By Nickel S.
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Extra resources for Convex analysis
Example text
24) can be readily applied to more conventional positioning systems such as radars and sonars (see the example below). Here again super resolution effects can be exploited. But the important factor to keep in mind is that the channel capacity cannot be exceeded. 3 The following is an example of a capacity calculation. Consider a simple polar radar system which measures range by means of pulses and angle using a phased array antenna. Suppose that the pulse bandwidth is small compared to the carrier frequency, the directivity of the antenna is high and the field of view is relatively small.
F, of K successive components can be written as px(X) = ( 2 , ~ ) . 10) where • is the covariance matrix. 12) where i and s range from 1 to n; j and t range from 1 to K. Now define an n by n matrix R ( k ) given by [R(k)],. 13) where k = j - t. +. = [R(j - ~)1,.. 14) 60 5. Coding Accordingly, ~K can be written as a block Toeplitz matrix R(0) R(-1) R(1) R(0) : : R(-(K- ... R ( K - 1) R ( K - 2) ... 1)) R ( - ( K - 2)) ... 15) Note that a T ( k ) = R ( - k ) . 17) i=1 where A~,Kis the ith eigenvalue o f ~ .
This corresponds to a radial-radial positioning system with one reference site at (0, 0) and one at (0, a). s7) with f~ being a constant of proportionality. 88) and T,, is defined by the correlation matrix 0 ~ " This class of devices is simply the class of radial-radial devices with one reference site at (0, 0) and the other at (0, a), with additive gaussian errors whose variance obeys the inverse square law. There will be no universal optimum, the optimum device will depend on the form o f p x ( X ).