MIMO Radar
4.3 THE MIMO VIRTUAL ARRAY
4.4.5 Phased Array versus Orthogonal Waveforms
We have shown that the transmit gain of an array antenna can be controlled by design of the MIMO signal correlation matrix. The two extreme cases are considered: the phased array (rank-1 matrix) and orthogonal waveforms (full-rank matrix). These matrices were given in (4.21) and (4.22).
The gain for these two cases can be computed using (4.30). The gains of the phased array, GPA, and of the radar using orthogonal waveforms, G⊥, are found to be
GPA
θ; ˜θ0, θ0
=
⎛
⎜⎝ETX(θ) aθ˜0
H
a(θ)2 M
⎞
⎟⎠
ERX(θ)b(θ0)Hb(θ)2 N
(4.31)
G⊥(θ;θ0) =
⎛
⎜⎝ETX(θ) aθ0
H a(θ)2 M2
⎞
⎟⎠
ERX(θ)b(θ0)Hb(θ)2 N
(4.32)
Melvin-5220033 book ISBN : 9781891121531 September 15, 2012 10:56 132
132 C H A P T E R 4 MIMO Radar
FIGURE 4-5 MIMO array factor on transmit. This example
corresponds to an array with 10 subarrays where each subarray has 20λ/2spaced elements.
−20 −15 −10 −5 0 5 10 15 20
−20
−15
−10
−5 0 5 10
Azimuth (deg)
Gain (dB)
Transmit Array Factor
Phased Array Orthogonal Waveforms
Recall thatθ0is the angle to which the beam is digitally steered and that ˜θ0is the direction to which the phased array steered the beam on transmit. The receiver is free to varyθ0
with digital processing, but ˜θ0is fixed.
As should be expected, the receive gains are identical between the phased array and orthogonal waveforms, but there are two key differences between the transmit gain terms.
First, since the phased array transmits a concentrated beam in the direction ˜θ0, it is unable to apply any digital steering of the transmit beam; the phased array has already decided in which direction to send energy. On the other hand, the radar that employs orthogonal signals is able to resteer the transmit beam to any angle (so long as the subarray pattern permitted energy to be radiated in that direction).
However, the cost of doing this is evident. The phased array realizes a transmit beam- forming gain that provides an increase in SNR by a factor of M, the number of transmitting subarrays. This benefit is lost by the radar that uses orthogonal signals.
These differences are illustrated in Figure 4-5, where the array factors of a MIMO radar employing orthogonal waveforms as well as a traditional phased array are presented.
The performance of an array antenna for use in a radar system is well quantified by considering three gain patterns: the steered response, the beampattern, and the point spread function. These describe the ability of the data collected by the system to be used to digitally form beams in desired directions with desired properties.
The steered response and the beampattern quantify the degree to which the antenna can be digitally steered to an angle of interest as well as the ability to reject returns from undesired angles. Given an angle of interest, the steered response describes the ability of the array to observe signals arriving from that direction when the array is steered to that direction of interest, while the beampattern describes the ability of the array to reject targets from other angles [14]. The distinctions between these patterns are summarized in Table 4-1.
Let GRX(θ;θ0)be the gain of the receive array in the directionθ when it is digitally steered to the angleθ0. The steered response evaluates this gain for the case whenθ =θ0, that is, the gain is evaluated in the direction that the array has been digitally steered. If the array is steered to the angleθ0, then the beampattern evaluated atθ describes how much
Melvin-5220033 book ISBN : 9781891121531 September 15, 2012 10:56 134
134 C H A P T E R 4 MIMO Radar
FIGURE 4-7 Two-way angular point spread functions. The phased array resolves targets in angle using only receive degrees of freedom. For the orthogonal
waveform cases, the filled configuration provides enhanced sidelobe
performance and the sparse configuration provides improved angular resolution.
Point Spread Functions
−10 −5 0 5 10
−60
−50
−40
−30
−20
−10 0
Azimuth (deg)
Relative Gain (dB)
Phased Array (Filled) Orth. Waveforms (Filled) Orth. Waveforms (Sparse)
The (angular) point spread function (PSF) quantifies the angular resolution capability of an array antenna system. Suppose that a target is present and located at some angleθ. To evaluate the PSF at angle θ0, a beamformer designed for this angle of interest θ0 is applied to data containing a target at angleθ. This quantifies the degree to which a target located atθwill obscure a target located atθ0.
Comparing the transmit PSF of the phased array to the orthogonal waveforms in Figure 4-6, we see that the phased array (regardless of spoiling) provides no angular resolution on transmit. If, on the other hand, orthogonal waveforms are used and the radar can preserve its transmit degrees of freedom, then angular resolution is possible on transmit.
The PSF is related to the ambiguity function that is familiar from the radar literature.
The standard radar ambiguity function describes the response due to a target with a partic- ular range and Doppler in nearby range and Doppler bins. This idea was extended to the MIMO case in [15], where an ambiguity function in terms of range, Doppler, and angle is developed.
Let us now reconsider the sparse configuration presented in Figure 4-2. The (two- way) PSF for the sparse array using orthogonal waveforms is presented in Figure 4-7.
As predicted by the virtual array analysis, the sparse array is able to provide enhanced resolution since it provides an effectively larger array than the filled configuration. Note that orthogonal waveforms are required to use the sparse configuration; otherwise, the sparsity would introduce undesirable grating lobes. The filled configuration provides improved sidelobe performance, since a taper is effectively applied to the aperture as a result of overlapping virtual phase centers.
These results demonstrate the ability of a radar to use orthogonal waveforms to im- prove performance. Even if truly orthogonal waveforms are not practical, this analysis may be repeated for a given set of waveforms using the appropriate signal correlation matrix to quantify the impact of this nonorthogonality with the framework that has been developed.
Melvin-5220033 book ISBN : 9781891121531 September 15, 2012 10:56 135