Capacity and Reliability Function per Fourth Moment Cost for WSSUS Fading Channels

This paper addresses the capacity of wide sense stationary uncorrelated scattering (WSSUS) fading channels. Associated with a given input signal we define a quantity called the “fourthegy” of the signal, relative to a given WUSSUS channel. The name is inspired by the fact that the measure is fourth order in the input signal amplitude. The fourthegy depends on the signal through its ambiguity function, and on the channel through a simple channel response function. The maximum possible mutual information for the channel per unit fourthegy is found. Roughly speaking, the fourthegy is a sum over time and frequency bins of the local signal energy squared. The fourthegy-to-energy ratio of directsequence spread spectrum signals is inversely propoportional to bandwidth. Therefore, for such signals, the capacity per unit energy (or the capacity per unit time for fixed power) tends to zero as the bandwidth increases. This does not happen to signals that are more bursty in time-frequency space, such as frequency-hopped signals or M-ary Frequency Shift Keyed signals. A similar result was found by Gallager and Medard for a less conventional channel model. Numerical evaluation of the bound shows it to be informative only for rather large bandwidths. I. WIRELESS CHANNEL MODELS A time-varying linear model of a wireless channel is adopted. The output y(t) of the channel is given by where u(t) is the input, h(t , T ) is the time-varying channel transfer function, and n( t ) is white Gaussian noise. It is assumed that h(t , T ) for fixed T is a wide-sense stationary (WSS) process, that h(t , T ) is a Gaussian random process. Uncorrelated scattering (US) is also assumed, meaning that the variables h( t , T ) for different values of T are uncorrelated. 11. CAPACITY A N D RELIABILITY FUNCTION PER UNIT COST Gallager[2] in his seminal work discussed energy limited channels, i.e., channels where the energy per degree of freedom is very small. Restricting the input to binary signals he computed the reliability function [I] per unit cost. The cost could be the energy or could be something else, but it is assumed that some input 0 has zero cost. Gallager showed that the reliability function per unit cost is given by, ‘This work was supported by a Motorola Fellowship, by the US Army Research Office under Grant DAAHO5-95-1-0246, and by the National Science Foundation under contract NSF NCR 9314253. where cost(u) is the cost associated with input U, and A = dPYIu=u/dPYIU=O is the likelihood ratio of the input U with respect to the 0 input. VerdQ’] considered capacity per unit energy cost and showed that where D(.ll .) is the Kullback-Liebler distance between measures. 111. CAPACITY AND RELIABILITY FUNCTION CALCULATIONS Consider signaling over a time interval of duration T , with T much larger than the coherence time and maximum delay spread of the channel, so that the capacity per unit time (or per unit energy or per other units) over the time interval is representative of the corresponding long-term capacity. Let a WUSSUS channel be given by (1) such that h is complexvalued and Gaussian with mean zero and autocorrelation function R H ( t s, T ) ~ ( T -U) = E[h(t , ~ ) h * ( s , U)], and suppose n ( t ) is white, complex-valued Gaussian noise with one-sided power spectral density 6’. We also constrain the input waveforms to have finite energy. Equation (1) can be written in the following manner where, given U, so,t ( t ) is a complex-valued, zero mean Gaussian random process independent of n( t ) with covariance function given by C(s, t ) = E [ s , , t ( s ) ~ ~ , ~ ( t ) ] . Let { X ; } ~ o be the eigenvalues of the covariance operator, E. Mercer’s theorem using the Karhunen-Loeve expansion yields a simple expression for D ( P ~ X = ~ I I P ~ I X = O ) and A in terms of the eigenvalues. Capacity per unit fourth-moment cost Define the fourthegy of an input signal U by Jc(u) = A?. ’ Another representation for the fourthegy Jc( U ) IS J C ( U ) = s, JI Ix(4 T ) I 2 4 H ( 4 T)dTdV, (5) where x ( r , v ) is the symmetric ambiguity function [5] of the signal u(t) which is defined as follows u(t 3T / 2 ) U * ( t ~ / 2 ) e ” ” ~ ~ d t , ((I;) ‘Admittedly the name “fourthegy” lacks luster, but we feel t ha t a postive sounding name, like energy, is needed, rather than a negative sounding name, like fourth moment cost. Our bounds show tha t a certain amount of fourthegy is needed per bit for diffuse \VUSSIJs fading channels, just as a certain amount of energy per bit is needed for either WUSSUS or additive Gaussian channels. 0-7803-5268-8/99/$10.00 @ 1999 IEEE 42 and where the channel response function +H(T , v) is given by Note that the fourthegy Jc(u) is fourth-order in the signal input, and that Jc(u) captures both time and frequency aspects of the signal. It can be shown that Jc(u) 5 G&JIu(t)l4dt and similarly that Jc(u) 5 G& IU(f)14a’f where U ( f ) is the Fourier transform of u(t). We can, thereafter, show the following