Prerequisites & Notation

Before You Begin

This chapter applies asymptotic random matrix tools to estimation and detection problems. The reader should be comfortable with the LMMSE estimator (Chapter 7), the LASSO and $\ell_1$ -minimization (Chapter 14), and basic facts about the spectra of Hermitian matrices. Familiarity with the Stieltjes transform (Book FSP, Chapter 21) is helpful but not strictly required — the key identities are re-derived here.

LMMSE estimator and the Wiener-Hopf equations(Review ch07)
Self-check: Can you write the LMMSE estimator of $\mathbf{x}$ from $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ and compute its MSE?
LASSO and the $\ell_1$ norm(Review ch14)
Self-check: Can you state the LASSO estimator and explain why the $\ell_1$ penalty promotes sparsity?
Structured sparsity and block-sparse recovery(Review ch15)
Self-check: Can you explain why $\ell_{2,1}$ regularization recovers block-sparse signals?
Eigenvalue decomposition and SVD
Self-check: Can you relate the singular values of $\mathbf{A}$ to the eigenvalues of $\mathbf{A}^H\mathbf{A}$ ?
Stieltjes transform of a probability measure
Self-check: Can you write the Stieltjes transform $m(z) = \int (x-z)^{-1}\,d\mu(x)$ and invert it via Plemelj's formula?
Convergence in distribution (weak convergence of measures)
Self-check: Can you distinguish almost-sure convergence of empirical spectral distributions from convergence in probability?

Notation for This Chapter

Key symbols used in this chapter. We work in a double asymptotic regime where matrix dimensions grow jointly at a fixed ratio, so ratios like $\beta = n_t/n_r$ will appear everywhere.

Symbol	Meaning	Introduced
$N, M$	Matrix dimensions (rows $\times$ columns); the asymptotic regime sends both to infinity with $M/N \to \delta$	s01
$\delta = M/N$	Aspect ratio (undersampling ratio in compressed sensing)	s01
$\rho = s/M$	Normalized sparsity (fraction of measurements equal to sparsity level)	s02
$\beta = n_t/n_r$	Ratio of transmit to receive antennas in MIMO	s01
$m_\mu(z)$	Stieltjes transform of the measure $\mu$	s01
$F_N(x)$	Empirical spectral distribution of an $N \times N$ matrix	s01
$\text{SNR}$	Signal-to-noise ratio per symbol	s01
$\eta_{\text{MP}}(\cdot; \cdot)$	Marchenko-Pastur $\eta$ -transform (limiting SINR functional)	s01
$R_\mu(z), S_\mu(z)$	R-transform and S-transform of a probability measure $\mu$	s03
$\rho^*(\delta)$	Donoho-Tanner phase transition curve	s02
$\mathbf{A}$	Measurement / sensing matrix (Gaussian, $M \times N$ )	s02
$\sigma^2$	Additive noise variance	s01

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