Question

In maximal ratio combining each branch is weighted with the complex conjugate of that branch's complex fading gain. However, in practice the receiver must estimate fading gains in order to estimate fading gains and multiply received signals by them. Assume that this estimation is based on pilot symbol insertion, in which case the weights become subject to complex Gaussian error. If an M-branch diversity system with Rayleigh fading and mean signal y on each branch is used, the pdf of the output SNR becomes.

(1-rho²)ᴹ⁻¹ e⁻ʸ/ʸ M-1 ( M-1) [ rho²γ ]ⁿ 1
___________ Σ _____ __

fᵧ (y) = y n = 0 ( n ) (1-rho²)γ n!

where rho² is the normalized correlation coefficient between the fading gain on a branch and its estimate, i.e., the weight.
Show that the pdf above can be written as a weighted sum of M ideal maximal ratio combining SNR pdfs.

Answer 1

The question addresses the performance of an M-branch maximal ratio combining system under imperfect channel estimation in wireless communications engineering. The output SNR's pdf is influenced by the estimation error, which allows it to be expressed as a weighted sum of ideal MRC SNR pdfs for varying branch counts.

Step-by-step explanation:

The question asked involves a concept from wireless communications engineering, specifically the performance of a diversity combining technique known as maximal ratio combining (MRC) under conditions of imperfect channel estimation. In maximal ratio combining, each branch is weighted by the complex conjugate of that branch's complex fading gain. When channel state information is estimated through pilot symbols, weights are subject to error modeled as complex Gaussian noise.

The pdf of the output SNR in an M-branch diversity system under Rayleigh fading can indeed be expressed as a weighted sum of M ideal MRC SNR pdfs. The normalized correlation coefficient (rho^2) between the actual fading gain and its estimate affects the output SNR. The presence of complex Gaussian error in the weights results in an SNR distribution where each term represents the contribution of an MRC with one fewer diversity branch than the previous term.

The expression given for the pdf, fᵧ(y), indicates a summation over 'n' terms, where 'n' varies from 0 to M-1, representing the influence of estimation error on the diversity branches. Each term in the summation reflects the performance of an 'n+1'-branch diversity system under ideal conditions, scaled by the error term. This demonstrates how an MRC system's performance degrades as a function of the estimation error represented by (1-rho^2).