A New LFM-Signal Detector Based on Fractional Fourier Transform
© Zhiping Yin and Weidong Chen. 2010
Received: 29 December 2009
Accepted: 24 June 2010
Published: 12 July 2010
A new LFM-signal detector formulated by the integration of the 4th-power modulus of the fractional Fourier transform is proposed. It has similar performance to the modulus square detector of Radon-ambiguity transform because of the equivalence relationship between them. But the new detector has much lower computational complexity in the case that the number of the searching angles is far less than the length of the signal. Moreover, it is proved that the new detector can be generalized to the integration of the n th-power ( ) modulus of the fractional Fourier transform via mathematical derivation. Computer simulation results have confirmed the effectiveness of the proposed detector in LFM-signal detection.
The detection of the Linear Frequency Modulation (LFM) signal is very important in many information systems, such as communications, radar, and sonar, for its wide use in these systems. In recent years, several time-frequency-based methods for LFM-signal detection have been proposed. Several intelligible detection algorithms using the short-time Fourier transform (STFT) or wavelet transform are mentioned in [1, 2]. However, the poor resolution according to the narrow or time-variant window used in the analysis limits their applications in practice. The discrete chirp-Fourier transform is employed in  to estimate the chirp rate of LFM-signal. But its rigorous constraint, that signal length must be prime number and the chirp rate must be integer, limits its application. Since the LFM-signal distributes as a straight line in the Wigner-Ville distribution (WVD) plane, the Radon-Wigner transform (RWT) algorithm computes integrals along the lines with different angles and positional offsets in the time-frequency plane to detect the LFM-signal [4, 5], and a two-dimensional search is necessary to track the straight lines in the time-frequency plane. The lines, whose integral values exceed a certain threshold, correspond to the LFM-signal. Similar to WVD, the ambiguity function (AF) of the LFM-signal distributes as a straight line passing through the origin in the AF plane. Therefore, Radon-ambiguity transform is also used to detect the LFM-signal, especially in the case that the chirp rate is the only parameter of interest, and two kinds of detectors (the envelope detector and the modulus square detector) have been studied in . For Radon-Wigner transform or Radon-ambiguity transform, both the calculation of the full time-frequency plane (or AF plane) and the transformation from the Cartesian coordinate to the polar coordinate are indispensable, whose high computational complexity severely hinders the usefulness of the two methods in the LFM-signal detection, especially in the case of long signal detection. Recently, as a new time-frequency analysis tool, the fractional Fourier transform (FRFT) attracts more and more attention in signal processing field [7, 8]. Several methods based on FRFT have been proposed to detect LFM-signal and estimate its parameters [9–12]. An effective method for parameter estimation and recovery of time-varying signals including LFM-signal by using FRFT is proposed in . According to the equivalence relationship between RWT and the fractional power spectrum [13, 14], a LFM-signal detection and parameters estimation method based on FRFT has been presented [10, 11], which achieves a near-RWT performance at much lower computational complexity.
Similar to WVD, ambiguity function also has a close relation to FRFT [12, 15, 16]. Then an LFM-signal detector using fractional autocorrelation, which corresponds to the envelope detector of Radon-ambiguity transform, is proposed . In this paper, we propose a new LFM-signal detector formulated by the integration of the 4th-power modulus of the fractional Fourier transform. The new detector is equivalent to the modulus square detector of Radon-ambiguity transform and performs similarly as well in LFM-signal detection. But the new detector has much lower computational complexity in the case of long signal detection, or in the case that the possible distribution area of the signal in the ambiguity plane is limited to a small-angle sector. Mathematical derivation proves that the new detector can be generalized to the integration of the n th-power ( ) modulus of the fractional Fourier transform.
The structure of this paper is as follows. In Section 2, a simple review of the definition of FRFT and its relation to AF are given. The modulus square detector of Radon-ambiguity transform is briefly introduced in Section 3. The new detector based on FRFT is proposed and its computational complexity compared with Radon-ambiguity transform is discussed in Section 4. Section 5 gives the mathematical derivation of the generalization of the new detector. Computer simulations are given in Section 6 to show the effectiveness of the proposed detector. Finally, conclusions are made in Section 7.
2. Definition of FRFT and Its Relation to AF
The standard Fourier transform is an operator that transforms a time domain signal into a frequency domain representation. In time-frequency plane, Fourier transform can be interpreted as a counterclockwise rotation operator from the time axis to the frequency axis. FRFT, as the generalization of the standard Fourier transform, can be considered as a counterclockwise rotation of the signal coordinate around the origin on the u axis with an arbitrary angle , and transforms a signal to an intermediate domain between time and frequency. Therefore, FRFT can be classified into the time-frequency analysis tools, and it is strongly related to other important time-frequency transforms, such as WVD and AF.
By combing AF and Radon transform, a new LFM-signal detection method called Radon-ambiguity transform has been established . It is more efficient than RWT in the case that the chirp rate is the only parameter of interest.
Equation (6) reveals the relationship between AF and FRFT as follows: the slice of AF at angle in the ambiguity plane is the Fourier transform of the fractional power spectrum of angle . Based on this relationship, an LFM-signal detection and chirp rate estimation method using FRFT, which has similar performance but lower computational complexity compared with Radon-ambiguity transform, is proposed in this paper.
3. The Modulus Square Detector of Radon-Ambiguity Transform
The detector's good performance is discussed in detail and its usefulness in LFM-signal detection has been well demonstrated by some numerical examples in .
4. The New Detector Based on FRFT
The equivalence between the new detector and the modulus square detector of Radon-ambiguity transform can be easily demonstrated from the derivation process. For LFM-signal, the new detector will reach its maximum when the angle is equal to –arc cot(K), where K is the chirp rate of the LFM-signal. According to the equivalence relationship between the new detector and the modulus square detector of Radon-ambiguity transform, they have the nearly same performance except the computational complexity in theory.
5. The Generalization of the New Detector
An interesting phenomena, that the effective detector is not only limited to the integration of the 4th-power modulus of the fractional Fourier transform, but can also be generalized to n th-power modulus for , is found in our simulation. In this section, the mathematical derivation is implemented to prove it.
where K is the chirp rate of the LFM-signal.
It can be seen that, for LFM-signal, the generalized detector will reach its maximum corresponding to the chirp rate. Therefore, the generalized detector can be regarded as an effective detector for LFM-signal. However, the 4th-power modulus form detector is most frequently used in practice for its lower computational complexity compared with other generalized forms.
6. Simulation Results
This paper presents a new detector of LFM-signal, which is the integration of the 4th-power modulus of its fractional Fourier transform. The new detector has equivalence relation to the modulus square detector of Radon-ambiguity transform, and performs similarly to Radon-ambiguity transform in LFM-signal detection under much lower computational complexity. Moreover, it has been proved mathematically that the new detector can be generalized to the integration of the n th-power modulus of the fractional Fourier transform for . Computer simulations have verified the effectiveness of the new detectors.
This work is supported by the China Postdoctoral Science Foundation under Grant 20090460726. The authors would like to thank the anonymous reviewers who give some valuable comments for improving the quality of this paper.
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