Data-aided frame timing acquisition for fractal modulation in an AWGN channel

doi:10.1016/j.sigpro.2005.05.010

Signal Processing

Volume 86, Issue 2, February 2006, Pages 310-318

https://doi.org/10.1016/j.sigpro.2005.05.010 Get rights and content

Abstract

We propose a frame timing acquisition algorithm for fractal modulation in additive white Gaussian noise (AWGN) channels. Our algorithm exclusively uses the data redundancy inherent in fractal modulation to locate the start time of all the sub-bands (start-of-frame). The acquisition functions are derived using the maximum-likelihood method and the start-of-frame that maximizes the function attained by a serial search algorithm. Monte-Carlo simulations are conducted to evaluate the mean acquisition time of our algorithm.

Introduction

In studies of fractal modulation, it is assumed that the start-of-frame of a fractal modulation is known, but a technique to detect it has heretofore not been developed. A fractal modulation modulates the same data into different time-frequency cells. The diversity property results in the reliable transmission of data in a channel whose duration and bandwidth are both unknown to the transmitter [1], [2]. This property has been used to study channel estimation, equalization design, and data transmission in a fading environment [3], [4], [5]. These applications assume that the start-of-frame of a fractal modulation is known. When the start-of-frame is correctly detected, the time-frequency cells containing the same data can be identified, and the diversity property can be applied.

This frame synchronization, however, cannot be obtained by a wavelet modulation synchronization algorithm. Although such an algorithm can be used to obtain the symbol timing of sub-bands, it cannot be used to find time-frequency cells that contain the same data [6], [7]. We, therefore, need to develop a new frame timing recovery method for detection of the start-of-frame instant of fractal modulation.

We assume that baseband transmission is used and that the time domain clock error is the major factor that degrades the performance of a receiver. We propose a data-aided maximum likelihood approach to derive a data-aided frame timing acquisition function for a fractal modulated signal that exclusively uses the data redundancy inherent in fractal modulation. A series search approach is proposed to avoid calculating the derivative of the irregular likelihood function and finding the zero of the derivative.

In Section 2, we review fractal modulation and demodulation. In Section 3, we derive a frame timing acquisition function using a maximum likelihood approach; a serial search algorithm is introduced in Section 3.1. Simulation results of the acquisition performance are shown in Section 4. Finally, in Section 5, we present our conclusion and indicate the direction of future work to improve frame timing acquisition.

Section snippets

Fractal modulation and demodulation

An orthogonal wavelet is the basic signal waveform of fractal modulation. In an orthogonal wavelet transform, basis functions are all dilations and translations of a single function called a mother wavelet $ψ (t)$ . An orthogonal wavelet transformation of a signal $x (t)$ , with the wavelet $ψ (x)$ , is described in terms of synthesis and analysis equations in which the inverse wavelet transform is $x (t) = \sum_{m} \sum_{n} x_{m, n} ψ_{m, n} (t),$ and the wavelet transform is $x_{m, n} = \int x (t) ψ_{m, n} (t) d t,$ where $ψ_{m, n} (t) = 2^{m / 2} ψ (2^{m} t - n),$ and m and n are

Frame timing acquisition in an AWGN channel

We use $d = [d_{0}, d_{1}, \dots, d_{L - 1}]^{T}$ as the information vector with independent $d_{i} \in {\sqrt{E_{b}},- \sqrt{E_{b}}}$ . A received waveform with transmission delay $τ$ can be written as $r (t; τ) = s (t - τ) + w (t) .$ Suppose that $w (t)$ in Eq. (15) is a white Gaussian noise with a zero mean and variance $σ^{2}$ . If a demodulation fractal basis ${\bar{ϕ}}_{i} (t)$ is used to extract the information bit $d_{i}$ by projecting the received signal $r (t)$ onto the basis, we have $r_{i} (\hat{τ}; τ) = \int r (t; τ) {\bar{ϕ}}_{i} (t - \hat{τ}) d t$ $= s_{i} (\hat{τ} - τ) + w_{i} (\hat{τ}),$ where $s_{i} (\hat{τ} - τ) = \int s (t - τ) {\bar{ϕ}}_{i} (t - \hat{τ}) d t,$ $w_{i} (\hat{τ}) = \int w (t) {\bar{ϕ}}_{i} (t - \hat{τ}) d t .$ $w_{i} ($

Simulation results

All the simulation programs are written in Matlab, and our wavelets $ψ_{m, n} (t)$ are approximated by discrete points. We use $T_{0}$ to denote the signaling interval (the time interval between two adjacent symbols) at the sub-band $m = 0$ , which is the bottom row in Fig. 1, and 128 points to represent the interval $T_{0}$ . In all the experiments, we use the Meyer wavelet, which has a length of $16 T_{0}$ . The Meyer wavelet $ψ (t)$ has $2, 048$ discrete points in the slowest sub-band. Also, we choose $β = 1$ for modulation.

Conclusion and future work

Because a technique to detect the start-of-frame of fractal modulation had yet to be developed, the diversity property of a fractal modulation could not be applied. Our frame timing acquisition algorithm uses a serial search algorithm to locate the beginning timing of all sub-bands in a delayed signal. The acquisition algorithm obtains the maximum-likelihood solution in AWGN channels. As our acquisition precision is proportional to the number of sampling points, by increasing the sampling rate,

Acknowledgements

We would like to express our gratitude to the Reviewers for many insightful suggestions.

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Signal Processing

Data-aided frame timing acquisition for fractal modulation in an AWGN channel

Abstract

Introduction

Section snippets

Fractal modulation and demodulation

Frame timing acquisition in an AWGN channel

Simulation results

Conclusion and future work

Acknowledgements

Wavelet based representations for a class of self similar signals with applications to fractal modulation

IEEE Trans. Inform. Theory

Emerging applications of multirate signal processing and wavelets in digital communications

Proc. IEEE

Performance analysis of fractal modulation transmission over fast-fading wireless channels

IEEE Trans. Broadcasting

Clock synchronization for wavelet-based multirate transmissions

IEEE Trans. Commun.

Timing recovery in digital synchronous data receivers

IEEE Trans. Commun.

Generalized serial search code acquisition: the equivalent circular state diagram approach

IEEE Trans. Commun.