By S. Uṇṇikrishṇa Pillai, C. S. Burrus (auth.), S. Uṇṇikrishṇa Pillai, C. S. Burrus (eds.)

This publication is meant as an advent to array sign method ing, the place the relevant pursuits are to use the on hand a number of sensor details in a good demeanour to notice and possi bly estimate the indications and their parameters found in the scene. some great benefits of utilizing an array as opposed to a unmarried receiver have prolonged its applicability into many fields together with radar, sonar, com munications, astronomy, seismology and ultrasonics. the first emphasis here's to target the detection challenge and the estimation challenge from a sign processing standpoint. lots of the contents are derived from available resources within the literature, even though a cer tain volume of unique fabric has been integrated. This e-book can be utilized either as a graduate textbook and as a reference publication for engineers and researchers. the fabric awarded the following might be simply understood via readers having a again flooring in uncomplicated chance concept and stochastic procedures. A prelim inary direction in detection and estimation conception, even though now not crucial, may well make the analyzing effortless. in reality this ebook can be utilized in a one semester path following likelihood idea and stochastic processes.

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L, and j = L + 1, ... ,K. 133) A special case of this general situation, where all O-k; k = 1, 2, ... 68) are real, is treated in [34]. K. :. :. 2M +K - 2. 3K or, equivalently, the minimum number of sensors must be at least [3K /2]. 134) where y is any arbitrary K x 1 vector. 135) for any y t= 0, thus proving the positive definite property of Go GJ or ~. 135) needs to be demonstrated only for a typical Yo E N (vt) , the null space of Vt. 134) also reduces to zero. To prove our claim it is enough to show that for such a typical Yo' the vector Ht Yo does not - 56belong to N(Vt).

K }, using the diagonal nature of H it is also easy to see that Ht Vj is linearly independent of the remaining VI' Z = L + 1, · .. ,K, Z fj. Further the pair Vj and Ht vi' j = L + 1, L +2, ... , K, is also linearly independent of each other. To see this note that because of the full row rank property of Vt, at least one of the vii; i = 1, 2, ... 136) must be nonzero for every Z. 133). Thus the matrix [~f I Htvj ] is of rank 2. This proves the linear inderendence of Vj and H vj • From the above discussion it follows that H v~ is linearly independent of vj , j = L + 1, L +2, "', K and hence H Vj ¢ N (Vt ), j Now for any Yo E N (vt) we have = L + 1, L +2, "', K.

85). Now let R(i) = [rei) +r*(i)]j2, i = 1,2,'" ,M-1. 93) we have R(i) = K +1 ~ cke j1l:d. 94) where by definition ck = Re [bk ], k = 1, 2, ... , K +1 . 95) - 43- R(O) K+I = Re[r(O)] = E k =1 Re[bk ] 2 +0"0 = K+I 2 Eck+O"o. 96) k =1 Then the hermitian Toeplitz matrix Ts formed from these R (i); i 1,2, ... 99) cK 0 Note that ck ; k =1, 2, ... 90) can be applied to estimate the actual angles of arrival. 100) ° 2, ••• , Ok' - 440 -20 dB -40 -60 0 30 60 90 120 150 180 150 180 Angle: degree (a) 0 -20 dB -40 -60 0 30 60 90 120 Angle: degree (b) Fig.