized interstellar medium (IISM) Real-valued passband 0 f c f c +W Demodulation Complex-valued baseband 0 W
3 Digital modulation alternatives Discovery options Focus on the complex baseband signal: Data symbols {B k } Amplitude modulation: { B k h(t k T s ), < k < } Orthogonal signaling: { h Bk (t k T s ), < k < } Multiple-symbol: Make additional assumptions about data symbol alphabet Symbol-by-symbol: Single symbol waveform h(t) multiplied by some unknown amplitude and phase Here we pursue the symbol-by-symbol option: Applies to all modulation alternatives Potentially forgos signal energy Time-frequency support for h(t) Received signal impairments Transmitter: Frequency W What should W and T be? What other properties should h(t) have? Temporarily consider only: White Gaussian noise Radio interference in the vicinity of the receiver Receiver: T Time How advantageous is it to know more about h(t)? Optimization infers specific and credible properties for W, T, and h(t) How does the receiver infer this knowledge?
4 Two orthonormal bases An orthonormal basis renders the reception finite-dimensional: Fourier series (time-limited signal) Sampling theorem (bandlimited signal) Finite-dimensional representation of h(t) Regardless of basis: Noise is completely random and isotropic Choice of basis: Transmitter and receiver must assume the same basis We choose the Fourier series Dimensionality of basis: Degrees of freedom (DOF) is K = W T Isotropic noise Radio-frequency interference How to best deal with interference depends on its characteristics Signal Matched filter looks in the signal direction Narrowband case: Isotropic noise Energy = K σ " Sensitivity depends on E s and σ......and not W, T, and the shape of h(t) Frequency W Narrowband interference Time Want signal energy uniformly distributed over 0 f W Want W large and T small T small
5 Interference Broadband interferer: Ideal signal design for interference To counter interference, the signal should be isotropic: Frequency W small Pulse-like interference T Time Want signal energy uniformly distributed over 0 t T Want T large and W small Isotropic signal Energy =! Interference Statistically completely random In-band interference energy is reduced by /K after matched filtering Spread spectrum: Want K = W T large Current and past searches will likely miss this signal Pseudo-random signal based on π Some environmental factors Binary expansion of π is history s most studied pseudo-random generator: Real and imaginary Magnitude Time-invariant Plasma dispersion Scattering Time-varying Doppler Turbulence Scintillation (fading)
6 Bandwidth stress test of the ISM High data rate. W T and /T large Spread spectrum. W T >> We choose spread spectrum: Suppresses interference Usually less affected by multipath Discovery is easier ISM bandwidth is free Plasma dispersion The ISM is conductive due to ionization in interstellar gas clouds: f = frequency in Hz D Homogeneous refractive index: ( ( ) ) / fp n = f Group delay: τ(f ) = D DM f Hz pc cm 3 s DM = column density of electrons, ranges between and 000 cm 3 pc Delay spread Delay spread vs f c Dispersion favors large f c : τ max rapidly as f c The delay spread τ max = range of group delays across f c f f c + W : τ max = τ(f c ) τ(f c + W ) W = MHz, DM =, 0, 00, 000
7 Relation of group delay and phase Typical case Frequency response of propagation: Delay changes linearly and phase quadratically Monochromatic phase shift: F(f ) = F(f ) e iφ(f ) π τ(f ) = dφ(f ) df f c = GHz, W = MHz, DM = 00 Phase after wrapping Impulse response Φ in radians h m Impulse response energy is spread uniformly over 0 t τ max but phase is chaotic t msec arg h m t msec f = m/t, T = msec, 0 m < 000 f in MHz DFT {e i φm } and τ max 0.8 msec
8 Fourier-series representation of h(t) Effect of delay spread on one component of h(t) Assuming φ(f ) linear for small f f 0 Fourier-series basis for an isolated pulse h(t) is a natural for characterizing dispersion: h(t) = E s w(t) K K m=0 c m e iπmt/t w(t) e i π f 0t F(f ) e i φ(f ) e i φ(f 0) w (t τ(f 0 )) e i π f 0t Effect of group delay on w(t) can be ignored if τ max << T Argues in favor of choosing T >> τ max Filter bank receiver processing Receiver de-spreading Filter bank: One channel (out of K ) Spread the interference without affecting the noise statistics: Y (t) w ( t) Sample t = 0 Y m Y m P m c m e i π mt/t Y m = Es K c m e i φm + N m 0 m < K P m = ( ) Es K + O m e i φm E O m = σ
9 Performance metric Values we encounter What increase in E s maintains fixed P FA and P D? Asymptote for large K : E s f (K ) if E s α f (K ) as K Central limit theorem and law of large numbers apply Algorithm E s?? Incoherent matched filter Maximum likelihood log K Energy estimation K Dispersion estimation K Energy penalty Energy vs power Algorithm E s P s Increase in energy required to maintain P FA and P D E s {, log K, K, K } At K = 0, E s {, 3.7, 0 3, 0 } Incoherent matched filter Maximum likelihood Energy estimation T log(w T ) log(w T ) T W W T T Dispersion estimation W T W
10 Known τ max : incoherent matched filter Isotropic noise again Assuming τ max (hence φ m ) is supplied by a genie: Signal P m Phase equalizer e i φm Matched filter K K m=0 Incoherent carrier phase Q Isotropic noise Energy = K σ " E s Detection based on energy estimation Isotropic noise again Estimating signal energy does not require knowledge of τ max or {c m }: Signal Y m or P m K m=0 Q Isotropic noise Energy = K σ " E s K
11 Partial equalization for restricted delay spread Maximum delay spread A priori knowledge of group delay Equalization for minimum DM Group delay max min Equalizer min Group delay W W f f h m t msec arg h m t msec Smaller delay spread max min W f For specific f c, W, and LOS the delay spread is bounded by τ max Ω Knowing Ω, we search over T > Ω Partial delay equalization The duration of the impulse response Ω < T. Restricted-delay spread energy estimation Maximum likelihood Find L orthonormal basis vectors that represent e i φm for any 0 τ max Ω < T P m Min DM delay equalizer Impulse response DFT Partial energy L Q k=0 L Ω T K Find that basis vector most likely to represent filter bank output: Qn = IMF for n th basis maxn Q n = threshold input E s τ W becomes independent of T Resulting energy penalty is small: E s log L
12 Nonlinear DOF reduction The first difference of phase is a slowly varying function of m: φ m = φ m+ φ m π T τ ( m T ) Orthonormal basis for e i φ m {e i φm, 0 m < K } is always less than one period of a complex exponential Φ Exp i Φ m L = 5 orthonormal basis functions m.0 DOF L = suffices for most purposes Nonlinear DOF reduction Estimation of dispersion A noisy estimate of e i φm can be formed from the filter bank output: A noisy estimate of φ m can be formed from the filter bank output: P m+ P m = ( ) ( ) Es K + O Es m+ K + O m e i φm Use the ML approach for P m+ P m, but with only L = basis vectors Like all autocorrelation algorithms: arg (P m+ Pm) = ( φ m+ + Θ m+ Θ m ) ( ) Es Θ m = arg K + O m Slope of φ m vs m is proportional to τ max mod π Es K, same as energy estimator Results from the noise-on-noise O m+ O m term Θ m uniform distribution on [0, π] unless E s K mod π nonlinearity is the killer
13 Phase estimation and unwrapping Conclusions regarding dispersion arg (P m+ P m) E s σ = α K K = 000 (30 db) α = 0, 3, and 0 db Histogram Direct estimate of τ max is too noisy Maximum likelihood detection requires a modest penalty in E s but is computationally expensive Energy estimation for impulse response 0 t τ max is low complexity but requires larger increase in E s Search parameters: Large f c to reduce E s penalty and computational burden Search over T > τ max, but T not so large that time-varying effects come into play A search over W is not necessary Summary Takeaways In white Gaussian noise, detector should use a matched filter In radio-frequency interference, signal optimally appears statistically like Bandlimited and time-limited white Gaussian noise Large W T ISM bandwidth stress test demonstrates tradeoff between computational burden and Carrier frequency fc Prior knowledge of dispersion measure DM Received signal energy penalty Interference rejection Optimization provides implicit design coordination Propagation impairments constrain search parameters The more a priori knowledge of the signal, the more sensitive its detection Communication engineering is immediately relevant to SETI Scattering and fading under study
14 Postscript Thanks to: SETI Institute: Samantha Blair, Gerry Harp, Jill Tarter, Rick Standahar and Kent Cullers National Aeronautics and Space Administration Further information My homepage:
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