In this paper, we establish the partial orders (POs) of rate-compatible polar codes under both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC). Firstly, we define the POs for rate-compatible polar codes under block rate matching. Additionally, we demonstrate that certain POs for mother code lengths remain valid under rate matching in the BEC. Finally, leveraging the existing POs in the BEC, we derive POs in the BMSC under block rate matching.

In this paper, we study the lossless analog compression fori.i.d.discrete-continuous mixed signals via the polarization-based framework. We prove that for discrete-continuous mixed source, the error probability of maximum a posteriori (MAP) estimation polarizes under the Hadamard transform, which extends the polarization phenomenon to analog domain. Building on this insight, we propose the partial Hadamard compression and develop the corresponding analog successive cancellation (SC) decoder. The proposed scheme consists of deterministic measurement matrices and non-iterative reconstruction algorithm, providing benefits in both space and computational complexity. Using the polarization of error probability, we prove that our approach achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdú.

In this paper, we define partial orders (POs) of polar codes based on the Bhattacharyya parameter and the bit-error probability, respectively. These POs are applicable to arbitrary binary memoryless symmetric channel (BMSC). Leveraging the extremal inequalities of polarization transformation, we derive new POs for BMSC based on the corresponding POs observed in the Binary Erasure Channel (BEC). We provide examples that demonstrate the inability of existing POs to deduce these novel POs. Furthermore, we establish upper bounds for the expansion parameter$\beta$if the polar codes constructed by$\beta- \mathbf{expansion}$method obey these POs.

Unequal error protection (UEP) codes can facilitate the transmission of messages with different protection levels. In this paper, we study the achievability bounds on UEP by the generalization of Gilbert-Varshamov (GV) bound. For the first time, we show that under certain conditions, UEP enhances the code rate comparing with time-sharing (TS) strategies asymptotically.

In this paper, we study the lossless analog compression for i.i.d. nonsingular signals. Through analyzing analog polarization under Hadamard transform, we propose efficient successive cancellation (SC) decoding algorithm over analog domain. Thanks to the polarization of Rényi information dimension (RID) and the absorption of discrete entropy, we prove that the proposed scheme achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdú.