### Abstract

Polynomial-phase signals (PPS's), i.e., signals parameterized as s(t) = A exp(j2πΣ
_{m=0}
^{M} a
_{m}t
^{m}), have been extensively studied and several algorithms have been proposed to estimate their parameters. From both the application and the theoretical points of view, it is particularly important to know the spectrum of this class of signals. Unfortunately, the spectrum of PPS's of generic order is not known in closed form, except for first- and second-order PPS's. The aim of this letter is to provide an approximate behavior of the spectrum of PPS's of any order. More specifically, we prove that: i) the spectrum follows a power law behavior f
^{-γ}, with γ = (M - 2)/(M - 1); ii) the spectrum is symmetric for M even and is strongly asymmetric for M odd; and iii) the maximum of the spectrum has an upper bound proportional to T
^{(M-I)/M} and a lower bound proportional to T
^{1/2}. These results are useful to predict the performance of the so-called high order ambiguity function (HAF) and the Product-HAH (PHAF), specifically introduced to estimate the parameters of PPS's, when applied to multicomponent PPS's.

Original language | English (US) |
---|---|

Pages (from-to) | 237-340 |

Number of pages | 104 |

Journal | IEEE Signal Processing Letters |

Volume | 5 |

Issue number | 9 |

DOIs | |

State | Published - 1998 |

Externally published | Yes |

### Fingerprint

### Keywords

- High-order ambiguity function
- Polynomial-phase signals

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Signal Processing

### Cite this

*IEEE Signal Processing Letters*,

*5*(9), 237-340. https://doi.org/10.1109/97.712109

**On the spectral properties of polynomial-phase signals.** / Scaglione, Anna; Barbarossa, Sergio.

Research output: Contribution to journal › Article

*IEEE Signal Processing Letters*, vol. 5, no. 9, pp. 237-340. https://doi.org/10.1109/97.712109

}

TY - JOUR

T1 - On the spectral properties of polynomial-phase signals

AU - Scaglione, Anna

AU - Barbarossa, Sergio

PY - 1998

Y1 - 1998

N2 - Polynomial-phase signals (PPS's), i.e., signals parameterized as s(t) = A exp(j2πΣ m=0 M a mt m), have been extensively studied and several algorithms have been proposed to estimate their parameters. From both the application and the theoretical points of view, it is particularly important to know the spectrum of this class of signals. Unfortunately, the spectrum of PPS's of generic order is not known in closed form, except for first- and second-order PPS's. The aim of this letter is to provide an approximate behavior of the spectrum of PPS's of any order. More specifically, we prove that: i) the spectrum follows a power law behavior f -γ, with γ = (M - 2)/(M - 1); ii) the spectrum is symmetric for M even and is strongly asymmetric for M odd; and iii) the maximum of the spectrum has an upper bound proportional to T (M-I)/M and a lower bound proportional to T 1/2. These results are useful to predict the performance of the so-called high order ambiguity function (HAF) and the Product-HAH (PHAF), specifically introduced to estimate the parameters of PPS's, when applied to multicomponent PPS's.

AB - Polynomial-phase signals (PPS's), i.e., signals parameterized as s(t) = A exp(j2πΣ m=0 M a mt m), have been extensively studied and several algorithms have been proposed to estimate their parameters. From both the application and the theoretical points of view, it is particularly important to know the spectrum of this class of signals. Unfortunately, the spectrum of PPS's of generic order is not known in closed form, except for first- and second-order PPS's. The aim of this letter is to provide an approximate behavior of the spectrum of PPS's of any order. More specifically, we prove that: i) the spectrum follows a power law behavior f -γ, with γ = (M - 2)/(M - 1); ii) the spectrum is symmetric for M even and is strongly asymmetric for M odd; and iii) the maximum of the spectrum has an upper bound proportional to T (M-I)/M and a lower bound proportional to T 1/2. These results are useful to predict the performance of the so-called high order ambiguity function (HAF) and the Product-HAH (PHAF), specifically introduced to estimate the parameters of PPS's, when applied to multicomponent PPS's.

KW - High-order ambiguity function

KW - Polynomial-phase signals

UR - http://www.scopus.com/inward/record.url?scp=0004629703&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0004629703&partnerID=8YFLogxK

U2 - 10.1109/97.712109

DO - 10.1109/97.712109

M3 - Article

AN - SCOPUS:0004629703

VL - 5

SP - 237

EP - 340

JO - IEEE Signal Processing Letters

JF - IEEE Signal Processing Letters

SN - 1070-9908

IS - 9

ER -