Equiripple MIMO Beampattern Synthesis using Chebyshev Approximation

This letter presents a method for synthesizing equiripple MIMO transmit beampatterns using Chebyshev approximation. The MIMO beampattern is represented as a non-negative real-valued trigonometric polynomial where the $\ell^{\text{th}}$ order polynomial coefficient is the sum of the $\ell^{\text{th}}$ order diagonal of the waveform correlation matrix. The optimal coefficients for a given equiripple beampattern design is then posed as a Chebyshev approximation problem which is efficiently solved using the Parks-McClellan algorithm from optimal Finite Impulse Response (FIR) filter design theory. The unique advantage of this synthesis method is that it provides a closed form method to generating MIMO correlation matrices that realize the desired equiripple beampattern. This correspondingly facilitates the design of waveform sets that closely approximate those correlation matrices. This method is demonstrated via two illustrative design examples; the first using traditional partial signal correlation methods and the second using transmit beamspace processing. Both examples realize equiripple beampatterns using constant envelope and spectrally compact waveform sets.

Gradient-Descent Based Optimization of Multi-Tone Sinusoidal Frequency Modulated Waveforms

This paper describes a gradient-descent based optimization algorithm for synthesizing Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveforms with low Auto-Correlation Function (ACF) sidelobes in a specified region of time delays while preserving the ACF mainlobe width. The algorithm optimizes the Generalized Integrated Sidelobe Level (GISL) which controls the mainlobe and sidelobe structure of the waveform's ACF. This optimization is performed subject to nonlinear constraints on the waveform's RMS bandwidth which directly controls the ACF mainlobe width. Since almost all of the operations of the algorithm utilize the Fast Fourier Transform (FFT), it is substantially more computationally efficient than previous methods that synthesized MTSFM waveforms with low ACF sidelobes. The computational efficiency of this new algorithm facilitates the design of larger dimensional and correspondingly larger time-bandwidth product MTSFM waveform designs. The algorithm is demonstrated through several illustrative MTSFM design examples.

Characterizing the Ambiguity Function of Constant-Envelope OFDM Waveforms

This paper investigates the radar Ambiguity Function (AF) properties of Constant Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM) waveforms employing Phase-Shift Keying (PSK). The CE-OFDM is in fact a special case of the Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveform which allows for applying many of the same mathematical techniques of the MTSFM model to the CE-OFDM model. This results in novel compact closed-form expressions for the spectrum, AF, and Auto-Correlation Function (ACF) of the CE-OFDM waveform. The mainlobe structure of the CE-OFDM's AF is characterized by the Ellipse of Ambigiuity (EOA) model. This produces precise closed-form expressions for the CE-OFDM's Root-Mean Square (RMS) bandwidth and the degree of range-Doppler coupling present in the waveform's AF mainlobe. These expressions show that a CE-OFDM waveform employing PSK as the symbol encoding scheme will possess a fixed RMS bandwidth for fixed modulation index $h$ and number of sub-carriers $L$. Additionally, we show that the EOA model predicts that a CE-OFDM waveform employing PSK encoding will almost always possess a ``Thumbtack-Like'' AF shape.

Gradient-Descent Based Optimization of Constant Envelope OFDM Waveforms

This paper describes a gradient-descent based optimization algorithm for synthesizing Constant Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM) waveforms with low Auto-Correlation Function (ACF) sidelobes in a specified region of time-delays. The algorithm optimizes the Generalized Integrated Sidelobe Level (GISL) which controls the mainlobe and sidelobe structure of the waveform's ACF. The operations of this Gradient-Descent GISL (GD-GISL) algorithm are FFT-based making it computationally efficient. This computational efficiency facilitates the design of large dimensional waveform design problems. Simulations demonstrate the GD-GISL algorithm on CE-OFDM waveforms employing Phase-Shift Keying (PSK) symbols that take on a continuum of values (i.e, $M_{\text{PSK}} = \infty$). Results from these simulations show that the GD-GISL algorithm can indeed reduce ACF sidelobes in a desired region of time-delays. However, truncating the symbols to finite M-ary alphabets introduces perturbations to the waveform's instantaneous phase which increases the waveform's ACF sidelobe levels.