⚡ electrical engineering

Sparsification of Precoding Codebooks for PAPR Reduction via Grassmannian Representations

This paper proposes a Grassmannian manifold-based sparsification method for precoding codebooks that significantly reduces the peak-to-average power ratio (PAPR) in uplink scenarios while preserving achievable rates and requiring no changes to existing feedback mechanisms.

Original authors: Joe Asano, Yuto Hama, Hiroki Iimori, Szabolcs Malomsoky, Naoki Ishikawa

Published 2026-04-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Joe Asano, Yuto Hama, Hiroki Iimori, Szabolcs Malomsoky, Naoki Ishikawa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a high-definition video stream from your phone to a cell tower. To make this happen fast and clear, your phone uses a technique called MIMO (Multiple-Input Multiple-Output), which is like having multiple "lanes" on a highway for data to travel simultaneously.

To make sure the data arrives correctly, the phone uses a Precoder. Think of the precoder as a traffic director. It tells the data which lanes to take and how to shape the signal so it doesn't crash into itself.

The Problem: The "Squashed" Signal

In modern 5G and future 6G networks, we want to send more data at once (multi-stream). However, there's a catch.

The traffic directors (precoders) currently used are dense. Imagine a traffic director who is constantly waving their arms, shouting, and moving in every direction to guide the cars. While this works well for guiding the cars, it creates a lot of chaos and "peaks" in the signal.

In technical terms, this creates a high PAPR (Peak-to-Average Power Ratio).

The Analogy: Imagine trying to fill a water balloon. If you pour water in steadily, it's easy. But if you have to suddenly blast a huge wave of water in for a split second (the "peak") to get the same amount of water in, the balloon might burst.
The Consequence: To handle these sudden "peaks," your phone's battery and amplifier have to be very powerful and inefficient, or they have to turn down the volume (power) to avoid breaking. This wastes battery and reduces the range of your signal.

The Solution: The "Sparse" Traffic Director

The authors of this paper propose a clever trick: Sparsification.

Instead of a traffic director waving everywhere, they want a director who only moves in a few specific, strategic spots. They call this a Sparse Precoder.

The Analogy: Instead of a chaotic dance, imagine a traffic director who stands perfectly still in just two or three key spots, pointing only where absolutely necessary. The signal becomes "sparse" (mostly zeros, with a few active points).
The Benefit: This creates a much smoother signal with lower peaks. It's like filling that water balloon with a steady, gentle stream instead of a violent splash. This saves battery and allows the phone to transmit at higher power without breaking.

The Challenge: Don't Lose the Way

Here is the tricky part: If you change the traffic director's style too much, the cars might get lost. The data needs to arrive at the exact same place it would have with the old, dense director.

The researchers faced a dilemma:

Old Way: Design brand new traffic directors from scratch. (Too hard, requires changing the whole global standard).
Their Way: Keep the same instructions (feedback index) the phone sends to the tower, but secretly change the style of the director on the phone to be "sparse."

How They Did It: The "Grassmann" Map

To solve this, the authors used a mathematical concept called the Grassmann Manifold.

The Analogy: Imagine the "direction" of the data isn't a point on a flat map, but a point on the surface of a sphere (or a complex, curved shape).
The Goal: They needed to find a "Sparse" point on this sphere that is as close as possible to the original "Dense" point.

They developed two methods to find this new point:

The Perfect Match (Unitary Transformation):
Sometimes, you can simply rotate the original traffic director's instructions. It's like taking a dense, chaotic dance routine and rotating it 90 degrees so that the dancers end up standing still in the right spots. If this works, the signal is perfectly preserved (zero loss), but the peaks are gone.
The Best Guess (Sparse PCA):
Sometimes, a perfect rotation isn't possible. In these cases, they use a technique called Sparse Principal Component Analysis (SPCA).
- The Analogy: Imagine you have a complex painting, and you need to recreate it using only 5 brushstrokes. You can't copy every detail, but you pick the 5 most important strokes that capture the essence of the painting.
- The Result: The new signal is 99.9% identical to the old one, but with a much smoother power profile.

The Results

The researchers tested this on simulated 5G/6G systems and found:

Battery Life: The signal peaks dropped by more than 1 dB. In the real world, this means your phone can transmit with more power without overheating or draining the battery, potentially giving you better coverage.
Speed: The data speed (achievable rate) stayed exactly the same. The "sparse" directors guided the cars just as well as the "dense" ones.
Compatibility: Because they kept the same "feedback index," this can be implemented on your phone without needing to change the global 5G/6G standards. The tower doesn't even know the phone is using a new, smarter style of traffic director.

Summary

In short, this paper teaches us how to make our cell phone signals smoother and more efficient by changing the internal "traffic director" to be less chaotic and more strategic. It's like switching from a frantic, all-over-the-place conductor to a calm, precise one who gets the same job done with half the effort.

1. Problem Statement

In the evolution toward 6G systems, uplink traffic is expected to surge, necessitating efficient multi-stream transmission. While 5G uplink currently restricts Discrete Fourier Transform Spread OFDM (DFT-s-OFDM) to single-stream transmission to maintain low Peak-to-Average Power Ratio (PAPR), extending this to multi-stream transmission is crucial for capacity. However, applying standard dense precoding matrices (designed for 5G) to multi-stream DFT-s-OFDM breaks the single-carrier property, leading to significantly increased PAPR. High PAPR forces Power Amplifiers (PAs) to operate with large back-offs, reducing efficiency and coverage.

Existing solutions either design entirely new codebooks (which requires changing feedback mechanisms and standards) or fail to preserve the achievable rate. The core challenge is to reduce PAPR in multi-stream DFT-s-OFDM by sparsifying precoders without altering the existing feedback mechanism or sacrificing spectral efficiency.

2. Methodology

The authors propose a framework that exploits the geometric property that precoding matrices lie on a Grassmann manifold ( $G(N_t, N_s)$ ). The goal is to map existing dense precoders ( $W$ ) to sparse precoders ( $P$ ) such that they represent the same (or nearly the same) subspace, thereby preserving the channel capacity while reducing PAPR.

The problem is formulated as minimizing the chordal distance between the original dense precoder and the new sparse precoder, subject to a sparsity constraint (specifically, $N_t$ non-zero elements per precoder matrix).

The paper develops a unified design algorithm with two distinct approaches:

A. Exact Sparsification via Unitary Transformation

Concept: If a sparse precoder $P$ exists that spans the exact same subspace as the dense precoder $W$ , there exists a unitary matrix $U$ such that $P = WU$.
Implementation: The authors derive a closed-form solution to find $U$ . They define a "sparsity pattern" (the set of indices for non-zero elements) and verify its feasibility by checking if the null space of specific submatrices of $W$ allows for the construction of an orthonormal basis.
Algorithm: Using a Gram-Schmidt orthogonalization process on the null spaces of the complement sets of the support, the method constructs $U$ sequentially. If successful, the resulting $P$ has zero distortion (chordal distance = 0).

B. Approximate Sparsification via Sparse Principal Component Analysis (SPCA)

Concept: When an exact unitary transformation is impossible for a given sparsity pattern, the method seeks the best approximation.
Implementation: The problem is reformulated as maximizing the inner product $\|W^H P\|_F^2$ , which is equivalent to an SPCA problem.
Algorithm: For a given sparsity pattern, the optimal non-zero entries are found by solving a Rayleigh quotient maximization problem (finding the eigenvector corresponding to the largest eigenvalue of a submatrix). The algorithm iterates through all candidate sparsity patterns to select the one that maximizes the preserved energy (minimizes distortion).

C. Unified Design Algorithm

The proposed system operates as follows:

The User Equipment (UE) selects an index from the original dense codebook (standard 5G NR codebook) based on channel feedback.
The network (or UE implementation) maps this index to a pre-calculated sparse precoder using the proposed algorithm.
The sparse precoder is applied to the signal.
Crucially, the feedback index remains unchanged, ensuring compatibility with existing 5G/6G standards.

3. Key Contributions

Standard-Compliant PAPR Reduction: A method to reduce PAPR in multi-stream DFT-s-OFDM without modifying the existing feedback mechanism or requiring new codebook indices.
Grassmannian Formulation: A novel problem formulation that treats precoder sparsification as a subspace approximation problem on the Grassmann manifold.
Two-Pronged Solution:
- A closed-form exact sparsification method using unitary transformations (achieving zero rate loss when feasible).
- An SPCA-based approximate method for cases where exact sparsification is not possible.
Unified Algorithm: A practical construction algorithm that prioritizes exact solutions and falls back to SPCA approximation only when necessary.

4. Numerical Results

The authors evaluated the method using 5G NR codebook specifications with uncorrelated Rayleigh fading channels.

Achievable Rate:
- For $(N_t, N_s) = (4, 2)$ , the proposed method achieved zero performance loss compared to the original dense codebook because exact unitary transformations were feasible for all indices.
- For $(N_t, N_s) = (8, 4)$ and cases requiring SPCA approximation, the Signal-to-Noise Ratio (SNR) loss was negligible (approx. 0.05 dB to 0.2 dB).
PAPR Reduction:
- The proposed sparse codebooks achieved a PAPR reduction of more than 1 dB at a Complementary Cumulative Distribution Function (CCDF) of $10^{-2}$ .
- This reduction allows for higher transmit power or improved PA efficiency.
Sparsification Distortion:
- The distortion (chordal distance) remained around 0.2 even as matrix dimensions increased, confirming that the sparse precoders effectively approximate the original subspaces.

5. Significance

This work addresses a critical bottleneck in the transition to 6G uplink: enabling multi-stream DFT-s-OFDM without incurring the PAPR penalties that typically render such transmission impractical.

Practicality: By preserving the feedback index, the solution can be implemented via a software update on User Equipment (UE) without requiring changes to the base station or the 3GPP standard specifications.
Efficiency: It bridges the gap between the theoretical benefits of multi-stream transmission and the hardware limitations of power amplifiers.
Theoretical Insight: It demonstrates that Grassmannian geometry provides a powerful framework for optimizing codebook structures, allowing for exact or near-exact subspace preservation under sparsity constraints.

In summary, the paper presents a viable, low-complexity, and standard-compliant pathway to enhance uplink capacity and coverage in future wireless systems by intelligently sparsifying precoding codebooks.