U.S. patent application number 15/923835 was filed with the patent office on 2018-07-26 for digital communication receiver using partial knowledge of the channel state information.
The applicant listed for this patent is Khalifa University of Science and Technology. Invention is credited to Arafat Jamil Al-Dweik, Mohammed Al-Mualla, Youssef Iraqi.
Application Number | 20180212806 15/923835 |
Document ID | / |
Family ID | 57352068 |
Filed Date | 2018-07-26 |
United States Patent
Application |
20180212806 |
Kind Code |
A1 |
Al-Dweik; Arafat Jamil ; et
al. |
July 26, 2018 |
DIGITAL COMMUNICATION RECEIVER USING PARTIAL KNOWLEDGE OF THE
CHANNEL STATE INFORMATION
Abstract
The present invention proposes a demodulator device, a receiver
and a demodulation method for M-ary amplitude shift keying systems
(MASK) that requires partial knowledge of the CSI, namely, the
channel attenuation coefficient. Therefore, the new demodulator,
receiver and demodulation method do not require the knowledge of
the channel phase shift. Consequently, no complicated channel
estimation techniques are required, and the system will be very
robust to the system impairments such as phase noise, I-Q
imbalance, etc. In this sense, the new technique is denoted as
semi-coherent demodulation (SCD). To reduce the complexity of the
new SCD, a suboptimal demodulator is derived which has much lower
complexity than the optimal while providing almost the same error
probability.
Inventors: |
Al-Dweik; Arafat Jamil; (Abu
Dhabi, AE) ; Iraqi; Youssef; (Abu Dhabi, AE) ;
Al-Mualla; Mohammed; (Abu Dhabi, AE) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Khalifa University of Science and Technology |
Abu Dhabi |
|
AE |
|
|
Family ID: |
57352068 |
Appl. No.: |
15/923835 |
Filed: |
March 16, 2018 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
15350948 |
Nov 14, 2016 |
9954699 |
|
|
15923835 |
|
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|
|
14816576 |
Aug 3, 2015 |
9509538 |
|
|
15350948 |
|
|
|
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H04L 5/0048 20130101;
H04L 27/06 20130101; H04B 1/10 20130101 |
International
Class: |
H04L 27/06 20060101
H04L027/06; H04L 5/00 20060101 H04L005/00; H04B 1/10 20060101
H04B001/10 |
Claims
1. A digital communication receiver for detecting signals
transmitted by a digital transmitter through a communication
channel, the channel having a channel attenuation |h| and a channel
phase shift having a multipath fading effect on the transmitted
signals, the receiver comprising a demodulator configured to
demodulate signals received by the receiver using channel
coefficients representing the channel attenuation only without any
knowledge of the channel phase shift.
2. The digital communication receiver as claimed in claim 1,
wherein the demodulator is robust to phase noise, large phase
variations and time-varying I-Q imbalance.
3. The digital communication receiver as claimed in claim 2,
wherein the demodulator uses a M-ary amplitude shift keying
technique, the transmitted signals being modulated by the
transmitter using said same technique before transmission using a
modulation order M equal or superior to 2.
4. The digital communication receiver as claimed in claim 3,
wherein the detected signals have a symbol Error Rate (SER)
intermediate in terms of performance between a coherent detection
and a non-coherent detection assuming a same spectral
efficiency.
5. The digital communication receiver as claimed in claim 4,
wherein the demodulator is less complex than a coherent
demodulator, and wherein the SER performance of the detected
signals using the demodulator is substantially similar to a SER
performance obtained using a coherent demodulator.
6. The digital communication receiver as claimed in claim 5,
wherein the channel is a multi-path fading channel.
7. A computer-implemented demodulation method comprising: receiving
from a digital communication receiver signals transmitted by a
digital transmitter through a communication channel, the channel
having a channel attenuation |h| and a channel phase shift having a
multipath fading effect on the transmitted signals; and
demodulating the signals received by the receiver using only
channel coefficients representing the channel attenuation without
any knowledge of the channel phase shift for detecting the
transmitted signals.
8. The demodulation method as claimed in claim 7, wherein the
demodulation method is robust to phase noise, large phase
variations and time-varying I-Q imbalance.
9. The demodulation method as claimed in claim 8, wherein the
demodulation method uses a M-ary amplitude shift keying technique,
the transmitted signals being modulated by the transmitter using
said same technique before transmission using a modulation order M
equal or superior to 2.
10. The demodulation method as claimed in claim 9, wherein the
detected signals have a symbol Error Rate (BER) intermediate in
terms of performance between a coherent detection and a
differentially coherent detection assuming a same spectral
efficiency.
11. The demodulation method as claimed in claim 10, wherein the
demodulation method is less complex than a coherent demodulation,
and wherein the SER performance of the detected signals using the
demodulation method is substantially similar to a SER performance
obtained using a coherent demodulation.
12. The demodulation method as claimed in claim 11, wherein the
channel is a multi-path fading channel.
13. A demodulator device for detecting signals transmitted by a
digital transmitter to a digital receiver through a communication
channel, the channel having a channel attenuation |h| and a channel
phase shift having a multipath fading effect on the transmitted
signals, the demodulator device being configured to communicate
with the digital receiver for demodulating signals received by the
receiver using channel coefficients representing the channel
attenuation only without any knowledge of the channel phase
shift.
14. The demodulator device as claimed in claim 13, wherein the
demodulation is robust to phase noise, large phase variations and
time-varying I-Q imbalance.
15. The demodulator device as claimed in claim 14, wherein the
demodulator device uses a M-ary amplitude shift keying technique,
the transmitted signals being modulated by the transmitter using
said same technique before transmission using a modulation order M
superior to 2.
16. The demodulator device as claimed in claim 15, wherein the
detected signals have a Bit Error Rate (BER) intermediate in terms
of performance between a coherent detection and a differentially
coherent detection assuming a same spectral efficiency.
17. The demodulator device as claimed in claim 16, wherein the
demodulator device is less complex than a coherent demodulator, and
wherein the BER performance of the detected signals using the
demodulator is substantially similar to a BER performance obtained
using a coherent demodulator.
18. The demodulator device as claimed in claim 17, wherein the
channel is a multi-path fading channel.
Description
CROSS REFERENCE TO RELATED APPLICATION
[0001] This application is a continuation of U.S. patent
application Ser. No. 15/350,948 filed on Nov. 14, 2016 which is a
continuation of U.S. patent application Ser. No. 14/816,576 filed
on Aug. 3, 2015 content of which is hereby incorporated by
reference.
FIELD OF THE INVENTION
[0002] The present invention generally relates to the field of
digital communications, and more particularly to a communication
system, digital receiver, a demodulator and a method of
demodulation using partial knowledge of the channel state
information.
BACKGROUND OF THE INVENTION
[0003] In the literature, there are three types of demodulation
techniques that are currently used in various wired and wireless
communications techniques [1], namely, coherent, non-coherent and
differentially coherent, which are denoted as CD, NCD and DCD,
respectively. Each of the mentioned demodulation techniques is used
for particular applications based on the channel and system
resources and requirements. Moreover, selecting a particular
modulation/demodulation method enables to trade-off the error
performance, complexity and spectral efficiency. For example, CD
provides low error probability given that the channel state
information (CSI) is known accurately at the receiver side.
However, accurate knowledge of the CSI requires invoking channel
estimation techniques, which might require high complexity signal
processing techniques and might affect the system spectral
efficiency as well. On the contrary, NCD does not require any
information about the CSI hence its a low complexity demodulator,
however the error probability is generally very high [2]. DCD is
different from CD and NCD because it requires the transmitter to
introduce memory in the transmitted sequence. The information
symbols extraction does not require the knowledge of the CSI at the
received side, however, it requires the CSI to be almost fixed over
two consecutive symbols. The probability of error and complexity
for the DCD is generally in between CD and NCD. The main
disadvantage of DCD is that it requires differential encoding at
the transmitter side, and the receiver should know the phase of the
first symbol in the received sequence [3], hence it requires some
sort of pilot symbols which degrades the spectral efficiency.
Moreover, it is very sensitive to phase noise and I-Q imbalance
impairments [4]. Therefore, DCD is not suitable for applications
where the received signal suffers from phase noise, large phase
variations, or time-varying I-Q imbalance.
SUMMARY OF THE INVENTION
[0004] Brief Explanation
[0005] The present invention presents a new class of demodulation
for M-ary amplitude shift keying systems (MASK) that requires
partial knowledge of the CSI, namely, the channel attenuation
coefficient. Therefore, the new demodulator does not require the
knowledge of the channel phase shift. Consequently, no complicated
channel estimation techniques are required, and the system will be
very robust to the system impairments such as phase noise, I-Q
imbalance, etc. In this sense, the new technique is denoted as
semi-coherent demodulation (SCD). To reduce the complexity of the
new SCD, a suboptimal demodulator is derived which has much lower
complexity than the optimal while providing almost the same error
probability.
[0006] Generally speaking, the CSI is composed of real and
imaginary components, which can be expressed as h=|h|
e.sup.j.theta., where |h| corresponds to the channel attenuation
and .theta. is the phase shift. In the proposed receiver, only |h|
is required to detect the transmitted symbols with low probability
of error. The proposed receiver can be used to increase the
spectral efficiency of most digital communications receiver and/or
reduce their error probability.
[0007] Main Features:
[0008] This patent describes an efficient new digital
communications receiver that has never been considered before. The
main features of the new receiver are: [0009] Low complexity,
because it does not require estimating the phase of the carrier at
the receiver side. [0010] Provides much better error performance as
compared to the non-coherent receivers. [0011] It is based on a
novel demodulation scheme that was never considered by other
researchers. [0012] Can be configured to provide high spectral
efficiency when higher order modulation is used. [0013] Enables the
design engineers to trade off complexity and error performance
based on the system requirements. [0014] Very robust to harsh
channels with severe phase noise, I-Q imbalance and carrier
frequency offsets. [0015] Can be incorporated in coherent systems
to provide low complexity blind channel estimation.
[0016] Benefits: [0017] Low complexity is an essential requirement
for most modern communication systems. Because most
state-of-the-art communications systems support mobility, most of
the communicating devices will be supported by a limited size and
energy batteries. Consequently, saving the processing power can
contribute significantly to prolong the battery life. [0018] High
spectral efficiency is one of the most critical parameters that
determine the network capacity. The demand for additional spectrum
has witnessed a substantial surge in the past few years, and it is
expected to keep growing due to the emerging broadband
applications, and the increasing number of subscribers who require
data support on their wireless devices. Therefore, developing
spectrally efficient communication systems is indispensable. [0019]
Due to the widespread of personal wireless systems, the
communications channel has become very hostile as compared to the
channels of classical communication systems. Most of the
communications sessions currently are being held from indoor
buildings or high speed transportation tools such as trains and
vehicles. In such environments, performing coherent detection is
extremely challenging because accurate channel estimation and
synchronization in such environments is infeasible. The proposed
system does not require phase estimation or synchronization, and
hence it will be very robust in severe channels. [0020] Blind
channel estimation is essential for coherent digital
communications. The new approach enables low complexity blind
channel estimation for coherent systems.
[0021] Limitations:
[0022] The bit error performance of the proposed technique is not
as good as the coherent systems in ideal channel conditions.
However such difference becomes very small in practical non-ideal
scenarios. Moreover, the performance difference can be
substantially reduced when diversity techniques are involved, which
makes the system very attractive for various applications.
[0023] Aspects of the Invention:
[0024] As a first aspect of the invention, there is provided a
digital communication receiver for detecting signals transmitted by
a digital transmitter through a communication channel, the channel
having a channel attenuation |h| and a channel phase shift having a
multipath fading effect on the transmitted signals, the receiver
comprising a demodulator configured to demodulate signals received
by the receiver using channel coefficients representing the channel
attenuation only without any knowledge of the channel phase
shift.
[0025] Preferably, the demodulator is robust to phase noise, large
phase variations and time-varying I-Q imbalance.
[0026] Preferably, the demodulator uses a M-ary amplitude shift
keying technique, the transmitted signals being modulated by the
transmitter using said same technique before transmission using a
modulation order M equal or superior to 2.
[0027] Preferably, the detected signals have a symbol Error Rate
(SER) intermediate in terms of performance between a coherent
detection and a non-coherent detection assuming a same spectral
efficiency.
[0028] Preferably, the demodulator is less complex than a coherent
demodulator, and wherein the SER performance of the detected
signals using the demodulator is substantially similar to a SER
performance obtained using a coherent demodulator.
[0029] Preferably, the channel is a multi-path fading channel. The
multi-path fading channel can be for example Rayliegh, Ricean or
Nakagami.
[0030] Preferably, the received signals have an energy .eta. and
the channel attenuation coefficients have a magnitude |h|.sup.2,
and wherein the demodulator is configured to equalize the energy of
the received signals .eta. using only the magnitude of the channel
coefficients |h|.sup.2 such that an equalized envelop of the
received signals is obtained according to the following equation in
which the multipath fading effect of the channel on the transmitted
signals is converted into an additive disturbance:
.zeta. = .eta. h 2 = v 2 h 2 = A i 2 + 1 h 2 [ ( h * A i * w + hA i
w * ) + w 2 ] . ##EQU00001## [0031] where .zeta. is: the energy
equalized signal [0032] where v is: the received signal [0033]
where A.sub.i.sup.2 is: the energy of the transmitted symbol [0034]
where w is: additive white Gaussian noise [0035] where (.)* denotes
the complex conjugate.
[0036] Preferably, the demodulator makes decisions for the
detection of transmitted signals according to the following
conditional probability distribution function (PDF):
P ( .zeta. | E i ) = ( .PSI. i + .sigma. w 4 ) .sigma. h 2 [ (
.zeta. - E i ) 2 .sigma. h 2 + 2 .PSI. i + .sigma. w 4 ] 3 / 2
##EQU00002## [0037] where .zeta. is a decision variable and where
.psi..sub.i =(.zeta.+E.sub.1.sigma..sub.w.sup.2.sigma..sub.w.sup.2
[0038] where E.sub.i is: the energy of the transmitted symbol
[0039] where .sigma..sub.w.sup.2 is: the noise variance [0040]
where .sigma..sub.h.sup.2 is: the variance of the fading
coefficients
[0041] Preferably, the demodulator makes decisions for the
detection of transmitted signals according to the following minimum
distance detector (MDD) equation:
A ^ i = arg min E i [ .zeta. - E i ] 2 , i = [ 0 , 1 , , M - 1 ] ,
##EQU00003## [0042] where A.sub.i is: the estimated symbol
amplitude [0043] where M is the modulation order.
[0044] Preferably, the MDD has a Signal to Error Ratio (SER) which
follows the following equation:
P S = 1 - 1 M + 1 M [ i = 1 M - 1 .lamda. i - i = 2 M .chi. i ] ,
##EQU00004##
where
.lamda. i = .DELTA. 1 - .delta. 2 [ i - 0.5 ] .GAMMA. 2 ( .delta. 2
[ i - 0.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 3 i + 1.5 ] .GAMMA.
+ 1 ##EQU00005## .chi. i = .DELTA. 1 + .delta. 2 [ i - 0.5 ]
.GAMMA. 2 ( .delta. 2 [ i - 1.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2
- 5 i + 3.5 ] .GAMMA. + 1 ##EQU00005.2##
where
.GAMMA. = .sigma. H 2 .sigma. w 2 E _ , and E _ = 1 M i = 0 M - 1 E
i ##EQU00006##
is the average power per symbol which is normalized to unity,
[0045] where P.sub.S is: the symbol error probability [0046] where
.chi..sub.i is: already defined above [0047] where .delta. is:
amplitude difference between adjacent symbols [0048] where .GAMMA.
is: average symbol energy
[0049] Preferably, the attenuation channel coefficients are
obtained by inserting pilot symbols within the transmitted signals
with a particular time spacing, the pilot symbols having a constant
modulus, |s|.sup.2=P, where P is a constant.
[0050] Preferably, the estimated value of the channel attenuation
magnitude |h|.sup.2 is obtained by computing {circumflex over
(.alpha.)}=.eta..sub.P/|s|.sup.2,
Where
[0051]
.eta..sub.P=|v.sub.P|.sup.2=|h|.sup.2|s|.sup.2+(h*s*w+hsw*)+|w|.su-
p.2.
[0052] As a further aspect of the invention, there is provided a
computer-implemented demodulation method comprising: [0053]
receiving from a digital communication receiver signals transmitted
by a digital transmitter through a communication channel, the
channel having a channel attenuation |h| and a channel phase shift
having a multipath fading effect on the transmitted signals; and
[0054] demodulating the signals received by the receiver using only
channel coefficients representing the channel attenuation without
any knowledge of the channel phase shift for detecting the
transmitted signals.
[0055] Preferably, the demodulation method is robust to phase
noise, large phase variations and time-varying I-Q imbalance.
[0056] Preferably, the demodulation method uses a M-ary amplitude
shift keying technique, the transmitted signals being modulated by
the transmitter using said same technique before transmission using
a modulation order M equal or superior to 2.
[0057] Preferably, the detected signals have a symbol Error Rate
(BER) intermediate in terms of performance between a coherent
detection and a differentially coherent detection assuming a same
spectral efficiency.
[0058] Preferably, the demodulation method is less complex than a
coherent demodulation, and wherein the SER performance of the
detected signals using the demodulation method is substantially
similar to a SER performance obtained using a coherent
demodulation.
[0059] Preferably, the channel is a multi-path fading channel. The
multi-path fading channel can be for example Rayliegh, Ricean or
Nakagami.
[0060] Preferably, the received signals have an energy 7.sub.7 and
the channel attenuation coefficients have a magnitude |h|.sup.2,
and wherein the demodulation method further comprises equalizing
the energy of the received signals 7.sub.7 using only the
manguitude of the channel coefficients |h|.sup.2 such that an
equalized envelop of the received signals is obtained according to
the following equation in which the multipath fading effect of the
channel on the transmitted signals is converted into an additive
disturbance:
.zeta. = .eta. h 2 = v 2 h 2 = A i 2 + 1 h 2 [ ( h * A i * w + h A
i w * ) + w 2 ] . ##EQU00007## [0061] where .zeta. is: the energy
equalized signal [0062] where v is: the received signal [0063]
where A.sub.i.sup.2 is: the energy of the transmitted symbol [0064]
where w is: additive white Gaussian noise [0065] where (.)* denotes
the complex conjugate.
[0066] Preferably, the demodulation method further comprises making
decisions for the detection of transmitted signals according to the
following conditional probability distribution function (PDF):
P ( .zeta. E i ) = ( .PSI. i + .sigma. w 4 ) .sigma. h 2 [ ( .zeta.
- E i ) 2 .sigma. h 4 + 2 .PSI. i + .sigma. w 4 ] 3 / 2
##EQU00008## [0067] where .zeta. is a decision variable and where
.PSI..sub.i=(.zeta.+E.sub.i).sigma..sub.w.sup.2.sigma..sub.h.sup.2.
[0068] where E.sub.i is: the energy of the transmitted symbol
[0069] where o.sub.w.sup.2 is: the noise variance [0070] where
.sigma..sub.h.sup.2 is: the variance of the fading coefficients
[0071] Preferably, the demodulation method further comprises making
decisions for the detection of transmitted signals according to the
following minimum distance detector (MDD) equation:
A ^ i = arg min E i [ .zeta. - E i ] 2 , i = [ 0 , 1 , , M - 1 ] ,
##EQU00009## [0072] where A.sub.i is: the estimated symbol
amplitude [0073] where M is the modulation order.
[0074] Preferably, the MDD has a Signal to Error Ratio (SER) which
follows the following equation:
P S = 1 - 1 M + 1 M [ i = 1 M - 1 .lamda. i - i = 2 M .chi. i ] ,
##EQU00010##
where
.lamda. i = 1 - .delta. 2 [ i - 0.5 ] .GAMMA. 2 ( .delta. 2 [ i -
0.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 3 i + 1.5 ] .GAMMA. + 1
##EQU00011## .chi. i = 1 + .delta. 2 [ i - 0.5 ] .GAMMA. 2 (
.delta. 2 [ i - 1.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 5 i + 3.5
] .GAMMA. + 1 . ##EQU00011.2##
where
.GAMMA. = .sigma. H 2 .sigma. w 2 E _ , and E _ = 1 M i = 0 M - 1 E
i ##EQU00012##
is the average power per symbol which is normalized to unity [0075]
where P.sub.S is: the symbol error probability [0076] where
.chi..sub.i is: already defined above [0077] where .delta. is:
amplitude difference between adjacent symbols [0078] where .GAMMA.
is: average symbol energy
[0079] Preferably, the attenuation channel coefficients are
obtained by inserting pilot symbols within the transmitted signals
with a particular time spacing, the pilot symbols having a constant
modulus, |s|.sup.2=P, where P is a constant.
[0080] Preferably, an estimated value of the channel attenuation
magnitude |h|.sup.2 is obtained by computing
.alpha.=.eta..sub.P/|s|.sup.2,
Where
[0081]
.eta..sub.P=|v.sub.P|.sup.2=|h|.sup.2|s|.sup.2+(h*s*w+hsw*)+|w|.su-
p.2.
[0082] As another aspect of the invention, there is provided a
demodulator device for detecting signals transmitted by a digital
transmitter to a digital receiver through a communication channel,
the channel having a channel attenuation |h| and a channel phase
shift having a multipath fading effect on the transmitted signals,
the demodulator device being configured to communicate with the
digital receiver for demodulating signals received by the receiver
using channel coefficients representing the channel attenuation
only without any knowledge of the channel phase shift.
[0083] Preferably, the demodulation is robust to phase noise, large
phase variations and time-varying I-Q imbalance.
[0084] Preferably, the demodulator device uses a M-ary amplitude
shift keying technique, the transmitted signals being modulated by
the transmitter using said same technique before transmission using
a modulation order M superior to 2.
[0085] Preferably, the detected signals have a Bit Error Rate (BER)
intermediate in terms of performance between a coherent detection
and a differentially coherent detection assuming a same spectral
efficiency.
[0086] Preferably, the demodulator device is less complex than a
coherent demodulator, and wherein the BER performance of the
detected signals using the demodulator is substantially similar to
a BER performance obtained using a coherent demodulator.
[0087] Preferably, the channel is a multi-path fading channel. The
multi-path fading channel can be for example Rayliegh, Ricean or
Nakagami.
[0088] Preferably, the received signals have an energy .eta. and
the channel attenuation coefficients have a magnitude |h|.sup.2,
and wherein the demodulator is configured to equalize the energy of
the received signals .eta. using only the manguitude of the channel
coefficients |h|.sup.2 such that an equalized envelop of the
received signals is obtained according to the following equation in
which the multipath fading effect of the channel on the transmitted
signals is converted into an additive disturbance:
.zeta. = .eta. h 2 = v 2 h 2 = A i 2 + 1 h 2 [ ( h * A i * w + h A
i w * ) + w 2 ] . ##EQU00013## [0089] where .zeta. is: the energy
equalized signal [0090] where v is: the received signal [0091]
where A.sub.i.sup.2 is: the energy of the transmitted symbol [0092]
where w is: additive white Gaussian noise [0093] where (.)* denotes
the complex conjugate.
[0094] Preferably, the demodulator device makes decisions for the
detection of transmitted signals according to the following
conditional probability distribution function (PDF):
P ( .zeta. E i ) = ( .PSI. i + .sigma. w 4 ) .sigma. h 2 [ ( .zeta.
- E i ) 2 .sigma. h 4 + 2 .PSI. i + .sigma. w 4 ] 3 / 2
##EQU00014## [0095] where .zeta. is a decision variable and where
.PSI..sub.i=(.zeta.+E.sub.i).sigma..sub.w.sup.2.sigma..sub.h.sup.2.
[0096] where E.sub.1 is: the energy of the transmitted symbol
[0097] where .sigma..sub.w.sup.2 is: the noise variance [0098]
where .sigma..sub.h.sup.2 is: the variance of the fading
coefficients
[0099] Preferably, the demodulator device makes decisions for the
detection of transmitted signals according to the following minimum
distance detector (MDD) equation:
A ^ i = arg min E i [ .zeta. - E i ] 2 , i = [ 0 , 1 , , M - 1 ] ,
##EQU00015## [0100] where A.sub.i is: the estimated symbol
amplitude [0101] where M is the modulation order.
[0102] Preferably, the MDD has a Signal to Error Ratio (SER) which
follows the following equation:
P S = 1 - 1 M + 1 M [ i = 1 M - 1 .lamda. i - i = 2 M .chi. i ] ,
##EQU00016##
where
.lamda. i = 1 - .delta. 2 [ i - 0.5 ] .GAMMA. 2 ( .delta. 2 [ i -
0.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 3 i + 1.5 ] .GAMMA. + 1
##EQU00017## .chi. i = 1 + .delta. 2 [ i - 0.5 ] .GAMMA. 2 (
.delta. 2 [ i - 1.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 5 i + 3.5
] .GAMMA. + 1 . ##EQU00017.2##
where
.GAMMA. = .sigma. H 2 .sigma. w 2 E _ , and E _ = 1 M i = 0 M - 1 E
i ##EQU00018##
is the average power per symbol which is normalized to unity [0103]
where P.sub.S is: the symbol error probability [0104] where
.chi..sub.i is: already defined above [0105] where .delta. is:
amplitude difference between adjacent symbols [0106] where .GAMMA.
is: average symbol energy
[0107] Preferably, the attenuation channel coefficients are
obtained by inserting pilot symbols within the transmitted signals
with a particular time spacing, the pilot symbols having a constant
modulus, |s|.sup.2=P, where P is a constant.
[0108] Preferably, an estimated value of the channel attenuation
magnitude |h|.sup.2 is obtained by computing
.alpha.=.eta..sub.P/|s|.sup.2,
Where
[0109]
.eta..sub.P=|v.sub.P|.sup.2=|h|.sup.2|s|.sup.2+(h*s*w+hsw*)+|w|.su-
p.2.
[0110] In an embodiment of the invention, the channel attenuation
coefficients .alpha. are obtained by: [0111] inserting pilot
symbols within the transmitted signals with a particular time
spacing for forming a transmitted frame with data symbols having
the following structure d=[d.sub.PSK.sup.{1}, s.sub.ASK.sup.{2}, .
. . d.sub.ASK.sup.{Q}, d.sub.ASK.sup.{Q+1}, d.sub.ASK.sup.{Q+2}, .
. . , d.sub.ASK.sup.{2Q}, d.sub.PSK.sup.{2Q+1}, . . . ], where the
pilot symbols have a constant modulus ||==1 .A-inverted., where C
is a constant, and where Q is a constant set a priori based on
configuration criteria; [0112] using least-squared estimation to
compute .alpha. such that a channel attenuation coefficient
obtained from an th pilot symbol is in accordance with the
following equation:
[0112] .alpha. ^ = r PSK 2 d PSK 2 = .alpha. + h * d PSK * w + h d
PSK w * + w 2 ##EQU00019##
where r.sub.PSK is a received signal that corresponds to a given
pilot symbol. [0113] forming the following sparse vector using the
computed .alpha.:
[0113] a=[{circumflex over (.alpha.)}.sup.{1}], 0.sup.{2}, . . . ,
0.sup.{Q}, {circumflex over (.alpha.)}.sup.{Q+1}, 0.sup.{Q+2}, . .
. , 0.sup.{2Q}, {circumflex over (.alpha.)}.sup.{2Q+1}, . . . ],
2Q+1=L; [0114] using interpolation to compute {circumflex over
(.alpha.)}.sup.{i} where = mod Q.noteq.1;
[0115] In an embodiment of the invention, the transmitted signals
are detected by computing =||.sup.2/, mod Q.noteq.1.
[0116] In an embodiment of the invention, once is obtained,
obtaining channel state information for all data symbols by: [0117]
compiling =/, mod Q.noteq.1; [0118] using interpolation to find ,
mod Q=1; [0119] constructing a vector h=[h.sup.{1}, h.sup.{2}, . .
. , h.sup.{L}].
[0120] In an embodiment of the invention, an entire received vector
comprising the data symbols is detected coherently by:
{circumflex over (d)}=HH.sup.Hr
where r=[r.sup.{1}, r.sup.{2}, . . . , r.sup.{L}],
H=diag{h.sup.{1}, h.sup.{2}, . . . , h.sup.{L}}, and () denotes the
Hermitian transpose operation.
[0121] In an embodiment of the invention, the configuration
criteria based on which Q is set comprises at least one of a
channel coherence time, a spectral efficiency, and an interpolation
error tolerance.
[0122] In an embodiment of the invention, the pilot symbols are
modulated by the transmitter using phase shift keying (PSK).
[0123] As a further aspect of the invention, there is provided a
digital communication system comprising a transmitter, a receiver
and a demodulator implementing the demodulation technique in
accordance with the various embodiments of the present
invention.
[0124] As a further aspect of the invention, there is provided a
computer readable medium embedding computer instructions configured
to execute the demodulation technique in accordance with the
various embodiments of the present invention.
DRAWINGS
[0125] The invention will now be described with reference to the
accompanying drawings, which illustrate a preferred embodiment of
the present invention without restricting the scope of the
invention's concept, and in which:
[0126] FIG. 1 illustrates the PDF of a conventional NCD decision
variables for M=4 and SNR=30 dB;
[0127] FIG. 2 illustrates the PDF of SCD decision variables for M=4
and SNR=30 dB;
[0128] FIG. 3 illustrates SER of CD, NCD and SCD using MLD,
M=2;
[0129] FIG. 4 illustrates SER of CD, NCD and SCD using MLD,
M=4;
[0130] FIG. 5 illustrates SER of CD, SCD and NCD using MLD and MDD,
M=2;
[0131] FIG. 6 illustrates SER of SCD for various vehicle speeds
with imperfect channel estimates, M=2;
[0132] FIG. 7 illustrates SER of CD and SCD using different
.sigma..sub.PN.sup.2 values;
[0133] FIG. 8 illustrates proposed blind CSI estimation using
amplitude-coherent detection using a simple frame structure, where
each frame is composed of two symbols, one ASK and PSK;
[0134] FIG. 9 shows the SER using the proposed blind channel
estimation algorithm compared to QPSK modulation under different
scenarios.
DETAILED DESCRIPTION OF THE INVENTION
[0135] MASK Modulation
[0136] In MASK modulation, the baseband representation of the
transmitted signal is given by
d=A.sub.i, i .di-elect cons.{0, 1, . . . , M-1} (1)
where M is the modulation order, the amplitudes A.sub.i .di-elect
cons. for coherent detection, while for NCD A.sub.i.gtoreq.0.
Without loss of generality, the amplitudes are selected such that
A.sub.i+1>A. Moreover, the amplitude spacing is usually assumed
to be uniform where A.sub.i+1-A.sub.i=.delta.. Since the average
symbol power is normalized to unity, then
1 M i = 0 M - 1 A i 2 = 1. ##EQU00020##
The transmitted amplitudes can be described by,
A.sub.i=i.times..delta., i .di-elect cons.{0, 1, . . . M-1},
(2)
where
.delta. = 6 ( 2 M - 1 ) ( M - 1 ) . ( 3 ) ##EQU00021##
[0137] Conventional NC MASK Detection
[0138] Assuming that the information symbols are transmitted over a
Rayleigh frequency-flat fading channel, the received signal can be
expressed as
v=hA.sub.i+w, i .di-elect cons.{0, 1, . . . , M-1} (4)
[0139] where the channel fading coefficient h is a complex normal
random variable h.about.(0, 2.sigma..sub.H.sup.2) and w.about.(0,
2.sigma..sub.w.sup.2) denotes the additive white Gaussian noise
(AWGN). To perform NCD, the energy of the received signal should be
computed,
.eta.=|v|.sup.2=|h|.sup.2|A.sub.i.sup.2|+(h*A*.sub.jw+hA.sub.iw*)+|w|.su-
p.2. (5)
where (.)* denotes the complex conjugate. The received signal
energy .eta. is the decision variable that will be fed to the
maximum likelihood detector (MLD). The conditional probability
distribution function (PDF) of .eta. can be expressed as
P ( .eta. | E i ) = 1 2 ( .sigma. w 2 + .sigma. h 2 E i ) exp ( - 1
2 .eta. .sigma. w 2 + .sigma. h 2 E i ) , ( 6 ) ##EQU00022##
[0140] where E.sub.i=A.sub.i.sup.2. The PDF in (6) for M=4 is shown
in FIG. 1 where it is clear that the only amplitude that can be
identified easily is the A.sub.0=0 case. For all other symbols, the
probability of error is very high. Consequently, NCD of MASK can
provide reliable symbol error rate (SER) only for the M=2 case,
which leads to low spectral efficiency.
[0141] Based on the PDF given in (6), the MLD can be expressed as
[5],
A ^ i = arg min A i .eta. ( .sigma. w 2 + .sigma. h 2 E i ) + ln (
.sigma. w 2 + .sigma. h 2 E i ) , i = [ 0 , 1 , , M - 1 ] . ( 7 )
##EQU00023##
[0142] It is worth noting that the MLD of NCD of MASK requires
accurate knowledge of .sigma..sub.w.sup.2 and .sigma..sub.h.sup.2.
The SER using MLD can be expressed as [5],
P S = 1 - 1 M + 1 M i = 1 M - 1 exp [ - ln ( .GAMMA. i + 1 + 1
.GAMMA. i + 1 ) 1 - .GAMMA. i + 1 .GAMMA. i + 1 + 1 ] - exp [ - ln
( .GAMMA. i + 1 + 1 .GAMMA. i + 1 ) .GAMMA. i + 1 + 1 .GAMMA. i + 1
- 1 ] ( 8 ) ##EQU00024##
where
.GAMMA. i = .sigma. h 2 .sigma. w 2 E i . ##EQU00025##
[0143] The SER of the NCD-MASK for M=2 using optimal MLD is
presented in FIG. 3. As it can be noted from the figure, the SER is
decreasing as a function of SNR, which implies that the NCD of MASK
where M=2 can provide reliable SER at high SNR values. As it can be
noted from the figure, the performance degradation of the NCD as
compared to the CD is equivalent to 10 dB. Unlike the M=2 case, the
M=4 SER depicted in FIG. 4 shows that NCD is prohibitively high to
be incorporated in any practical application. As it can be seen
from the figure, the NCD suffers from an error floor at SER
.about.0.3. The SER for the CD in Rayleigh fading channels is
reported in [6].
[0144] The New Semi-Coherent Demodulator
[0145] To eliminate the impact of the multiplicative fading we
introduce the new SCD, which can be obtained by equalizing the
received symbols energy using only the magnitude of the channel
coefficients. The equalized envelop .zeta. can be expressed as
.zeta. = .eta. h 2 = v 2 h 2 = A i 2 + 1 h 2 [ ( h * A i * w + hA i
w * ) + w 2 ] . ( 9 ) ##EQU00026##
[0146] As depicted in (9), the multiplicative effect of multipath
fading has been converted into an additive disturbance. The process
of computing |h|.sup.2 for practical systems will be presented in
the following sections.
[0147] The conditional PDF of the decision variable .zeta. is given
by
P ( .zeta. | E i ) = ( .PSI. i + .sigma. w 2 ) .sigma. h 2 [ (
.zeta. - E i ) 2 .sigma. h 4 + 2 .PSI. i + .sigma. w 4 ] 3 / 2 ( 10
) ##EQU00027##
where 105 =(.zeta.+E.sub.i).sigma..sub.w.sup.2.sigma..sub.h.sup.2.
As it can be noted from FIG. 2 that considers the M=4 case, the
conditional PDFs are well separated and the tails of both PDFs
decay rapidly as a function of .zeta., which implies that the SER
will be much lower than that of the conventional non-coherent
detectors. Moreover, it can be noted that the peak values of the
PDF is inversely proportional to the transmitted amplitude, which
is uncommon in most conventional systems.
[0148] The SER of the SCD using MLD is presented in FIGS. 3 and 4
for M=2 and 4, respectively. As it can be noted from both figures,
the SER of the SCD significantly outperforms NCD. Moreover, SCD
managed to provide reliable SER for M>2, which implies that it
can provide higher spectral efficiency as compared to NCD.
Furthermore, it can be noted from FIG. 3 that the SER degradation
of SCD as compared to CD is about 5.5 dB at SER of 10.sup.-3, and
it is about 11 dB when M=4.
[0149] Based on the PDF given in (10), it can be shown that the
optimum detector has high complexity, and it requires the knowledge
of .sigma..sub.w.sup.2 and .sigma..sub.h.sup.2. Consequently,
suboptimal solutions should be considered. Towards this goal, it is
straightforward to show that in high SNR scenarios, an efficient
suboptimal detector for SCD can be expressed as
A ^ i = arg min E i [ .zeta. - E i ] 2 , i = [ 0 , 1 , , M - 1 ] ,
( 11 ) ##EQU00028##
[0150] which corresponds to the minimum distance detector (MDD).
The SER based on MDD can be expressed as
P S = 1 - 1 M + 1 M [ i = 1 M - 1 .lamda. i - i = 2 M .chi. i ] , (
12 ) ##EQU00029##
where
.lamda. i = .DELTA. 1 - .delta. 2 [ i - 0.5 ] .GAMMA. 2 ( .delta. 2
[ i - 0.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 3 i + 1.5 ] .GAMMA.
+ 1 ##EQU00030## B i = .DELTA. 1 + .delta. 2 [ i - 0.5 ] .GAMMA. 2
( .delta. 2 [ i - 0.5 ] .GAMMA. ) 2 + 2 .delta. 2 [ 2 i 2 - 5 i +
3.5 ] .GAMMA. + 1 . ##EQU00030.2##
where
.GAMMA. = .sigma. H 2 .sigma. w 2 E _ , and E _ = 1 M i = 0 M - 1 E
i ##EQU00031##
is the average power per symbol which is normalized to unity for
all systems.
[0151] The SER for M=2 is shown in FIG. 5 using MLD and MDD.
Because the MLD and MDD have equal SER for the CD, only one curve
is presented. As it can be noted from the figure, the SER
degradation of SCD as a result of using MDD is negligible while it
is substantial for the NCD case. Consequently, the MDD for SCD
offers near-optimal SER with low complexity.
[0152] Partial CSI Estimation
[0153] As it can be noted from (9), the partial CSI required for
the SCD is the channel attenuation coefficient |h|.sup.2. The most
straightforward approach to obtain |h|.sup.2 is to insert pilot
symbols within the information symbols with a particular time
spacing. The spacing between the pilot symbols can be optimized
based on the channel variations in the time domain. For quasi
static and slowly varying channels, the number of pilots is
insignificant and hence, the spectral efficiency degradation
becomes negligible. The main requirement for the pilot symbols is
to have a constant modulus, |s|.sup.2=P, where P is a constant.
Therefore, the energy of the received signal when a pilot symbol s
is transmitted can be expressed as,
.eta..sub.P=|v.sub.P|.sup.2=|h|.sup.2|s|.sup.2+(h*s*w*)+|w|.sup.2.
[0154] The estimated value of |h|.sup.2.alpha. can be obtain by
computing {circumflex over (.alpha.)}=.eta..sub.P/|s|.sup.2.
[0155] The channel variations over time can be described using
Jake's model [7]. Assuming that the channel is Rayleigh fading with
L.sub.h independent multipath components, the time correlation
between the channel coefficients can be expressed as,
E[h.sub.nh.sub.m]=.beta..sub.tJ.sub.0(2.pi.f.sub.dT.sub.s(n-m)),
(14)
where T.sub.s is the symbol period, .beta..sub.l is the normalized
power of the lth multipath component where
.SIGMA..sub.l=0.sup.L.sup.h .beta..sub.l, J.sub.0(.) is the Bessel
function of the first kind and zero order and f.sub.d is the
maximum Doppler shift. Therefore, the time variation over few
consecutive symbols is small. In broadband communications, the
ratio of number of pilot symbols to the information symbols is one
of the main factors that determine the system spectral efficiency.
Typically, the pilot spacing in the time domain is about 4 symbols
[8]-[13].
[0156] Numerical Results
[0157] In the previous parts, the SER performance was obtained
under ideal channel conditions and perfect channel estimation.
Therefore, this section presents the SER in the presence of
mobility, channel estimation errors, and phase noise.
[0158] The SER of the SCD in the presence of mobility is presented
in FIG. 6 using M=2. The channel is assumed to be Rayleigh
frequency-nonselective with time correlation that follows the
Jake's model. The bit rate is set to 2 Mbps and the carrier
frequency is 2.4 GHz. The pilot symbols are inserted periodically
with a separation of 4 data symbols. The channel attenuation at the
non-pilot symbols is obtained using linear interpolation. As it can
be noted from the figure, the SER degradation is less than 3 dB for
a vehicle speed (V) of 120 km/h and it is about 2 dB when V=60
km/h. Although such degradation is generally acceptable at such
high speeds, the SER can still be improved using more accurate
interpolation techniques.
[0159] The SER of SCD and CD in the presence of phase noise (PN) is
presented in FIG. 7 The received signal in the presence of PN can
be expressed as
v=he.sup.j.phi.A.sub.i+w, i .di-elect cons.{0, 1, . . . , M-1}
where .phi. is a function of the phase noise power, and it is
typically modeled as a random jitter
.phi..about.(0,.sigma..sub.PN.sup.2) [14], where
.sigma..sub.PN.sup.2 is measured in rad.sup.2. As it can be noted
from the figure, the SER of SCD is independent of the PN, which is
expected because the SER depends only on the magnitude of the
channel response. On the contrary, CD is sensitive to PN
particularly at high values of .sigma..sub.PN.sup.2. It is worth
noting that PN can be caused by the transmitter and receiver
frequency jitter, timing and frequency synchronization, and channel
estimation error, therefore, large PN values might be experienced
in particular system and channel conditions [14].
[0160] The new receiver for digital communication systems proposed
is based on a novel demodulation technique that requires only
partial knowledge of the channel state information, which
simplifies the channel estimation process. The error rate
performance of the new receiver is substantially lower than that of
the conventional non-coherent demodulators. The proposed system
enables high spectral efficiency implementation of digital
communication systems by exploiting the pilots for joint data
transmission and channel estimation.
[0161] Blind CSI Estimation Using Amplitude-Coherent Detection
[0162] In this section, we propose a low complexity blind channel
estimation technique using ACD. As it can be noted from
aforementioned discussion, the partial CSI required for the ACD is
the channel attenuation coefficient .alpha.. The most
straightforward approach to obtain .alpha. is to insert pilot
symbols within the information symbols, and then use least-squared
estimation to compute .alpha.. The main requirement for the pilot
symbols is to have a known amplitudes at the detector side.
Therefore, without loss of generality, we assume that the pilot
symbols satisfy ||.sup.2==1 .A-inverted.. Since MPSK has constant
modulus, we assume that all pilot symbols are MPSK modulated.
[0163] If the pilot and data symbol during the lth signaling
interval are denoted by and , respectively, then the transmitted
frame has generally the following structure,
d=[d.sub.PSK.sup.{1}, s.sub.ASK.sup.{2}, . . . , d.sub.ASK.sup.{Q},
d.sub.ASK.sup.{Q+1}, d.sub.ASK.sup.{Q+2}, . . . ,
d.sub.ASK.sup.{2Q}, d.sub.PSK.sup.{2Q+1}, . . . ]. (15)
[0164] The value of Q depends on the channel coherence time,
spectral efficiency, interpolation error tolerance, etc.
[0165] Using least square estimation, the channel attenuation
factor obtained from the lth pilot symbol can be expressed as,
.alpha. ^ = r PSK 2 d PSK 2 = .alpha. + h * d PSK * w + hd PSK w *
+ w 2 ( 16 ) ##EQU00032##
where r.sub.PSK is the received signal that corresponds to a given
pilot symbol. Similar to conventional coherent systems, the channel
estimates can be used to form the following sparse vector
a=[{circumflex over (.alpha.)}.sup.{1}], 0.sup.{2}, . . . ,
0.sup.{Q}, {circumflex over (.alpha.)}.sup.{Q+1}, 0.sup.{Q+2}, . .
. , 0.sup.{2Q}, {circumflex over (.alpha.)}.sup.{2Q+1}, . . . ],
2Q+1=L. (17)
[0166] Then, interpolation can used to compute {circumflex over
(.alpha.)}.sup.{i} where = mod Q.noteq.1. Finally, the data symbols
can be detected by computing =||.sup.2/, mod Q.noteq.1.
[0167] As it can be noted from the aforementioned discussion, the
pilot symbols design and channel estimation approach used are
generally similar to those used in coherent detection. However, it
is interesting to note that once is obtained, then the full CSI can
be obtained for all data symbols where =/, mod Q.noteq.1. Then,
interpolation can be used to find , mod Q=1, which allows
constructing the vector h=[h.sup.{1}, h.sup.{2}, . . . ,
h.sup.{L}]. Consequently, the entire received vector can be
detected coherently
{circumflex over (d)}=HH.sup.Hr
where r=[r.sup.{1}, r.sup.{2}, . . . , r.sup.{L}],
H=diag{h.sup.{1}, h.sup.{2}, . . . , h.sup.{L}}, and ( ) denotes
the Hermitian transpose operation. Therefore, if the pilot symbols
are regular MPSK data-bearing symbols, then the data can be
recovered and utilized. In this sense, the data and pilot symbols
exchange their roles recursively to estimate the CSI and detect the
data with low complexity and no power or spectrum penalties. FIG. 8
shows the proposed technique using a simple frame structure, where
each frame is composed of two symbols, one ASK and PSK. The the
interpolation is not required in such scenarios because
h.sup.{l}.apprxeq.h.sup.{l+1}.
[0168] Because both d.sub.PSK and d.sub.ASK symbols are bearing
data, none of them should be referred to as pilot symbol. Moreover,
the ratio between the number of PSK and ASK symbols is channel and
system dependent. However, PSK SER is typically lower than ACD.
Therefore, the number of PSK symbols can be increased to provide
lower SER as long as the separation between ASK symbols is small
enough to provide accurate channel estimation.
[0169] It is important to note that using A.sub.0=0 for channel
estimation with ACD should be avoided since the channel coefficient
is undefined in such scenarios. A simple solution to resolve this
matter is to use A.sub.m=(m+1).times..delta., m .di-elect cons.{0,
. . . , M-1}. For an average power
1 M m = 1 M E m = 1 ##EQU00033##
and equally spaced constellation points, the amplitude separation
can be defined as s.sub.m+1-s.sub.m .delta., where
.delta. = 6 ( M + 1 ) ( 2 M + 1 ) . ##EQU00034##
Therefore, P.sub.S can be expressed as
P S = 1 - 1 M + 1 M i = 1 M - 1 A i - B i + 1 A i = .DELTA. 1 -
.delta. 2 [ i + 0.5 ] .GAMMA. 2 ( .delta. 2 [ i + 0.5 ] .GAMMA. ) 2
+ 2 .delta. 2 [ 2 i 2 + i + 0.5 ] .GAMMA. + 1 B i = .DELTA. 1 +
.delta. 2 [ i - 0.5 ] .GAMMA. 2 ( .delta. 2 [ i - 0.5 ] .GAMMA. ) 2
+ 2 .delta. 2 .GAMMA. 2 i 2 - i + 0.5 ] + 1 ( 5 ) ##EQU00035##
[0170] It is worth noting that the SER when A.sub.0>0 is higher
than the case where A.sub.0=0 due to the loss of power efficiency.
Such limitation can be avoided by setting A.sub.0=0, however, CSI
over the entire frame has to be recovered from nonuniformly spaced
samples [15].
[0171] Numerical Results of the Blind Detection Technique
[0172] FIG. 9 shows the SER using the proposed blind channel
estimation algorithm compared to QPSK modulation under different
scenarios. The results are obtained for N=1 and M=4 regardless of
the modulation type. The transmitted frames for QPSK modulation
with pilot estimation are composed of 5 symbols with QPSK
modulation, 4 data symbols and one pilot, and all symbols have
equal power. The proposed hybrid frame comprises of 5 symbols as
well, 4 QPSK modulated and one with 4-ASK modulation. The channel
is assumed to be quasi-static Rayleigh fading where the channel
parameters remain fixed over one frame period, while they changes
randomly at different frame periods. As it can be noted from the
figure, the pure ACD with perfect channel estimation exhibits the
highest SER. Such performance is obtained because MASK is less
power efficient than QPSK even under imperfect channel estimation
conditions. However, the SER of the hybrid frame using the proposed
channel estimation exhibits a 4 dB improvement over MASK with ACD
at SER=10.sup.-3. The SER of conventional QPSK with pilot based
channel estimation leads the proposed system by about 3.5 dB.
However, such SER performance improvement is obtained at the
expense of spectral efficiency. Moreover, the SER degradation is
caused partially by channel estimation errors and the higher SER of
the MASK symbol used within the frame. Nevertheless, the proposed
blind channel estimation demonstrated a highly reliable results
with 100% spectral efficiency. Moreover, the performance gap could
be less under more severe channel models such as the ones with
strong phase noise.
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