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.