Optimal Power Allocation for OFDM-based Ranging Using Random Communication Signals

The International Telecommunication Union (ITU) has recently endorsed a global vision for 6G, identifying Integrated Sensing and Communication (ISAC) technology as one of six critical use cases for next-generation networks [1]. To date, ISAC has been widely recognized as a key enabler for a number of emerging 6G applications, including autonomous vehicles, Internet of Things (IoT) networks, unmanned aerial vehicle (UAV) networks, and the low-altitude economy, which require simultaneous communication and sensing functionalities[2, 3, 4].

The ISAC system is a unified framework that provides both wireless communication and radar sensing functions. ISAC significantly enhances spectrum utilization and mitigates spectrum conflicts between radar and communication systems. Furthermore, it reduces the size and energy consumption of the equipment. Due to its high spectrum efficiency and low hardware costs, ISAC technology has garnered substantial attention from both academia and industry [5, 6, 7, 8, 9, 10]. ISAC waveforms, as the basis of ISAC technology, have been studied extensively [11, 12, 13, 14]. In general, waveform design methodologies for ISAC can be categorized into sensing-centric design, communication-centric design, and joint design approaches [3]. For the sensing-centric design, the priority of sensing functionality is higher than that of the communication counterpart. In such cases, one typically focuses on maximizing sensing performance or implementing basic communication capabilities over existing sensing waveforms or infrastructures[15],[16]. Communication-centric schemes, on the other hand, rely on the existing communication waveforms and standard-compatible protocols, such as single-carrier (SC) signals, orthogonal time frequency space (OTFS) signals and orthogonal frequency division multiplexing (OFDM) signals [17],[18]. Different from the above two methods, the joint design strategy creates new ISAC waveforms, aiming to achieve an optimized balance between sensing and communication capabilities[19],[20].

The communication-centric approach is expected to be more advantageous for ISAC networks due to its lower implementation complexity[5],[6]. Towards that end, numerous researches have been centering on communication-centric ISAC transmission, with a particular focus on waveform design and performance analysis [21, 22, 23, 24]. However, a critical challenge arises because communication data symbols must inherently exhibit randomness to carry meaningful information, which can adversely affect sensing accuracy. This phenomenon has recently been characterized as the deterministic-random tradeoff (DRT), a fundamental limitation in ISAC system design [25]. As a result, developing optimal communication-centric waveforms that mitigate sensing performance degradation becomes a crucial research objective. Against this backdrop, our recent work [26] took an initial step towards answering this fundamental question: What is the optimal communication-centric ISAC waveform under random signaling? In [26], we established the superiority of OFDM modulation in achieving the lowest ranging sidelobe. Specifically, we demonstrated that among all communication waveforms with a cyclic prefix (CP), OFDM is the only globally optimal waveform that achieves the lowest ranging sidelobe for both quadrature amplitude modulation (QAM) and phase shift keying (PSK) constellations. This holds true in terms of both the expectation of the integrated sidelobe level (EISL) and the expected sidelobe level at each lag of the periodic auto-correlation function (P-ACF). Furthermore, we proved that for communication waveforms without a CP, OFDM serves as a locally optimal waveform for QAM/PSK constellations, achieving a local minimum of the EISL of the aperiodic auto-correlation function (A-ACF).

Although the superiority of OFDM modulation over other waveforms in reducing ranging sidelobe level has been demonstrated in [26], it generally assumed a uniform power distribution across transmitted symbols, leaving the optimal power allocation (PA) strategies for random ISAC signaling unexplored. Numerous studies have investigated PA schemes for OFDM-based ISAC systems [27, 28, 29, 30, 31]. In [27], the authors proposed three distinct PA schemes aimed at maximizing the mutual information (MI) of radar sensing under different scenarios. In [28], the authors designed an optimization algorithm to obtain the PA design over subcarriers, thus the sensing MI of the OFDM-based ISAC waveform is improved. The authors in [29] conceived a power minimization-based joint subcarrier assignment and power allocation (PM-JSAPA) approach for ISAC systems. Similarly, a joint subcarrier and PA method for the integrated OFDM with interleaved subcarriers (OFDM-IS) waveform was proposed in [30]. The proposed method minimizes the autocorrelation sidelobe level of the integrated OFDM-IS waveform, using a cyclic minimization algorithm to solve the optimization problem. In [31], the authors proposed two designs for a radar-communication spectrum sharing problem by maximizing the output signal-to-interference-plus-noise ratio (SINR) at the radar receiver while maintaining certain communication throughput and power constraints. Although extensive studies on OFDM PA have been conducted for deterministic signaling, none have rigorously addressed the issue of PA design for random signaling.

In this paper, we investigate the PA design for random OFDM communication signals, in the context of target range estimation under a monostatic ISAC setup. In this setup, the ISAC transmitter (Tx) sends out an OFDM communication signal modulated with random symbols for dual purposes of delivering information to communication users and multi-target ranging. The signal is received by communication users, and simultaneously reflected from distant targets back to a sensing receiver (Rx) that is colocated with the ISAC Tx. As a result, despite its inherent randomness, the sensing Rx has full knowledge of the ISAC signal. To evaluate the ranging performance under matched filtering algorithms, we adopt the auto-correlation function (ACF) of the ISAC signal as a performance indicator, considering both periodic and aperiodic convolutions for signals with and without CP, respectively. For clarity, we summarize our main contributions as follows:

• •

  We develop a generic framework to analyze the ranging performance of OFDM-based random ISAC signals. Specifically, we analyze P-ACF and A-ACF for the considered signal by deriving their closed-form expressions.

• •

  We prove that the uniform PA scheme achieves the lowest EISL for both P-ACF and A-ACF. In particular, it also achieves the lowest average sidelobe level at every lag of the P-ACF.

• •

  We derive a closed-form expression for P-ACF with PA and frequency zero-padding, where the latter is a classical method to improve the range resolution of the matched filtering output. We demonstrate that in the case of frequency zero-padding, uniform PA does not yield the lowest possible ranging sidelobe. To address this, we formulate an optimization problem and develop the Projected Gradient Descent (PGD) algorithm to conceive a PA scheme that minimizes the ranging sidelobe.

• •

  By comparing ranging performances, we observe that while the PA scheme obtained via the PGD algorithm achieves a lower sidelobe level, it results in a slight increase in the mainlobe width compared to the uniform PA scheme. Building upon this observation, we introduce a new constraint to limit the mainlobe width and employ the Successive Convex Approximation (SCA) algorithm to solve the corresponding optimization problem. Our results indicate the existence of a tradeoff between the mainlobe width and the sidelobe level.

The remainder of this paper is organized as follows. Section II introduces the system model of the ISAC system under consideration and the corresponding performance metrics. Section III derives the closed-form expressions for both P-ACF and A-ACF of random ISAC signals using OFDM modulation and PA schemes. The optimal PA strategies of P-ACF and A-ACF with OFDM modulation for random ISAC signals are discussed in Section IV. Section V conceives optimal PA strategies for P-ACF with frequency zero-padding and OFDM modulation in random ISAC signals. Section VI presents simulation results to validate the theoretical analysis. Finally, Section VII concludes the paper.

Notations: Matrices are denoted by bold uppercase letters (e.g., $\mathbf{U}$), vectors are represented by bold lowercase letters (e.g., $\mathbf{x}$), and scalars are denoted by normal font (e.g., $N$); The $n$th entry of a vector $\mathbf{s}$, and the $(m,n)$-th entry of a matrix $\mathbf{A}$ are denoted as $s_{n}$ and $a_{m.n}$, respectively; $\otimes$ and $\operatorname{vec}\left(\cdot\right)$ denote the Kronecker product and the vectorization, $\left(\cdot\right)^{T}$, $\left(\cdot\right)^{H}$, and $\left(\cdot\right)^{*}$ stand for transpose, Hermitian transpose, and the complex conjugate of the matrices; $\ell_{p}$ norm is written as $\left\|\cdot\right\|_{p}$, and $\mathbb{E}(\cdot)$ represents the expectation operation; The notation Diag($\mathbf{a}$) denotes the diagonal matrix obtained by placing the entries of $\mathbf{a}$ on its main diagonal.

In [26], the closed-form expressions of P-ACF and A-ACF without PA have been theoretically analyzed. In this section, we systematically investigate the closed-form expressions of both P-ACF and A-ACF with PA in OFDM systems.

In this section, we analyze the optimal PA based on OFDM modulation for both the P-ACF and A-ACF cases.

In this section, we analyze the P-ACF for the OFDM scheme incorporating PA and frequency zero-padding. Zero-padding refers to the process of appending zeros to the end of a signal or data sequence to extend its length. Specifically, zero-padding in the frequency domain involves adding zero values to the high-frequency portion of the signal’s spectrum, thereby increasing the overall length of the spectrum. By doing this, the resolution of the time-domain signal is enhanced through increasing the number of sampling points. Consequently, the accuracy of the range estimation may be improved.

According to (20), when the OFDM modulation and PA are applied, the P-ACF is the inverse discrete Fourier transform (IDFT) of the sequence $|c_{n}|^{2}$ up to a factor $\frac{1}{N}$. Through frequency zero-padding with $N(L-1)$ null subcarriers, where $L\geq 1$ represents the padding factor, we generate an augmented sequence $\hat{\mathbf{c}}$ with an extended length of $NL$ samples, yielding

where $\mathbf{0}_{N(L-1)}$ represents the all-zero vector with length $N(L-1)$.

Analogous to the derivation in (20), we construct an enhanced P-ACF incorporating both PA and frequency-domain zero-padding. The modified P-ACF, denoted as $\hat{r}_{k}$, is expressed as

which is the IDFT of the sequence $\hat{\mathbf{c}}$ up to a factor $\frac{1}{NL}$.

The normalized $\text{EISL}_{\text{ZP}}$ can be denoted as

Through comprehensive theoretical analysis and numerical simulations, we found that the uniform PA scheme ceases to be optimal for minimizing the normalized $\text{EISL}_{\text{ZP}}$ when frequency zero-padding is implemented. To address this limitation, we formulate the following constrained optimization problem to derive a suitable PA strategy that can minimize the normalized $\text{EISL}_{\text{ZP}}$.

Notably, the optimization problem in (48) requires minimizing a fractional objective function over the simplex. To solve this constrained problem, we adopt the PGD algorithm to obtain a locally optimal solution. As outlined in Algorithm 1, the PGD method enhances the conventional Gradient Descent (GD) by introducing a projection step, which ensures that each iteration remains within the feasible region [36], [37]. The gradient of the objective function $f(\mathbf{p})$ at an arbitrary point $\mathbf{p}$ is given by

where

Moreover, the projector over the simplex can be expressed as

where $\mathbf{1}_{N}$ represents the all-one vector with length $N$ and $\mathcal{B}$ denotes the feasible region of the optimization variable $\mathbf{p}$.

The average sidelobe levels achieved by the proposed PA scheme, obtained via Algorithm 1, are illustrated in Fig. 5 and Fig. 6. In these figures, “optimal PA” denotes the PA scheme derived from Algorithm 1. The results indicate that the optimal PA offers considerable sidelobe suppression compared to the “uniform PA”. However, this improvement is accompanied by a slight broadening of the mainlobe, which in turn leads to a degradation in range resolution performance.

To mitigate the adverse effects of PA on mainlobe width, we propose to add a new constraint in the original optimization problem (48). The enhanced formulation is given by:

The newly added constraint $\mathbb{E}(|\hat{r}_{q}|^{2})\leq\frac{1}{2}\mathbb{E}(|\hat{r}_{0}|^{2})$ ensures that the average sidelobe level at delay $q$ remains within 3 dB of the average mainlobe level $\mathbb{E}(|\hat{r}_{0}|^{2})$. This condition is equivalent to $10\text{log}(\frac{\mathbb{E}(|\hat{r}_{q}|^{2})}{\mathbb{E}(|\hat{r}_{0}|^{2})})\leq-3\text{dB}$.

Compared to the original formulation in (48), the introduction of the third constraint in (53) makes it challenging to project $\mathbf{p}$ onto the feasible region. As a result, the PGD algorithm is no longer directly applicable for solving the modified optimization problem. Upon analyzing the problem in (53), we observe that although all three constraints are convex, the objective function remains non-convex. Given these conditions, the Successive Convex Approximation (SCA) algorithm becomes a suitable alternative. SCA is a powerful framework for handling non-convex optimization problems, as it addresses non-convex optimization problems through iterative minimization of convex surrogate functions [36, 38, 39]. The core procedure of the SCA algorithm lies in constructing an appropriate surrogate function $f_{su}(\mathbf{p},\mathbf{p}^{(r)})$ and determining a suitable step size $\alpha^{(r)}$. In Algorithm 2, the surrogate function is defined as

which is the first-order Taylor expansion of the objective function near the $r$th iteration point. Furthermore, the step size $\alpha^{(r)}$ at each iteration is determined via the exact line search method.

In this section, we further validate our conclusions and theoretical analysis by presenting numerical results. All the simulation results are attained by averaging over $1000$ random realizations.

First, we focus on the P-ACF with PA of 16-QAM and 64-QAM. In Fig. 1, the sidelobe performances of 16-QAM and 64-QAM constellations for CP-OFDM with uniform PA are shown. It can be observed that the theoretical values align closely with the simulation results, confirming the accuracy of the derivations in Section III-A. Additionally, a 3 dB performance gain in the peak-to-sidelobe level ratio is achieved by doubling $N$. In Fig. 2, the sidelobe performances of the P-ACF for 64-QAM with two PA schemes, uniform PA and random PA, are compared. It is clear that the uniform PA scheme achieves a much lower sidelobe level than the random PA scheme.

In addition to the P-ACF, the sidelobe level performance of the A-ACF with PA is shown in Fig. 3 for the 16-QAM and 16-PSK constellations. We examine the normalized EISL without CP under different PA schemes. We observe that the theoretical results match perfectly with the numerical simulations for both the average and random PA schemes, confirming the accuracy of the theoretical derivations in Section III-B. Additionally, the normalized EISL for the uniform PA scheme is significantly lower than that of the random PA scheme in both the 16-QAM and 16-PSK constellations.

To verify the theoretical formulation of the normalized $\text{EISL}_{\text{ZP}}$ in (47), we present comparative simulations of computed $\text{EISL}_{\text{ZP}}$ values for different constellation schemes and PA methods in Fig. 4, where the proposed “optimal PA” scheme utilizes the power vector $\mathbf{p}$ obtained through Algorithm 1. The close agreement between theoretical predictions and simulation measurements validates our analytical model. Notably, under the “optimal PA” scheme, the normalized $\text{EISL}_{\text{ZP}}$ for 16-PSK is significantly lower than that of the “uniform PA” case. For 16-QAM, the reduction is more modest, but the “optimal PA” still achieves a lower normalized $\text{EISL}_{\text{ZP}}$ compared to the uniform counterpart.

Next, we examine the sidelobe performances of 16-QAM and 16-PSK constellations with frequency zero-padding and different PA schemes. Fig. 5 and Fig. 6 display the P-ACF of 16-QAM and 16-PSK under OFDM signaling across different PA schemes and values of $N$. For both 16-QAM and 16-PSK, the “optimal PA” configuration achieves a lower sidelobe level than that of the “uniform PA”. Analyzing these figures further, while the “optimal PA” reduces the sidelobe level, it also slightly increases the mainlobe width compared to the “uniform PA” scheme.

In Fig. 7, we compare the convergence results of Algorithm 2 for 16-QAM and 16-PSK under different 3 dB mainlobe width constraints, where the limiting positions $q$ are set to $\frac{L}{2}$, $\frac{L}{2}+2$ and $\frac{L}{2}+4$, respectively. The “uniform PA” scheme and the “unconstrained mainlobe width” scheme serve as baseline references, where the latter refers to the PA vector obtained from Algorithm 1 without any constraint on mainlobe width. It is observed that as the 3 dB constraint becomes more stringent (i.e., smaller $q$), the convergence result increasingly resembles that of the “uniform PA” scheme. Conversely, when the constraint is relaxed (i.e., larger $q$), the convergence result tends to align more closely with the “unconstrained mainlobe width” case.

Combining Figs. 7, 8, and 9, we observe that as the PA scheme approaches the “uniform PA” scheme, the mainlobe width becomes narrower, yet the sidelobe level increases. This trend holds for both 16-QAM and 16-PSK constellations. These results suggest a tradeoff between mainlobe width and sidelobe level, allowing for a flexible selection of PA schemes based on specific performance requirements.

In this paper, we investigated optimal PA for OFDM-based random ISAC signals. Our findings demonstrated that the uniform PA achieves the global minimum EISL for both periodic and aperiodic autocorrelation functions for all constellations. Additionally, we proved that this scheme achieves the lowest sidelobe at every P-ACF lag. For frequency zero-padding scenarios, where uniform PA is suboptimal, we employed efficient optimization algorithms to design tailored PA schemes that balance the trade-off between sidelobe suppression and mainlobe width. The effectiveness of our theoretical framework was further validated through comprehensive simulations. Future research should explore the optimal signal basis and PA schemes for two-dimensional ambiguity function (2D AF) design.