Randomized Isometric Linear-Dispersion Space-Time Block Coding for the DF Relay Channel

This paper presents a randomized linear-dispersion space-time block code for decode-and-forward synchronous relays. The coding matrices are obtained as a set of columns (or rows) of randomly generated Haar-distributed unitary matrices. With respect to independent and identically distributed (i.i.d.)-generated codes, this particular isometric structure reduces the intersymbol interference generated within each relay. The gain over i.i.d. codes in terms of spectral efficiency is analyzed for both the LMMSE and the ML receivers under the assumption of frequency-flat quasi-static fading. In this setting, the spectral efficiency is a random quantity, since it depends on the random coding matrices. However, it is proven that the spectral efficiency converges in probability to a deterministic quantity when the dimensions of the matrices tend to infinity while keeping constant their ratio, i.e., the coding rate α. Consequently, when the random coding matrices are large enough, the presented system behaves as a deterministic one. This result is achieved by means of the rectangular R-transform, a powerful tool of free probability theory which allows determining the distribution of the singular values of a sum of rectangular matrices.