Ultra-wideband frequency position modulation using nonlinear compressed sensing
A Frequency Position Modulation system for encoding signals for transmission. A signal's discrete frequency support is used to represent symbols. The signal can be non-uniformly spread over many GHz of instantaneous bandwidth, resulting in a communications system that is resilient to interference and difficult to intercept. The FPM symbols are recovered using adaptive projections that use an analog polynomial nonlinearity paired with an analog-to-digital converter that is sampling at a rate at that is only a fraction of the instantaneous bandwidth of the signal. In the presence of partial band interference, nonlinearities generated by the transmitter of are exploited by the receiver to help unambiguously recover tones that could otherwise be lost. The nonlinearities are generated by driving the power amplifier of the transmitter into saturation to induce distortions at a desired level.
1. A method for modulating a digital signal for transmission, comprising:
generating a plurality of tones, each tone being centered at a different frequency,
generating at least second and third order intermodulation terms by driving the plurality of tones through a power amplifier at a power level that drives the power amplifier into saturation,
said tones and said second and third order intermodulation terms being the elements of a frequency position modulation constellation,
wherein the symbol s(t) to be transmitted is
s
(
t
)
=
(
∑
ω
m
∈
U
k
cos
(
ω
m
t
)
)
︸
s
^
(
t
)
+
∑
p
=
1
P
g
p
(
s
^
(
t
)
)
for
0
≤
t
<
T
s
,
wherein the nonlinearity-generated intermodulation terms are defined by g p (ŝ(t) according to
∑
p
=
1
P
g
p
(
s
^
(
t
)
)
=
∑
p
=
1
P
∫
τ
1
…
∫
τ
p
h
(
τ
1
,
…
,
τ
p
)
∏
i
=
1
p
s
^
(
t
-
τ
i
)
ⅆ
τ
1
…
ⅆ
τ
p
.
where h(Σ 1 , . . . , Σ p ) is the multidimensional system response of the transmitter, wherein ω m is an m-th tone from a set of tones that forms a k-th symbol U k , and T s is a symbol rate.
2. A method for modulating a digital signal for transmission, comprising:
generating a plurality of tones, each tone being centered at a different frequency;
generating at least second and third order intermodulation terms by driving the plurality of tones through a power amplifier at a power level that drives the power amplifier into saturation, said tones and said second and third order intermodulation terms being the elements of a frequency position modulation constellation, and
receiving the signal, filtering and mixing the signal down to baseband according to
y
(
t
)
=
[
(
h
CMB
(
t
)
*
s
(
t
)
)
∑
ω
n
∈
U
LO
cos
(
ω
n
t
)
]
*
h
AA
(
t
)
+
n
(
t
)
using local oscillator frequencies ω n ∈U LO with U LO being defined by
U
LO
=
⋃
i
=
1
N
pos
2
ω
L
+
BW
N
pos
(
2
i
-
1
)
,
where BW is the contiguous bandwidth over which frequency position modulation symbols are transmitted, N pos is the number of tones in one of the frequency position modulation symbols, ω L is the lowest transmitted frequency within the bandwidth BW, h AA (t) represents an anti-aliasing filter, *represents the convolution operator, y(t) is a reconstructed signal, s(t) is the symbol to be transmitted, h CMB (t) represents a comb filter, ω N is an n th local oscillator frequency within the set of U LO local oscillator frequencies, and n(t) is a noise term representing receiver noise.
3. The method according to claim 2 , wherein the frequency support of h CMB (t) is permuted on a symbol-by-symbol basis using a frequency hopping pattern known to both the transmitter and receiver.
4. A transmitter for modulating a digital signal for transmission and for transmitting the signal, comprising:
a tone generation module adapted to generate a plurality of tones, each tone being centered at a different frequency; and
a power amplifier configured to transmit and amplify the tones,
wherein in operation, at least second and third order intermodulation terms are generated by first applying a digital nonlinear transform and then driving the plurality of tones through the power amplifier at a power level that drives the power amplifier into saturation,
said tones and said second and third order intermodulation terms being the elements of a frequency position modulation constellation,
wherein the symbol s(t) to be transmitted is
s
(
t
)
=
(
∑
ω
m
∈
U
k
cos
(
ω
m
t
)
)
︸
s
^
(
t
)
+
∑
p
=
1
P
g
p
(
s
^
(
t
)
)
for
0
≤
t
<
T
s
.
wherein the nonlinearity-generated intermodulation terms are defined by g p (ŝ(t) according to
∑
p
=
1
P
g
p
(
s
^
(
t
)
)
=
∑
p
=
1
P
∫
τ
1
…
∫
τ
p
h
(
τ
1
,
…
,
τ
p
)
∏
i
=
1
p
s
^
(
t
-
τ
i
)
ⅆ
τ
1
…
ⅆ
τ
p
,
wherein h(τ 1 , . . . , τ p ) is the multidimensional system response of the digital nonlinear transform, ω m is an m-th tone from a set of tones that forms a k-th symbol U k , and T s is a symbol rate.
5. A transmitter for modulating a digital signal for transmission and for transmitting the signal, comprising:
a tone generation module adapted to generate a plurality of tones, each tone being centered at a different frequency; and
a power amplifier configured to transmit and amplify the tones,
wherein in operation, at least second and third order intermodulation terms are generated by first applying a digital nonlinear transform and then driving the plurality of tones through the power amplifier at a power level that drives the power amplifier into saturation,
said tones and said second and third order intermodulation terms being the elements of a frequency position modulation constellation,
in combination with a receiver configured to receive and demodulate the transmitted signal,
wherein in operation, the receiver receives the transmitted signal, and filters and mixes the transmitted signal down to baseband according to
y
(
t
)
=
[
(
h
CMB
(
t
)
*
s
(
t
)
)
∑
ω
n
∈
U
LO
cos
(
ω
n
t
)
]
*
h
AA
(
t
)
+
n
(
t
)
using local oscillator frequencies ω n ∈U LO with U LO being defined by
U
LO
=
⋃
i
=
1
N
pos
2
ω
L
+
BW
N
pos
(
2
i
-
1
)
,
where BW is the contiguous bandwidth over which frequency position modulation symbols are transmitted, N pos is the number of tones in one of the frequency position modulation symbols, ω L is the lowest transmitted frequency within the bandwidth BW, h AA (t) represents an anti-aliasing filter, *represents the convolution operator, y(t) is a reconstructed signal, s(t) is the symbol to be transmitted, h CMB (t) represents a comb filter, ω N is an n th local oscillator frequency within the set of U LO local oscillator frequencies, and n(t) is a noise term representing receiver noise.
6. The transmitter according to claim 5 , wherein the frequency support of h CMB (t) is permuted on a symbol-by-symbol basis using a frequency hopping pattern known to both the transmitter and receiver.