Extending the Multitone Sinusoidal Frequency Modulation Signal Model by Fourier Expansions With Arbitrary Periods

In this article, we revisit and generalize the multitone sinusoidal frequency modulation (MTSFM) expansion, which has been used in the past decade for various radar and sonar applications, such as optimal transmit waveform design and modeling nonlinear frequency modulation. In its original form, the MTSFM describes a phase-modulated signal on a time interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> by the Fourier expansion of its instantaneous frequency or modulation function. The period of this expansion is chosen equal to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> resulting in the sinusoidal expansion functions being orthogonal on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula>. In this article, we extend the MTSFM to expansions with arbitrary periods and, hence, nonorthogonal sinusoidal expansion functions. When applied to represent a specific signal, these expansions have different convergence rates and differentiability characteristics and, hence, result in signals with different spectral compactness and efficiency. Also, we expand the time-dependent phase of the signal instead of the instantaneous frequency or modulation function, which allows classically (in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> sense) for the representation of less smooth signals. We will illustrate these aspects by three examples of representing a classical MTSFM expansion, an up-chirp, and a pseudorandomly phase-modulated pulse by generalized MTSFM expansions, for which we calculate quantities such as root-mean-square (RMS) bandwidth, frequency response, autocorrelation function, and approximation accuracy. Also, we provide tangible examples of using the generalized expansions in a gradient-based optimizer to synthesize signals for optimal integrated sidelobe ratio under RMS bandwidth constraint. To this end, we derive general and easy-to-use expressions for the derivatives of the objective and constraint functions.