Zero-Forcing Precoding for Frequency Selective MIMO Channels with $H^infty$ Criterion and Causality Constraint

We consider zero-forcing equalization of frequency selective MIMO channels by causal and linear time-invariant precoders in the presence of intersymbol interference. Our motivation is twofold. First, we are concerned with the optimal performance of causal precoders from a worst case point of view. Therefore we construct an optimal causal precoder, whereas contrary to other works our construction is not limited to finite or rational impulse responses. Moreover we derive a novel numerical approach to computation of the optimal perfomance index achievable by causal precoders for given channels. This quantity is important in the numerical determination of optimal precoders.

💡 Research Summary

The paper addresses the design of causal linear‑time‑invariant (LTI) precoders for frequency‑selective multiple‑input multiple‑output (MIMO) channels that suffer from inter‑symbol interference (ISI). The authors adopt a worst‑case (H∞) performance criterion: they seek a precoder G(z) that exactly inverts the channel H(z) (i.e., H(z) G(z)=I for all z in the unit disc) while minimizing the supremum norm ‖G‖∞. This norm directly bounds the worst‑case transmit‑energy amplification and the worst‑case noise enhancement when the channel is imperfectly known, making it a natural robustness measure.

A key theoretical contribution is the reformulation of the existence of a causal right‑inverse in terms of a Toeplitz operator T_{H*}. Theorem 1 shows that a causal inverse with norm bounded by δ⁻¹ exists if and only if the operator inequality ‖T_{H*}u‖₂ ≥ δ‖u‖₂ holds for all u in the Hardy space H². From this, the optimal performance index is defined as

 γ_opt(H) = inf{‖G‖∞ | G∈H∞, H G = I}.

The authors prove that γ_opt(H) equals the reciprocal of

 ρ(H) = inf_{‖u‖{H²}=1}‖T{H*}u‖_{H²}.

Thus the problem reduces to computing the smallest singular value of the infinite‑dimensional Toeplitz operator associated with H*.

Because direct computation of ρ(H) is infeasible, the paper introduces a convergent numerical scheme. The Hardy space is approximated by the finite‑dimensional subspace of polynomials of degree ≤ N, denoted P_N. Restricting both the domain and range of T_{H*} to P_N yields a finite matrix Γ_{H,N} whose entries are the Fourier coefficients of H*. Theorem 7 establishes that

 ρ_N(H) = σ_min(Γ_{H,N})

forms a monotonically decreasing sequence converging to ρ(H) as N→∞. Consequently, γ_opt(H) can be approximated arbitrarily well by

 γ_opt(H) ≈ 1/σ_min(Γ_{H,N})

for sufficiently large N. This provides a practical way to evaluate the fundamental performance limit of any causal precoder for a given channel.

Having obtained γ_opt(H), the authors construct an explicit optimal precoder. Using the theory of Schur functions and block‑operator realizations (Lemma 3), they show that any contractive analytic matrix function F(z) can be represented as

 F(z) = D + C z (I − zA)⁻¹ B,

where the block matrix