Quick Estimation of Periodic Signal Parameters From 1-Bit Measurements

Abstract

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Measurement of the Parameters of Multiple Sinusoids Based on Binary Data

“…Conversely, when ϵ increases, estimation variance is reduced at the price of a larger bias. As indicated in [34], the choice ϵ = 1 √ N provides reasonable results.…”

“…The measurement method aims at minimizing numerical calculations that are known to be time-consuming, in favor of ordi-nary matrix inversion methods that are efficiently performed even by a moderately complex microprocessor system. The method extends the binary quantile-based estimator (BQBE) published in [34] and is called in the following frequency BQBE (FBQBE). With respect to BQBE, FBQBE does not assume that the signal is periodic, nor that the component frequencies in the signal are known.…”

“…Also define Λ = [λ 1 • • • λ K ] T as the vector of unknown frequency components, with T as the transpose operator. The estimation procedure uses a modified version of the BQBE described in [34], based on the assumption of known component frequencies, and described in the following to ease readability. It will be explained in the case of a single signal component, i.e.…”

“…Observe that the value of ϵ controls the tradeoff between the estimator bias and variance [34]. In fact, a very small value of ϵ guarantees a reduced bias because indices in I j in (6) refer to very similar values of λ n .…”

“…Thus, the BQBE estimator tends to cancel the effect of unmodeled signal components when it is used to estimate the sinewaves one-at-a-time. In fact, as shown in [34], the matrix (H T H) −1 in ( 9) is approximately diagonal and, as shown by ( 8), H T contains the outcomes of sampled cosine and sine waves having approximately the same frequency λ. Thus, when (H T H) −1 H T is multiplied by Z, the effects of all components having a frequency different from λ tend to vanish.…”