Achievable Rate of Linear Holographic MIMO With Arbitrary Aperture-Length

The continuous aperture of Holographic MIMO enables us to encode and transmit information spatially. This paper investigates the achievable rate of linear Holographic MIMO with arbitrary aperture-length using the finite blocklength information theory. Specifically, we first employ the prolate spheroidal wave functions to expand the received wavenumber band-limited electromagnetic field. This orthogonal representation enables two schemes to convey information related to the normal additive white Gaussian noise (AWGN) channel and the non-normal AWGN channel, namely the NA and NNA schemes, respectively. Then we derive the accurate achievable rates and the converse bounds of the two schemes by extending the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa \beta $ </tex-math></inline-formula> bound in finite blocklength information theory. Moreover, we derive an approximate closed-form expression of the achievable rate in the large aperture-length regime based on normal approximation. The approximation indicates that for a given space efficiency, the error probability decreases rapidly as the aperture length L increases, with the rate of decline determined by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q\left ({{O\left ({{\sqrt {L}}}\right)}}\right)$ </tex-math></inline-formula>. Finally, we obtain the asymptotic results when the aperture-length tends to infinity. Numerical results demonstrate that the NA scheme outperforms the NNA scheme when the blocklength is small, while the NNA scheme excels in the large blocklength regime. The accuracy of the approximation and the validity of the asymptotic results are verified.