APTBM: Amplitude-Phase-Time Block Modulation

• APTBM is a modulation technique that enforces block-level amplitude, phase, and energy constraints to support efficient PA operation while mitigating nonlinear distortion.
• It utilizes a two-stage reconstruction framework with coarse phase compensation and fine trust-region based projection to ensure robust signal recovery.
• Simulation and testbed results demonstrate up to 4 dB IBO reduction and a 59.1% PA efficiency gain, validating its practical benefits in modern communication systems.

Amplitude-Phase-Time Block Modulation (APTBM) is a modulation technique developed to improve power amplifier (PA) efficiency in communication systems by enabling operation at reduced input back-off (IBO) levels while mitigating nonlinear distortion. Unlike conventional schemes such as @@@@1@@@@ and PSK, APTBM imposes block-level amplitude, phase, and time-domain constraints that permit constraint-guided signal reconstruction, counteracting PA-induced nonlinearities and enabling efficient, robust transmission (Xia et al., 20 Dec 2025).

1. APTBM Symbol Structure and Constraints

APTBM encodes information into blocks of two consecutive time-domain symbols. Each $n$-th block is

$$\mathbf{c}_n = \begin{bmatrix} a_n \ b_n \end{bmatrix} \in \mathbb{C}^2, \quad n=1,\dots,N$$

where the global transmit vector is $$\mathbf{s} = [\mathbf{c}_1^T, \dots, \mathbf{c}_N^T]^T \in \mathbb{C}^{2N}$$.

Each block satisfies strict amplitude and phase constraints,

$$|a_n|^2 + |b_n|^2 = P \qquad \angle a_n + \angle b_n = 2\phi_n$$

where $P$ denotes fixed block energy and $\phi_n \in [0,2\pi)$ is the initial block phase that encodes information bits.

Block symbols $(a_n, b_n)$ are generated as follows. Take $\mathbf{S}_n = [S_{1,n}, S_{2,n}, S_{3,n}]^T$ from $L$ points on the Poincaré sphere of radius $P$, and an initial phase $\phi_n \in \{2\pi m/M\}_{m=0}^{M-1}$ to define: $$\begin{aligned} a_n &= e^{j\phi_n} \sqrt{\frac{P+S_{1,n}}{2}} \exp(-j\theta_n) \ b_n &= e^{j\phi_n} \sqrt{\frac{P-S_{1,n}}{2}} \exp(+j\theta_n) \ \theta_n &= \frac{1}{2}\arctan\left(\frac{S_{3,n}}{S_{2,n}}\right) \end{aligned}$$ Thus, information is embedded in the block-energy sphere point, the initial phase, and the relative phase splitting between $a_n$ and $b_n$. This contrasts with single-symbol legacy modulations, as both block total power and strict phase sum are enforced at the modulation stage.

2. Power Amplifier Nonlinearity and Distortion Model

PA nonlinearity is modeled as a memoryless per-sample mapping with AM–AM and AM–PM characteristics: $\mathbf{y} = f(\mathbf{s})$ At the block level,

$$\tilde{\mathbf{c}}_n = f(\mathbf{c}_n) = \mathbf{c}_n + \mathbf{d}_{\mathrm{dom}}(\mathbf{c}_n) + \mathbf{d}_{\mathrm{res}}(\mathbf{c}_n)$$

where $\mathbf{d}_{\mathrm{dom}}$ denotes dominant (major, structured) distortion and $\mathbf{d}_{\mathrm{res}}$ is the residual (minor, complex) distortion.

The modified Rapp model provides

$$|f(s)| = \alpha(|s|) |s|, \quad \angle f(s) = \angle s + \varphi(|s|)$$

with

$$\alpha(r) = \frac{g_0 r}{ [1+ (g_0 r/A_{\mathrm{sat}})^{2 q_0}]^{1/(2q_0)} }, \quad \varphi(r)= \frac{ \alpha_0 r^{q_1} }{ 1 + (r/\beta_0)^{q_2} }$$

Dominant distortion,

$$\mathbf{d}_{\mathrm{dom}}(\mathbf{c}_n) = (\alpha(|\mathbf{c}_n|)-1) \odot \mathbf{c}_n + j\, \varphi(|\mathbf{c}_n|) \odot \mathbf{c}_n$$

can be substantially canceled with block-level reconstruction; residual distortion,

$$\mathbf{d}_{\mathrm{res}}(\mathbf{c}_n) = f(\mathbf{c}_n) - \mathbf{c}_n - \mathbf{d}_{\mathrm{dom}}(\mathbf{c}_n)$$

remains after dominant terms are removed.

3. Two-Stage Reconstruction Framework

The block-based, two-stage algorithm is designed for robust signal recovery in the presence of severe PA nonlinearity.

Stage I: Coarse Reconstruction

1. Phase Compensation Using an offline AM–PM curve $\varphi(r)$, a nominal phase shift based on average input power is subtracted from each received $\tilde{\mathbf{c}}_n$:

$$\psi_n = \angle \tilde{\mathbf{c}}_n - \varphi( \mathbb{E}\{|x(t)|^2\} ), \qquad \breve{\mathbf{c}}_n = |\tilde{\mathbf{c}}_n| \odot e^{-j\psi_n}$$

2. Amplitude Reconstruction Let $(\breve a_n,\breve b_n)^T$ denote $\breve{\mathbf{c}}_n$. Compute

$$P_{d,n} = \frac{|\breve a_n|^2 - |\breve b_n|^2}{|\breve a_n|^2 + |\breve b_n|^2}, \qquad \xi_n = \frac{1}{1+e^{\tan(\frac{\pi}{2}P_{d,n})}}$$

Enforce the block energy:

$$\tilde a_n = \xi_n |\breve a_n| + (1-\xi_n) \sqrt{P-|\breve b_n|^2}$$

$$\tilde b_n = (1-\xi_n)|\breve b_n| + \xi_n\sqrt{P-|\breve a_n|^2}$$

Final compensated output:

$$\check{\mathbf{c}}_n = [\tilde a_n, \tilde b_n]^T \odot e^{j\psi_n} \approx \mathbf{c}_n + \mathbf{d}_{\mathrm{res}}$$

Stage II: Fine Reconstruction

To correct the residual, solve for each block: $$\min_{\mathbf{c}_n \in \mathbb{C}^2} \| \check{\mathbf{c}}_n - \mathbf{c}_n \|^2 \quad \text{s.t.} \quad |a_n|^2 + |b_n|^2 = P, \quad \angle a_n + \angle b_n = 2\phi_n$$ These constraints are equivalent to

$$\mathbf{c}_n^H\mathbf{c}_n = P, \qquad \Re\{\mathbf{c}_n^H \mathbf{d}_n\} = 0$$

with $$\mathbf{d}_n = [-j e^{j\phi_n}|a_n|,\ -j e^{j\phi_n}|b_n|]^T$$.

Since $\phi_n$ is not known, estimate from $\check{\mathbf{c}}_n$: $$\hat\phi_n = \arg \left( \frac{\check a_n}{|\check a_n|} + \frac{\check b_n}{|\check b_n|} \right)$$ Then iteratively perform variable substitution. At each iteration $k$, freeze $\mathbf{d}_n$ as $\check{\mathbf{d}}_n^{(k)}$, solve: $$\min_{\mathbf{c}_n} \|\check{\mathbf{c}}_n^{(k)}-\mathbf{c}_n\|^2\quad \text{s.t.}\ \mathbf{c}_n^H\mathbf{c}_n=P,\ \Re\{\mathbf{c}_n^H\check{\mathbf{d}}_n^{(k)}\}=0$$ This is recast to a trust-region subproblem for $\mathbf{z} \in \mathbb{R}^4$ with one linear constraint: $$\min_{\mathbf{z}} \|\mathbf{w} - \mathbf{z}\|^2 \quad \text{s.t.}\ \mathbf{z}^T \mathbf{z} = P,\ \mathbf{u}^T\mathbf{z} = 0$$ with real vectors $\mathbf{w}$, $\mathbf{u}$ specified from previous outputs. The closed-form solution is: $$\mathbf{z}^\star = \frac{1}{1-\lambda} \mathbf{U}_\perp \mathbf{U}_\perp^T \mathbf{w},\qquad \lambda = 1 - \sqrt{ \frac{ \mathbf{w}^T \mathbf{U}_\perp \mathbf{U}_\perp^T \mathbf{w} }{P} }$$ where $\mathbf{U}_\perp$ is an orthonormal basis of the nullspace of $\mathbf{u}^T$. The output is mapped back to the complex pair $\mathbf{c}_n$.

Empirically, the iterative projection converges in $K\approx2$ steps to a fixed point with minimal residual distortion (Xia et al., 20 Dec 2025).

4. Information Embedding and Modulation Capabilities

Information encoding in APTBM exploits three intertwined degrees of freedom per block: (i) the position on the block-energy sphere, (ii) the global phase $\phi_n$ (carrying $\log_2 M$ bits), and (iii) the internal phase split $\theta_n$. The modulation alphabet is implemented as points on the surface of the Poincaré sphere, enabling multi-dimensional information representation in both amplitude and phase. This construction preserves block total power and phase sum, facilitating robust constraint-guided reconstruction downstream.

A distinguishing feature is the enforcement of both the block-wise energy constraint and the sum-phase constraint, unlike QAM and PSK where such constraints are absent or applied only individually at the symbol level. This architecture directly supports the subsequent signal reconstruction stages specific to PA distortion mitigation.

5. Performance Characteristics and Evaluation

Table: Simulation and Testbed Outcomes with Two-Stage APTBM Reconstruction

Evaluation Domain	IBO Reduction	PA Efficiency Gain	Testbed Frequency Range
Simulations (AWGN/PA)	up to 4 dB	up to 59.1%	—
mmWave (38 GHz)	2 dB	33.9%	sub-6 GHz (5.4 GHz), 38 GHz
sub-6 GHz (5.4 GHz)	0.3–0.5 dB	not specified	5.4 GHz

Numerical simulations indicate that the two-stage algorithm for APTBM enables up to 4 dB IBO reduction over previous APTBM baselines at a BER of $10^{-3}$, resulting in a 59.1% improvement in PA efficiency. Error vector magnitude (EVM) and BER are driven below $10^{-4}$. Testbed experiments, including sub-6 GHz and mmWave frequencies with real hardware, validate the practical gains: 2 dB IBO reduction and 33.9% PA efficiency improvement at 38 GHz, and up to 0.5 dB IBO reduction at 5.4 GHz, using only software/algorithmic processing and without hardware modifications (Xia et al., 20 Dec 2025).

6. Algorithmic Implementation Details

The two-stage signal reconstruction algorithm operates blockwise, leveraging both block structure and PA characteristics.

• Stage I:
  • Phase compensation utilizes an offline AM–PM curve.
  • Amplitude reconstruction enforces the block total energy via a nonlinear mixing function of symbol amplitudes.
• Stage II:
  • The nonlinear projection is solved via variable substitution, resulting in a sequence of trust-region subproblems.
  • Each subproblem admits a closed-form solution by projecting onto the intersection of a hypersphere and a hyperplane in $\mathbb{R}^4$.
  • Practical implementations converged in $K\approx2$ iterations per block.

The framework's efficiency is enhanced by explicit use of the block-structure constraints at every step, contrasting previous approaches that only applied constraints heuristically or statistically at the recovery stage.

7. Significance and Limitations

APTBM, when combined with the two-stage reconstruction methodology, enables substantially lower IBO for PAs while maintaining acceptable BER and high PA efficiency. The explicit enforcement of APTBM block constraints proves essential for decomposing and mitigating nonlinear distortion. Numerical and hardware evaluations confirm robust, repeatable gains over prior methods (Xia et al., 20 Dec 2025).

A plausible implication is that applying similar constraint-guided reconstruction techniques to other block-modulation forms might further generalize the approach's benefits relating to nonlinearity mitigation. However, the approach presupposes precise knowledge of the block structure at the receiver and the availability of accurate PA AM–AM/AM–PM profiles, which may limit immediate deployment in certain legacy or highly dynamic environments.