Multipath Effect on Covariance Based MIMO Radar Beampattern Design

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2 Multipath Effect on Covariance Based MIMO Radar Beampattern Design peaks related to change in phase variance which inclined by false paths are provided in section IV. Section V is focused on conclusion and references are presented at the end of the paper. II. COVARIACEBASEDMIMORADARS Assume that we have a collection of transmitter antenna which are located at known coordinates x i = x,i, x 2,i, x 3i = (x, y, z) in some spherical coordinate along the z-axis. In the presented study and in all of the examples and formulas of current paper, it is assume that these transmitter antennas are along the z-axis. It is assume that each transmitter antenna is driven by a specific signal on the carrier frequency of f c or with wavelength of λ and complex envelope of s i t, i =,,. At a specific point in space at a distance of r and direction of k(θ, ϕ) from the transmitter's antenna, each radiated signal s i (t) gives rise to a "signal" the far field at radius r, with complex envelope given by y i t, r, θ, ϕ = 4πr s i t r 2π ej λ x i T k(θ,ϕ) () c Where, in this equation, k is a unit vector in the (θ, ϕ)direction. At the far field, these signals add linearly and the radiated powers P i add linearly as well. At this point assume that the i-th element location is on the z-axis at coordinate z i. The signal at position (r, θ, ϕ) resulting from all of the transmitted signals at far field will be: y t, r, θ, ϕ = y i t, r, θ, ϕ = 4πr i= The power density is of the entire signals then given by P y r, θ, ϕ = 4πr 2 k= l= i= < s k t s l t > e j s i t r 2πz i c ej λ 2π z k z l λ And it is known that the complex signal cross-correlation is defined by R kl =< s k t s l t > (4) With defining the direction vector as below a θ = e j 2πz λ sin θ,..., e j 2πz λ The normalized power density P(θ, ϕ) of signals, in (W/ster), is P θ, ϕ = 4π k= l= R kl e j2π λ z k z l sin (θ) sin θ si n (θ) sin θ T (5) Recognizing that (6) is quadratic form in the Hermitian matrix R which is the cross-correlation matrix of signals, this can be written compactly as P θ, ϕ = 4π a θ Ra θ (7) This normalized power density P(θ, ϕ) is exactly the beampattern which we wish to find[3]. First we wish to show some examples of beampatterns produce from such a cross-correlation matrix and further in this paper we will examine effect of multipath on these beam-patterns. Fig. shows the beampattern produced by signal cross-correlation matrix of (8), (9) and () respectively (blue, brown and green plots in this figure). It is noticeable that these figures are beam-patterns of -element uniform linear array (ULA) with half-wavelength spacing. (6) (2) (3) (8) (9) In general case the elements of the signal cross-correlation matrix are complex values except the diagonal elements that they are real. This general case is related to MIMO radars but in the case of phased array radar, all 44 P a g e

3 Multipath Effect on Covariance Based MIMO Radar Beampattern Design the transmitter signals are correlated with each other and then all the elements in R, are equal to (blue one in Fig. ). III. SIGALMODELIPRESECEOFMULTIPATH In this section we'd like to take multipath into consideration. Then suppose that the transmitted signal from a specific antenna at a desire point in the space with the distance of r and direction of f(θ, ϕ) would be in the following form: m y i t, r, θ, ϕ = A M 4πr s im t r m e jτ im () c m= This is a complete general form for multipath effect. In this equation m shows the maximum number of paths that the transmitted signals have to receive to desire point it is said maximum number because actually in real, it is possible that different signals come from different paths since we have said here maximum number of paths. A m shows the amplitude of receive signal that in general is a complex value and also can be zero to show that one path doesn't exist for one antenna. S im shows the m th path for i th antenna, r m denotes the distance of m th path and τ im is the phase shift which occurs during m th path to i th antenna. This equation has many unknown parameters and then working with that is complex, so for the following we take bellow simplifier assumption into consideration. Each multipath occurs from a path that have almost the distance of line of sight path. It means that r m = r. Actually this occurs when the obstacles which cause the multipath be close to transmitter antennas With the above assumption we can write () in the following way: m y i t, r, θ, ϕ = A M 4πr s im t r c ejτ im (2) m= Since we take A m complex value, we eliminate dependence of phase shift to multipath and include it's influence into A m then (2) equation can be written as It is noticeable that we take m y i t, r, θ, ϕ = A M 4πr s im t r c ejτ i (3) m= A m =, m = (4) Which shows the normalize amplitude for line of sight signal. ow with the above expression, sum of the all the transmitted signals with their multipath can be written as follow: y t, r, θ, ϕ = i= y i t, r, θ, ϕ The power density is of the entire signals then given by That can be written as follow: P y r, θ, ϕ = 4πr 2 m m = j= m= k= l= m m 4πr m m= i= A m s i t r c ejτ i (5) < A m s k t A j s l t > e j τ k τ l P y r, θ, ϕ = 4πr 2 A j A m < s k t s l t > e j τ k τ l (7) j= m= k= l= Then the normalized power density P(θ, ϕ) of signals, in (W/ster), is P θ, ϕ = 4π m j= A j m m= A m k= l= R kl e j(τ k τ l ) Just like above recognizing that (8) is quadratic form in the Hermitian matrix R which is the cross-correlation matrix of signals, this can be written compactly as P θ, ϕ = 4π m j= A j m m= A m a θ Ra θ (9) (8) 6 45 P a g e

4 Multipath Effect on Covariance Based MIMO Radar Beampattern Design So this expression denotes beam-pattern in presence of multipath effect. In next section with some numerical examples we investigate this effect and its influence on the desire beam-pattern. IV. UMERICALEXAMPLES In this section we wish to consider effect of multipath on our desire beam-pattern. We will examine this effect in two manners. First we would like to investigate this effect on location of target or directions that we wish to have peak in our beam-pattern, second we'd like to examine effect of multipath on amplitude of beampattern on desired directions and false peaks that occur during this effect. In the following we will examine these cases. A. Multipath effect on phased array As it stated on previous section for a phased array radar we deal with a matric cross correlation of signals which all the elements are equal to one. Fig. shows multipath effect on such a case. Beampattern # with multipath Beampattern #2 with multipath Beampattern of (8) Beampattern of (9) Beampattern of () Angle (deg) Fig..Beam-pattern of phased array respect to (7). The blue one is corresponds to phases array without any multipath and black one and red one is related to phased array pattern with multipath. Brown and green plots are corespond to (9) and () expressions respepectly This figure depicts all we need. First of all it should mention that we consider two assumption in this figure. First we assume that each signal has two path to desire point of interest in other words one path is line of sight and our desire path, and there is only one another path for each signal to that point of interest amplitude of false path have taken equal to line of sight case and it inclined only a random phase with normal distribution with zero mean and π radians variance to signal of false path. Second assumption is that we take the total power at the point of interest identical for both multipath case and without multipath case. It is noticeable that these figures are beam-patterns of -element uniform linear array (ULA) with half-wavelength spacing. As it seen from this figure, in both cases of multipath side-lobe levels have increase related to phased array case without multipath. Multipath can also change the exact location of beam-pattern peak from our desire direction, the case which has occurred due to red plot in Fig. and at the other hand as it is seen from black plot from this figure it can cause false peaks in our beam-pattern. In next sections we will consider this effect in more details. B. Random phase and amplitude effect on peak location of desire beampattern In this section we wish to consider that how multipath with different number of paths, random amplitude and random phase can affect our desire beam-pattern in direction, amplitude (side-lobe level) and false peaks. In this section and following sections we will take our desire beam-pattern just like the case shown in Fig. which is in green color. Since it has one peak and has no side-lobe we can examine multipath effect more easily and more accurate. In Fig. 2 we have considered the case that there is only two paths from each transmitter antenna to a desire point of interest which one of them is line of sight and our desire path and one of them is false path. and 46 P a g e

5 error (deg) error (deg) Multipath Effect on Covariance Based MIMO Radar Beampattern Design also we have assume that the false path is inclined the signals with random amplitude of uniform distribution between and ( is normalized amplitude for the line of sight signal amplitude) and random phase with normal distribution with zero mean and variance of between and π radians. Then we wish to consider mean error of peak power direction of the beam-pattern refer to change in phase variance between and π radians Fig. 2. mean error of beam-pattern peak direction in presence one path of multipath For a special case, in Fig. 3 we have shown mean error of above case but with signal amplitude of false path equal to. That means equal to amplitude of line of sight signal. Both of these plots have drawn for 3 iterations in each point Fig. 3. mean error of beam-pattern peak direction in presence of one path of multipath with constant amplitude of multipath signal From these figures it is apparent that as phase variance of multipath signal increase error of beampattern peak direction will increase as well. This is just done for amplitude of signal which is obtained by comparing Fig. 2 and 3. As it is seen from these figures with constant amplitude equal to line of sight signal, this figure has two times bigger error in estimating the exact angle to focusing power. C. Effect of number of path in multipath on peak location of desire beam-pattern In this section we are interested in examine the effect of number of paths to our previous section result. For this purpose we consider the case which there is elements half wave length uniform array just like the previous section. We also consider the case which all false paths inclined random amplitude with uniform distribution between to ( is one for the line of sight path) and random phase with normal distribution of zero mean and variable variance between to π, to the transmitted signals. But here we consider three false paths instead of one false path in previous section. In other words in this case including line of sight path, totally we have four paths to desire point of interest. Like the previous section, Fig. 4 is related to general case with random amplitude of multipath signal and Fig. 5 is related to specific case which all the false paths have identical amplitude equal to one for line of sight signal. That means actually they are counterparts of Fig. 2 and 3 with three false paths. 47 P a g e

6 error (deg) error (deg) error (deg) Multipath Effect on Covariance Based MIMO Radar Beampattern Design Fig. 4.mean error of beam-pattern peak direction in presence of three path of multipath Fig. 5. mean error of beam-pattern peak direction in presence of three path of multipath with constant amplitude of multipath signal As it seen from these figures error in focusing the beam-pattern to a desire direction will increase as the paths of multipath increase. This is an important parameter in operational environment that how many source of multipath we may have. Answer to this question will help us to use suitable signals to decrease influence of multipath and achieve our goal. Fig. 6 shows mean error in beam-pattern peak direction related to change into number of multipath paths. For this figure we have taken paths of multipath amplitudes random with uniform distribution between and ( means normalize value of line of sight amplitude signal) and the phase which these paths inclined to signals has taken random variables with normal distribution of zero mean and variance of π. This figure has plotted for to number of false paths Fig. 6. mean error of beam-pattern peak direction in presence of multipath related to number of false paths umber of false paths 48 P a g e

7 probability of false peak (%) probability of false peak (%) Multipath Effect on Covariance Based MIMO Radar Beampattern Design As it seen from this figure as number of false paths increase, mean error of beam-pattern peak location will increase as well. In this section we investigated effect of multipath to beam-pattern peak direction. At next section we will consider effect of multipath on false peak that produce due to this effect. D. Multipath effect on false peaks In this section we will investigate multipath effect on total shape of beam-pattern and we will examine this effect on producing false peaks that may occur during this effect as it shown previously in Fig.. like latter sections, in this section we ignore power difference at the points of interest due to multipath. For comparison reasons we will take total power at desire points all identical. In here we consider the case which there is elements half wave length uniform array just like the previous sections. We also consider the case which all false paths inclined random amplitude with uniform distribution between to ( is one for the line of sight path) and random phase with normal distribution of zero mean and variable variance between to π, to the transmitted signals. As in this section we are looking for false peaks, we have to define a standard to recognizing false peaks. We define this standard as be at least half of the main beam. In the other words, if one side-lobe be at least as bigger as half of the main beam amplitude, we name this side-lobe as false peak. E. Random phase and amplitude effect on false peaks In this section we wish to consider that how multipath with random amplitude and random phase can affect our desire beam-pattern and producing false peaks. In Fig. 7 we have considered the case that there is only two paths from each transmitter antenna to a desire point of interest which one of them is line of sight and our desire path and one of them is false path, and also we have assume that the false path is inclined the signals with random amplitude of uniform distribution between and ( is normalized amplitude for the line of sight signal amplitude) and random phase with normal distribution with zero mean and variance of between and π radians. Then we wish to see in how many cases of our examination multipath produces at least one false peak related to change in phase variance between and π radians Fig. 7. mean error of beam-pattern peak direction in presence one path of multipath For a special case, in Fig. 8 we have shown percentage of error of above case but with signal amplitude of false path equal to. That means equal to amplitude of line of sight signal. Both of these plots have drawn for 3 iterations in each point Fig. 8. mean error of beam-pattern peak direction in presence of one path of multipath with constant amplitude of multipath signal 49 P a g e

8 probability of false peak (%) probability of false peak (%) Multipath Effect on Covariance Based MIMO Radar Beampattern Design From these figures it is obvious that as phase variance of multipath signal increase percentage of having false peak will increase as well. This is just done for amplitude of signal which is obtained by comparing Fig. 7 and 8. As it is seen from these figures with constant amplitude equal to line of sight signal, this figure has five times bigger error in estimating the exact angle to focusing power. F. Effect of number of paths in multipath on false peaks of desire beam-pattern In this section we are interested in examine the effect of number of paths to our previous section results. For this purpose we consider the case which there is elements half wave length uniform array just like the previous sections. We also consider the case which all false paths inclined random amplitude with uniform distribution between to ( is one for the line of sight path) and random phase with normal distribution of zero mean and variable variance between to π, to the transmitted signals. But here we consider three false paths instead of one false path in previous section. In other words in this case including line of sight path, totally we have four paths to desire point of interest. Like the previous section, Fig. 9 is related to general case with random amplitude of multipath signal and Fig. is related to specific case which all the false paths have identical amplitude equal to one for line of sight signal. That means actually they are counterparts of Fig. 7 and 8 with three false paths Fig. 9.mean error of beam-pattern peak direction in presence of three path of multipath Fig.. mean error of beam-pattern peak direction in presence of three path of multipath with constant amplitude of multipath signal As it seen from these figures, false peak error probability will increase as the paths of multipath increases. This is an important parameter in operational environment that how many source of multipath we may have. Answer of this question will help us to use suitable signals to decrease influence of multipath and achieve our goal. Fig. shows false peak error probability in beam-pattern related to change into number of multipath paths. For this figure we have taken paths of multipath amplitudes random with uniform distribution between and ( means normalize value of line of sight amplitude signal) and the phase which these paths inclined to 5 P a g e

10 Multipath Effect on Covariance Based MIMO Radar Beampattern Design [] B. Friedlander, Waveform design for MIMO radars, IEEE Trans. Aerosp. Electron. Syst., vol. 43, pp , Jul. 27. [2] Y. Yang and R. S. Blum, MIMO radar waveform design based on mutual information and minimum mean-square error estimation, IEEE Trans. Aerosp. Electron. Syst., vol. 43, no., pp , Jan. 27. [3] Y. Yang and R. S. Blum, Minimax robust MIMO radar waveform design, IEEE J. Sel. Topics Signal Process., vol., no., pp , Jun. 27. [4] E. Fishler, A. Haimovich, R. Blum, L. Cimini, D. Chizhik, and R. Valenzuela, Performance of MIMO radar systems: Advantages of angular diversity, in Proc. 38th Asilomar Conf. Signals, Syst. Comput., ov. 24, vol., pp [5] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela, MIMO radar: An idea whose time has come, in Proc. IEEE Radar Conf., Apr. 24, pp [6] A. M. Haimovich, R. S. Blum, and L. J. Cimini, MIMO radar with widely separated antennas, IEEE Signal Process. Mag., vol. 25, no., pp. 6 29, Jan. 28. [7] J. Li and P. Stoica, MIMO radar with colocated antennas, IEEE Signal Process. Mag., vol. 24, no. 5, Sep P a g e

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