# Bandwidth Efficiency Optimization for Constant Envelope Multicarrier Modulation with Power Allocation and Bit Loading

Manchuan Fang, Ye Liu, Chung Shue Chen

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Manchuan Fang, Ye Liu, Chung Shue Chen. Bandwidth Efficiency Optimization for Constant Envelope Multicarrier Modulation with Power Allocation and Bit Loading. IEEE/CIC International Conference on Communications in China (ICCC), Aug 2025, Shanghai, China. hal-05171478

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Bandwidth Efficiency Optimization for Constant
Envelope Multicarrier Modulation with Power
Allocation and Bit Loading

Manchuan Fang*, Ye Liu*, Chung Shue Chen†

*Dept. of Communications and Networking, Xi'an Jiaotong-Liverpool University, China

†Nokia Bell Labs, Paris-Saclay Center, 91300 Massy, France

Email: Manchuan.Fang21@student.xjtlu.edu.cn, ye.liu02@xjtlu.edu.cn, chung_shue.chen@nokia-bell-labs.com

**Abstract**—This research presents an enhanced Constant Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM) scheme optimized for systems requiring high spectral efficiency. While multi-carrier modulation techniques like OFDM enable high data rates, their practical implementation is hindered by the peak-to-average power ratio (PAPR) issue, which degrades power efficiency and increases hardware complexity. CE-OFDM overcomes this limitation by generating constant-amplitude signals through phase modulation, thereby improving amplifier efficiency and reducing nonlinear distortion. We study a frequency-dependent power allocation scheme where the amplitudes of high-frequency subcarriers are suppressed more than those of low-frequency subcarriers, thereby reducing bandwidth and improving system efficiency. Additionally, as high-frequency subcarriers are allocated less power, bit-loading is enabled to better align channel conditions. The modulation order (M-value) for M-ary quadrature amplitude modulation (QAM) is dynamically adjusted in response to variations in the signal-to-noise ratio (SNR), ensuring maximal bandwidth efficiency with a guaranteed error performance. Numerical results demonstrate substantial improvements in spectral efficiency while maintaining acceptable bit error rates (BER), making CE-OFDM more viable for 5G/6G applications where both power efficiency and spectral utilization are critical.

**Keywords**—Constant envelope OFDM, bandwidth efficiency, power allocation, adaptive bit-loading, PAPR, signal processing.

# I. INTRODUCTION

The rapid advancement of wireless communications has intensified the demand for high-speed, reliable data transmission. However, signal propagation in complex environments introduces challenges including multi-path fading, delay spread, and interference, degrading system performance, posing significant challenges to achieving these goals [1]–[3]. Orthogonal Frequency Division Multiplexing (OFDM) has emerged as a cornerstone technology in addressing these challenges. OFDM is a prominent implementation that divides a broadband channel into multiple orthogonal sub-channels, enabling parallel transmission, which enhances spectral efficiency and reduces inter-symbol interference (ISI) [1]. This technique has been widely adopted in various communication standards, including digital broadcasting, wireless local area networks (WLANs), and cellular networks [4]. Nevertheless, OFDM has relatively

large peak power, resulting in undesirable PAPR (peak-to-average power ratio). To transmit these OFDM signals with high PAPR without distortion, the transmission end requires a high-power amplifier. Meanwhile, due to the irregularity of the occurrence of large peak power, it demands greater power back-off in amplifiers, limiting the energy efficiency, which is unfavorable to practical systems.

Constant Envelope OFDM (CE-OFDM) was proposed to eliminate PAPR by converting OFDM signals into phase-modulated, constant-envelope waveforms, theoretically achieving 0 dB PAPR [1]. The instantaneous power of the signal is kept as constant, and the peak power is equal to the average power, eliminating the energy loss caused by excessive peak power. Specifically, CE-OFDM can be considered as mapping OFDM signal to the unit circle [2]. This approach allows efficient nonlinear power amplification, which can greatly save power consumption, making it particularly attractive for battery-constrained devices, and is of great significance for both military and civilian applications [5]. However, one main issue of CE-OFDM is its bandwidth expansion. The constant-envelope nature of CE-OFDM signals results in a broader spectral occupancy compared to traditional OFDM, which would be a drawback in spectrum-constrained environments.

Prior to CE-OFDM, power allocation optimization had been considered to improve the spectral efficiency and reliability of OFDM systems. For example, [6] proposed an optimization that dynamically adjusted the modulation order and power allocation of subcarriers. Specifically, the channel state information (CSI) was utilized to adapt the transmission power and modulation order of each subcarrier, allocating higher power and more bits to subcarriers with high signal-to-noise ratios (SNR) while reducing power and bits on subcarriers with low SNR. This approach not only improved the spectral efficiency but also significantly enhanced the communication quality and reliability in high-noise environments. There are many related bandwidth allocation and power control methods which however typically involve two steps: first, determining the number of subcarriers allocated to each user based on their average SNR; second, calculating the optimal subcarrier distribution among users. When each user experiences flat fading channels, we can then minimize the total required power [7]. For example, water-filling would be used for the power allocation [8] and adaptive bandwidth techniques can

This work is supported in part by Xi'an Jiaotong-Liverpool University Research Development Fund (RDF-21-01-045) and the French ANR under grant ANR-24-CE25-2514 (WARM-M2M, https://warm-m2m.univ-ubs.fr).
<figure><img src="image_2.png" /><figcaption>Fig. 1. CE-OFDM system framework</figcaption></figure>

further help to reduce interference and enhance the overall resource utilization efficiency [9].

To date, few research studies specifically targeted power allocation and bit loading in CE-OFDM systems. However, these techniques, extensively studied in conventional OFDM frameworks, offer valuable insights and potential methodologies. By leveraging these established approaches, this study aims to extend and adapt power allocation and bit loading methods to CE-OFDM systems, thereby filling the potential research gap and presents a new optimized CE-OFDM scheme for systems which require enhanced power efficiency and high spectral efficiency. Noticing that the bandwidth of an angle modulation signal can be positively related to the amplitude of high frequency component of the modulating signal, we propose a frequency-dependent power allocation scheme for the OFDM signal, where less power is allocated to subcarriers with higher frequencies, aiming at suppressing high-frequency-subcarrier amplitudes, albeit at the cost of reduced information capacity. To mitigate the loss of overall data rate, dynamic bit-loading is introduced to better align channel conditions. As high-frequency sub-carriers are allocated minimal power, different modulation orders are applied on each subcarrier with different carrier frequency based on the SNR conditions, to optimize the bandwidth-error performance trade-off and enhance overall resource utilization efficiency.

The rest of the paper is organized as follows. In Section II, we describe the system model for CE-OFDM system. In Section III, we present the proposed power allocation for bandwidth control. In Section IV, we present the details of bit loading. In Section V, we show the simulation result and comparison. Finally, we conclude the paper in Section VI.

## II. SYSTEM MODEL AND PROBLEM FORMULATION

The system modeling is grounded in the classical CE-OFDM framework as shown in Fig. 1.

During each T-second block interval, an $N_{\text{DFT}}$-point inverse discrete Fourier transform (IDFT) calculates a block of time samples $\{x[n]\}$. The sampling rate is therefore $F_0 = N_{\text{DFT}}/T$.

### A. Conjugate Symmetric Zero-Padding

To obtain a real-valued OFDM sequence, the input to the IDFT is a conjugate symmetric, zero-padded data vector:

$$ \begin{cases} [0, X[1], X[2], \dots, X[N_{\text{QAM}}], \\ 0_{1 \times N_{\text{zp}}}, 0, X^*[N_{\text{QAM}}], \dots, X^*[2], X^*[1]] \end{cases}, \quad (1) $$

where:

* $\{X[k]\}_{k=1}^{N_{\text{QAM}}}$ are M-QAM data symbols, M is the modulation order,

* $0_{1 \times N_{\text{zp}}}$ is an $N_{\text{zp}}$-element row vector of zeros (such that $N_{\text{zp}}$ is the number of zeros padded), and

* $N_{\text{DFT}} = 2N_{\text{QAM}} + N_{\text{zp}} + 2$.

The zeros at index $k=0$ and $k = N_{\text{QAM}} + N_{\text{zp}} + 1$ are used to maintain conjugate symmetry, and the remaining $N_{\text{zp}}$ zeros achieve the effect of oversampling the time-domain sequence. The oversampling factor is defined as:

$$ C_{os} \triangleq \frac{N_{\text{DFT}}}{N_{\text{DFT}} - N_{\text{zp}}}. \quad (2) $$

Using this convention, the output of the IDFT can be expressed as [1]:

$$ \begin{aligned} x[n] &= \sum_{k=0}^{N_{\text{DFT}}-1} X[k] e^{j2\pi kn/N_{\text{DFT}}} \\ &= 2 \sum_{k=1}^{N_{\text{QAM}}} \Re\{X[k]\} \cos\left(\frac{2\pi kn}{N_{\text{DFT}}}\right) - \Im\{X[k]\} \sin\left(\frac{2\pi kn}{N_{\text{DFT}}}\right), \end{aligned} \quad (3) $$

for $n = 0, 1, \dots, N_{\text{DFT}} - 1$, where $j = \sqrt{-1}$.

### B. Phase Modulation

The high PAPR OFDM sequence $\{x[n]\}$ is phase-modulated to obtain a 0 dB PAPR sequence:

$$ s[n] = e^{jC \cdot x[n]}, \quad (4) $$

where C is a scaling constant.

### C. Digital-to-Analog Converter (DAC)

The discrete-time samples are then passed through a digital-to-analog (D/A) converter, and the result is amplified and transmitted into the channel. The time-domain representation of the phase-modulated signal is:

$$ s(t) = A e^{j(2\pi hm(t)+\theta)}, \quad 0 \le t < T, \quad (5) $$

where A is the signal amplitude, $\theta$ is an arbitrary phase offset, $h$ is a constant that is linearly related to $C$, $T = N_{\text{DFT}}/F_0$ is the block duration, and $m(t)$ is the OFDM signal that contains the digital information, i.e.,

$$ m(t) = C \sum_{k=1}^{N_{\text{QAM}}} \Re\{X[k]\} \cos\left(\frac{2\pi kt}{T}\right) - \Im\{X[k]\} \sin\left(\frac{2\pi kt}{T}\right). \quad (6) $$

The message signal can be more compactly written as

$$ m(t) = C_{\text{norm}} \sum_{k=1}^{N} I[k] q_k(t), \quad (7) $$

where:

$$ I[k] = \begin{cases} \Re\{X[k]\}, & k = 1, 2, \dots, N_{\text{QAM}} \\ -\Im\{X[k - N_{\text{QAM}}]\}, & k = N_{\text{QAM}} + 1, \dots, N \end{cases} \quad (8) $$

$$ q_k(t) = \begin{cases} \cos\left(\frac{2\pi kt}{T}\right), & k = 1, 2, \dots, N_{\text{QAM}} \\ \sin\left(\frac{2\pi(k - N_{\text{QAM}})t}{T}\right), & k = N_{\text{QAM}} + 1, \dots, N \end{cases} \quad (9) $$
$$C_{\text{norm}} = \sqrt{\frac{2}{N\sigma_I^2}}, \qquad (10)$$

and $\sigma_I^2 = \frac{2(M-1)}{3}$, such that $C_{\text{norm}}$ gives the average power of the QAM symbols [10].

### D. AWGN Channel Effect

The message signal is transmitted through a single-path Additive White Gaussian Noise (AWGN) channel, and the baseband representation of the received signal is<fnref n="1" />:

$$r(t) = s(t) + w(t), \qquad (11)$$

where $w(t)$ is complex-valued Gaussian noise with zero mean and power spectral density $\Phi_{ww}(f) = N_0$ (flat over the signal bandwidth).

### E. Phase Demodulator

The phase demodulator processes the received signal $r(t)$ to extract the phase information. Define $\phi(t) = 2\pi hm(t)$ such that the CE-OFDM signal is $s(t) = Ae^{j(\phi(t)+\theta)}$. The phase demodulator output $\hat{\phi}(t)$ is thus given by:

$$\hat{\phi}(t) = \angle r(t) = \angle(s(t) + w(t)) \\ = \phi(t) + \theta + \xi(t), \qquad (12)$$

where $\angle$ denotes the phase extraction operator, $\xi(t) = \arctan\left[\frac{A_w(t) \sin(\phi_w(t) - \phi(t) - \theta)}{A + A_w(t) \sin(\phi_w(t) - \phi(t) - \theta)}\right]$, $A_w(t) = |w(t)|$ is the envelope of $w(t)$, and $\phi_w(t) = \arctan[w(t)]$ is the phase of $w(t)$.

The DFT component projects $\hat{\phi}(t)$ onto each of the orthonormal basis functions $q_k(t)$, where the k-th output of the projection can be evaluated as:

$$\hat{Q}[k] = \frac{1}{T} \int_{0}^{T} \hat{\phi}(t) q_k(t) = \pi h C_{\text{norm}} I[k] + N[k], \qquad (13)$$

assuming high carrier-to-noise ratio, where $N[k]$ can be approximated as a zero-mean Gaussian random variable with variance $\frac{N_0}{2TA^2}$.

## III. BANDWIDTH CONTROL VIA SUBCARRIER POWER ALLOCATION

While embedding the OFDM signal into the phase of a carrier completely eliminates the high PAPR issue of OFDM, a potential consequence is the enlarged bandwidth consumption due to the modulation of the angle of a carrier signal as compared to double-sideband-type of modulations. It is well-known that in angle modulation, when the modulating signal is a simple sinusoidal function (i.e., tone modulation) such that the modulated signal is of the form:

$$s_{\text{tone}} = A \cos(2\pi f_c t + \beta \sin(2\pi f_m t)), \qquad (14)$$

where $f_c$ is the frequency of the carrier signal and $f_m$ is the frequency of the modulating sinusoid, then the bandwidth of the modulated signal is positively related to $\beta$ and $f_m$ [11].

In fact, Carson's rule estimates that such an angle modulated signal has a bandwidth approximately equal to

$$B \approx 2(\beta + 1)f_m, \qquad (15)$$

which clearly shows that the bandwidth of an angle modulation signal increases as $\beta$ or $f_m$ increases.

An OFDM signal is constructed by a linear combination of sinusoidal functions. On the other hand, in the extreme case where the OFDM signal consists of only one subcarrier, and within that subcarrier only the in-phase or the quadrature channel is used, then CE-OFDM reduces to the tone-modulated angle modulation. In that case, the bandwidth of $s(t)$ is directly proportional to the amplitude of the digital symbol and the frequency of the subcarrier. Keeping the frequency of the subcarrier unchanged, the smaller the energy of the symbol, the smaller the bandwidth that will be consumed by $s(t)$.

The analysis of the bandwidth of $s(t)$ is involved when multiple subcarriers are considered. However, given the above discussion, it is conjectured that one can control the bandwidth of CE-OFDM signal by applying power allocation factors to the OFDM subcarriers such that less power is allocated to a subcarrier that has a higher frequency. More specifically, let $\alpha_k$ be the power allocation factor for the k-th subcarrier that has frequency $k\Delta f$, $k = 1, 2, ..., N_{QAM}$, such that $\alpha_n$ decreases as $n$ increases. To keep the conjugate symmetry property of the data symbol vector, $\alpha_k$ will be applied to both the k-th subcarrier and the $N_{DFT} - k + 1$ subcarrier. Then, when power allocation is considered, instead of using $m(t)$ as the modulating signal, we use

$$m_{pa}(t) = C_{\text{norm}} \left[ \sum_{k=1}^{N_{\text{QAM}}} \alpha_k I[k] q_k(t) + \sum_{k=1}^{N_{\text{QAM}}} \alpha_k I[N_{\text{QAM}} - k + 1] q_{N_{\text{QAM}} - k + 1}(t) \right] \qquad (16)$$

to control the bandwidth of $s(t)$. In particular, we use two power allocation strategies, i.e.,

$$\alpha_k = \frac{C_1}{f_k}, \quad k = 1, 2, ..., N_{\text{QAM}} \qquad (17)$$

and

$$\alpha_k = \frac{C_2}{f_k^2}, \quad k = 1, 2, ..., N_{\text{QAM}}, \qquad (18)$$

where $C_1$ and $C_2$ are constants that keep the total transmission power constant before and after power allocation.

## IV. BIT LOADING

When power allocation is applied to the OFDM subcarriers, the SNRs of the subcarriers may differ significantly such that using the same QAM modulation scheme for all subcarriers may become inefficient. We consider applying bit loading to optimize the spectral efficiency by allocating different numbers of bits to different subcarriers based on their SNRs. More specifically, higher-order modulation schemes (e.g., 64-QAM) are used on subcarriers with higher SNRs, while lower-order

<footnote n="1">¹Here, we assume that the channel does not incur any phase shift, since a non-zero phase shift does not affect the error performance [1].</footnote>
schemes (e.g., 4-QAM) are used for subcarriers with lower SNR. This ensures that subcarriers with better SNR conditions would carry more bits, thereby increasing the overall system performance [6], [8]. The steps to perform the bit loading are detailed as follows:

* **SNR Calculation after Power Allocation:** The SNR of the k-th subcarrier after power allocation is calculated as:

$$ \text{SNR}_{\text{pa}}[k] = \text{SNR}[k] \cdot \alpha_k^2 \quad (19) $$

where $\text{SNR}[k]$ is the SNR of the k-th subcarrier before power allocation and can be obtained from (13).

* **Threshold-based Bit Allocation:** The system uses predefined SNR thresholds to determine the modulation order for each subcarrier:

$$ M_k = \begin{cases} 4, & \text{if } \text{SNR}_k < \text{threshold}_1, \\ 8, & \text{if } \text{threshold}_1 \le \text{SNR}_k < \text{threshold}_2, \\ 16, & \text{if } \text{threshold}_2 \le \text{SNR}_k < \text{threshold}_3, \\ 32, & \text{if } \text{threshold}_3 \le \text{SNR}_k < \text{threshold}_4, \\ 64, & \text{if } \text{SNR}_k \ge \text{threshold}_4, \end{cases} \quad (20) $$

where $M_k$ is the modulation order for the k-th subcarrier, and $\text{threshold}_i$ is the predefined SNR threshold(s).

* **Bit Calculation:** The number of bits allocated to each subcarrier is determined by the modulation order:

$$ b_k = \log_2(M_k). \quad (21) $$

Then, the total number of transmitted bits of one CE-OFDM signal is simply the summation of $b_k$ over all k, where $k = 1, 2, ... N_{\text{QAM}}$.

$$ N_{\text{Total}} = \sum_{k=1}^{N_{\text{QAM}}} b_k \quad (22) $$

Note that the above supports different threshold scheme for different power allocation coefficient. The choice of threshold scheme affects the modulation order allocation and thus the overall system performance. The bit loading scheme could improve the spectral efficiency but may increase BER for subcarriers with poor SNR. The above performance trade-off can be optimized by selecting appropriate thresholds.

# V. SIMULATION AND NUMERICAL RESULTS

## A. Simulation Setup

Table I presents the system parameters. Note that the number of subcarriers is 128, while the number of IFFT points is set to 512 to ensure that the oversampling factor $C_{\text{os}}$ is four. Conventionally, one sets the oversampling factor to be greater than or equal to four to ensure phase continuity in CE-OFDM; insufficient oversampling can lead to phase transitions [12].

The performance metrics for evaluation includes the required bandwidth, resulting bandwidth efficiency, and BER. Spectral analysis involves computing the energy spectral density (ESD) of the modulated signals via FFT-based spectral estimation. The 99% effective bandwidth is determined by integrating the ESD until the cumulative energy reaches 99%

<table><caption>Table I: Simulation Parameters for CE-OFDM</caption><thead><tr><th>Variable</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td>N<sub>s</sub></td><td>No. of symbols transmitted per run</td><td>1</td></tr><tr><td>N<sub>DFT</sub></td><td>No. of points in IFFT</td><td>512</td></tr><tr><td>T<sub>used</sub></td><td>Duration of each CE-OFDM symbol</td><td>1.28 × 10<sup>-3</sup> s</td></tr><tr><td>N</td><td>Total no. of subcarriers</td><td>128</td></tr><tr><td>N<sub>QAM</sub></td><td>No. of subcarriers used for QAM modulation</td><td>64</td></tr><tr><td>N<sub>zp</sub></td><td>Zero-padding length added to match FFT size</td><td>382</td></tr><tr><td>R<sub>s</sub></td><td>Symbol rate used for R<sub>b</sub> calculation</td><td>781.25 bps</td></tr><tr><td>2πh</td><td>Coefficient for phase modulation</td><td>Less than 0.9</td></tr><tr><td>C<sub>os</sub></td><td>Oversampling factor</td><td>4</td></tr><tr><td>Q<sub>bandwidth</sub></td><td>Percentage of bandwidth</td><td>99%</td></tr><tr><td>M<sub>k</sub></td><td>Array of modulation order used</td><td>[4,8,16,32,64]</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Fig. 2. BER for M-QAM without power allocation.</figcaption></figure>

[1]. When calculating the effective bandwidth, the FFT spectrum may be asymmetrical about the carrier frequency $f_c$. However, experiments shown that $[f_c - W, f_c + W]$ covers 99% of the signal energy with small errors. The effective bandwidth is then used to calculate the spectral efficiency, defined as the ratio of total transmitted information bits to the 99% energy bandwidth. The simulations span a range of SNR levels, enabling the investigation of BERs against different SNR values and the search of SNR thresholds for bit loading.

The value of $2\pi h$ is set to be less than or equal to 0.9. The reason is that when $2\pi h$ equals one or more, phase ambiguity arises when performing demodulation (i.e., noise-free sample values may exceed the range of $[-\pi, \pi]$).

Fig. 2 shows the BER performance without power allocation optimization, where $2\pi h = 0.9$. On the other hand, Fig. 3 shows the BER performance for the same $2\pi h$, where power allocation via (17) or (18) is applied, respectively. As expected, the power allocation would yield higher BER as compared to no power allocation. It is clear that the BER with power allocation via (18) is the worst among the three schemes. This also confirms our previous discussion on the trade-off between
<figure><img src="image_4.png" /><figcaption>Fig. 3. BER for M-QAM with power allocation, where $2\pi h = 0.9$.</figcaption></figure>

<table><caption>Table II: SNR threshold for bit loading to achieve a BER of $10^{-3}$, where $2\pi h = 0.5$.</caption><thead><tr><th>Modulation</th><th>power allocation using (17)</th><th>power allocation using (18)</th></tr></thead><tbody><tr><td>4-QAM</td><td>12.1 dB</td><td>24.9 dB</td></tr><tr><td>8-QAM</td><td>16.9 dB</td><td>28.0 dB</td></tr><tr><td>16-QAM</td><td>19.0 dB</td><td>31.9 dB</td></tr><tr><td>32-QAM</td><td>21.6 dB</td><td>34.8 dB</td></tr><tr><td>64-QAM</td><td>24.9 dB</td><td>38.2 dB</td></tr></tbody></table>

the bandwidth efficiency and error performance.

When the value of $2\pi h$ reduces, the SNR thresholds that are required to achieve a certain BER for various QAM schemes increases, since the SNR of a subcarrier reduces if $2\pi h$ reduces (see (5)). The comparison between Fig. 3 and Table II shows that larger SNR is required to achieve a BER of $10^{-3}$ when $2\pi h$ is reduced from 0.9 to 0.5.

## B. Numerical Results

Table III presents the measured 99% effective bandwidth for the CE-OFDM signals, where three different power allocation schemes (including without power allocation) are considered and $2\pi h = 0.9$. We can observe that the bandwidth consumption is the highest when no power allocation is used. The bandwidth consumption when the power allocation in (17) is used reduces the bandwidth consumption by about 37% as compared to no power allocation, and the more aggressive power allocation scheme in (18) reduces the bandwidth consumption by about 67% as compared to no power allocation.

Table IV compares the bandwidth efficiency which is given by:

$$ \eta = \frac{R_b}{B}, \qquad (23) $$

where:

<table><caption>Table III: Average bandwidth of CE-OFDM (P. A. stands for power allocation), where $2\pi h = 0.9$.</caption><thead><tr><th>System configuration</th><th>99% bandwidth (MHz)</th></tr></thead><tbody><tr><td>CE-OFDM without P. A.</td><td>0.169</td></tr><tr><td>CE-OFDM using P. A. via (17)</td><td>0.106</td></tr><tr><td>CE-OFDM using P. A. via (18)</td><td>0.055</td></tr></tbody></table>

* $R_b$ is the data rate (bps) that is calculated as

$$ R_b = R_s \cdot N_{Total}, \qquad (24) $$

where $R_s$ is the symbol rate (bits/second) defined in Table I and $N_{Total}$ is given by (22).

* B is the signal bandwidth in Hz.

The table demonstrates how power allocation techniques along (i.e., without bit loading) can effectively compress the signal spectrum while maintaining energy concentration. We can see that the bandwidth efficiency of power allocation via (18) is always higher than that via (17) and also that of no power allocation optimization. However, this improvement comes with higher SNR requirements (see in Table II).

<table><caption>Table IV: Bandwidth efficiency comparison for various P. A. schemes without bit loading, where $2\pi h = 0.9$.</caption><thead><tr><th rowspan="2">Modulation scheme</th><th colspan="3">Bandwidth efficiency (bps/Hz)</th></tr><tr><th>No P.A.</th><th>P.A. via (17)</th><th>P.A. via (18)</th></tr></thead><tbody><tr><td>4-QAM</td><td>0.5892</td><td>0.9563</td><td>1.8831</td></tr><tr><td>8-QAM</td><td>0.8874</td><td>1.4016</td><td>2.6479</td></tr><tr><td>16-QAM</td><td>1.1846</td><td>1.9023</td><td>3.7483</td></tr><tr><td>32-QAM</td><td>1.4836</td><td>2.3514</td><td>4.3973</td></tr><tr><td>64-QAM</td><td>1.7787</td><td>2.8704</td><td>5.6420</td></tr></tbody></table>

Fig. 4 illustrates the spectral efficiency of the CE-OFDM system after bit-loading, where we present different values of $2\pi h$ and BER thresholds for comparison. The SE curves tend to stabilize at high SNR values. This is because the channel conditions are good when SNR is high enough and the highest modulation order would be selected. In addition, we can see that when the power allocation in (18) is used, much higher spectral efficiency can be obtained (when the SNR is high enough). This is because the power allocation via (18) reduces the bandwidth to a small value at the cost of higher SNR compensation. The nonlinear growth of the two curves reflects the nature of wireless communication systems: In low SNR region, the system is limited by noise and the efficiency improvement is slow, while in high SNR area, the system is limited by bandwidth resources and the efficiency growth tends to level off.

In addition, it is observed that a smaller value of $h$ leads to less bandwidth consumption (also see [1]). As a result, we observe from Fig. 4 that a smaller value of $h$ gives rise to an increase of spectral efficiency. In particular, when the power allocation in (18) is used, the $2\pi h = 0.5$ setting always achieves a higher spectral efficiency as compared to the $2\pi h = 0.9$ case. Also, for the power allocation in (18), the $2\pi h = 0.5$ case appears to achieve a maximum spectral efficiency of about 8.9 bps/Hz, which is about 62% higher
<figure><img src="image_5.png" /><figcaption>Fig. 4. Spectral efficiency with power allocation and bit-loading.</figcaption></figure>

than that of the $2\pi h = 0.9$ case. The power allocation scheme in (18) generally achieves much higher spectral efficiency than the power allocation scheme in (17) when SNR is high. The spectral efficiency of the $2\pi h = 0.9$ case when the BER threshold is $10^{-3}$ is similar to that of the $2\pi h = 0.8$ case when the BER threshold is $10^{-4}$. In other words, one seems to be able to trade BER performance with spectral efficiency.

Fig. 5 illustrates the BER of the CE-OFDM system after bit-loading, where $2\pi h = 0.9$. The blue curve refers to the result based on power allocation via (18) and the red curve refers to that via (17). As mentioned previously, the BER thresholds for bit-loading are different for the two power allocation schemes. For the blue curve, the high-frequency subcarriers are severely attenuated, resulting in effective signal energy loss and weakened anti-noise ability. High-frequency information is overly suppressed, especially at low SNR, where noise dominates high-frequency subcarriers, causing demodulation errors. For red curve, the power attenuation is relatively gentle, such that high-frequency signal components are more robust to noise as compared to the blue-curve case.

The thresholds in Fig. 3 give the simulated minimum SNR requirement to ensure that the BER of a single sub-carrier (by averaging over all subcarriers) is less than or equal to $10^{-3}$ or even lower. According to (20), if the threshold is not satisfied, the lowest modulation order is selected. Thus, the BER as obtained by averaging over all subcarriers is dominated by the weakest subcarrier. When the SNR increases beyond threshold values, we can see in Fig. 5 that the BERs become lower than $10^{-3}$. The faster decrease of the red curve is due to the fact that the thresholds for power allocation using (17) are generally at least 10 dB lower than that for power allocation using (18), enabling more subcarriers to meet the target BER at a medium SNR. The two curves in Fig. 5 also show that high spectral efficiency is achieved at the expense of low BER reliability. Different trade-off may be achieved by modifying the strategy of power allocation and bit-loading.

## VI. CONCLUSION

We studied an enhanced CE-OFDM scheme optimized for systems requiring high spectral efficiency. Results have shown

<figure><img src="image_6.png" /><figcaption>Fig. 5. BER Performance for system after power allocation and bit-loading, where $2\pi h = 0.9$.</figcaption></figure>

the feasibility of bandwidth reduction in CE-OFDM through the proposed power allocation. The numerical study of CE-OFDM performances optimization via power allocation and bit-loading revealed interesting observations on the trade-off between the bandwidth efficiency and error performance. The adaptive CE-OFDM system has showed significant bandwidth saving compared to conventional CE-OFDM. Other schemes for bandwidth control, bandwidth efficiency optimization, and performance trade-off may be subjects for further research.

## ACKNOWLEDGEMENT

The authors would like to thank Prof. Emmanuel Boutillon for his valuable comments.

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