MIMO Capacity of Radar as a Communications Channel
Patrick Bidigare
Veridian Ann Arbor R&D Center
3300 Plymouth Rd.
Ann Arbor, MI 48130
Patrick.Bidigare@veridian.com
some way and then retransmitting these back to the radar system.
The radar system will receive the RF tag signals along with radar
echoes from the clutter and thermal noise. Since we are
concerned with the channel capacity of a multiple receiver radar,
the details of pulse modification method is somewhat irrelevant,
however it is helpful to keep a specific method in mind. The
conceptual RF tag model shown in figure 1 uses a digital
convolver (a finite impulse response filter) to convolve data onto
incident pulses.
ABSTRACT
An RF tag is a wireless communication device that can embed
information into a SAR or GMTI radar collection by receiving
radar pulses, modifying these and transmitting them back to
the radar. This allows the radar system to be used as a
communications channel. This radar/tag channel may be
viewed as a spatio-temporal MIMO channel, where the inputs
are the pulses transponded by the tag and the outputs are all
the pulses received by all the radar receiver channels. The
interference in this channel is composed of thermal noise and
clutter returns and has a rich covariance structure that has
been studied extensively in the STAP community.
This presentation explores the dependence of radar/tag
channel capacity on the number of spatial receive channels
available on the radar system. We derive a formula for the
capacity of a multiple receiver radar system and then calculate
via simulation these capacities in the specific case of the
Veridian DCS radar and the BAE SYSTEMS digital RF tag.
Figure 1: Digital RF Tag Device Model
The effect of the digital convolver is to create a series of
weighted time replicas of the incident radar waveform. On
reception, the radar system pulse compresses the signal returns.
This also has the effect of compressing the individual pulse
replicas produced by the tag device as shown in figure 2.
1. INTRODUCTION
RF tags communications is an area of research that lies in the
intersection of GMTI STAP and MIMO communications, and an
understanding of the results in both areas is necessary. As we
will demonstrate in this paper, RF tags can be cast as special case
of a MIMO channel to which channel capacity formulas and
space-time coding techniques apply. Unlike ordinary MIMO
channels, the interference present in the RF tags channel can be
highly colored both spatially and temporally and has been
studied extensively in the STAP literature.
The difficulty of isolating independent transmit and receive
channels often leads to tag designs in which a single antenna is
multiplexed for both transmit and receive.
This paper represents the third in a series on RF tags
communications. In [8], an adaptive signal processing algorithm
was developed for suppressing interference while preserving tag
signals in multiple receiver radar systems. In [7], the Shannon
channel capacity for a single receiver radar was derived and the
dependence of capacity on PRF, resolution and range were
investigated. This paper finishes the story by calculating the
capacity for an arbitrary multiple receiver radar. We show for a
real radar system the advantage that multiple receivers makes on
the capacity of the channel.
2.
2.1
Figure 2: Tag Signal Illustration
2.2
Single Receiver Radar Channel
To begin with, we consider a radar system with a single
receive channel. For completeness, we will consider to tag signal
models: One in which the tag retransmits all received pulses (AP)
and the other in which every other pulse (EOP) is transponded.
These are illustrated in figure 3.
RF TAGS COMMUNICATIONS
CHANNEL
Signal Model
An RF tag can embed communications signals in a radar data
stream by receiving the incident radar pulses, modifying these in
1
Y  HX  Z . In much of the MIMO literature, the channel
matrices can be arbitrarily sized with arbitrary complex-valued
elements. The interference is usually taken to be spatially (and
temporally) white with a continuous stream of time samples
available from the channel.
All Pulse Model
A multiple receiver radar channel can be cast as a specific type
of MIMO channel. The RF tags MIMO channel is shown in
figure 5.
We model a single receiver radar as a set of Nbins identical
parallel discrete complex Gaussian channels corresponding to the
tag signal bearing range bins. Here Nbins might be the number
of taps in the tag FIR convolver, or more generally the number of
radar range bins available for embedding information. The
complex sample Ym , k  X m , k  Z m , k in the kth range bin of the
Pulse #1 Ha ( )
1
Z11
Ha 2 ( )
X1
Z 21
Ha 3 ( )
Y11
Y21
Z 31
Y31
Pulse #2 Ha 1 ( )
Z12
Ha 2 ( )
X2
Z
Ha 3 ( ) 22
Y12
Y22
Z 32
Y32
Pulse #3 Ha 1 ( )
Z13
Ha 2 ( )
X3
Z 23
Ha 3 ( )
Y13
Z 33
Y23
Spatial-Temporal Receivers
Figure 3: Single Receiver RF Tag Channels
Temporal Transmitters
Every Other Pulse Model
Y33
mth pulse is the sum of the tag contribution X m, k plus the
Figure 5: RF Tags MIMO Channel
interference Z m , k from clutter and thermal noise. We model the
For RF tags, the transmitters are temporally separated and
correspond to the repeated pulses. The receive channels
separated both spatially and temporally. The signal received on
pulse m , receiver n is denoted Yn , m .
interference as independent from range bin to range bin, and
denote
its
pulse-to-pulse
autocorrelation


by RZ (m1 , m2 )  E Z m2 ,k Z m1,k .
2.3
*
The RF tags MIMO channel is always an injective mapping
from the temporal input symbols X m to the space-time output
Multiple Receiver Radar Channel
The single receiver channel model generalizes easily to the
case of a multiple receiver radar.
In this context, it
mathematically takes the form of an injective multiple input
multiple output (MIMO) channel.
A conventional MIMO
channel is shown in figure 4.
Y1
H 21
Z1
H 31
Y2
H 12
X2
Z2
H 22
Y3
H 32
Spatial Receivers
Spatial
Transmitters
H 11
X1
Z3
Figure 4: Conventional MIMO Channel
The channel consists of a number of spatially separated
transmitters and receivers with a channel transfer matrix H and
an interference source Z .
The input-output relation is
symbols Yn , m . Unlike the conventional MIMO formulation, the
interference contaminating our received signals is highly
correlated spatially and temporally in exactly the same way it is
in the GMTI STAP problem.
Finally, while a conventional MIMO channel is assumed to
have a continuous stream of time samples, our RF tags MIMO
channel has a finite number of range bins into which it can code
information.
3.
INJECTIVE MIMO CHANNELS
A general expression for the capacity of a MIMO channel was
derived in [6]. In the case of an injective MIMO channel (one in
which the linear mapping defined by H is injective), a more
compact expression for the capacity can be given.
Given an injective MIMO channel Y  HX  Z , the
minimum variance unbiased estimator of X given Y is

W  H  R Z1 H

1
H  R Z1 Y which is a sufficient statistic [1, ch
2], [3, ch. 5] (receiver) for X . Consequently the cascaded
channel X  Y  W has the same capacity as the original
X  Y . This channel is equivalent to a bijective MIMO
channel

with

1
Z
U H R H
X

Y
H
identity
1

channel
matrix
and
interference
H R Z as shown in figure 6.
(H*RZ-1H)-1H*RZ-1
W
Continuous Time-Frequency Support
X
Z
FREQUENCY
W
U
I(X;Y) = I(X;W)
Z
Conventional MIMO:
1
Z
(H*RZ-1H)-1H*RZ-1
BWRF
Figure 6: Injective MIMO Channel Equivalence
TIME
Formulas for the spectral efficiency of such a channel are
well-known. Form an informed transmitter (which knows the
interference correlation and can spectrally optimize its
transmitted signal), the maximum spectral efficiency is given by
N TX
 (  i )   i
C IT   log
i
i 1





RF Tags:
Radar Time-Frequency Support
FREQUENCY
BWRF
Radar Pulses
Range Gate Receive Window
where  is chosen to satisfy the energy constraint
1
NTX
NTX
   i 

TIME
E.
Figure 7: Time Frequency Support Regions
n1
Here N TX is the number of transmit channels, E is the

energy per transmission and 1 ,  ,  NTX
the
equivalent

1
Z
RU  H R H
*

bijective
1
channel
 are the eigenvalues of
covariance
matrix
.
For an uninformed transmitter, the optimal coding scheme is
to transmit a spectrally white signal. The corresponding
maximum spectral efficiency is then
N
 E  i
CUT   log
i 1
 i
TX




4. RF TAGS MIMO CHANNEL
The formulas for the spectral efficiency of an injective MIMO
channel will be used for the RF tags MIMO channel. In order to
apply these formulas we first need to interpret spectral efficiency
in the case of RF tags. We then formulate the RF tags channel
matrix H and energy constraint E . Finally, we determine the
space-time interference covariance matrix R Z as a function of
the radar parameters.
4.1
Spectral Efficiency
In a conventional MIMO channel, an RF band is sampled
directly at (or above) the Nyquist rate. Here the full timefrequency support is available for communications. In contrast,
the pulsed operation of a radar and front end processing allows
only a small fraction of the time frequency support to be utilized
for communications. This is illustrated in figure 7.
In conventional MIMO, the spectral efficiency is the number
of information bits transmitted per Nyquist sample. In the RF
tags case, it is the number of information bits sent per range bin
per pulse. The channel capacity of a conventional MIMO
channel is obtained by multiplying the spectral efficiency times
the bandwidth, while for RF tags we multiply by PRF  N bins
for the all-pulse model or
PRF
 N bins for the every-other pulse
2
model.
4.2
Energy Constraint
The total tag energy transmitted per pulse is given by the Friis
transmission equation [2, ch. 2]
Per Bin
Tag Energy
Pulse
Width
Radar Antenna
Effective Area
 Gtag
1

2
4R
4
2
N bins  Etag  ERPtag  Tp 
Range Bins
Used
Tag Effective
Radiated Power
Rx
Propagation
Loss to Radar
For maximum spectral efficiency, this energy is equally
divided among all range bins as shown above.
4.3
Channel Matrix
In deriving the formula for the channel matrix, we note that
signals transmitted on one pulse do not affect those received on
other pulses. On a given pulse, each radar receiver hears the
same tag signal but with a slightly different one-way delay. If the
receive antennas are identical and the radar transmitted pulses
have small fractional bandwidth, then it is appropriate to model
the signal received by the antennas as identical up to a phase
factor [5].
The case of a uniform linear array is depicted in figure 8, with
physical phase center displacement vector d .
Aft
Center
patches shown in figure 9.
Fore
k̂
d

M
N
= # Pulses
= # Receivers
Figure 8: Steering Vector to RF Tag
Figure 9: Antenna Illumination & Clutter Patch Model
If k̂ denotes the unit vector toward the RF tag device and 
is the wavelength, then the spatial steering vector is
We model the return from each clutter patch as a complex,
zero mean random variable whose variance (energy) is given by
the radar range equation [2]
 1 
a( )  e2i 
e4i 
where  
kˆ  d

Patch
Energy
Pulse
Width
E patch ( )  P  T p  G Tx ( ) 
Transmit
Power
is the spatial frequency: the receiver to
receiver phase increment due to the position of the RF tag.
4.4
Antenna
Effective Area
2
Rx
 G ( )
1

4
4R 2
(10)
This variance corresponds to the zero-lag sample of an
autocorrelation sequence. For a single stationary clutter patch,
the autocorrelation between two pulses and/or receivers is
while the channel matrix for the every-other-pulse (EOP) tag
H EOP
Clutter
Reflectivity
Propagation
Loss to Radar
0 
a( )

 0
a( )   I M M  a( )
 


M
and is given by
2
0
 0
a( )
0

 0
0

a( )
 0
 

TX Antenna
Gain
 R rg sin( graze ) 
The channel matrix for a tag that transponds all pulses (AP) is
case is MN 
1
 0
4R 2
Patch
Area
MN  M and given by
H AP
Propagation
Loss to Patch






2R( , δ) 

R patch ( , δ)  E patch ( ) exp   2i
(11)




where R( , δ)  δ  kˆ ( ) is the change in range to the
clutter patch and δ is the displacement vector between the
virtual antenna phase center positions (Fig. 7)
Interference
The interference in the channel is the sum of the contributions
from clutter and thermal noise. We treat the thermal noise as
both spatially and spectrally white. The variance of the thermal
noise (per range bin per pulse) is given by Boltzmann’s equation
Thermal
Energy
The total clutter correlation between pulses (because the
patches
are
assumed
independent)
is
given
by
Temperature
Ethermal  k  T  Fn
Boltzmann’s
Constant
Receiver
Noise Figure
Figure 10: Range Change Between Receivers and/or Pulses
Determines Correlation Coefficient.
Rclutter (δ)   R patch ( , δ) . This correlation is shown as a

(9)
The clutter contribution to a range bin sample is a coherent
sum of the returns from a large number of independent clutter
function of displacement δ for the Veridian DCS radar in figure
11.
Noise Figure
Fn  2dB
Standoff Ranges
R  15 km  45 km
Resolutions
 rg  3cm  3m
Tag
Pulse
Frequencies
Repetition
Number of Range Bins
Nbins  200
Effective Radiated Power
ERPtag  0.4W
Median
Clutter
Sectional Density
Figure 11: Clutter Correlation vs. VPC Displacement for
Veridian DCS Radar.
For an aircraft traveling at a velocity V AC , physical antenna
PRF  200 Hz  5KHz
Cross
CCSD  15 dB
The antenna structure of the DCS is shown in figure 12. It
consists of three receivers whose apertures have a –25dB Taylor
weighting applied. The middle phase center is used for transmit
while all three channels are independently received.
displacement d and pulse repetition interval PRI , the virtual
phase center displacement between phase centers n1 and n 2 on
pulses
m1
m2
and
 m,n  m V AC PRI  n
respectively
is
given
by
d
, where m  m 2  m1 and
2
-25dB Taylor
Weighting
n  n 2  n1 .
The entries of the interference matrix are thus
R Z m ,n ,m ,n   Rclutter ( m,n )
1
1
2
2
  (m) (n) E thermal
5. CAPACITY SIMULATIONS
Radar
The channel capacity of a radar system depends on a dozen
different radar, tag and clutter parameters.
This can
unnecessarily obfuscate the interpretation of the radar channel
capacity. To simplify things, we will calculate the radar channel
capacity for the Veridian DCS radar system operating with a
representative RF tag design. We will show the dependence of
the capacity on the radar PRF, standoff range and resolution.
The channel capacity behavior will exhibit the same basic
properties for all radar system and RF tag designs, although the
actual capacity numbers will change.
Parameter
Symbol & Value(s)
Transmit Power
PTxradar  3.5kW
Peak Tx/Rx Gain
G radar  29 dBi
Azimuth Beamwidth
 az  5.25 
Elevation Beamwidth
 el  6.25
Wavelength
  .03m
Pulse Width
T p  45 .51s
Figure 12: DCS Receiver Apertures.
5.1
Single Channel Spectral Efficiencies
To appreciate the spectral efficiency for a multiple receiver
radar, it is useful to first examine capacities of a single receiver
system. Figure 13 shows the dependence of spectral efficiency
on radar standoff range. We fix the resolution at .3m and look at
the spectral efficiency of the uninformed transmitter, every-other
pulse channel for three standoff ranges.
Figure 12: Spectral Efficiency vs. PRF
In the high PRF, thermal dominated region, spectral efficiency
(and hence capacity) decreases with increasing range. This is
because the thermal noise level is independent of range while the
2
tag energy is proportional to 1 / R .
We see a somewhat
surprising result in the low PRF clutter dominated region where
spectral efficiency increases with increasing range. This occurs
because the dominating clutter energy is proportional to 1 / R 4 .
The spectral efficiency is less sensitive to changes in range in the
thermal dominated region than in the clutter dominated region of
the log-compressive characteristics of channel capacity at higher
signal to noise ratios. More properties of the single receiver
spectral efficiencies may be found in [7].
5.2
Multiple Channel Spectral Efficiencies
Finally, we compare the spectral efficiency of a single channel
radar to that of a multiple channel radar. We consider three
cases. The first (green) case is that of a single receive aperture
where we transmit and receive on only the center DCS antenna.
The second (blue) case is using the center antenna for transmit
and all three independently for receive. This is not a fair
comparison yet because the three independent apertures will
naturally have more receive gain than the single aperture. To
account for this, we introduce a third (red) case where all three
apertures are beamformed into a single receiver with maximum
gain toward the RF tag. These cases are illustrated in figure 13.
Three Independent
Receive Apertures
Single Simple
Receive Aperture
Capacity = 320Kbps
Capacity = 1.6Mbps
Capacity =
20Kbps
 2
 1
One VPC
displacement
flown b/t pulses
Figure 14: Spectral Efficiency of Multiple Receiver DCS / RF
tags channel.
As a final point, it is interesting make the following contrast.
For a conventional MIMO channel with additive white noise, the
spectral efficiency depends only on the singular values of the
channel matrix [6]. In the case of RF tags, the singular values do
not change as more radar receivers are added, however because
of the space-time color of the interference, the spectral efficiency
of a multiple receiver radar can outperform that of a single
receiver radar.
6. REFERENCES
[1] T. Cover and J. Thomas, Elements of Information Theory.
Three Apertures Beamformed
into Single Receiver
Figure 13: DCS Aperture Configurations Compared
The spectral efficiencies for these three cases are shown as a
function of PRF in figure 14. Note that the spectral efficiency of
the 3 receiver case in uniformly higher than both the single
receiver and the beamformed “long antenna” cases. At very low
or very high PRFs, the efficiency of the 3 receiver and long
antenna cases are nearly identical. The advantage of a multiple
receiver radar exists at intermediate values of the PRF. We
notice that the spectral efficiency of the 3 receiver case peaks at
PRF200 and PRF 400. These points correspond to the
PRFs in which the platform two or one virtual phase center
displacements respectively between adjacent pulses. For PRFs at
or above the DPCA condition, the spectral efficiency of a
multiple phase center radar is near it’s maximum.
Wiley, 1991.
[2] C. Balanis, Antenna Theory: Analysis & Design (2nd
edition) Wiley, 1997.
[3] S. Kay, Fundamentals of Statistical Signal Processing
(Volume 1: Estimation Theory) Prentice Hall, 1993.
[4] R. Gallager, Information Theory and Reliable
Communication. Wiley, 1968.
[5] J. Ward, Space-Time Adaptive Processing for Airborne
Radar. Lincoln Laboratory Technical Report 1015
[6] Bliss et. al. Environmental issues for MIMO capacity. IEEE
Trans. on SP, 50(9): 2129-2142, Sept. 2002.
[7] P. Bidigare, “The Shannon Channel Capacity of a Radar
System”, Conference Record of the Thirty Sixth Asilomar
Conference on Signals, Systems and Computers, Nov 2003.
[8] P. Bidigare & M. Nayeri, “RF Tags: Radar as a
Communications Channel”, Proceedings of the 2002 10th
ASAP Workshop