Aerospace and Electronic Systems Magazine December 2017 - 57
Frazer
2. For the case where the radar range is the same for each layer rF,1
= rE,1 and rF,2 = rE,2, then the respective elevation of departure for
each mode will be such that φF ,1 > φE ,1 and φF ,2 > φE ,2. For different
range but same layer the lesser range has the greater elevation of
departure whereby φE ,1 > φE ,2 and φF ,1 > φF ,2.
We note that the two-layer representation of the ionosphere is
a simplification. The ionosphere is a dispersive birefringent media
that exists within the Earth's magnetic field [45]. A more realistic
representation might include the E-layer, and F-layer low-ray, Flayer high-ray and with each layer having ordinary and extraordinary ray paths that are caused by the Earth's magnetic field. For example, in some of our subsequent results, we observe and identify
using a co-sited ionospheric sounder four propagation modes over
a one-way path including, 1E (one-hop propagation via E-layer),
1F2l (one-hop propagation via F2-layer low-ray), 1F2h-o (one-hop
propagation via F2-layer ordinary high-ray), and 1F2h-x (one-hop
propagation via F2-layer extraordinary high-ray) [45]. We shall
adopt this notation to describe ionospheric propagation modes
throughout the article.
NOTATION, SIGNAL MODEL, AND PROCESSING
The multiple-input single-output (MISO) based transmitter and receiver scheme that we use in the Mode Selection Experiment consisted of K transmitter antenna array elements concurrently transmitting K different radar waveforms u(t ) = u1 (t ),..., uk (t ),..., u K (t )
of equal energy. This is the element-space waveform set case
where there is a one-to-one mapping between waveforms and array
elements. Other configurations, such as beam-space, which map
one waveform per beam in a multibeam arrangement, and subarray space, which map one waveform per antenna subarray, were
not used in the experiment and so are not considered here [46]
although, particularly in the subarray case, have practical application in MIMO radar systems.
A single radar receiver (hence MISO) is located to receive the
propagated waveform set.
ing over the domain S of anticipated scatterer extent S ∈ (ν ,τ ). The
transmitter beamformer is then
y (ν ,τ ) = w H z (ν ,τ )
(4)
for some beamformer weight w. We seek to investigate how we
can create various w to demonstrate the mode-selectivity aspect of
Mode-Selective OTHR over a one-way propagation path.
The particular waveform set should be selected such that the
cross-ambiguity between the K members of the waveform set does
not adversely interact with the expected target and clutter scatterer
distribution in delay and Doppler. This means choosing (for the
discrete-time representation of u(t) assumed in our radar signal
processing and with j = −1 in this equation)
χ k, l (ν ,τ ) = uk (t ) ∗ ul* (t − τ )e − j 2π tν < for k ≠ l
(5)
S
over the domain of Doppler-delay space S that one anticipates scatterer response to be present and is some small value that specifies
the upper acceptable limit of cross-ambiguity between waveforms.
See the full discussion in [24] and the experimental examples in
[39] which includes an example where there is significant crosscoupling between scatterer responses from different members of
the waveform set.
ADAPTIVE MODE SELECTION
The adaptive spatial processor used is the well-known minimum
variance distortionless response (MVDR) beamformer [47]. The
MVDR beamformer is defined for a preserved steer direction θ as
wo =
R −1s(θ )
s (θ ) R −1s(θ )
H
(6)
where
s(θ ) = [1, e jθ ,..., e j ( K −1)θ ] T
(7)
K
Z (t ) = η0 ak (θ d )uk (t − τ 0 )e j 2πν 0 t
(1)
This choice of wo ensures that total beamformer output energy
where η0 is a random complex scattering coefficient that is assumed identical for all waveforms, τ0 and ν0 are the delay and Doppler shift of the one-way propagation channel and,
w oH R w o is minimized while the preserved steer direction gain
w oH s(θ ) = 1. In practice, the covariance matrix R is estimated from
a region of Doppler-delay space comprising P Doppler cells by Q
delay cells. The sample covariance matrix is given by
k =1
a K (θ d ) ≡ [a1 (θ d ),..., aK (θ d )] T
(2)
is the K-variate transmit antenna array manifold (steering) vector
for the direction-of-departure θd that will illuminate the propagation path.
The received radar return Z(t) is processed as for conventional
radar receive processing (matched filter, etc.) K times, once for
each waveform in the waveform set as the reference waveform. Let
z (ν ,τ ) = z1 (ν ,τ ),..., zk (ν ,τ ),..., z K (ν ,τ )
(3)
be the outputs of the K-respective discrete-time matched filters indexed in Doppler ν and delay (or equivalently range) τ and extendDECEMBER 2017
1 P Q
Rˆ =
z(ν ,τ )z H (ν ,τ )
PQ i =1 j =1
(8)
This estimated covariance matrix Rˆ is used in place of the exact
covariance R in (6). Reduction of achieved signal to noise ratio
improvement using the beamformer due to the use of Rˆ in place of
R is discussed in the established work of [48].
Throughout the article we use the term MVDR mode-selective
beamformer to indicate the use of the algorithm detailed in this
section. In all cases we select s(θ)from either an estimate (for example, from (9)) of a desired θ and knowledge of the array manifold determined using an off-line calibration procedure or, alterna-
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