---
title: 'Hadamard Coded Modulation: Principles & Applications'
url: https://www.emergentmind.com/topics/hadamard-coded-modulation-hcm
type: topic
---

# Hadamard Coded Modulation: Principles & Applications

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) [1406.2897][1404.1148], to wireless and massive IoT multicarrier platforms [2605.14482], and near-capacity coded systems with nonlinearities [1703.03141].

## 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 [1404.1148]. The WHT provides $O(N \log N)$ complexity, integer-only operations, and no twiddle-factor or complex arithmetic requirements [2605.14482][1404.1148].

## 2. HCM Modulation, Transmission, and Demodulation

In canonical HCM, the modulator performs the following steps:

1. **Bit Mapping:** Map input bits to $\mathbf{u} \in \mathbb{R}^N$ (e.g., $M$-PAM/OOK).
2. **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.
3. **Cyclic Prefix, Interleaving, or Shaping:** A cyclic prefix is optionally added for ISI mitigation, and pulse shaping (e.g., sinc) may be applied [1406.2897][1404.1148][2605.14482].
4. **Transmission:** The waveform is launched (direct modulation or via a nonlinear mapping in certain HCM variants [1703.03141]).

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.

\[
\mathbf{v} = \frac{1}{N}[H_N^T \mathbf{y} - \overline{H}_N^T \mathbf{y}] + \mbox{(DC compensation vector)}
\]

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 $N$ [1406.2897][1404.1148]:

\[
\textrm{PAPR}_{\textrm{HCM}} = 2
\]

OFDM, by contrast, exhibits PAPR scaling as $\mathrm{O}(\log N)$ in expectation, up to $N$ 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 $P_\mathrm{max}/2$. For higher efficiency, DC-reduced HCM (DCR-HCM) removes the constant offset from each block:

\[
\mathbf{x}' = \mathbf{x} - \min(\mathbf{x})
\]

The information rate penalty is negligible for large $N$ (rate loss $\approx 0.8\%$ for $N=128$ [1404.1148]). DCR-HCM can achieve a $\approx 3$ dB reduction in average optical power at equal BER versus standard HCM in clipping-limited regimes [1404.1148].

In multi-carrier or nonlinear-channel settings, applying a memoryless nonlinearity $f(\cdot)$ 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 $3.9$ dB at $N=1024$ [1703.03141].

## 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 $\pi$ (found via binary LP) spreads ISI uniformly over all codewords. At the receiver, the inverse permutation is applied before IFWHT [1404.1148][1406.2897].
- **Cyclic Prefix (CP):** Applied for block-wise convolutional invariance and ISI suppression (as in CP-OFDM, CP-WHTDM) [2605.14482].
- **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 $\mathcal{O}(B N)$ rather than $\mathcal{O}(N^2)$ when the effective channel has bandwidth $B \ll N$. For QPSK,

  \[
  \eta_\mathrm{QPSK}(r, \tau) = \frac{1}{\sqrt{2}}\left[\tanh\left(\frac{\sqrt{2}\mathrm{Re}(r)}{\tau}\right) + j\tanh\left(\frac{\sqrt{2}\mathrm{Im}(r)}{\tau}\right)\right]
  \]

The efficacy of interleaving and MMSE equalization for ISI suppression in dispersive VLC is documented in [1404.1148][1406.2897]. 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 [2605.14482].

## 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                   | $1.4\times10^{-2}$  | $2.0\times10^{-2}$  |
| OFDM (1-tap MMSE)           | $>2$ | 18,432             | 12,288              | $\approx 10^{-1}$   | $>0.4$              |
| AFDM + CD-MAMP              | 2    | 22,528             | 20,480              | $2.1\times10^{-2}$  | —                   |
| OTFS + CD-MAMP              | 2    | 49,152             | 32,768              | $4.7\times10^{-2}$  | —                   |

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 [1406.2897][1404.1148]. In coded/numerical channel settings, HCM–AMP approaches within $1.5$ dB of AWGN capacity [1703.03141]. Complexity always scales as $O(N\log N)$ 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 [2605.14482][1404.1148].
- **Power Efficiency:** DC-reduced variants achieve up to 50% average power savings in optical systems [1404.1148], with natural dimming control via DC level manipulation [1406.2897].
- **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 [1404.1148][2605.14482].
- **Spectral Efficiency:** HCM is compatible with OOK, low-order PAM, and QAM; rate loss due to forced-zero first row becomes negligible as $N$ grows.
- **Dimming Support (VLC):** DC control is built-in, in contrast to PWM/OOK hybrids required in OFDM [1406.2897].

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 [2605.14482].

## 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 [1406.2897][1404.1148].
- **IoT and Low-Power Terminals:** Zero-multiplier transmitters, real-only arithmetic, and minimal silicon footprint [2605.14482].
- **Massive Multiuser Wireless:** Efficient hardware for large blocklengths and high-mobility environments [2605.14482].
- **Capacity-Approaching Channels:** With tailored nonlinearities and approximate message passing, HCM can approximate channel-coding ensemble performance while reducing PAPR [1703.03141].

A current research limitation is pilot and channel estimation design in the WHT domain [2605.14482]. For channels with extremely high frequency selectivity ($L \approx N$), equalization complexity rises, and OFDM may outperform HCM in static or quasi-static settings. The search for optimal interleavers in large $N$ regimes also poses algorithmic challenges [1404.1148].

---

**References:**
- [2605.14482] "WHTDM: Walsh-Hadamard Transform Division Multiplexing for Doubly-Selective Channels"
- [1406.2897] "Hadamard Coded Modulation for Visible Light Communications"
- [1703.03141] "Orthogonal Transform Multiplexing with Memoryless Nonlinearity: a Possible Alternative to Traditional Coded-Modulation Schemes"
- [1404.1148] "Hadamard Coded Modulation: An Alternative to OFDM for Optical Wireless Communications"

Source: https://www.emergentmind.com/topics/hadamard-coded-modulation-hcm