Formas de adquirir la residencia para efectos fiscales

Vec{A (θ)} + e (4.12)

where e = Vec {E}, and the operator ⊗ is the Kronecker product. The following, well- known relationship between the vectorization operator and the Kronecker product was used [10]: if A is an arbitrary N× K matrix and B is an arbitrary K × M matrix, then we have

Vec{AB} = (IM⊗ A) Vec {B}

=^B^T⊗ IN

Vec{A} (4.13)

We can define the MIMO steering vector, s(θ), for an angle θ in terms of the MIMO channel matrix, A(θ)

s(θ)= ^R^T_φ⊗ IN



Vec{A (θ)} (4.14)

The covariance matrix of the noise vector is Re = E ^ee^H

= R^T_φ⊗ IN

(4.15)

These definitions allow us to write the MIMO signal model after matched filter pro-cessing as

z(θ) =√

γ e^iψs(θ) + e (4.16)

where the MIMO steering vector is given in (4.14) and the noise covariance matrix in (4.15).

If the radar transmits orthogonal waveforms each of unit-power, then the MIMO signal correlation matrix is an identity matrix (R_φ = I). For this special case, the data vector is

z_⊥(θ) =√

γ e^iψVec{A (θ)} + e (4.17) where the covariance matrix of the noise is spatially white. By transmitting orthogonal waveforms, the radar retains its transmit degrees of freedom in that it has a full observation of the full MIMO steering matrix, A(θ).

It is obvious from this signal model and the MIMO steering vector of (4.14) that the MIMO signal correlation matrix, R_φ, characterizes the performance of a MIMO radar.

4.4.2 MIMO Signal Correlation Matrix

As is evident from the definition in (4.11) and the preceding discussion, the MIMO signal correlation matrix, R_φ, is the M × M matrix that describes the (zero-lag) cross- and

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128 C H A P T E R 4 MIMO Radar

auto-correlations of the M transmitted waveforms, {φm(t)}. The element in row m and column m^describes the response of waveform m when the matched filter corresponding to waveform m^is applied. This element of the matrix may be written

(Rφ)m,m^

=

 _∞

−∞φm(t) φm^∗(t) dt (4.18) Since a phased array is a special case of MIMO radar, it also possesses a MIMO signal correlation matrix. Letφ0(t) be the radar waveform used by a phased array system.

Each subarray will transmit this signal but with a phase shift applied to steer the beam in a particular direction. To steer a beam in the direction ˜θ0, the transmitted signals may be written

_PA(t)= a ^∗θ˜0

φ0(t) (4.19)

where a(θ) is the transmit steering vector corresponding to the direction θ. To satisfy the energy constraint of (4.3), the signalφ0(t) must be normalized so that

 _∞

−∞|φ0(t)|² dt = 1

M (4.20)

The signal correlation matrix is found to be R_φ/PA= 1

Ma^∗θ˜0

a^∗θ˜0

H

(4.21) This steering angle, ˜θ0, must be chosen by the phased array before transmitting since this is a form of analog beamforming. As the phased array scans its transmit beam from dwell to dwell, the signal correlation matrix will change, but it will remain rank-1.

Note that the correlation matrix is scaled so that the trace (the sum of the diagonal elements) of the correlation matrix is 1. This is required so that the signal-to-noise ratio remains γ regardless of the structure of the MIMO correlation matrix. In effect, this enforces a constant transmit power between designs to allow reasonable comparisons.

Consider now the case where each subarray transmits one of a suite of orthogonal waveforms; the correlation matrix is full-rank. If orthogonal waveforms are used, each with equal power, then the correlation matrix is a scaled identity matrix,

R_φ/⊥= 1

MIM (4.22)

Another example is when the array is spoiled on transmit. To cover a larger area, the transmit beamwidth may be increased by applying a phase taper across the array or by transmitting out of a single subarray. In the latter case, the transmitted waveforms are considered to be identically zero for all but one subarray. This approach is referred to as spoiling on transmit since it effectively spoils the transmit beampattern of the phased array by trading peak gain for a wider coverage area. The signal correlation matrix of the spoiled phased array, R_φ/Spoil, is rank-1, just as in the unspoiled phased array case. In the context of synthetic aperture radar (SAR), as well as in synthetic aperture sonar, an array that uses a single element on transmit and multiple elements on receive is referred to as a Vernier array.

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4.4 MIMO Radar Signal Processing 129

Consider the case of three transmitters (M = 3). The MIMO signal correlation ma-trices for the phased array (steered to broadside), the spoiled phased array, and the radar using orthogonal waveforms are

By considering the extreme cases of the phased array (rank-1 correlation matrix) and the radar using orthogonal waveforms (full-rank correlation matrix), the effect of the signal correlation matrix on the energy transmitted can be observed. As is well-known, a phased array radar transmits most of its energy in a particular direction. Of course, to accomplish this without violating laws of physics, energy is not transmitted in other directions. On the other hand, a radar transmitting orthogonal waveforms emits energy in all directions with similar power (subject to the pattern of the subarray).

An example of the radiated power as a function of angle comparing the phased array and the radar using orthogonal waveforms is presented in Figure 4-4. As expected, the phased array transmits a set of waveforms that cohere in a desired direction and destruc-tively interfere in other directions. The spoiled phased array uses a single (omnidirectional) element, so no beamforming gain is provided. Similarly, the radar using orthogonal wave-forms transmits uncorrelated wavewave-forms. Unlike the phased array, these signals do not cohere in any preferred direction.

There is a continuum of signal correlation matrices between these extremes of com-pletely uncorrelated and perfectly correlated waveforms. It will be shown that, by choosing the correlation among the signals transmitted by a MIMO radar, peak gain can be traded for a larger access area through transmit resteering.

−80 −60 −40 −20 0 20 40 60 80

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