Aerospace and Electronic Systems Magazine May 2018 - 36

FDA-OFDM for Integrated Navigation, Sensing, and Communication Systems



rm  



am jψ m     j 2π  f m  t − c0  
 
e e
m = 0 rm  

M −1

E ( r ,θ , t ) = 

(2)

where c0 is the speed of light, am is the amplitude and ψm is the
phase of the excitation which gives rise to the radiated electric
field. The rm = r − md sin θ is the range for the mth element, with r,
d, and θ being the range for the first element, element spacing and
azimuth angle, respectively.
In amplitude sense, the distance differences between individual
antenna elements can be ignored, namely, rm ≈ r. For the standard
FDA, namely, Δfm = mΔf, with Δf being the frequency increment,
(2) can be reformulated as


r 

1 j 2π f0  t − c0  M −1
E ( r ,θ , t ) ≈ e
 ame jψ m e jmγ FDA
r
m=0

(3)

with γFDA = 2πΔf (t − r/c0) + 2πf0d sin θ/c0. Note that, due to the fact

that r  ( M − 1) d sin θ and f 0  Δf , the term m2πΔfd sin θ/c0 is ignored in γFDA. The term am and e jψ m can be combined together as the
weight, namely, wm = ame − jψ m. Equation (3) can then be rewritten as


r 

1 j 2π f0  t − c0  H
E ( r ,θ , t ) ≈ e
w a ( r ,θ , t )
r

(4)

where w = [w1 w2 ... wM−1]T is the weighting vector and
T

a ( r ,θ , t ) = 1 e jγ FDA ... e j ( M −1)γ FDA  is the FDA steering vector, with
H
and T being the conjugate transpose and transpose operators, respectively. Suppose the following parameters: M = 32, Δf = 3 kHz,
and r = 120 km, Figure 2 compares the FDA transmit beampattern
with that of a phased-array. It is seen that the FDA electronic field
(array factor) depends on both the range, angle and time, which
provides a potential radar sensing application for joint range and
angle estimation of targets [23].

COMMUNICATION-EMBEDDED FDA-OFDM WAVEFORM
Analogous to a standard OFDM scheme, the FDA-OFDM chirp
approach can be regarded as a parallel stream of multiple chirp signals with orthogonal carriers, each modulated and radiated by one
of the FDA elements according to the communication information.
The FDA-OFDM chirp waveform with Nc carriers and Ns symbols
can be expressed as

x (t ) =

N s −1 N c −1

 c
k =0 n =0

k, n

e

(

jπ 2 f n t + kr t 2

)e jφ rect  t − kTw 


k, n



Tw



(5)

Beampattern comparisons of FDA and phased-array: (a) ContinuousFDA with time dependent beampattern. (b) Phased-array with time
independent beampattern.

element at the kth symbol, Tw is the symbol duration including an
elementary symbol duration T and a guard interval cyclic prefix
duration Tg corresponding to the maximum expected time delay
and rect(t/Tw) describes a rectangular window with duration Tw.
In order to avoid coupling interference between individual carriers, fn should hold as fn = i/T = iΔf, i = 0, 1, ..., Nc − 1, with Δf
being the frequency separation. Note that fn is not always equal to fi
= i Δf. This is because that, according to the FDA-OFDM scheme,
fn should be randomly selected from the frequency offset set {i
Δf}, i = 0, 1, ..., Nc − 1. Accordingly, the spectrum of each carrier
signal is
1

where ck,n denotes the amplitude and phase state representing information data for transmission at the nth carrier in the kth symbol,
fn = f0 + Δfn is the hopping frequency according to the OFDM coding sequence, kr is the chirp rate, ϕk,n is the initial phase of the nth
36

Figure 2.

kr

e

jπ sgnkr  4 − jπ ( f − i T )

e

2

kr

(6)

with sgn[ ] and f being the signum function and Fourier frequency
variable, respectively. Obviously, it is different from the stan-

IEEE A&E SYSTEMS MAGAZINE

MAY - JUNE 2018



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