My colleague Dr. Apurva Mody (of BAE Systems, IEEE 802.22, and the WhiteSpace Alliance) and I have received a patent on a CSP-related invention we call tunneling. The US Patent is 9,755,869 and you can read it here or download it here. We’ve got a journal paper in review and a 2013 MILCOM conference paper (My Papers ) that discuss and illustrate the involved ideas. I’m also working on a CSP Blog post on the topic.
In this post we discuss ways of estimating -th order cyclic temporal moment and cumulant functions. Recall that for , cyclic moments and cyclic cumulants are usually identical. They differ when the signal contains one or more finite-strength additive sine-wave components. In the common case when such components are absent (as in our recurring numerical example involving rectangular-pulse BPSK), they are equal and they are also equal to the conventional cyclic autocorrelation function provided the delay vector is chosen appropriately.
The more interesting case is when the order is greater than . Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order greater than is . So our estimation task is to estimate -th order temporal moment and cumulant functions for using a sampled-data record of length .
So why do I obsess over cyclostationary signals and cyclostationary signal processing? What’s the big deal, in the end? In this post I discuss my view of the ultimate use of cyclostationary signal processing (CSP): Radio-Frequency Scene Analysis (RFSA). Eventually, I hope to create a kind of Star Trek Tricorder for RFSA.
Let’s discuss an application of cyclostationary signal processing (CSP): time-delay estimation. The idea is that sampled data is available from two antennas (sensors), and there is a common signal component in each data set. The signal component in one data set is the time-delayed or time-advanced version of the component in the other set. This can happen when a plane-wave radio frequency (RF) signal propagates and impinges on the two antennas. In such a case, the RF signal arrives at the sensors with a time difference proportional to the distance between the sensors along the direction of propagation, and so the time-delay estimation is also commonly referred to as time-difference-of-arrival (TDOA) estimation.
Consider the diagram shown to the right. A distant transmitter emits a signal that is well-modeled as a plane wave once it reaches our two receivers. An individual wavefront of the signal arrives at the two sensors at different times.
The line segment AB is perpendicular to the direction of propagation for the RF signal. The angle is called the angle of arrival (AOA). If we could estimate the AOA, we can tell the direction from which the signal arrives, which could be useful in a variety of settings. Since the triangle ABC is a right triangle, we have
When , the wavefronts first strike receiver 2, then must propagate over meters before striking receiver 1. On the other hand, when , each wavefront strikes the two receivers simultaneously. In the former case, the TDOA is maximum, and in the latter it is zero. The TDOA can be negative too, so that azimuthal degrees can be determined by estimating the TDOA.
In general, the wavefront must traverse meters between striking receiver 2 and striking receiver 1,
Assuming the speed of propagation is meters/sec, the TDOA is given by
In this post I’ll review several methods of TDOA estimation, some of which employ CSP and some of which do not. We’ll see some of the advantages and disadvantages of the various classes of methods through inspection, simulation, and application to collected data.