1
Proof of Proposition 1 in
“A Joint Design Approach for Spectrum Sharing
between Radar and Communication Systems”
We state Proposition 1 in [1] as follows:
Proposition 1 ( [1]). Suppose that {Rxl } is initialized by {Rxl } ≡ R0x . Then, the optimal value of (PR ) in every
+
+
0
iteration of the proposed algorithm could be achieved by {Rnxl } such that for any l, l0 ∈ N+
L (or l, l ∈ NL̃ \ NL ),
it holds that Rnxl = Rnxl0 .
Proof: The proposition can be proved using induction. We focus on the proof for l, l0 ∈ N+
L in the following.
+
The proof for l, l0 ∈ N+
\
N
is
similar.
L
L̃
From the proposition, we know that {Rxl } is initialized such that R0xl = R0xl0 = R0x , ∀l, l0 ∈ N+
L . We need
to show that the optimal value of (PR ) in the n-th iteration is also achieved by {Rnxl } such that Rnxl = Rnxl0 ,
n
∀l, l0 ∈ N+
L . Because {Rxl } is obtained via several inner iterations of solving (P̃R ) in [1], it suffices to show that
the above property could be passed on between the iterations of solving (P̃R ).
n(i−1)
Suppose that, in the (i − 1)-th inner iteration, the optimal value of (P̃R ) is achieved by {Rxl
} such that
n(i−1)
n(i−1)
ni∗ } is obtained by solving (P̃ ) with {R̄ } =
.
During
the
i
-th
iteration,
{R
Rxl
= Rxl0 , ∀l, l0 ∈ N+
R
xl
xl
L
PL
n(i−1)
ni∗ , Rni , ∀l ∈ N+ is also feasible and achieves the same radar
{Rxl
}. We will show that Rni
≡
1/L
R
x
l=1 xl
xl
L
SINR as {Rni∗
does.
Based
on
the
concavity
of
C
(R
,
·)
,
we
have
l
xl
xl }l∈N+
L
L
X
ni
Cl (Rni∗
xl , ·) ≤ LCl Rx , · .
l=1
P
ni
ni∗
ni
For the communication transmission power, it trivially holds that L
l=1 Tr(Rxl ) = LTr(Rx ). Therefore, {Rx }
is also feasible. The objective f˜(Rxl ) of (P̃R ) in the i-th iteration is affine w.r.t. Rxl with coefficient depending
n(i−1)
n(i−1)
n(i−1)
˜
on R̄xl = Rxl
. Given that Rxl
= Rxl0 , ∀l, l0 ∈ N+
L , the affine functions f (·) for are identical for any
+
ni
ni∗
l ∈ NL . Therefore, {Rx } achieves the same objective value as {Rxl }l∈N+L does.
Proposition 1 is proved.
R EFERENCES
[1] B. Li, H. Kumar, and A. P. Petropulu, “A joint design approach for spectrum sharing between radar and communication systems,” in
submitted to IEEE International Conference on Acoustics, Speech and Signal Processing, March 2016.