Hadamard Coded Modulation: Principles & Applications

Updated 10 June 2026

Hadamard Coded Modulation (HCM) is a modulation scheme that maps data onto orthogonal Walsh–Hadamard codewords, ensuring low peak-to-average power ratio and simplified real-valued processing.
HCM incorporates techniques such as cyclic prefixing, DC bias removal, and interleaving to mitigate ISI and nonlinear distortions, making it robust across VLC, wireless, and IoT platforms.
With an O(N log N) complexity and competitive performance, HCM offers enhanced power efficiency and hardware simplicity, achieving significant average power savings in optical and high-mobility scenarios.

Hadamard Coded Modulation (HCM) is a class of coded-modulation schemes in which data symbols are mapped onto orthogonal codewords of the Walsh–Hadamard transform (WHT), replacing the conventional Fourier-based transform stages of OFDM with a fast, real-valued, sign-only, unitary transform. HCM leverages the spreading and orthogonality properties of Hadamard codewords for peak-to-average power ratio (PAPR) reduction, simple hardware implementation, and robustness to certain transmission channel impairments. In recent practice, HCM has been realized across a range of physical-layer systems, from visible light communications (VLC) (Noshad et al., 2014, Noshad et al., 2014), to wireless and massive IoT multicarrier platforms (Hao et al., 14 May 2026), and near-capacity coded systems with nonlinearities (Zhidkov, 2017).

1. Signal Model and Walsh–Hadamard Transform Foundation

HCM relies on the Walsh–Hadamard matrix $\mathbf{H}_N \in \mathbb{R}^{N \times N}$ (entries $\pm1/\sqrt{N}$ ), which is real, orthogonal, and can be generated recursively via the Sylvester construction. Data is organized into a length- $N$ vector (QAM, PAM, or OOK symbols depending on context). The core transformations are:

$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$

For modulation, the data vector $\mathbf{a}$ (possibly zero-forced in the DC bin) is transformed by the WHT, yielding the time-domain or channel input block. In binary VLC-oriented HCM, a (bipolar) Hadamard matrix $H^b_N = H_N - \overline{H}_N$ is used, and OOK or small-amplitude M-PAM symbols are common (Noshad et al., 2014). The WHT provides $O(N \log N)$ complexity, integer-only operations, and no twiddle-factor or complex arithmetic requirements (Hao et al., 14 May 2026, Noshad et al., 2014).

2. HCM Modulation, Transmission, and Demodulation

In canonical HCM, the modulator performs the following steps:

Bit Mapping: Map input bits to $\mathbf{u} \in \mathbb{R}^N$ (e.g., $M$ -PAM/OOK).
Hadamard Coding: Compute the transmit vector, e.g.,

$\mathbf{x} = H^b_N (2\mathbf{u} - 1)/2 + P/2 \mathbf{1}$

where the DC offset sets nonnegativity for IM/DD links.

Cyclic Prefix, Interleaving, or Shaping: A cyclic prefix is optionally added for ISI mitigation, and pulse shaping (e.g., sinc) may be applied (Noshad et al., 2014, Noshad et al., 2014, Hao et al., 14 May 2026).
Transmission: The waveform is launched (direct modulation or via a nonlinear mapping in certain HCM variants (Zhidkov, 2017)).

At the receiver, after photodetection (IM/DD) or downconversion, the received samples are processed by the inverse WHT (IFWHT), optionally followed by DC bias removal or memory-enhanced equalization.

$\pm1/\sqrt{N}$ 0

Standard PAM or QAM hard or soft demodulation is then applied.

3. PAPR Reduction, DC Bias Removal, and Nonlinear Robustness

A defining advantage of HCM over OFDM is its strict PAPR bound. For any HCM block, the PAPR is at most 2, independent of blocklength $\pm1/\sqrt{N}$ 1 (Noshad et al., 2014, Noshad et al., 2014):

$\pm1/\sqrt{N}$ 2

OFDM, by contrast, exhibits PAPR scaling as $\pm1/\sqrt{N}$ 3 in expectation, up to $\pm1/\sqrt{N}$ 4 in the worst case.

In direct-detection optical or VLC applications, this affords HCM the ability to operate at higher average optical power, as signal peaks remain within the nonlinear operating region of LEDs up to $\pm1/\sqrt{N}$ 5. For higher efficiency, DC-reduced HCM (DCR-HCM) removes the constant offset from each block:

$\pm1/\sqrt{N}$ 6

The information rate penalty is negligible for large $\pm1/\sqrt{N}$ 7 (rate loss $\pm1/\sqrt{N}$ 8 for $\pm1/\sqrt{N}$ 9 (Noshad et al., 2014)). DCR-HCM can achieve a $N$ 0 dB reduction in average optical power at equal BER versus standard HCM in clipping-limited regimes (Noshad et al., 2014).

In multi-carrier or nonlinear-channel settings, applying a memoryless nonlinearity $N$ 1 after WHT (e.g., piecewise-chaotic map) approaches the random-coding limit. If used as a precoder for OFDM, HCM reduces the Nyquist-sampled PAPR by up to $N$ 2 dB at $N$ 3 (Zhidkov, 2017).

4. Equalization, Interleaving, and ISI Mitigation

HCM is not diagonalized by time-invariant channels as in OFDM, so symbol-level interference can arise in dispersive links. Several approaches have demonstrated effective mitigation:

Interleaving: Permuting each symbol block (“symbol-length interleaving”) by an optimal permutation $N$ 4 (found via binary LP) spreads ISI uniformly over all codewords. At the receiver, the inverse permutation is applied before IFWHT (Noshad et al., 2014, Noshad et al., 2014).
Cyclic Prefix (CP): Applied for block-wise convolutional invariance and ISI suppression (as in CP-OFDM, CP-WHTDM) (Hao et al., 14 May 2026).
Iterative Equalization: In highly dispersive or doubly-selective channels, cross-domain memory approximate message passing (CD-MAMP) is employed. CD-MAMP alternates linear estimation steps (exploiting the WHT domain channel’s approximately banded structure) with nonlinear denoising, yielding complexity $N$ 5 rather than $N$ 6 when the effective channel has bandwidth $N$ 7. For QPSK,

$N$ 8

The efficacy of interleaving and MMSE equalization for ISI suppression in dispersive VLC is documented in (Noshad et al., 2014, Noshad et al., 2014). In high-mobility radio, WHTDM with CD-MAMP outperforms OFDM by an order of magnitude in BER at 120 km/h and remains robust to 500 km/h and large delay spreads (Hao et al., 14 May 2026).

5. Performance Metrics and Complexity

Performance and implementation characteristics are summarized below.

Scheme	PAPR	FM Adds (per 1024)	FM Mults (per 1024)	BER (at 120 km/h)	BER (at 500 km/h)
WHTDM + CD-MAMP	2	12,288	0	$N$ 9	$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 0
OFDM (1-tap MMSE)	$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 1	18,432	12,288	$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 2	$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 3
AFDM + CD-MAMP	2	22,528	20,480	$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 4	—
OTFS + CD-MAMP	2	49,152	32,768	$\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 5	—

In VLC experiments, HCM and DCR-HCM achieve BERs two to three orders of magnitude lower than ACO-OFDM or DCO-OFDM at high optical power (Noshad et al., 2014, Noshad et al., 2014). In coded/numerical channel settings, HCM–AMP approaches within $\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 6 dB of AWGN capacity (Zhidkov, 2017). Complexity always scales as $\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 7 in the transform stages, with zero or minimal multiplier cost in HCM/WHTDM (multipliers required only in OFDM or nonlinearities).

6. Comparative Architecture and Implementation Trade-Offs

Arithmetic Structure: HCM (including WHTDM) is natively real-valued and sign-only in its core transform, obviating the need for complex multiplies and coefficient storage, and lending itself to parallel and low-power hardware implementations (Hao et al., 14 May 2026, Noshad et al., 2014).
Power Efficiency: DC-reduced variants achieve up to 50% average power savings in optical systems (Noshad et al., 2014), with natural dimming control via DC level manipulation (Noshad et al., 2014).
PAPR Robustness: The strict PAPR ceiling enables operation closer to physical transmitter boundaries before nonlinear distortion impairs system performance.
Interference Control: Interleaving (for dispersive/ISI channels) and iterative detection (for doubly selective or “random” channel matrices) deliver effective mitigation with minimal additional complexity (Noshad et al., 2014, Hao et al., 14 May 2026).
Spectral Efficiency: HCM is compatible with OOK, low-order PAM, and QAM; rate loss due to forced-zero first row becomes negligible as $\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 8 grows.
Dimming Support (VLC): DC control is built-in, in contrast to PWM/OOK hybrids required in OFDM (Noshad et al., 2014).

The primary trade-off is that while HCM/WHTDM outperforms OFDM in high-mobility, ISI-laden, or peak-limited scenarios, OFDM retains some BER advantage in quasi-static, frequency-flat channels due to its perfect channel diagonalization (Hao et al., 14 May 2026).

7. Application Domains and Limitations

HCM is well-suited for scenarios where low PAPR, hardware simplicity, low power, or inherently nonlinear (IM/DD) transmission is critical:

Visible Light Communications: Enabling high-illumination downlink data with minimal nonlinear distortion and built-in dimming control (Noshad et al., 2014, Noshad et al., 2014).
IoT and Low-Power Terminals: Zero-multiplier transmitters, real-only arithmetic, and minimal silicon footprint (Hao et al., 14 May 2026).
Massive Multiuser Wireless: Efficient hardware for large blocklengths and high-mobility environments (Hao et al., 14 May 2026).
Capacity-Approaching Channels: With tailored nonlinearities and approximate message passing, HCM can approximate channel-coding ensemble performance while reducing PAPR (Zhidkov, 2017).

A current research limitation is pilot and channel estimation design in the WHT domain (Hao et al., 14 May 2026). For channels with extremely high frequency selectivity ( $\mathbf{x} = \mathbf{H} \mathbf{a}, \qquad \mathbf{a} = \mathbf{H} \mathbf{x}$ 9), equalization complexity rises, and OFDM may outperform HCM in static or quasi-static settings. The search for optimal interleavers in large $\mathbf{a}$ 0 regimes also poses algorithmic challenges (Noshad et al., 2014).

References:

(Hao et al., 14 May 2026) "WHTDM: Walsh-Hadamard Transform Division Multiplexing for Doubly-Selective Channels"
(Noshad et al., 2014) "Hadamard Coded Modulation for Visible Light Communications"
(Zhidkov, 2017) "Orthogonal Transform Multiplexing with Memoryless Nonlinearity: a Possible Alternative to Traditional Coded-Modulation Schemes"
(Noshad et al., 2014) "Hadamard Coded Modulation: An Alternative to OFDM for Optical Wireless Communications"