1

Leveraging Edge Caching in NOMA Systems with
QoS Requirements

arXiv:1801.07430v1 [] 23 Jan 2018

José Armando Oviedo and Hamid R. Sadjadpour
Department of Electrical Engineering, University of California, Santa Cruz
Email: {xmando, hamid}@soe.ucsc.edu

Abstract—Non-Orthogonal Multiple Access (NOMA) and
caching are two proposed approaches to increase the capacity
of future 5G wireless systems. Typically in NOMA systems,
signals at the receiver are decoded using successive interference
cancellation in order to achieve capacity in multi-user systems.
The leveraging of caching in the physical layer to further improve
on the benefits of NOMA is investigated, which is termed cacheaided NOMA. Specific attention is given to the caching cases
where the users with weaker channel conditions possess a cache
of the information requested by a user with a stronger channel
condition. The probability that any of the users is in outage
for any of the rates required for this NOMA system, defined
as the ”union-outage,” is derived for the case of fixed-power
allocation, and the power allocation strategy that minimizes the
union-outage probability is derived. Simulation results confirm
the analytical results, which demonstrate the benefits of cacheaided NOMA on reducing the union-outages probability.

I. I NTRODUCTION
Data caching and Non-Orthogonal Multiple Access
(NOMA) are being heavily investigated for implementation
in the next generation of wireless systems, namely 5G [1],
[2]. NOMA has relied on superposition coding (SC) at the
transmitter, and successive interference cancellation (SIC) at
the receiver [3], [4] in order to achieve capacity in multi-user
wireless systems. Given the advanced storage and processing
capabilities of current and future user terminals, caching can
then be leveraged at the receiver side of a user with weaker
channel strength in a NOMA system.
Consider a NOMA system with caching enabled user terminals. While a user with stronger channel can use SIC to decode
and remove the interfernce caused by the signal intented for
the user with weaker channel, the user with weaker channel
can take advantage of caching if it has already cached the
information requested by the user with stronger channel. Since
the header of the signal contains the modulation and rate
information for each signal to be decoded, and can contain a
unique file identifier for the data, the user with weaker channel
can then recreate the signal transmitted for the stronger user
and subtract it from the composite signal, similar to how
successive interference cancellation works. The difference here
is that the rate at which the information of the other user is sent
does not have to be within the capacity of the user containing
the cache. Thus, this technique called cache-aided NOMA
(CA-NOMA) creates two interference free channels for the
two users by taking advantage of two proposed techniques in
the emerging 5G wireless standards.

However, since the capacity of the weaker user’s channel
increases, this creates different problems related to the limits
of what the stronger user can successfully decode using SIC.
With regular NOMA, the capacity at which the weaker user’s
information is sent is always upper-bounded by the capacity at
which the stronger user can perform SIC. With caching, this
is no longer the case.
The rest of the paper is organized as follows: section II
covers the relevant previous work on NOMA and caching,
section III details the system model used in our analysis,
section IV outlines the definition of the union-outage event
that occurs in NOMA and covers the analysis for satisfying a
minimum QoS, section V compares our analysis to simulation
results, and we share our concluding thoughts and propose
future work in section VI.
II. P REVIOUS W ORK ON NOMA
The basic concept of NOMA was described in [3], which
showed that the approach of using SC at the transmitter and
SIC at the receiver with greater signal-to-interference-plusnoise ratio (SINR) achieves capacity of the channel. It is
known [4] that for every orthogonal approach to scheduling
multiple users in a point-to-multipoint transmission, there
always exists a non-orthogonal approach that will outperform
it. The exact power allocations and associated expected capacity gains were obtained in [5] for the 2-user SISO channel
with completely random user-pairing. Then the results were
extended in [6] to the general K-user SISO systems to find
the expected capacity gains for optimal user-pairing (according
to [7]), and closed-form expressions were found for the power
allocation coefficients that allow all users to achieve capacities
greater than or equal to their respective OMA capacities.
Similarly, in [8] the authors use a dynamic power allocation
strategy that always outperforms OMA in order to improve
the outage capacity performance of the network.
Caching at the edge of networks, including small Base
Stations (BSs) and device-to-device communications for future
5G wireless networks is described in [9], where it was shown
that caching is an energy efficient solution to help the network
improve coverage probabilities while maximizing spectral efficiency. In [10], it is demonstrated that using decentralized
coded caches at user terminals increases the throughput capacity of the network, while alleviating the burden of network
traffic caching at BSs and helper nodes.
In CA-NOMA, caching is used at the user terminals (users)
in order to assist in eliminating interference from channels

2

that cannot otherwise take advantage of SIC due to weaker
channel conditions. We show that downlink capacity and
outage probabilites improve when a user possesses a cache
of information requested by a user with a stronger channel,
then derive the fundamental power allocation constraints, and
find the optimum power allocation that minimizes the unionoutage probability of the NOMA system.
III. CA-NOMA S YSTEM M ODEL

AND

C APACITY

Consider a two-user orthogonal multiple access (OMA)
single-input-single-output (SISO) system. Let the total transmit time period be T , where users are allocated equal nonoverlapping time slots of length T /2, and allocated the total
transmit SNR ξ for that slot1 . For each user k = 1, 2, if the
BS is transmitting signal xk (E[|xk |2 ] = 1), the channel gain
between the BS and user-k is hk (|hk |2 ∼ Exponential( β1 )),
and the receiver has noise zk ∼ CN (0, 1), then the received
signal at user k is given by
p
(1)
yk = ξhk xk + zk .
Since the time duration is
bps/Hz is given by
Ckoma =

1
2,

then the capacity of user k in

1
log2 (1 + ξ|hk |2 ).
2

(2)

Let the user with weaker channel be user-1, so that |h1 |2 <
|h2 |2 . In a two-user SISO NOMA system, where the BS is
transmitting the two signals using SC, the signal xi is carrying
the information for user-i. The power allocation coefficients
for users 1 and 2 are 1 − a and
pa, respectively.
√The SC signal
transmitted by the BS is then (1 − a)ξx1 + aξx2 , thus the
raw received signal at user-i is
p
p
ri = hi ( (1 − a)ξx1 + aξx2 ) + zi , i = 1, 2.
(3)
Since NOMA allows user-2 to remove the interference received by user-1’s signal, the received signal after using SIC
at user-2 is
p
y2 = aξh2 x2 + z2 .
(4)

However, in a CA-NOMA enabled system where user-1 caches
user-2’s information, user-1’s receiver can decode user-2’s
information and subtract its signal from the composite received
signal, and thus has its information carried on the signal
p
y1 = (1 − a)ξh1 x1 + z1 .
(5)
This gives rise to the capacities

C1 (a) = log2 (1 + (1 − a)ξ|h1 |2 )
2

C2 (a) = log2 (1 + aξ|h2 | ).

(6)
(7)

The capacity for user-2 to decode and subtract user-1’s signal
using SIC is


(1 − a)ξ|h2 |2
(8)
C2→1 (a) = log2 1 +
aξ|h2 |2 + 1
1 The same OMA formulation can be made in the case of frequency division
instead of time division.

When using regular NOMA, it is always true that if |h1 |2 <
|h2 |2 → C1 (a) < C2→1 (a). However, since CA-NOMA
allows user-1 to subtract user-2’s interference, this relationship
of capacities no longer holds for CA-NOMA. Thus the effects
of CA-NOMA on outage performances is investigated.
IV. CA-NOMA

FOR

ACHIEVING

A

M INIMUM Q O S R ATE

Suppose a downlink system requires a minimum rate R0
in order to schedule any user’s information to be transmitted
at the base-station. In a two-user CA-NOMA system, it is
required that there are no outage states in the system, which
includes the achievability of performing SIC. This gives rise
to the following outage events


2 R0 − 1
2
C1 (a) < R0 =⇒ Aout
=
|h
|
<
(9)
1
1
(1 − a)ξ


2 R0 − 1
2
C2 (a) < R0 =⇒ Aout
=
|h
|
<
(10)
2
2
aξ


2 R0 − 1
2
.
C2→1 (a) < R0 =⇒ Aout
=
|h
|
<
2
2→1
ξ(1 − a2R0 )
(11)
If any of these outage events occurs, then the system will
fail to perform a successful NOMA transmission. The union
outage probability, which is a function of a, is then computed
as
out
out
pU-out (a) = Pr{Aout
1 ∪ A2 ∪ A2→1 }

out
out
= 1 − Pr{Aout
1 ∪ A2 ∪ A2→1 }

out
out
= 1 − Pr{Aout
1 ∩ A2 ∩ A2→1 }.

(12)

The most common and simplest approach to power allocation in NOMA systems is fixed-power allocation. However,
instead of using an arbitrary fixed-power value, it is shown in
the following subsection that there are fundamental restrictions
on how fixed-power allocation should be used in CA-NOMA
systems, and by extension all NOMA systems.
A. Fixed-power allocation conditions for successful CANOMA
Suppose a fixed-power NOMA system is employed. The
power allocation region required must be fundamentally explored. This power allocation region must be such that all
of the events are feasible. It should be noted that there are
channel condition cases such that it is not possible for a user to
achieve the minimum rate required by the system even with the
optimum power allocation strategy, and thus no scheme exists
that can guarantee meeting the system QoS requirement.
By feasibility, it is meant that the power allocation region
out
out
must be restricted such that the events Aout
1 , A2 , and A2→1
occur with a probability less than 1. Therefore, the following
property of NOMA, and by consequence CA-NOMA, is
obtain.
Property 1. The channel for user-2 to perform SIC is always
in outage ∀a > 2−R0 and |h2 |2 > 0.
Proof: The channel h2 is in outage if
C2→1 (a) < R0

(13)

3

=⇒ξ|h2 |2 (1 − a2R0 ) ≤ 2R0 − 1.
However, since ∀a > 2−R0 , ξ(1 − a2R0 ) < 0
=⇒ |h2 |2 <

={|h2 |2 :

(14)

2 R0 − 1
< 0.
ξ(1 − a2R0 )

(15)

According to property 1 of CA-NOMA, for any power
allocation a ≤ 2−R0 , there exists a non-zero probability that
C2→1 (a) > R0 . Otherwise, SIC will fail, causing the entire
purpose of using NOMA to fail.
It is clear that there is a trade-off between C2 (a) and
C2→1 (a) that depends on a, which can be evaluated using
out
Aout
2 and A2→1 as follows
2 R0 − 1
2 R0 − 1
≶
aξ
ξ(1 − a2R0 )
1
≶ a.
=⇒
1 + 2 R0
1
1+2R0

Aout
2

(17)
(18)
Aout
2→1

This condition says that if a <
⊂
=⇒
, then
out
out
out
A2 ∩ A2→1 = A2 . This implies that the probability in (12)
can be simplified to
pU-out (a) = 1 −

Pr{Aout
1

∩

Aout
2 }.

(19)

Equation (18) also leads to the following.

2 1
2
Lemma 1. Let Aout
2,oma = |h2 | : 2 log2 (1 + ξ|h2 | ) < R0 .
1
Then ∀a ≤ 1+2R0 ,
out
out
Aout
2→1 ⊆ A2,oma ⊆ A2 .

Proof: Define the functions B2 (a) =
R0

(20)
2R0 −1
aξ

and

2 −1
B2→1 (a) = ξ(1−a2
R0 ) . B2 (a) is a montonically decreasing
function of a ∈ (0, 1), while B2→1 (a) is a monotonically
increasing function of a ∈ (0, 2−R0 ). If a ≤ 1+21R0 , then

B2 (a) ≥ B2 ( 1+21 R0 ) = B2→1 ( 1+21R0 ) ≥ B2→1 (a)
Let Aout∗
= {|h2 |2 : |h2 |2 < B2 ( 1+21R0 )}, so B2 ( 1+21R0 ) ≤
2
1
B2 (a) ⇒ Aout∗
⊆ Aout
2
2 , ∀a ≤ 1+2R0 . However, the event
Aout∗
2

2

2

={|h2 | : |h2 | <

={|h2 |2 : |h2 |2 <

={|h2 |2 : |h2 |2 <

B2 ( 1+21R0 )}
(2R0 −1)(1+2R0 )
}
ξ
4R0 −1
}
ξ

log2 (1 + ξ|h2 |2 ) < R0 } = Aout
2,oma

out
out
Therefore, since Aout∗
= Aout
2
2,oma ⇒ A2,oma ⊆ A2 .
out∗
2
2
Similarly, let A2→1 = {|h2 | : |h2 | < B2→1 ( 1+21R0 )}, so
1
out∗
B2→1 ( 1+21 R0 ) ≥ B2→1 (a) ⇒ Aout
2→1 ⊆ A2→1 , ∀a ≤ 1+2R0 .
Note that the event

which is impossible since |h2 |2 > 0, therefore making the
capacity impossible to be greater than R0 . Hence,


1 + ξ|h2 |2
∀|h2 |2 > 0, log2
< R0 .
(16)
1 + aξ|h2 |2
Remark 1. In general, if any user in a NOMA system must
decode a signal in the presence of interference, where the
interference cannot remove by SIC or caching, it will have
this type of fundamental power allocation restriction due to the
interference terms. Thus, in a NOMA system with minimum
QoS rate requirement R0 , the power allocation coefficient
of signals that will cause irremovable interference at certain
receivers is fundamentally restricted by a function of the
minimum rate R0 , independent of the channel gains.

1
2

2
2
1
Aout∗
2→1 = {|h2 | : |h2 | < B2→1 ( 1+2R0 )}
(
)
2 R0 − 1
= |h2 |2 : |h2 |2 <
2R0
ξ(1 − 1+2
R0 )

= {|h2 |2 : |h2 |2 <

= {|h2 |2 : |h2 |2 <
= {|h2 |2 :

1
2

(2R0 −1)(1+2R0 )
}
ξ
4R0 +1
}
ξ

log2 (1 + ξ|h2 |2 ) < R0 } = Aout
2,oma

out
out
out
Therefore, since Aout∗
2→1 = A2,oma ⇒ A2→1 ⊆ A2,oma . Hence,
1
out
out
out
A2→1 ⊆ A2,oma ⊆ A2 , ∀a ≤ 1+2R0 .
With lemma 1, it is clear that fundamentally the power
allocation region of a is restricted to the set (0, 1+21 R0 ] in
order to minimize the union-outage probability according to
equation (19), which is done in the following subsection.

B. Outage Probabilities
The joint p.d.f. of |h1 |2 and |h2 |2 for users paired randomly
x1 +x2
is given2 [6] by f|h1 |2 ,|h2 |2 (x1 , x2 ) = β22 e− β , for x1 , x2 ≥
out
out
out
out
0. Since Pr{Aout
1 ∩ AR2 } = Pr{A1 } − Pr{A1 ∩ A2 }, and
1
2R0 −1
2 0 −1
a < 2 ⇒ (1−a)ξ < aξ , then computing
Z ∞ Z ∞
2(2R0 −1)
2
2 − x1 +x
− (1−a)βξ
out
β
dx
dx
=
e
,
Pr{A1 } = R
e
2
1
2
2 0 −1
x1 β
(1−a)ξ
(21)
and
Pr{Aout
1

∩

Aout
2 }

=

Z

2R0 −1

2R0 −1
aξ
2R0 −1
(1−a)ξ

= −2e− a(1−a)βξ + 2e−

Z

x2
2R0 −1
(1−a)ξ

2(2R0 −1)
(1−a)βξ

2 − x1 +x
2
β
e
dx1 dx2
2
β

+ e−

2(2R0 −1)
aβξ

− e−

2(2R0 −1)
(1−a)βξ

,
(22)

gives the probability of union-outage as
out
out
pU-out (a) = 1 + Pr{Aout
1 ∩ A2 } − Pr{A1 }

= 1 + e−

2(2R0 −1)
aβξ

2R0 −1

− 2e− a(1−a)βξ .

(23)

The following lemma provides a tight approximation for the
optimum power allocation coefficient that minimizes equation
(23).
Lemma 2. The power allocation coefficient amin ∈ (0, 1+21R0 ]
that minimizes pU-out (a) is given approximately by
o
n
1
1
−A1 +A2 3
1
2 3
,
−
,
amin ≈ min A30 + A1 +A
R
2
2
1+2 0
(24)
R0

where A0 = 2
q
4
A0 1 − 27
A0 .

−1
βξ ,

A1 =

2
3
27 A0

− 32 A20 + A0 , and A2 =

2 Since the possession of a cache is independent of the order of the channel
gains, the distribution of the channel gains is the same as in [6]

4

Proof: Using the power series for t ≪ 1
∞ k
X
t
k=0

k!

= 1 + t + O(t2 ),

(25)

pU-out (a) can be approximated as

2


2 R0 − 1
2 R0 − 1
pU-out (a) ≈1 + 1 −
−2 1−
aβξ
a(1 − a)βξ
2A0
A20
.
(26)
= 2 +
a
1−a
Taking the derivative of (26) with respect to a, equating it to
0, and solving for a gives the following
d
2A0
2A2
=0
pU-out (a) = − 30 +
da
a
(1 − a)2
=⇒ 0 = a3 − A0 a2 + 2A0 a − A0 .

(27)
(28)

2
Let A1 = 27
A30 − 32 A20 + A0 and A3 = 2A0 − 13 A20 . Then
equation (30) looks like b3 + A3 b = A1 . Set the numbers q
and s to be defined by 3sq = A3 and s3 − q 3 = A1 . Note that
3
b = s − q solves equation (30). Therefore, since s = A
3q ,
 3
A3
− q 3 = A1
(31)
3q
1
A33
(32)
=⇒0 = q 6 + A1 q 3 − 27
q

 13
4
−A1 + A21 + 27
A33


(33)
=⇒q =
2

1
−A1 + A2 3
=
,
(34)
2
q
q
4
4
where A21 + 27
A33 = A0 1 − 27
A0 = A2 . Then s is found
by
1

1
A1 + A2 3
s = (q 3 + A1 ) 3 =
.
(35)
2


The value of a that makes equation (27) true is
A0
A0
+b=
+s−t
3
3
1 
1

−A1 + A2 3
A0
A1 + A2 3
−
.
=
+
3
2
2

10 -1

10 -2

10 -3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

a = power allocation coefficient

The solution to the cubic is as follows [12]. Let a = b + A30 .
Then the cubic equation becomes


3
2


A0
A0
A0
− A0
0= b+
− A0 b +
+ 2A0 b +
3
3
3
(29)

3
3
2 2
2
1 2
(30)
=⇒b + b 2A0 − 3 A0 = 27 A0 − 3 A0 + A0 .

a=

p out = outage probability

et =

Outage probability vs. power allocation coefficient
10 0

(36)
(37)

Therefore, since a ≤ (1 + 2R0 )−1 , then
o
n
 13
1
−A1 +A2 3
1
2
.
amin ≈ min A30 + A1 +A
−
,
2
2
1+2R0
(38)
Lemma 2 provides a close approximation to the optimum
power allocation coefficient that minimizes the outage prob-

Fig. 1. Outage probabilities are plotted vs. a; ξ = 20 dB, β = 2, R0 = 2
bps/Hz

ability when using fixed-power allocation strategies in CANOMA systems. Furthermore, amin becomes strictly less than
1
as ξ increases. This is beneficial because user-2’s strong
1+2R0
channel and ability to achieve C2 (a) > R0 allows the system
to then allocate more power to the weaker user to also achieve
the minimum rate, while making the success of performing
SIC guaranteed if C2 (a) > R0 .
V. C OMPARISON

OF THEORETICAL AND SIMULATION
RESULTS

The plot in figure 1 demonstrates the different outage
probabilities for each user. Note that the result for pU-out (a)
in equation (23) exactly matches the simulation for a ∈
(0, 1+21R0 ], which is the interval of interest, given Lemma
1. When a ∈ ( 1+21R0 , 2−R0 ), pU-out (a) is dominated by
Pr{Aout
2→1 }, which is undesirable since SIC must happen in
order to have a functioning NOMA system. When a ≥ 2−R0 ,
pU-out (a) = Pr{Aout
2→1 } = 1, which simply means that SIC is
impossible, causing the NOMA system to surely fail.
In figure 2, the plot of pU-out (a) vs. ξ is demonstrated for
the minimizing amin , and compared with the union-outage
probabilities of OMA and regular NOMA, where regular
NOMA has power allocation coefficient a∗ that minimizes
its union-outage probability. It can be seen that the unionoutage probability using CA-NOMA is superior to both regular
NOMA and OMA. The difference is most noticible when
10 < ξ < 30 dB. However, as ξ increases, regular NOMA
performs almost as well. This makes sense because a very
large ξ will make outage very unlikely, given that regular
NOMA still allows the flexibility of finding an optimum a∗ .
Figure 3 demonstrates the tightness of the approximation
of amin given by lemma 2. It is worth noting that there is a
threshold where amin < 1+21R0 that is a function of ξ and 2R0 .
As expected, the value of amin decreases as ξ increases, which
allows the system to take advantage of the strong channel to
help the weak channel have greater probability of achieving
the minimum QoS rate. From the results in figures 2 and 3, it

5

Outage probability vs. transmit SNR

10 0

along different degrees-of-freedom) can have their interference
eliminated at receivers if a content is cached. Furthermore, the
distribution of cached content based on its popularity can be
used to determine how often CA-NOMA can be fully utilized
on top of regular NOMA in multi-user systems.

p out = outage probability

10 -1

10 -2

R EFERENCES
10 -3

p

U-out
oma
U-out
noma
p
u-out

10 -4

(a

min

)

p

0

*

(a )

10

20

30

40

50

= transmit SNR [dB]

Fig. 2. Outage probabilities are plotted vs. ξ, with a = amin ; β = 2, R0 = 2
bps/Hz

Minimizing power allocation coefficient vs. transmit SNR

0.2

a min

a min = minimizing power allocation coefficient

0.18

a min approx

0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

10

20

30

40

50

= transmit SNR [dB]

Fig. 3. amin approximation compared to actual simulated value; R0 = 2
bps/Hz, β = 2

is clear that the union-outage performance using CA-NOMA
is better than the performance of OMA and regular NOMA.
VI. C ONCLUSION

AND

F UTURE W ORK

Using the two candidate technologies of NOMA and
caching at the user terminals is demonstrated to help a system
improve probability of achieving the mimimum QoS rate
required by a system. Using these results, it is clear that any
NOMA system can take advantage of users that possess caches
of content of other users in the system. The resulting CANOMA system can further be investigated to determine how
the power allocation coefficients are affected when the number
of simultaneous NOMA users is more than 2. Furthermore,
performance of CA-NOMA can be improved upon by fundamentally determining the power allocation coefficients of users
by utilizing their channel gains in their derivation similar to
[6].
The benefits of CA-NOMA can be extended to MIMO
systems, where a users in different clusters (and thus allocated

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