Compressive sensing of sparse radio frequency signals using optical mixing

We demonstrate an optical mixing system for measuring properties of sparse radio frequency (RF) signals using compressive sensing (CS). Two types of sparse RF signals are investigated: (1) a signal that consists of a few 0.4 ns pulses in a 26.8 ns window and (2) a signal that consists of a few sinusoids at different frequencies. The RF is modulated onto the intensity of a repetitively pulsed, wavelength-chirped optical field, and time-wavelength-space mapping is used to map the optical field onto a 118-pixel, one-dimensional spatial light modulator (SLM). The SLM pixels are programmed with a pseudo-random bit sequence (PRBS) to form one row of the CS measurement matrix, and the optical throughput is integrated with a photodiode to obtain one value of the CS measurement vector. Then the PRBS is changed to form the second row of the mixing matrix and a second value of the measurement vector is obtained. This process is performed 118 times so that we can vary the dimensions of the CS measurement matrix from 1×118 to 118×118 (square). We use the penalized ℓ1 norm method with stopping parameter λ (also called basis pursuit denoising) to recover pulsed or sinusoidal RF signals as a function of the small dimension of the measurement matrix and stopping parameter. For a square matrix, we also find that penalized ℓ1 norm recovery performs better than conventional recovery using matrix inversion.