Principles of Modern Radar - Volume 2 1891121537

10.1 Introduction 457where [w] ∗ m = w∗ m is the conjugate of the m-th element of w, and [x] m = x m is the m-thelement of x. Optimization problems typically involve the expected value of the outerproduct; the outer product of the data vector is the P × P matrix⎡x 1 x1 ∗ x 1 x2 ∗ ··· x 1 x ∗ ⎤Pxx H x 2 x1 ∗ x 2 x2 ∗ ··· x 2 x ∗ P= ⎢⎣.. . ..⎥(10.3). ⎦x P x1 ∗ x P x2 ∗ ··· x P x ∗ PThe Kronecker product conveniently describes a collection of multidimensional observations.For example, given vectors s s ∈ C M×1 and s t ∈ C N×1 , the Kronecker product iss t ⊗ s s = [ [s t ] 1 s T s [s t ] 2 s T s ··· [s t ] N s T s] T∈ CNM×1(10.4)where [s t ] m is the m-th element of s t . Lastly, the Schur product is tantamount to elementwisemultiplication of two vectors of the same length. For example, given vector s s ∈ C Mx1and the window function (taper) a s ∈ C M×1 , the Schur product isa s ⊙ s s = [ [a s ] 1 [s s ] 1 [a s ] 2 [s s ] 2 ··· [a s ] M [s s ] M] T(10.5)The notation : is taken from the MATLAB programming language. It is best understoodby example. For instancea = [n 1 : : n 2 ] (10.6)is a row vector with the first element equal to n 1 , subsequent elements incremented by, and the last element equal to n 2 .If = 1, it is usually dropped from (10.6), viz. [n 1 :1:n 2 ] = [n 1 : n 2 ]. For example, a = [−3.25:1.25 : 1.75] = [ −3.25 −2 −0.75 0.5 1.75 ] .10.1.5 Variable NamesFrequently used variables are as follows:ω c = center frequency (rads)f d = Doppler frequency (Hz)˜f d = normalized Doppler frequency (unitless)φ,θ = azimuth and elevation (rads)φ c = cone angle (rads)λ o = center wavelength (m)f sp = spatial frequency (unitless)M = number of channelsN = number of pulsesL = number of available range binsT = pulse repetition interval (s)c = velocity of propagation (speed of light, m/s)= receiver noise power, single channel/pulse (watts)σn 2σs2= signal power, single channel/pulse (watts)v p = platform velocity vector (m/s; m/s; m/s)s s = spatial steering vectorv s = hypothesized spatial steering vector