## Optimum Time-Frequency Distribution for Detecting a Discrete-Time Chirp Signal in White Gaussian Noise

### BibTeX

@MISC{Yasotharan_optimumtime-frequency,

author = {A. Yasotharan and T. Thayaparan},

title = {Optimum Time-Frequency Distribution for Detecting a Discrete-Time Chirp Signal in White Gaussian Noise},

year = {}

}

### OpenURL

### Abstract

In the continuous-time domain, Maximum-Likelihood (ML) detection of a chirp signal in white Gaussian noise can be done via the line-integral transform of the classical Wigner distribution. The line-integral transform is known variously as the Hough transform and the Radon transform. For discrete-time signals, the Wigner-type distribution defined by Claasen and Mecklenbrauker has become popular as a signal analysis tool. Moreover, it is commonly believed that ML detection of a discrete-time chirp signal in white Gaussian noise can be done via the line-integral transform of the Wigner-Claasen-Mecklenbrauker distribution. This belief is false and results in loss of performance. We derive a Wigner-type distribution for discrete-time signals whose line-integral transform can be used for ML detection of discrete-time chirp signals in white Gaussian noise. We provide simulated Receiver Operating Curves for the Wigner-Claasen-Mecklenbrauker distribution based method and the new ML-equivalent method and demonstrate the suboptimality of the former. I.

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(Show Context)
Citation Context ...ution based method and the new ML-equivalent method and demonstrate the suboptimality of the former. I. Introduction For a continuous-time signal r(t), the classical Wigner distribution is defined as =-=[1]-=- ∫ Wr(t, ω) = r(t + τ/2)r ∗ (t − τ/2)e −jωτ dτ, (1) where t is time and ω is frequency. In [1], the Wigner distribution was shown to have many properties that make it a useful signal analysis tool. Su... |

33 |
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(Show Context)
Citation Context ...irp signal via the hypothesis test (3) is known variously as the correlator method and the dechirp-Doppler method. A study of the use of time-frequency distributions for detecting signals is found in =-=[4]-=-. For a discrete-time signal r(n), the Wigner distribution defined by Claasen and Mecklenbrauker [5] has become popular as a signal analysis tool. Their definition of Wigner distribution is (n, θ) =2 ... |

16 |
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(Show Context)
Citation Context ...med, erroneously, that ML detection of the discrete-time chirp signal s(n) is equivalent to the hypothesis test n max c1,c2 ∑ n W CM r (n, c1 + c2n) H1 > < H0 γ. (11) However, it has been observed in =-=[6]-=- that the W CM r -based method (11) incurs a 3 dB loss due to nonlinearity. 1 In Appendix IV, we provide simulated Receiver Operating Curves for the ML method (9) and the W CM r -based method (11) and... |

9 |
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(Show Context)
Citation Context ... maximum on the left hand side (LHS) of (3) is less than the threshold γ then H0 is considered true and if the maximum on the LHS of (3) is greater than the threshold γ then H1 is considered true. In =-=[2]-=-, the hypothesis test (3) was shown to be approximately equivalent to H1 is the Signal-plus-Noise Hypothesis r(t) =s(t)+w(t), max ω0,m ∫ Wr(t, ω0 + mt) H1 > < H0 γ (4) for chirp signals of large durat... |

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(Show Context)
Citation Context ...rived time-frequency distribution may be considered a Wigner-type distribution. It turns out that the Wigner-type time-frequency distribution derived in this paper is the same as that derived by Chan =-=[7]-=- in an effort to solve the problem of aliasing in the Wigner distribution (6). Nevertheless, the optimality property of this distribution for detection of a discrete-time signal was not observed in [7... |

3 |
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(Show Context)
Citation Context ... to be approximately equivalent to H1 is the Signal-plus-Noise Hypothesis r(t) =s(t)+w(t), max ω0,m ∫ Wr(t, ω0 + mt) H1 > < H0 γ (4) for chirp signals of large duration. This equivalence was shown in =-=[3]-=- to be exact and valid even for finite-duration signals. More specifically, it was shown in [3] that ∫ 1 −j(ω0t+ ∣ r(t)e 2 mt2 2 ∫ ) dt ∣ = Wr(t, ω0 + mt)dt, (5) where the quantity on the right hand s... |

3 |
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(Show Context)
Citation Context ...unknown parameters b0, b1 and b2, with the background being additive white Gaussian noise. Discrete-time chirp signals arise directly in pulse Doppler radars when a target is moving with acceleration =-=[8]-=-. Discrete-time chirp signals also arise in synthetic aperture radars and inverse synthetic aperture radars. A discrete-time chirp signal may also arise as a sampled-version of a continuous-time chirp... |

1 |
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(Show Context)
Citation Context ...d-version of a continuous-time chirp signal. This is the case, for example, in electronic counter measures to LFM radar and sonar. Many situations where chirp signals occur in nature are described in =-=[9]-=-. For the above discrete-time detection problem, define ∣ ∣ N−1 ∑ ∆r(c1,c2) = ∣ n=0 1 −j(c1n+ r(n)e 2 c2n2 ) (7) 2 . (8) ∣ The classical Maximum-Likelihood (ML) method is then equivalent to the hypoth... |