MUSIC for single-snapshot spectral estimation: Stability and super-resolution

Abstract

This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The singlesnapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.

Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction

“…In this case, since we can recover the full signal of the superposition of the underlying sinusoids, we can use the single-snapshot MUSIC algorithm [27] to recover the underlying frequencies precisely.…”

“…For this case, we have obtained a bound on the recovery error for the superposition signal (Theorem 1 of our paper). We can further recover the frequencies using the single-snapshot MUSIC algorithm by choosing the R smallest local minimum of surrogate criterion function R ( ω ) in [27]. In [27], the authors provided the stability result of recovering frequency using the single-snapshot MUSIC algorithm (Theorem 3 of [27]).…”

“…We can further recover the frequencies using the single-snapshot MUSIC algorithm by choosing the R smallest local minimum of surrogate criterion function R ( ω ) in [27]. In [27], the authors provided the stability result of recovering frequency using the single-snapshot MUSIC algorithm (Theorem 3 of [27]). Specifically, the error in surrogate criterion function R ( ω ) is upper bounded by the Euclidean norm of the observation noise multiplied by a constant C , where the constant C depends on the largest and smallest nonzero singular values of the involved Hankel matrix.…”

“…Specifically, the error in surrogate criterion function R ( ω ) is upper bounded by the Euclidean norm of the observation noise multiplied by a constant C , where the constant C depends on the largest and smallest nonzero singular values of the involved Hankel matrix. Moreover, the recovered frequency deviates from the true frequency in the order of noise standard deviation when the noise is small (Remark 9 of [27]). We remark that this stability result from [27] is applicable without imposing separation condition on frequencies.…”

Robust Spectral Compressed Sensing via Structured Matrix Completion

“…Our incoherence condition naturally requires certain separation among all frequency pairs, as when two frequency spikes are closely located, µ 1 gets undesirably large. As shown in [43,Theorem 2], a separation of about 2/n for line spectrum is sufficient to guarantee the incoherence condition to hold. However, it is worth emphasizing that such strict separation is not necessary as required in [26], and thereby our incoherence condition is applicable to a broader class of spectrally sparse signals.…”