The Chebyshev Polynomial Series Frequency Modulation Model for Waveform Design and Analysis

Abstract

Polynomial phase signals (PPS) are a staple of waveform design and analysis in sonar, radar, and communications fields. They also find application in the modeling of bioacoustic emissions, especially those of echolocating animals such as bats and odontocetes. This work presents a novel PPS waveform formulation that exploits some special properties of Chebyshev polynomials, such as orthogonality, recurrence relations, and equivalence to trigonometric functions. The result is the Chebyshev Polynomial Frequency Modulation (CPSFM) family of waveforms, which prove useful in the modeling of bioacoustic signals and the approximation of non-polynomial-phase signals such as hyperbolic chirps. We demonstrate that the CPSFM model admits compact analytic expressions for fundamental continuous-time signal processing functions such as the Fourier transform, the convolution and correlation operations, and the ambiguity function. Derivations for these expressions using CPSFM are presented, along with their application to the analysis of biosonar emissions of Mexican free-tailed bats.

Figures (6)

• Figure 1: Ambiguity function for HFM waveform and 3 CPSFM approximations. The second-order CPSFM, which is an LFM, performs poorly, with substantially degraded SNR and range resolution characteristics. The fourth-order CPSFM shows good Doppler tolerance.
• Figure 2: AF Doppler cuts at $\nu_0=1.124$ for true HFM and three CPSFM approximations. Initial and terminal frequencies are 100kHz and 200kHz, and duration is 2ms. The chosen Doppler coefficient corresponds to relative velocity of 20m/s.
• Figure 3: Spectrograms of two segments (a) and (b) of a recording of the echolocation transmissions of a swarm of Tadarida brasiliensis. Note that the axes are un-normalized time and frequency $\{t, f\}$, as opposed to normalized $\{x, g\}$, to make the figures easier to interpret.
• Figure 4: (a) Fourth-order Chebyshev series fit for the instantaneous frequency for CPSFM models for two calls. Each call is about 7.5ms in duration and sweeps through roughly an octave of bandwidth. Coefficients of the resulting Chebyshev series are $\{a_n\} =${34.9629 -13.1297 -1.0060 0.6347 -0.6599} and $\{b_n\} =${38.8165 -11.9987 2.6602 -0.2573 -1.0253}. (b) Spectra of the two model waveforms, computed via M-GBF expansions.
• Figure 5: Auto- and cross-correlation functions of two model waveforms. CCF peak is 11.6 dB attenuated relative to ACF, and is much less compact. Lower right plot shows the CCF of waveform (a) versus pseudo-random noise whose spectrum magnitude is the same as that of waveform (a), which shows a 14.3 dB attenuation in peak response.