Abstract
We consider a communication system whereby T-seconds time-limited codewords are transmitted over a W-Hz band-limited additive white Gaussian noise channel. In the asymptotic regime as , it is known that the maximal achievable rates with such a scheme converge to Shannon’s capacity with the presence of degrees of freedom. In this work we study the degrees of freedom and the achievable information rates for finite values of . We use prolate spheroidal wave functions to obtain an information lossless equivalent discrete formulation and then we apply Polyanskiy’s results on coding in the finite block-length regime. We derive upper and lower bounds on the achievable rates and the corresponding degrees of freedom and we numerically evaluate them for sample values of . The bounds are asymptotically tight and numerical computations show the gap between them decreases as increases. Additionally, the possible decrease from in the available degrees of freedom is upper-bounded by a logarithmic function of .
Highlights
Wireless communication technologies that use radio waves impose tight requirements on the used spectrum as adjacent radio bands may be used by other users or technologies
This is naturally related to the available degrees of freedom when sending time-limited codewords over a band-limited channel, which is the maximal number of independent data symbols that can be transmitted to the receiver
In the following we evaluate the degrees of freedom and the data rates for different SNR levels as L depends on the SNR; for example, since the water-filling algorithm is used for Equation (10), L can be increased by increasing the “water level” in the water-filling algorithm, which means that the SNR level must be increased
Summary
Wireless communication technologies that use radio waves impose tight requirements on the used spectrum as adjacent radio bands may be used by other users or technologies. Wyner made some assumptions to avoid these issues in [3] and in the third and fourth models in [2] He derived the channel capacity of the different models by relating the continuous time to discrete time models as Shannon did in [1], but by using the prolate spheroidal wave functions (PSWFs) and their property that as 2WT → ∞, the first 2WT PSWFs form asymptotically a complete orthonormal (CON) set for the time-limited and approximately band-limited signals. We use the ideal low pass filter as a model for the channel to force the transmitter to confine its transmitted information and energy in the allocated band and model good receiver designs: Given any practical low pass filter, one can implement a sharper low pass filter that is closer to the ideal one In such a model, there are no issues when it comes to infinite noise power and/or infinite capacity, inter-codeword interference is inevitable.
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