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The uncertainty principles of random signals related to the linear canonical transform

This paper establishes Heisenberg and Donoho-Stark uncertainty principles for random signals within the linear canonical transform (LCT) framework, demonstrating that the LCT's parameters provide greater flexibility in characterizing signal concentration than the traditional Fourier transform.

Original authors: Jia-Yin Peng, Bing-Zhao Li

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jia-Yin Peng, Bing-Zhao Li

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 describe a complex, shifting cloud of smoke. Sometimes the smoke is thick in one spot, and sometimes it spreads out. Now, imagine that this smoke isn't just a single cloud, but a random cloud that changes its shape every time you look at it, like a cloud that behaves differently depending on the wind or the temperature.

This paper is about a mathematical rulebook for understanding these "random clouds" (random signals) when we look at them through a special, flexible lens called the Linear Canonical Transform (LCT).

Here is the breakdown of the paper's main ideas using simple analogies:

1. The Problem: The "Foggy" Trade-off

In the world of signals (like sound or radio waves), there is a famous rule called the Uncertainty Principle. Think of it like trying to take a photo of a speeding car.

If you use a fast shutter speed, you get a sharp picture of the car's position (time), but you can't tell how fast it's going (frequency).
If you use a slow shutter speed to see the speed (frequency), the car looks like a blur, and you lose its exact position.

You can't have perfect clarity in both at the same time. This paper asks: What happens if the car itself is unpredictable? What if the car is a "random signal" that changes its behavior every time you try to measure it?

2. The Tool: The "Magic Zoom Lens" (LCT)

Usually, scientists use a standard lens called the Fourier Transform to look at signals. It's like a fixed zoom lens that only shows you the speed or the position, but not both well.

The authors use a Linear Canonical Transform (LCT). Think of the LCT as a magic, adjustable zoom lens.

You can twist a dial on this lens (changing parameters $a, b, c, d$ ).
Depending on how you twist it, the lens can show you a mix of time and frequency, or focus on specific patterns.
The paper shows that because this lens is adjustable, it gives us more freedom to manage the trade-off between time and frequency than the standard lens.

3. The First Discovery: The "Random Fog" Rule (Heisenberg Uncertainty)

The authors developed a new version of the uncertainty principle for these random clouds.

The Old Rule: For a predictable signal, the product of "blur in time" and "blur in frequency" has a hard floor. You can't get sharper than a certain limit.
The New Rule: For random signals, this limit isn't just a fixed number. It depends on two things:
1. How "random" the signal is (its internal chaos).
2. How you set the dials on your magic lens (the LCT parameters).

The Analogy: Imagine trying to guess the average shape of a cloud that changes every second. The paper proves that the "fuzziness" of your guess depends on how much the cloud is jiggling (randomness) and which specific angle you are looking at it from (the LCT settings). If you pick the right angle, you can actually reduce the fuzziness more than you could with the standard lens.

4. The Second Discovery: The "Puzzle Piece" Rule (Donoho-Stark Uncertainty)

The second part of the paper deals with a different kind of limit. Imagine you have a puzzle, but some pieces are missing.

The Question: Can a random signal be squeezed into a tiny box in "time" AND a tiny box in "frequency" at the same time?
The Answer: No. The paper proves that if you try to force a random signal to be very small in both areas, it will fail.
The Twist: The size of the boxes you can use depends on the probability of the signal behaving a certain way. It's not just about the size of the box; it's about how likely the signal is to be in that box.

The Analogy: If you have a random cloud, you can't say, "It will definitely be in this tiny corner of the sky AND this tiny corner of the wind map." The paper gives a formula for the smallest possible "corner" you can claim, which changes based on how the random cloud behaves and how you set your magic lens.

5. The Payoff: Fixing Broken Signals

The paper ends by showing how these rules help fix broken signals.
Imagine you are trying to reconstruct a song, but parts of the recording are missing or covered in static (noise).

Because the authors know the exact limits of how "small" the signal can be in both time and frequency (thanks to the rules above), they can prove that if the missing parts aren't too big, the original song can be uniquely and safely recovered.
They show that using the adjustable LCT lens makes it easier to recover these signals compared to using the standard lens, because the LCT offers more flexibility to fit the signal into the available data.

Summary

In short, this paper takes the famous rule that "you can't know everything at once" and updates it for unpredictable, random signals. It introduces a flexible, adjustable lens (LCT) that allows scientists to:

Calculate exactly how blurry a random signal will be.
Determine the smallest possible "box" a signal can fit into.
Prove that we can reconstruct missing parts of random signals more effectively by using this flexible lens.

The key takeaway is that randomness adds a new layer of complexity, but by using the right mathematical tools (the LCT), we can still predict and control these signals better than before.

Technical Summary: Uncertainty Principles of Random Signals Related to the Linear Canonical Transform

Problem Statement
Uncertainty principles are fundamental tools in harmonic analysis and signal processing, characterizing the trade-off between a signal's localization in the time domain and its frequency domain. While the Heisenberg Uncertainty Principle (HUP) and the Donoho-Stark Uncertainty Principle (DSUP) are well-established for deterministic signals and have been extended to random signals within the Fourier Transform (FT) domain, there is a gap in the theoretical framework regarding random signals in the Linear Canonical Transform (LCT) domain. Real-world signals often exhibit both non-stationarity and random variations due to environmental perturbations or measurement noise. The LCT, which generalizes the FT and the Fractional Fourier Transform (FrFT), offers greater flexibility through parameter selection for analyzing such signals. However, the uncertainty bounds for random signals specifically within the LCT framework had not been rigorously formulated.

Methodology
The authors approach this problem by formulating the LCT of random signals on a probability space $(\Omega, \mathcal{F}, P)$ . A random signal is defined as $f(t, \eta) = \psi(t, \eta)e^{j\theta(t, \eta)}$ , where $\eta$ represents the random parameter. The methodology proceeds in three main stages:

Mathematical Formulation: The paper defines the LCT, FrFT, and FT for random signals, establishing the necessary definitions for mean values and variances (duration) in the LCT domain.
Derivation of HUP: Using the properties of the LCT kernel and the Parseval theorem, the authors derive an identity for the variance of the LCT frequency. By applying the Cauchy-Schwarz inequality to the time-domain and LCT-domain variances, they establish the Heisenberg Uncertainty Principle for random signals.
Derivation of DSUP: The authors employ operator theory, specifically defining time-limiting and frequency-limiting operators in the LCT domain. They calculate the Hilbert-Schmidt norm of the composition of these operators. Using the relationship between the operator norm and the Hilbert-Schmidt norm, they derive the Donoho-Stark Uncertainty Principle, which relates the measures of the time and frequency sets to the probability of their intersection.

Key Contributions and Results

Heisenberg Uncertainty Principle (HUP) for Random Signals in LCT:
The paper establishes that the product of the expected time variance and the expected LCT frequency variance is bounded by:
$E_\eta[\sigma^2_t(\eta)]E_\eta[\sigma^2_u(\eta)] \geq \frac{b^2}{4} + \left[ \int_\Omega \int_{-\infty}^{\infty} |(t - \mu_t)(at + b\partial_t\theta(t, \eta) - \mu_u)| \psi^2(t, \eta) dt dP \right]^2$
Here, $a$ and $b$ are parameters from the LCT matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ . The result explicitly shows that the uncertainty bound depends on the LCT parameter $b$ and the phase gradient of the random signal.
Donoho-Stark Uncertainty Principle (DSUP) for Random Signals in LCT:
The authors prove that a random signal cannot be simultaneously concentrated on arbitrarily small sets in both the time and LCT frequency domains. Specifically, if a signal is $\epsilon$ -concentrated on sets $T$ and $U$ with associated random event sets $B_T$ and $B_U$ , the following inequality holds:
$\frac{1}{2\pi|b||U||T|P(B_T \cap B_U)} \geq (1 - \epsilon_{B_T \times T} - \epsilon_{B_U \times U})^2$
This result indicates that the simultaneous concentration is constrained not only by the Lebesgue measures of the sets but also by the probability of the intersection of the random events and the LCT parameter $b$ .
Special Cases:
The paper demonstrates that the derived results for the LCT domain naturally reduce to known results in the FrFT domain (by setting specific LCT parameters) and the FT domain (by setting $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ ).
Signal Recovery:
Based on the DSUP, the paper presents stability results for the recovery of random signals from incomplete or missing observations in the LCT domain. It establishes conditions under which a unique and stable reconstruction is possible, provided the product of the set measures and the intersection probability satisfies specific bounds relative to the LCT parameter $b$ .

Significance and Claims
The paper claims that its findings provide a unified theoretical framework for analyzing random signals with both non-stationary and random characteristics. The primary significance lies in the explicit dependence of the uncertainty bounds on the LCT parameters. The authors state that this dependence indicates the LCT offers greater flexibility than the standard Fourier transform in controlling the balance between time and frequency localization.

The work is presented as having both theoretical and practical significance, particularly for:

Characterizing the time-frequency localization of random signals.
Facilitating the recovery of random signals from incomplete or partially missing observations.

The paper does not propose new experimental setups or specific application scenarios beyond these theoretical characterizations and recovery guarantees, maintaining a focus on the mathematical generalization of uncertainty principles to the stochastic LCT domain.