Coding for Fading Channels with Imperfect CSI at the Transmitter and Quantized Feedback

TL;DR

This work extends the Schalkwijk–Kailath (SK) paradigm to practical fading channels where the transmitter has imperfect channel state information and feedback is quantized. It develops three SK-type schemes: (i) a modulo-lattice augmented SK for quasi-static fading with I-CSIT and QFC, (ii) a two-path AF-relay–assisted SK exploiting relay signaling, and (iii) a frequency-domain SK design that converts arbitrary multi-path fading into parallel MIMO-like sub-channels via DFT/IDFT. Each scheme provides an explicit finite-blocklength lower bound on capacity under distortion and feedback constraints, supported by numerical results that compare against benchmarks and highlight gains from feedback and CSI refinement. The multi-path case with noiseless feedback introduces an OFDM-style approach with water-filling power allocation across sub-channels, offering a new SK-type alternative when feedback is memoryless. Overall, the paper broadens the applicability of SK-type coding to realistic wireless settings, paving the way for high-reliability, low-latency communication under imperfect CSIT and constrained feedback.

Abstract

The classical Schalkwijk-Kailath (SK) scheme for the additive Gaussian noise channel with noiseless feedback is highly efficient since its coding complexity is extremely low and the decoding error doubly exponentially decays as the coding blocklength tends to infinity. However, how to extend the SK scheme to channel models with memory has yet to be solved. In this paper, we first investigate how to design SK-type scheme for the 2-path quasi-static fading channel with noiseless feedback. By viewing the signal of the second path as a relay and adopting an amplify-and-forward (AF) relay strategy, we show that the interference path signal can help to enhance the transmission rate. Besides this, for arbitrary multi-path fading channel with feedback, we also present an SK-type scheme for such a model, which transforms the time domain channel into a frequency domain MIMO channel.

Figures (9)

• Figure 1: The quasi-static fading channel with perfect CSI at the transceiver (P-CSI-TR) and noiseless feedback (NF)
• Figure 2: The quasi-static fading channel with I-CSIT and QFC
• Figure 3: Rate versus blocklength $N$ for $\text{SNR} =10$, $\varepsilon=10^{-6}$, $h=0.9$ and $\widetilde{P}=10$
• Figure 4: Rate versus distortion $D$ for $N =100$, $\text{SNR} =10$, $\sigma_z=10^{-3}$, $\varepsilon=10^{-6}$, $h =0.9$ and $\widetilde{P}=10$
• Figure 5: Rate versus $\sigma_z$ for $\text{SNR} =4$, $h =1$, $D=0$, $\widetilde{P}=10$, $N=200$ and $\varepsilon=10^{-4}$

Theorems & Definitions (17)

• Proposition 1
• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Remark 1
• Definition 4
• Definition 5
• Definition 6
• Theorem 2