ON SUPERLUMINAL PROPAGATION AND INFORMATION VELOCITY Akhila Raman Berkeley, CA. Email: [email protected] Feb 19,2001.

Abstract: This paper examines some of the recent experiments on superluminal propagation. The meaning of information velocity from the perspective of a digital communication system is analyzed. It is shown that practical digital communication systems use bandlimited signals to transmit information consisting of random bits which cannot be predicted by looking at previously transmitted bits and that information is not limited to discontinuous step-functions alone. These random bits are pulse-shaped by using bandlimited pulse (e.g. gaussian pulse), in a causal fashion (meaning waveform during any given bit is obtained by convolution of present and past bits with the causal pulse, but cannot have information about future bits!). If these pulse-shaped random bits were not considered as information, if somehow the future random bits could be predicted by, say looking at the shape of the first bit, then there would be no need for elaborate communication systems like cellular systems and data modems! We could transmit the first bit waveform, turn off the transmitter and expect the receiver to correctly detect all future random bits, which clearly is impossible! It is shown that it is possible to achieve information velocity greater than speed of light, in media with zero dispersion and positive index of refraction less than unity in the frequency range of interest, and of sufficient length to make the time gained during medium transmission in comparison to vaccum transmission, more than the duration of several bits. It is shown that while signal causality is preserved from the perspective of an LTI system, Einstein causality is not preserved and hence what this means to relativistic causality violations is analyzed. It is further shown that in cases where the index of refraction or the group index is negative, the observed negative shift of the peak of the output pulse with respect to the input peak is merely due to the fact that the pulse was predictable. 1. Introduction: Let us consider the case of a Linear Time Invariant(LTI) system which is characterized by its causal impulse response h(t). The term causality is used throughout this paper to indicate signal and system causality, which requires that a signal g(t) or system h(t) is zero for t f 0 . Thus we see that the frequency response of the medium in f > f 0 has practically no effect on the delay of X(t). If the medium is not linear-phase, which is usually the case in dispersive wireless channels, then it introduces varying delay(dispersion) for each frequency component of X(t) in f ≤ f 0 , thus distorting X(t) . Again, this distortion can be mitigated as mentioned before. The Fourier Transform notations used in this paper are as follows1: ∞ j 2πft g (t ) = ∫ G ( f )e df −∞ ∞ − j 2πft G ( f ) = ∫ g (t )e dt −∞ This is emphasized because some papers on superluminal propagation2 have used an alternate notation with "f" replaced by "-f". Hence the notation used in this paper has to be borne in mind. 2. Superluminal propagation analysis in an LTI system: Let g (t ) = e −πt − ∞ ≤ t ≤ ∞ 2

(1)

be a gaussian signal whose standard deviation σ t = 1 / 2π Let G(f) be the Fourier Transform of g(t) which is also gaussian. −πf 2 G(f)= F[ g(t) ] = e (2) −∞ ≤ f ≤ ∞ And itsstandard deviation σ f = 1 / 2π For t > t 0 where t 0 = 3σ t , g (t ) drops to less than 39 dB of g(0)=1 and can be approximated to zero. Similarly, for f > f 0 where f 0 = 3σ f , G(f) can be approximated to zero. This is justified because more than 99.995% of the signal energy is contained in

the range t ≤ t 0 and f ≤ f 0 respectively in either domain! Thus we see that the gaussian signal is both time-limited and band-limited to a high degree. Note that the choice of t 0 = 3σ t is arbitrary and is assumed only as an example, it can be any multiple of σ t . A time-limited gaussian baseband pulse p(t) is formed by truncating g(t) for t > t 0 , and time-shifting it by t0. −π (t −t0 ) 2 p(t ) = g (t − t 0 ) = e for 0 ≤ t ≤ 2t 0 and is zero elsewhere. (3) Its Fourier Transform is given by P( f ) = F [ p(t )] = (G ( f ) ⊗ (2t 0 sin c(2 ft 0 ))) z 0 − j 2πft0 where z 0 = e is a linear phase term Where ⊗ denotes convolution.

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Though P(f) appears complicated, P( f ) is very nearly equal to G ( f ) for f ≤ f 0 , and contains more than 99.995% of total signal energy in that frequency range. P(f)= G(f) * z0 =0

for

f ≤ f 0 ; z0 is a linear phase term, does not affect magnitude.

for

f > f0

since P( f ) < 39 db of G(0)=1

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Let us consider the simple case of a binary phase-shift-keying(BPSK) system. The transmitted baseband signal X(t) is formed by convolving the random binary impulse train representing the symbols with the gaussian pulse p(t). We shall analyze BPSK in the baseband only, using the equivalence of bandpass and lowpass systems, since we can always shift it to any desired frequency range by multiplying the lowpass signal by the carrier frequency fc>f0. In general, number of symbols M=2b, where b = number of bits. For BPSK, one symbol equals one bit. Throughout this paper, we will use the term "symbol" to mean random binary information for the case of BPSK. N −1

N −1

n =0

n =0

X (t ) = p(t ) ⊗ ∑ a nδ (t − nTs ) =∑ a n p (t − nTs )

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where Ts =symbol duration. Choose Ts = t 0 = 3σ t an = binary i.i.d. random variable; takes the value ± 1 ; a n = 0 for n τ d and distortion is negligible, then it can be clearly shown that FTL information transfer occurred. 3. Negative group velocity experiments: In experiments demonstrating negative group velocity2,6, it has been argued that this is due to pulse reshaping mechanism and that no new information is available in the pulse

peak, which is not already present in its weak forward tail. Here another theory is presented which could also explain the observed negative shift in the pulse peak. Let us consider a gaussian pulse p(n) which is input to a discrete time LTI system with causal impulse response h(n), yielding an output signal y(n). h(n) = 2δ (n) − δ (n − n0 ) (18) where δ (n) is the discrete time impulse function. n0 ≥ 0 is an integer. We can see that h(n)=0 for n ω 0 . ω 0