Compressive sensing theory can be applied to reconstruct
the signal with far fewer measurements than what is usually considered
necessary, while in many scenarios, such as spectrum detection and modulation
recognition, we only expect to acquire useful characteristics rather than the
original signals, where selecting the feature with sparsity becomes the main
challenge. With the aim of digital modulation recognition, the paper mainly
constructs two features which can be recovered directly from compressive samples.
The two features are the spectrum of received data and its nonlinear
transformation and the compositional feature of multiple high-order moments of
the received data; both of them have desired sparsity required for
reconstruction from subsamples. Recognition of multiple frequency shift keying,
multiple phase shift keying, and multiple quadrature amplitude modulation are
considered in our paper and implemented in a unified procedure. Simulation
shows that the two identification features can work effectively in the digital
modulation recognition, even at a relatively low signal-to-noise ratio.
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1. Introduction
Constantly increasing volume of
data transmitted through the mobile communication networks and the needs of
users to increase the data rates lead to rapid development of mobile
communication systems. The future fifth generation (5G) wireless communication
tends to achieve a remarkable breakthrough both in data rate and spectral
efficiency [1]. With demand for large
data size and high data rate, vast spectrum resources are required urgently.
However, most spectrum resources below 2G are fixedly occupied by other
industries, although they have not been fully utilized. Considering this, the
purpose of spectrum sensing in mobile communication networks is to share
spectrum resources with other industries, without interfering with their normal
operations. Moreover, spectrum sensing can be also applied to coordinate the
public resources, which is a revolutionary change of the fixed spectrum
allocation system [2].
On account of the rearrangement
function needed in spectrum sensing and the fact that modulation recognition
can provide reliable parameters for it, digital modulation recognition is of
great importance in the whole system [3]. The goal of digital
modulation recognition is to identify the modulation format of an unknown
digital communication signal. For modulation classification, two general
classes of classical methods exist: likelihood-based and feature-based methods,
respectively [4, 5]. Based on the
likelihood function of the received digital signal, the former method makes the
decision by comparing the likelihood ratio with a threshold. In the
feature-based method, several features are usually chosen and the decision is
made jointly.
However, in traditional sensing
process, two approaches are based on Shannon-Nyquist sampling theorem and the
data scale to deal with can be enormous with a quite wide band especially in
the cooperation networks. These years, researchers have brought compressive
sensing (CS) in, which can solve the problem of high sampling rate caused by
Shannon-Nyquist sampling theorem. It is declared that if the signal has a
sparse representation in a fixed basis, we can reconstruct the sparse domain of
the signal by solving an optimization algorithm, using samplings far fewer than
dimensions of the original signal, and the original signal can be obtained by a
simple matrix operation [6, 7].
In many CS conditions, we expect
to acquire some signal characteristics rather than recovering the original
signal, since reconstructing signals allows for lots of extra operations, which
results in higher complexity in both time and space. Researchers have already
carried out much related valuable work in reconstructing signal characteristics
based on CS [8, 9]. Inspired by these
researches, we are devoted for finding identification features which can be
used in the digital modulation recognition and simultaneously have sparsity,
meaning they can be reconstructed directly by compressive samples.
In this paper, we propose a
feature-based method based on CS for digital modulation recognition. We
construct two identification features and use compressive samples to recover
them directly, without recovering the original signals. One identification
feature is the spectrum of received data and its nonlinear transformation,
which is based on the feature proposed by [10], and the other is a
compositional feature of multiple high-order moments of the received data.
These two features can be used to identify various kinds of modulation modes,
and, in this paper, we only focus on multiple frequency shift keying (MFSK), multiple
phase shift keying (MPSK), and multiple quadrature amplitude modulation (MQAM).
Simulations would be carried out to indicate that the performance of our method
can be effective and reliable, with lower complexity and better antinoise
property than traditional ones [11, 12].
The rest of this paper is
organized as follows. Section 2 would present the
system model adopted throughout the work, both the signal model and compressive
sensing model. In Section 3, we construct two
identification features and analysis sparsity of them. In Section 4, we build the linear
relationships between identification features and compressive samples and give
a brief introduction of the recovery method. Then, the whole recognition
flowchart will be shown in Section 5. Simulations and
analysis are present in Section 6. And, finally, we draw
the conclusion in Section 7.
2. System Model
2.1. Signal Model
In a spectrum sensing scenario, we
assume that a wide band received signal has one of the following modulation
modes: where represents the amplitude of the received
signal, represents the symbol period, represents the
impulse response of pulse shaping low-pass filter, in which we choose
rectangular pulse in this paper, and stands for additive Gaussian white
noise (AWGN). In the whole process, the timing offset and the carrier offset
are both assumed to be zero, and the form of is chosen as follows: where and ,
respectively, stand for the carrier frequency and order of the chosen
modulation mode, is the carrier spacing, and are a set of discrete
levels. It is worth noting that there is no need of pulse shaping for MFSK,
while in order to unity the form as (1), we
regard in MFSK as 1, which would not influence the use of it.
2.2. Compressive Sampling Model
According to the theory of CS, the
compressive sampling process can be modeled analytically as where is
the -length sampling vector of the received signal at a rate no
lower than Nyquist sampling rate. Represents the subsampling
measurements. is a real-value measurement matrix of size ,
which complies with the restricted isometry property (RIP), such as Gaussian
matrix, partial Fourier transform matrix, or others. In order to reconstruct
the th power of signal in Section 4.1, we adopt a special
measurement matrix, with the value of “1” randomly located in each row and
other elements being zero. Owing to the randomization of row elements, the
matrix satisfies the RIP requirement as well as Gaussian matrix. Furthermore,
the two-valued property of the matrix can enormously simplify the matrix
operations, which make it possible to establish linear relationship between the
nonlinear reconstruction targets with the compressive samples.
To classify the modulated signal based on, we will
firstly build the linear relationships between and the identification
features and then reconstruct the features directly with compressive samples by
solving an optimization algorithm, which would then be used to do digital
modulation recognition.
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3. Construction of the Identification Features
To achieve the goal of classifying modulation types
accurately, identification features should be chosen with distinguishing
details for each modulation type firstly. Secondly, identification features
should have desired sparsity, in order to be constructed by compressive samples
based on the theory of CS. According to these two requirements, we propose and
construct the following two identification features.
3.1. Feature 1: Spectrum of the Signal’s th Power
Nonlinear Transformation
Referring to [9], we calculate
the th power of the received signal; that is, where (). In the
following, it will be shown that the spectrum of specific power order of signal
level presents the recognizable characters for certain modulation types and
orders. is the noise caused by nonlinear transformation of .
Then, we calculate the spectrum of,
which is represented as, with ranging from 0 to a larger number. We
have the following relationship: where represents the -point IFFT
matrix.
For different kinds of modulation modes, the results are
quite different, which can be used to do the recognition, and we call this
feature the spectrum feature below.
For MFSK, according to (1) and (2), the spectrum of it can
be calculated as follows: where stands for the impulse function.
Obviously, there is impulse in the spectrum of MFSK, and the number of these
impulses just corresponds to order. As a contrast, the spectrum for MPSK can be
presented as follows: where stands for the convolution operation. The
spectrum of MPSK comes out to be a monotone decreasing sine function, with no
impulse. The calculation process of spectrum of MQAM is similar to MPSK, and
their consequences are also similar. Figure 1 shows the results
of different modulation types by. From (a) and (b), we can see that there is
apparent impulse for MFSK, just as we analyze in theory, quite different from
that in (c) and (d), which represent the results of MPSK and MQAM,
respectively. That is to say, we can distinguish MFSK from others by the
spectrum of the signal, and the number of pulses indicates the order of MFSK.
Figure 1: Spectrum of different modulation modes.
For MPSK, when, from the
expression, we can see that there is no impulse in this condition. However, when,
for, we choose rectangle filter. Since, becomes a constant, and the
Fourier transform of it is an impulse. Based on it, we compare the results of, and,
referring to Figure 2. It can be seen that the
impulse firstly appears when, with varying from small to big, which can be
used to determine the order of MPSK.
Figure 2: Spectrum of the power of signal
modulated by MPSK.
As for MQAM, owing to similarity
of the signal constellation, the property of MQAM is similar with QPSK, shown
as Figure 3. It can be easily seen
that the impulse firstly appears when, and it has no relationship with the
specific order of it.
Figure 3: Spectrum of the power of signal
modulated by MQAM.
The sparsity of this feature is in inverse proportion with,
which represents length of the signal as well as the IFFT size, on condition
that there appears the impulse.
To sum up, the only remaining problem is to distinguish
QPSK and MQAM. Therefore, we construct another feature for it.
3.2. Feature 2: A Composition of Multiple High-Order
Moments of the Signal
For a digital modulated
communication signal, the mixed moments of order are defined as (6), at a zero delay vector
[10]: where the superscript denotes
conjugation and means calculating the mean value.
In our system model, we intend to acquire and as
recognition parameters, which we call the high-order moment feature.
With carrier known, symbols in the
digital signals can be regarded as points in the signal constellation [13, 14]. Since points in
digital signals of linear modulations are of equal probabilities, when the data
size is large enough, we can use points in the signal constellation to calculate
the theoretical values of and, as Table 1 shows.
Table 1: Theoretical values of and.
Referring to Table 1, of different
modulation formats are of different theoretical times compared to, which is the
square value of.
We define the identification characteristic in
We take QPSK and 16QAM as
examples. According to (11), the theoretical
values of QPSK and 16QAM, respectively, come out to be 1 and 0.68. If we
get the identification characteristic of a signal, we can then identify
the modulation format by comparing with a suitable decision threshold.
Since high-order moment is a kind
of statistics, we need sample several times. Then, to obtain and, we
construct matrixes as follows: where represents conjugate transpose, represents
transpose, and stacks all columns of a matrix into a vector. For, the
element of matrix at row, column, is, are elements in the signal. When,
meaning diagonal elements, the values are equal to based on the definition
of high-order moments. However, when, the value comes out to be zero for the un-correlation
between symbols of the signal. For, corresponds to the element of. When,
the relationship is that corresponds to the element of. According to
(10), is the desired value,
so the diagonal elements of are equal to. Other elements are zero for
the un-correlation between symbols of the signal. The theoretical figures of and are
shown as Figure 4.
Figure 4: and of the signal modulated by
16QAM.
It is obvious that and in
Figure 4 are
sparse. For, all diagonal elements are nonzero, meaning the sparsity degree of
it is. For, the elements of are nonzero, meaning the sparsity degree of it
is.
4. Recovery of the Identification Features with
Compressing Samples
In this section, we introduce the approaches of
recovering the two identification features based on CS. We firstly build the
linear relationships between compressive samples and the defined features and
then give a brief introduction of the reconstruction algorithm and the
practical selection strategy for the measurement matrix.
4.1. Linear Relationships between Compressive Samples and
the Identification Features
4.1.1. Linear Relationships between Compressive Samples
and the Spectrum Feature
It is obviously that the power of
the signal is a nonlinear transformation. To get linear relationship between
compressive samples and the spectrum of the signal’s power nonlinear
transformation, we choose the special measurement matrix proposed in
Section 2. According to the nature
of this certain-form matrix, we can easily get the following relationship based
on (3): and is the
measurement matrix for feature 1. Then, referring to (5), we obtain where the
sensing matrix we needed is.
4.1.2. Linear Relationships between Compressive Samples
and the High-Order Moment Feature
For this identification feature,
the sampling matrix can be chosen as any one as long as it satisfies the
restricted isometry property (RIP):(i): according to (3) and the nature of
transpose, we get the following relationship, and stands for the
measurement matrix for : Take the average of both sides: We
use to represent , simultaneously refer to (12), and then get Next, we
apply the property to transform (19) to (20). It is worth noticing that,
for is a real-value matrix: where can be regarded as the sensing
matrix, with the scale of. (ii): since the sparsity degree of is far fewer
than that of, the dimension of signal needed and scale of measurement can also
be very low. We represent the measurement for as, while the only
difference of it from is the dimension. Similar to (17), there is Then,
according to, we can transform the two-dimensional relationship into
one-dimensional relationship: We can obtain Take the average of both sides: Based
on (13), we get the
relationship: where denotes. And then we have where the sensing
matrix is.
4.2. Reconstruction of Identification Features
Zy, Rz21, and Rz40 can be
calculated by the sampling value. With sensing matrixes and measurement vectors
known, the reconstruction of the sparse vectors can be regarded as the signal
recovery problem by solving the NP-hard puzzle as follows, taking as an example:
This can be transformed into a linear programming problem: which is
called -norm least square programming problem and is proved to be convex
that there exists a unique optimum solution. I > o is a weighting
scalar that balances the sparsity of the solution induced by the -norm
term and the data reconstruction error reflected by the -norm LS term.
In Section 4.1, we have mentioned
recovering three recognition features by using measurement matrixes, and,
respectively. However, practically, only using as the compressive
measurement may meet the requirement of recovering all of the features. The reasons
are that and differ in the dimension but are both designed with the
constraint of RIP property only. From the other aspect, the primary requirement
of constructing matrix is also the RIP condition.
5. Modulation Recognition with the Identification
Features
Given a received communication
signal modulated by MFSK, MPSK, or MQAM, we firstly get compressive samples
using measurement matrixes present in Section 2. In this process, due to
difference of sparsity we have analyzed in Section 3, various features may
apply various length of the signal, and this can be decided based on actual
situations. According to the approaches proposed above, the identification
features can be easily obtained. Then, we can recognize the modulation format
effectively referring to the flowchart shown in Figure 5, and specific steps are
listed in the following.
Figure 5: The process of digital modulation
recognition.
Step1. Reconstruct the spectrum feature
when with compressive samples. If there is impulse in the recovered
spectrum, the modulation mode can be identified as MFSK, and the number of
impulses indicates the order of it. However, if there is no impulse in the
feature, the communication signal is modulated by MPSK or MQAM, and then
Step 2 should be
conducted.
Step2. Reconstruct the spectrum feature
when with compressive samples and observe value of when the impulse
firstly appears. If when the impulse appears, the modulation mode can be
regarded as QPSK or MQAM, and then we go to Step 3. However, if when
the impulse appears, the signal is modulated by MPSK and this value of is
the order of it.
Step3. Reconstruct and of the
signal with compressive samples, get average values of the diagonal as and,
respectively, and then calculate based on (11). Compare with the
calculated boundary values shown in Table 1 and determine the
modulation type.
6. Numerical Results
This section presents the simulation results of our
feature-based recognition method. We firstly generate a stream of signals
modulated by MPSK, MFSK, or MQAM. All the signals share the same bit rate kbit/s
and the carrier frequency kHz, and the carrier spacing for MFSK is kHz.
For the two proposed features, the observation time is various because data volumes
needed by the two features are all different. The performance of reconstruction
is closely related to the signal-to-noise ratio (SNR), which is set as a
variable in our simulation, and simulations at every SNR are carried out for
500 times.
As mentioned above, information we
need to capture in feature 1 is whether there are impulses and the number of
them, rather than accurate numerical values. Therefore, we apply correct
detection rate of pulse to evaluate the performance of reconstruction of
spectrum feature, which is shown in Figure 6. We set a decision
threshold which equals two-thirds of the biggest reconstructed value, and if
there is no other value larger than the threshold, the biggest value would be
regarded as the impulse. In this scenario, the compressive ratio is set as 0.3,
which means. We calculate the detection rate for MFSK signal on, BPSK on, QPSK
and MQAM on, and 8PSK on, respectively. It is obvious that the detection rate
varies a lot with. The reason is that the power of signal is a nonlinear
transform, meaning that the uniformly distributed noise is magnified, and the
degree of magnification extends as the increasing of . Therefore,
detection rate of impulse when is the worst one.
Figure 6: Correct detection rate of impulse in
reconstructed feature 1.
Figure 7 shows the mean
square error (MSE) of reconstructed feature 2 with respect to the theoretical
ones. That is,
Figure 7: MSE of reconstructed and with
different compressive ratio.
We give the MSE of reconstructed and,
respectively, with the compressive ratio chosen as 0.3 and 0.45. From
Figure 7, we can see
that the performance of reconstruction of is closely related to the
compressive ratio, while the performance of reconstruction of is
relatively perfect even at a low compressive ratio. Moreover, we can easily get
the conclusion that when the compressive ratio is suitable, the precision of
feature 2 is high enough as long as the SNR is higher than 10 dB.
Figure 8 shows the correct
classification rate of different modulation modes at relatively low SNR.
Difference of the correct classification comes from various performance of
reconstruction of features, which has been shown in Figures 6 and 7. MFSK has high
recognition rate, larger than even when SNR = −6 dB. For MPSK, the correct
recognition rate declines as increases. However, for QPSK and MQAM, the
performance is quite different, and we give the following analysis.
Figure 8: Correct classification rate of different
modulation modes.
According to [14], we have the fact that of
just the signal and mixture of noise and signal are of the same value, so the
main cause of the error comes from.
As for, we have the following
proof stating the variation of the value in noisy condition and noiseless
condition. To describe this clearly, M2,1(y0), M2,1(v) , and M2,1(y) are,
respectively, used to replace while being in the following condition
of signal only, noise only, and the mixture of noise and signal:
is zero-mean random measure
noises with Gaussian distribution, which is independent from . According
to the nature of expectation, we know that
Therefore, we can obtain the
following relationship: meaning is the sum of signal power and noise
power.
From (11) and (27), we can obtain the
relationship of the theoretical and the actual: where denotes noise
power and denotes signal power.
To sum up, is added by the
power of noise, and, as a consequence, the identification
parameter becomes smaller; thus, QPSK may be recognized as 16QAM.
Therefore, the correct recognition rate of 16QAM is much higher than QPSK when
SNR is lower than 10 dB, as shown in Figure 8.
7. Conclusion
To solve the problem of high sampling rate for digital
modulation recognition in spectrum sensing, we have proposed a feature-based
method to identify the modulation formats of digital modulated communication
signals using compressive samples and have greatly lowered the sampling rate
based on CS. Two features are constructed in our method, one of which is the
spectrum of signal’s the power nonlinear transformation, and the other is
a composition of multiple high-order moments of the signal, both with desired
sparsity. By these two features, we have applied suitable measurement matrixes
and built linear relationships referring to them. The method successfully
avoids reconstructing original signals and uses recognition features to
classify signals directly, declining the algorithm complexity effectively.
Simulations show that correct recognition rates are different for different
modulation types but are all relatively ideal even in noisy scenarios. In
actual situations, the method can be decomposed aiming at variable demands,
and, for further work, we tend to improve the performance of the whole method
continuously, especially the noise elimination in the classification of QPSK
and MQAM.
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Source: https://www.hindawi.com/journals/misy/2016/9754162/
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