# US 11,451,419 B2 · Integrating volterra series model and deep neural networks to equalize nonlinear power amplifiers

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- Number: US 11,451,419 B2
- Document ID: US11451419B2
- Status: Granted
- Kind: B2
- Application number: 17/234,102
- Filed: 2021-04-19
- Published: 2021-08-05
- Granted: 2022-09-20
- Earliest claimed: 2019-03-15
- Assignees: The Research Foundation for The State University
- Applicants: The Research Foundation for The State University of New York
- Inventors: Xiaohua Li; Robert Thompson
- Web page: https://patentgrep.com/patent/US11451419B2

## CPC

- H04L 25/03165: {using neural networks}
- G06N 3/04: Architecture, e.g. interconnection topology
- G06N 3/045: Combinations of networks
- G06N 3/0464: Convolutional networks [CNN, ConvNet]
- G06N 3/08: Learning methods
- G06N 3/09: Supervised learning
- H03F 1/32: Modifications of amplifiers to reduce non-linear distortion (by negative feedback H03F1/34)
- H03F 3/20: Power amplifiers, e.g. Class B amplifiers, Class C amplifiers (H03F3/26 - H03F3/30 take precedence)
- H04B 1/16: Circuits
- G06N 3/044: Recurrent networks, e.g. Hopfield networks
- G06N 3/048: Activation functions
- H03F 2200/451: the amplifier being a radio frequency amplifier
- H04L 25/0224: {using sounding signals}
- H04L: TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION (arrangements common to telegraphic and telephonic communication H04M)
- G06N: COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
- H03F: AMPLIFIERS
- H04B: TRANSMISSION

## Abstract

The nonlinearity of power amplifiers (PAs) has been a severe constraint in performance of modern wireless transceivers. This problem is even more challenging for the fifth generation (5G) cellular system since 5G signals have extremely high peak to average power ratio. Non-linear equalizers that exploit both deep neural networks (DNNs) and Volterra series models are provided to mitigate PA nonlinear distortions. The DNN equalizer architecture consists of multiple convolutional layers. The input features are designed according to the Volterra series model of nonlinear PAs. This enables the DNN equalizer to effectively mitigate nonlinear PA distortions while avoiding over-fitting under limited training data. The non-linear equalizers demonstrate superior performance over conventional nonlinear equalization approaches.

## Claims

### Claim 1

A distortion-compensating processor, comprising:
an input port configured to receive a distorted signal y(n) representing information x(n) distorted by a channel h;
at least one automated processor, configured to:
decompose the distorted signal y(n) as a Volterra series expansion
y
⁡
(
n
)
=
∑
d
=
0
D
∑
k
=
0
P
b
k
⁢
d
⁢
x
⁡
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
;
and
implement an equalizer operating on separate terms of the Volterra series expansion to produce an output z(n) having reduced distortion with respect to the distorted signal y(n),
the equalizer comprising a deep neural network comprising a plurality of neural network hidden layers trained with respect to distortion of the channel h to provide an input-output response and
z
⁡
(
n
)
=
∑
k
=
1
P
∑
d
1
=
0
D
…
⁢
∑
d
k
=
0
D
f
d
1
,
…
,
d
k
⁢
∏
i
=
1
k
r
⁡
(
n
-
d
i
)
;
and
an output port configured to present the output,
wherein:
r(n) is a response of the channel h to the nonlinearly distorted radio frequency signal y(n);
p is a respective nonlinearity order;
d is a memory depth parameter;
D is a total memory length;
P is a total nonlinearity order;
k is a nonlinear order; and
bkd are nonlinear response parameters.

### Claim 2 (depends on 1)

The distortion-compensating processor according to claim 1, wherein the distorted signal is received from a radio receiver.

### Claim 3 (depends on 1)

The distortion-compensating processor according to claim 1, wherein the distorted signal is distorted by amplification by a radio frequency power amplifier and transmission through a radio frequency communication channel.

### Claim 4 (depends on 1)

The distortion-compensating processor according to claim 1, wherein:
the undistorted radio frequency signal x(n) is distorted by an analog process to produce the nonlinearly distorted radio frequency signal y(n), which passes through the channel h;
r
⁡
(
n
)
=
∑
ℓ
=
0
L
h
ℓ
⁢
y
⁡
(
n
-
ℓ
)
+
v
⁡
(
n
)
is a response signal sequence, wherein r(n) is stacked together into M+1 dimensional vectors r(n)=[r(n), . . . , r(n−M)]T, where (⋅)T denotes transpose, such that r(n)=HG(n)×(n)+v(n);
is a set of finite-impulse response channel coefficients;
is an equalization delay;
v(n) is an additive white Gaussian noise component signal sequence;
H is an (M+1)×(M+L+1) dimensional channel matrix
H
=
[
h
o
…
h
L
⋱
⋱
h
o
…
h
L
]
;
G(n)=diag{Vy(n)ejψy(n), . . . , Vn(y-M-L)ejψy(n-M-L)} is an (M+L+1)×(M+L+1) dimensional diagonal matrix of the nonlinear responses,
x(n)=[x(n), . . . ,x(n−M−L)]T, and
v(n)=[v(n), . . . ,v(n−M)]T;
fT=G′(n)[ƒ0, . . . , ƒM], where [ƒ0, . . . , ƒM]H≈[0, . . . , 1, . . . , 0] is computed by the equalizer to equalize the channel h;
G
′
(
n
)
=
1
V
y
⁡
(
n
-
d
)
⁢
e
-
j
⁢
ψ
y
⁡
(
n
-
d
)
is computed by the equalizer to equalize the analog process;
{circumflex over (r)}(n) is a resulting sequence after linear channel equalization;
z(n)=ƒT r(n)≈x(n−d) represents the output with equalization delay d; and
z
⁡
(
n
)
=
∑
d
=
0
D
∑
p
=
0
P
g
k
⁢
d
⁢
r
^
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
is a Volterra series model of the equalizer with coefficients gkd such that z(n)≈x(n−) for some equalization delay ,
wherein:
coefficients gkd are determined according to
min
{
g
kd
}
∑
n
=
L
N
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
L
)
-
∑
d
=
0
D
∑
k
=
1
P
g
k
⁢
d
⁢
r
^
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
❘
"\[RightBracketingBar]"
2
with training symbols x(n) and received samples {circumflex over (r)}(n),
z(n)=arg min∀x(n)|{circumflex over (r)}(n)−Vyejψyx(n)|2 is a maximum likelihood estimation operator for a nonlinear equalization output signal sequence having reduced distortion,
Vy is an amplitude of a signal sequence y(n),
ψy is a phase change of the signal sequence y(n); and
the deep neural network equalizer is trained to determine the channel coefficients , analog process responses Vy, ψy, and the channel equalizer ƒT.

### Claim 5 (depends on 4)

The distortion-compensating processor according to claim 4, wherein:
vector
⁢
a
=
[
g
0
⁢
0
,
g
0
⁢
1
,
…
,
g
P
⁢
D
]
T
,
vector
⁢
x
=
[
x
⁡
(
0
)
;
…
,
x
⁡
(
N
-
L
)
]
T
,
vector
⁢
B
=
[
r
^
(
L
)
r
^
(
L
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
L
)
❘
"\[RightBracketingBar]"
…
r
^
(
L
-
D
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
L
-
D
)
❘
"\[RightBracketingBar]"
P
-
1
⋮
⋮
r
^
(
N
)
r
^
(
N
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
N
)
❘
"\[RightBracketingBar]"
…
r
^
(
N
-
D
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
N
-
D
)
❘
"\[RightBracketingBar]"
P
-
1
]
,
and
⁢
min
{
g
kd
}
∑
n
=
L
N
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
L
)
-
∑
d
=
0
D
∑
k
=
1
P
g
k
⁢
d
⁢
r
^
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
❘
"\[RightBracketingBar]"
2
≈
min
a

x
-
Ba

2
,
having solution a=B+x, where B+=(BHB)−1B is the pseudo-inverse of the matrix B.

### Claim 6 (depends on 1)

The distortion-compensating processor according to claim 1, wherein the Volterra series expansion comprises at least fifth order terms, and the deep neural network comprises at least two convolutional network layers.

### Claim 7 (depends on 1)

The distortion-compensating processor according to claim 1, wherein the deep neural network comprises at least three one-dimensional layers, each layer having at least 10 feature maps, and a fully connected layer subsequent to the at least three layers.

### Claim 8 (depends on 1)

The distortion-compensating processor according to claim 1, wherein the distorted signal comprises a transmitted orthogonal frequency multiplexed radio frequency signal.

### Claim 9 (depends on 1)

The distortion-compensating processor according to claim 1, further comprising a demodulator, configured to demodulate the output as the set of symbols representing the information.

### Claim 10

A method of compensating for a distortion, comprising:
receiving a distorted signal y(n) representing information x(n) distorted and communicated through a channel h;
decomposing the distorted signal y(n) with a Volterra processor to produce a Volterra series expansion of form
y
⁡
(
n
)
=
∑
d
=
0
D
∑
k
=
0
P
b
k
⁢
d
⁢
x
⁡
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
;
and
equalizing the distorted signal y(n) with an automated nonlinear equalizer, comprising a deep neural network having a plurality of neural network hidden layers trained with respect to distortion of the channel h to have an input-output response
z
⁡
(
n
)
=
∑
k
=
1
P
∑
d
1
=
0
D
…
⁢
∑
d
k
=
0
D
f
d
1
,
⁢
…
,
d
k
⁢
∏
i
=
1
k
r
⁡
(
n
-
d
i
)
,
which receives the Volterra series expansion, and produces an output z(n) having reduced distortion with respect to the distorted signal y(n),
wherein:
r(n) is a response of channel h to the nonlinearly distorted radio frequency signal y(n);
p is a respective nonlinearity order;
d is a memory depth parameter;
D is a total memory length;
P is a total nonlinearity order;
k is a nonlinear order; and
bkd are nonlinear response parameters.

### Claim 11 (depends on 10)

The method according to claim 10, wherein the input comprises a radio frequency orthogonal frequency multiplexed signal amplified and distorted by a radio frequency power amplifier, received though a radio receiver.

### Claim 12 (depends on 10)

The method according to claim 10, further comprising:
computing a maximum likelihood estimation for a nonlinear equalization z(n)=arg min∀x(n)|({circumflex over (r)}(n)−Vyejψyx(n)|2, to produce the output z(n)=ƒT r(n)≈x(n−d), with equalization delay d;
training the deep neural network to determine the channel coefficients of the channel h, having a channel amplitude response Vy, a channel phase response ψy, and a channel equalization response ƒT;
approximating G′(n) with a Volterra series model
z
⁡
(
n
)
=
∑
d
=
0
D
∑
k
=
0
P
g
k
⁢
d
⁢
r
^
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
,
with coefficients gkd designed such that z(n)≈x(n−) for equalization delay , and estimated according to
min
{
g
kd
}
∑
n
=
L
N
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
L
)
-
∑
d
=
0
D
∑
k
=
1
P
g
k
⁢
d
⁢
r
^
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
❘
"\[RightBracketingBar]"
2
,
with training symbols x(n) and received samples {circumflex over (r)}(n);
wherein:
channel h, produces response
r
⁡
(
n
)
=
∑
ℓ
=
0
L
h
ℓ
⁢
y
⁡
(
n
-
ℓ
)
+
v
⁡
(
n
)
from y(n);
is a set of finite-impulse response channel coefficients;
is an equalization delay;
v(n) is an additive white Gaussian noise component;
r(n) is stacked together into M+1 dimensional vectors r(n)=[r(n), . . . , r(n−M)]T, where (⋅)T denotes transpose, such that r(n)=HG(n)×(n)+v(n);
H is an (M+1)×(M+L+1) dimensional channel matrix
H
=
[
h
o
…
h
L
⋱
⋱
h
o
…
h
L
]
;
G(n)=diag{Vy(n)ejψy(n), . . . , Vn(y-M-L)ejψy(n-M-L)} is an (M+L+1)×(M+L+1) dimensional diagonal matrix which consists of the nonlinear responses, x(n)=[x(n), . . . , x(n−M−L)]T, and v(n)=[v(n), . . . , v(n−M)]T;
fT=G′(n)[ƒ0, . . . , ƒM], where [ƒ0, . . . , ƒM]H≈[0, . . . , 1, . . . , 0] represents a channel equalization;
G
′
(
n
)
=
1
V
y
⁡
(
n
-
d
)
⁢
e
-
j
⁢
ψ
y
⁡
(
n
-
d
)
represents an analog processor equalization; and
{circumflex over (r)}(n) is a linearized representation of r(n).

### Claim 13 (depends on 12)

The method according to claim 12, wherein:
vector
⁢
a
=
[
g
0
⁢
0
,
g
0
⁢
1
,
…
,
g
P
⁢
D
]
T
,
vector
⁢
x
=
[
x
⁡
(
0
)
;
…
,
x
⁡
(
N
-
L
)
]
T
,
vector
⁢
B
=
[
r
^
(
L
)
r
^
(
L
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
L
)
❘
"\[RightBracketingBar]"
…
r
^
(
L
-
D
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
L
-
D
)
❘
"\[RightBracketingBar]"
P
-
1
⋮
⋮
r
^
(
N
)
r
^
(
N
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
N
)
❘
"\[RightBracketingBar]"
…
r
^
(
N
-
D
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
N
-
D
)
❘
"\[RightBracketingBar]"
P
-
1
]
,
and
⁢
min
{
g
kd
}
∑
n
=
L
N
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
L
)
-
∑
d
=
0
D
∑
k
=
1
P
g
k
⁢
d
⁢
r
^
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
^
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
❘
"\[RightBracketingBar]"
2
≈
min
a

x
-
Ba

2
,
having solution a=B+x, where B+=(BHB)−1B is the pseudo-inverse of the matrix B.

### Claim 14 (depends on 10)

The method according to claim 10, wherein:
the Volterra series expansion comprises at least fifth order terms; and
the deep neural network comprises at least three layers, each layer having at least 10 feature maps, and a fully connected layer subsequent to the at least three layers.

### Claim 15 (depends on 10)

The method according to claim 10, further comprising demodulating the output z(n) as the set of information symbols.

### Claim 16

A non-linear distortion-compensating processor,
comprising:
an input configured to receive a non-linearly distorted radio frequency signal r(n) representing information x(n) distorted by a radio frequency communication system to produce a distorted signal y(n) communicated through a channel h;
at least one automated processor, configured to:
decompose the non-linearly distorted radio frequency signal r(n) received from the channel h as a Volterra series expansion
y
⁡
(
n
)
=
∑
d
=
0
D
∑
k
=
0
P
b
k
⁢
d
⁢
x
⁡
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
;
process the Volterra series expansion with a deep neural network comprising a plurality of neural network hidden layers trained with respect to the distortion of the radio frequency communication system comprising the non-linear distortion and channel distortion, to equalize the distortion, the process having an input-output response
z
⁡
(
n
)
=
∑
k
=
1
P
∑
d
1
=
0
D
…
⁢
∑
d
k
=
0
D
⁢
f
d
1
,
⁢
…
,
d
k
⁢
∏
i
=
1
k
r
⁡
(
n
-
d
i
)
;
and
demodulate z(n) to extract the information x(n) modulated in the non-linearly distorted radio frequency signal r(n),
wherein:
p is a respective nonlinearity order;
d is a memory depth parameter;
D is a total memory length;
P is a total nonlinearity order;
k is a nonlinear order; and
bkd are nonlinear response parameters.

### Claim 17 (depends on 16)

The non-linear distortion-compensating processor according to claim 16, wherein the non-linearly distorted radio frequency signal is an orthogonal frequency multiplexed signal which is distorted by amplification by a power amplifier and transmission over the communication channel h.

### Claim 18 (depends on 16)

The non-linear distortion-compensating processor according to claim 16, wherein the deep neural network comprises at least three convolutional network layers, each layer having at least 10 feature maps, and a fully connected layer subsequent to the at least three layers.

### Claim 19 (depends on 16)

The non-linear distortion-compensating processor according to claim 16, wherein the deep neural network is trained with data comprising symbol-specific pairs of the information x(n) and the corresponding non-linearly distorted radio frequency signal r(n) representing the non-linearly distorted information y(n) communicated through the channel h.

### Claim 20 (depends on 16)

The non-linear distortion-compensating processor according to claim 16, wherein the response r(n) of the channel h is
r
⁡
(
n
)
=
∑
ℓ
=
0
L
h
ℓ
⁢
y
⁡
(
n
-
ℓ
)
+
v
⁡
(
n
)
;
and
the at least one automated processor is further configured to:
compute a maximum likelihood estimation for a nonlinear equalization z(n)=arg min∀x(n)|{circumflex over (r)}(n)−Vyejψy x(n)|2, to produce output z(n)=ƒT r(n)≈x(n−d), with equalization delay d;
train the deep neural network equalizer to determine a set of finite-impulse response channel coefficients of the channel h, having a channel amplitude response Vy, a channel phase response ψy, and a channel equalization response ƒT; and
approximate G′(n) with a Volterra series model
z
⁡
(
n
)
=
∑
d
=
0
D
∑
k
=
0
P
g
k
⁢
d
⁢
r
ˆ
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
ˆ
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
,
with gkd designed such that z(n)≈x(n−) for equalization delay , and coefficients gkd estimated according to
min
{
g
kd
}
∑
n
=
L
N
❘
"\[LeftBracketingBar]"
x
⁡
(
n
-
L
)
-
∑
d
=
0
D
∑
k
=
1
P
g
k
⁢
d
⁢
r
ˆ
(
n
-
d
)
⁢
❘
"\[LeftBracketingBar]"
r
ˆ
(
n
-
d
)
❘
"\[RightBracketingBar]"
k
-
1
❘
"\[RightBracketingBar]"
2
,
with training symbols x(n) and received samples {circumflex over (r)}(n);
wherein:
is an equalization delay;
v(n) is an additive white Gaussian noise component;
r(n) is stacked together into M+1 dimensional vectors r(n)=[r(n), . . . , r(n−M)]T, where (⋅)T denotes transpose, such that r(n)=HG(n)×(n)+v(n);
H is an (M+1)×(M+L+1) dimensional channel matrix
H
=
[
h
o
…
h
L
⋱
⋱
h
o
…
h
L
]
;
G(n)=diag {Vy(n)ejψy(n), . . . , Vn(y-M-L)ejψy(n-m-L)} is an (M+L+1)×(M+L+1) dimensional diagonal matrix which consists of the nonlinear responses, x(n)=[x(n), . . . , x(n−M−L)]T, and v(n)=[v(n), . . . , v(n−M)]T;
fT=G′(n)[ƒ0, . . . , ƒM], where [ƒ0, . . . , ƒM]H≈[0, . . . , 1, . . . , 0] represents a channel equalization, and
G
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V
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j
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ψ
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represents an analog processor equalization; and
{circumflex over (r)}(n) is a linearized representation of r(n).