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FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT .
Estimating the Frequency Response of a Receiver
Blind Identification
Yu Deyue
September 2012
Master’s Thesis in Electronics
Master’s Program in Electronics/Telecommunications
Examiner: Niclas Björsell
Supervisor: Efrain Zenteno
2
Yu Deyue Estimating the Frequency Response of a Receiver
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Abstract
Consider a receiver that has an unknown impulse response in the form of linear time-invariant system,
which is driven by an input signal and random noise with an unknown distribution. If the unknown
impulse response of the receiver could be identified, then its equivalent frequency response can be
used for calibration, using deconvolution aiming to regain the original signal under measurement.
Hence, the estimation of the transfer function of the receiver is of great importance from both
theoretical and practical applications. This thesis is devoted to the identification of the receiver’s
unknown finite impulse response (FIR form), by observing only the output signal through it. More
specifically, not only the amplitude and phase but also the orders of the finite impulse response of the
receiver are unknown. A blind algorithm is tested for the identification of the unknown
communication channel of the receiver in simulation. Further, in a practical implementation performed
in this thesis such blind algorithm is combined with a technique to compute the receiver’s transfer
functions. The word blind indicates that there is no restriction in the input signal set, it is allowed to be
non-stationary with unknown statistical model, and hence this algorithm is particularly suitable for de-
reverberation technology. The implementation of the blind algorithm is demonstrated through the
Matlab simulation, and verified through experimental measurements, where several of the
impairments of hardware will be considered in the analysis.
Yu Deyue Estimating the Frequency Response of a Receiver
ii
Acknowledgment
I would like to thank the supervisor, Ph.D. Mr Efrain Zenteno for his constructive comments, careful
reviews and scientific supervision.
My thanks also give to all the people in ITB/Electronics at the University of Gävle, for their
contributions of making a pleasant working environment.
Yu Deyue Estimating the Frequency Response of a Receiver
iii
Table of contents
Abstract .................................................................................................................................................... i
Acknowledgment .................................................................................................................................... ii
Table of contents .................................................................................................................................... iii
1 Introduction ..................................................................................................................................... 1
1.1 Problem statement .................................................................................................................... 1
1.2 Thesis goal ............................................................................................................................... 4
2 Theory ............................................................................................................................................. 5
2.1 Modulation ............................................................................................................................... 5
2.1.1 Digital carrier modulation ................................................................................................ 5
2.2 Blind channel identification ..................................................................................................... 6
2.2.1 SIMO (two-output) ........................................................................................................... 6
2.2.2 SIMO (more than two-output) .......................................................................................... 8
2.2.3 SISO to SIMO equivalence .............................................................................................. 9
2.3 System identification .............................................................................................................. 10
3 Simulation ..................................................................................................................................... 11
3.1 Channel order known ............................................................................................................. 12
3.1.1 Noiseless channel ........................................................................................................... 13
3.1.2 Noisy channel ................................................................................................................. 18
3.1.3 Consider the different feature input ................................................................................ 24
3.2 Channel order unknown ......................................................................................................... 25
4 Measurements ................................................................................................................................ 29
4.1 Hardware device introduction ................................................................................................ 29
4.2 Measurement set-up ............................................................................................................... 29
4.3 Noise reduction ...................................................................................................................... 30
4.4 Receiver identification ........................................................................................................... 31
4.4.1 Received data.................................................................................................................. 32
4.4.2 Existing channel ............................................................................................................. 33
Yu Deyue Estimating the Frequency Response of a Receiver
iv
4.4.3 Order selection................................................................................................................ 35
4.4.4 Identified receiver models .............................................................................................. 37
4.4.5 Validation of the identified receiver models .................................................................. 39
5 Discussion ..................................................................................................................................... 42
6 Conclusions ................................................................................................................................... 43
7 Future work ................................................................................................................................... 44
References ............................................................................................................................................. 45
Yu Deyue Estimating the Frequency Response of a Receiver
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Yu Deyue Estimating the Frequency Response of a Receiver
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1 Introduction
In digital signal processing applications, when a signal is measured through a receiver, the measured
signal becomes shaped by the frequency response of the receiver (or its time equivalent impulse
response). Such frequency response can be represented by a linear system, that is, a figure conveying
both amplitude and phase distortions, so any signal will suffer when it is applied to such a system. The
estimated frequency response of the receiver can be used for calibration, deconvolving signal that is
observed to regain the original signal under measurement, so the estimation of frequency response of
receiver compensate for the effect introduced in the receiver at measurement.
The problem of deconvolving any signal that observed through one or more unknown multichannel
arises in data communication applications. It is more often, the unknown input signals change rapidly
and become more unrealistic. In this thesis, an algorithm is tested for the blind identification of
multichannel outputs FIR systems of the receiver using the measured data and without requiring any
knowledge of the input signals statistics. But this blind algorithm requires some assumptions on the
input. However, the statistical model of the input could be unknown or there may not be enough data
samples to do a reasonably and accurate estimation. The oversampling is certainly necessary in the
output. The algorithm that tested in this thesis (blind identification of frequency response of receiver)
is based on the eigenvalue decomposition of a correlation matrix [1]. The input signal could be
obtained by deconvolution of the identified receiver’s FIR with the received signal.
1.1 Problem statement
The output is observed from an unknown linear time-invariant system with unknown input .
See the Figure.1.1 below. The problem is to identify the or inverse or, equivalently, so it could
recover the input .
Figure 1.1 An input passes through an unknown linear time-invariant system to the output.
In the measurement, we use the discrete output data to identify the unknown linear system. The output
data may have some possible small perturbations, which may affect the blind identification [2].
unknown
Yu Deyue Estimating the Frequency Response of a Receiver
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The output is observed from an unknown linear time-invariant system with an input
could be expressed mathematically as a convolution operation. If the FIR channels have an order ,
the output of the specific th channel is expressed by
∑
Where is the input sequence that with arbitrary statistical characteristic. is the element
of that is white Gaussian noise with zero mean and unknown variance, which represents the
perturbations that appear in the measurement process. The unknown channel response that is the
element of , is the number of elements of the channel , it is also called channel order.
is the number of output channels [1].
Normally, the estimated linear system could be very well modeled by rational system transfer
functions, in this thesis the focus will be in transfer functions that are completely described by zeros.
Hence, becomes the FIR system. It should be noted that may be a non-causal or non-minimum
phase. It is impossible to restore the input signal if the is instable. The Figure.1.2 below shows
the restoration scheme.
Figure 1.2 The input signal restoration scheme. The unknown system is identified by using blind identification
algorithm. The inverse of identified system is used for recovering the input signal.
Recently, the blind identification is popular to be implemented in single-input multiple-output (SIMO)
system in data communication applications. SIMO systems should not be limited to the multiple
physical receivers or sensors. They require a high hardware complexity given by a larger number of
receivers. For this reason, a single-input single-output (SISO) system is used instead of SIMO system.
However, the SISO system could be equivalently transformed into SIMO systems when the output
signal is oversampled, see Figure.1.3. It should be noticed that all the SIMO systems for blind
identification are transformed from SISO system. The blind identification is straight forward if the
order of the receiver is known. On the other hand, the blind identification becomes more intricate if
the order of the receiver is unknown. We need to find a method to estimate the proper order of the
unknown receiver, and this method applies the root-mean-square-error (RMSE) concept for the order
selection of the identified system.
unknown
identified
Yu Deyue Estimating the Frequency Response of a Receiver
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(c)
(d)
Figure 1.3 (a) The samples of a single output channel that needs be oversampled. (b) The samples of single
output channel data is transformed into multichannel data. The output data is formed by matrix. is
the number of multichannel, is the number of element in each sub-channel. (c) Example of oversampled
output data that will be transformed into six-multichannel outputs system. (d) Example of six-multichannel
outputs system that is transformed from a single output channel.
0 2 4 6 8-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
X
Y
Channel 1
Channel 3
Channel 2
Channel 6
Channel 5
Channel 4
Yu Deyue Estimating the Frequency Response of a Receiver
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The SISO system is equivalently transformed into SIMO system, the example is shown in Figure.1.3.
When a single channel output is oversampled, then it could be transformed into multiple output
channels.
1.2 Thesis goal
This thesis will be a research study on the blind algorithm that can be used for the estimation of the
frequency response of a receiver (or its time equivalent impulse response), where the impulse response
of the receiver will be assumed to have a finite impulse response (FIR) form. The work ends with the
implementation of the blind algorithm that is demonstrated through the Matlab simulation with further
comparisons, and then verified through experimental measurements, where several of the impairments
of hardware will be considered in the analysis.
Yu Deyue Estimating the Frequency Response of a Receiver
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2 Theory
2.1 Modulation
Modulation is the performance of an operation that adapts the shape of a communication signal to the
physical channel on which the signal transmission will be done. The channel that used as a medium for
information transmission could be electrical wires, optical fibers, radio link, etc. All the transmission
channels should have a finite bandwidth that limits the transmitted symbols. The modulation could be
currently divided into two classes: baseband modulation and carrier modulation. The modulation
technique will give a high data rates with a small bandwidth are desired because the user of the
transmission channel would like to employ the channel as efficiently as possible. This results the
modulation extremely high spectrum efficiency [3].
2.1.1 Digital carrier modulation
In the most modern communication systems, the communication signals are transmitted in digital
words. In digital modulations, a mapping of a discrete information sequence is made on the amplitude,
phase or frequency of a continuous signal. It should be noticed that the signal transmission at both the
transmitter and the receiver is done in baseband. The carrier frequency defines the propagation
properties of the radio signal going through the specific channel and its behavior within the
environment. The input signal is called the complex envelope of the band-pass signal that can be
written in its general form as follows.
∑
Where is the amplitude, is the element of frequency, is the number of samples. is the
equivalent lowpass signal of the transmitted signal. The communication signal that is modulated and
transmitted through the channel is written as
{ }
Where is the carrier frequency [4].
Yu Deyue Estimating the Frequency Response of a Receiver
6
2.2 Blind channel identification
2.2.1 SIMO (two-output)
From Equation (1.1), we know when it is the single-input two-output channel setting. This is
the simplest setting for blind identification, which is the fundamental principle of single-input
multiple-output channel setting, the Figure is shown followed.
Figure 2.1 Single-input two-output channel setting. is the input signal. is the observed output signal
that is used for blind identification only. equal zero if the noise is ignored. is white Gaussian
noise with zero mean and unknown variance. is the unknown channel. is the estimated channel.
In Figure.2.1, when the two unknown sub-channels are noiseless, the two estimated sub-channels are
identified up by multiplying a constant arbitrary factor [1]
(2.5)
When the noise is free, the outputs and
with as the convolution operator. Then,
[ ]
[ ]
Yu Deyue Estimating the Frequency Response of a Receiver
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yielding
These above equations show that the outputs of each channel are related by their channel responses. If
there are adequate data samples of outputs , the linear equations of estimated channel response can
be obtained by solving equations [1]
[ ] [
]
Where
[ [ ] [ ] [ ]
[ ] [ ]]
Let
[ ]
And
[
]
Then
is the null space of . There are many vectors that satisfy the equation above, an unique solution is
to find the minimum ‖ ‖ such that
Consider
are eigenvalues of .
So
Then
Let is the minimum eigenvalue of , is minimum eigenvector, we have [5]
So is the minimum eigenvector.
To avoid a trivial solution, the estimated channel response is obtained by solving
‖ ‖
Yu Deyue Estimating the Frequency Response of a Receiver
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is the estimated channel impulse responses in this equation. The solution is the eigenvector that
corresponds to minimum eigenvalue of . In single-input multiple-output system when the outputs
have channels, we can form pairs of equations that can be combined into a larger linear
system [1].
2.2.2 SIMO (more than two-output)
From Equation (1.1), when , it is single-input multiple-output channel setting. See
Figure.2.2. The FIR channel is considered. For channels case, there are pair’s equations.
The blind identification of the FIR systems is based only on the multiple channel outputs. There is a
basic idea behind this blind identification that is to exploit different instantiations of the same input
signal by multiple FIR output channels [6].
Figure 2.2 Single-input multiple-output channel setting. is the observed output signal that is used for blind
identification only. is the identified time-invariant system, is the number of the multiple output channels.
For the SIMO FIR system, the solution of blind identification algorithm is expressed in Equation
(2.24), and the data matrix is defined by [7]
[
]
is used to form the correlation matrix .
Deterministic
Blind
Identification
,
, ...,
Yu Deyue Estimating the Frequency Response of a Receiver
9
2.2.3 SISO to SIMO equivalence
The blind identification of multiple output channels do require many receivers or sensors, in order to
reduce the consumption, a single input channel with a single output channel is used. A communication
signal in a single input channel corresponding to a single physical receiver could be transformed into a
multichannel FIR system identification if the signal is oversampled. The system is shown in Figure.2.3
below.
Figure 2.3 Single-input single-output-multichannel setting. is the input signal. is the observed
output signal that is used for blind identification only. The special channel with length .
The data from a channel of the communication signal corresponding to a signal physical
receiver is transformed into a multichannel FIR system if the sampling rate is higher than the
baud rate. This is done through an example shown in Figure.2.3. The channel lasts for
neighbor bauds, the signal has a oversampling rate of . The data sample of one baud period can
form the new vectors [ ] , we have
=-1
=-1
=1
=1
Yu Deyue Estimating the Frequency Response of a Receiver
10
The arbitrary element that is expressed by
The above system transforms a scalar communication signal to the vector output of multichannel
system, which converts the single-output signal to vector stationary output signals [6].
2.3 System identification
System identification is a process of determining the transfer function of an unknown system or
channel. It is a general term that uses the statistical method to find the mathematical model of the
system from the measured data. For example, in the digital communication system, an input signal
passed directly through an unknown linear system to the output, the unknown channel may attenuate
the input signal and a phase shift may occur at the output. By calculating the equation
where is the input, is the output, is the linear system.
The unknown linear system could be obtained, which can be used to design the inverse system
for input signal recovery [8].
Yu Deyue Estimating the Frequency Response of a Receiver
11
3 Simulation
In this chapter, the implementation of the blind algorithm is demonstrated through the Matlab
simulations. Several SIMO FIR systems are employed for blind identification.
In theory, the solution of the blind identification algorithm is the eigenvector that corresponds to
minimum eigenvalue of correlation matrix. In this chapter the algorithm of blind identification is
tested in Matlab simulation to show how to blind identify an unknown linear time-invariant channel
and then it is employed in the receiver’s finite impulse response identification. Although the blind
identification is only based on the output of the channel that is oversampled, it does need an input
signal. The input communication signal is generated as a complex envelope analytical signal defined
by
∑
Where
: is the amplitude of the element.
: is the frequency of the element.
: is the phase of the element.
The input communication signal is modulated (upconverted) before it is applied into the channel
corresponding to a receiver. This output is then transformed to a SIMO representation to use the blind
algorithm described in Section 2.2. The blind identification algorithm is tested using different SIMO
FIR systems. The results of blind identification in Matlab simulation are presented in this chapter. The
root-mean-square-error (RMSE) is employed as the performance measure of the channel identification
[6], it is defined by
‖ ‖√
∑‖
‖
[∑
]
where is the order of channel, is the element of the exact impulse response, is the
element of estimated channel, is an arbitrary factor.
Yu Deyue Estimating the Frequency Response of a Receiver
12
The Matlab simulations are conducted to evaluate the performance of the blind identification
technique. An input signal is created, modulated and applied to the specific channel (the channel is
known for simulations). When using the oversampled output signal into the blind identification
technique two conditions are considered: noiseless and noisy situation. See Figure.3.1 below.
Figure 3.1 Scheme of Matlab simulations for blind identification.
The blind identification algorithm that is tested in this thesis is an eigenvector based algorithm, the
solution of the algorithm is the eigenvector corresponding to the minimum eigenvalue of correlation
matrix , so for some simulations, the eigenvalues of the correlation matrix are presented. The
comparisons of the identified channel using different SIMO FIR systems and exact channel are
presented as well. RMSE is calculated and plotted in the Figures to measure the performance of the
blind identification.
Through these simulations the channel to test is represented as:
[ ].
3.1 Channel order known
When the order of the identified channel is known, the blind identification becomes straight forward.
Several SIMO FIR systems are tested to measure the performance of the blind identification. In this
section, the blind identification is taken in both noiseless and noisy situation with the specific input
signal.
Input Modulation H
(known) Output
Noiseless
Noiseless
Noisy
Noisy
Blind
Identification
(order known)
Blind
Identification
(order unknown)
Yu Deyue Estimating the Frequency Response of a Receiver
13
3.1.1 Noiseless channel
In a noiseless situation, the SIMO FIR systems are constructed from the signal Equation (3.1),
with equal amplitude ( = , constant and zero phases ( ). The output of the single channel
FIR system must be oversampled so that gives an accurate blind identification. The eigenvector
corresponding to the minimum eigenvalue of the matrix is the estimated impulse response of the
channel. In order to find the minimum eigenvalue, all the eigenvalues of correlation matrix that
formed by using the two-multichannel outputs FIR systems are plotted together in Figure.3.2 below.
Figure 3.2 Magnitude of the eigenvalues of a correlation matrix formed by two-multichannel outputs FIR system
in noise free situation (channel order known).
All the eigenvalues of correlation matrix formed by using two-multichannel outputs FIR system are
indicated in Figure.3.2. This Figure plots the magnitude of all eigenvalues. The blind algorithm for
channel estimation search for the minimum eigenvalue (zero in the ideal case), thus the magnitude of
the eigenvalues can be seen as a metric for the performance of this method. The number of the
eigenvalues is equal to the channel order. It is easily to see that the first eigenvalue has the minimum
error value ( ) that corresponding to the minimum eigenvector, which is selected as the
0 2 4 6 8 10 1210
-4
10-3
10-2
10-1
100
101
X: 1
Y: 0.0001667
Index of eigenvalues
Ma
gn
itu
de
Yu Deyue Estimating the Frequency Response of a Receiver
14
solution of the estimated channel. The other eigenvalues have larger magnitude compare to the first
one.
3.1.1.1 Two-multichannel outputs FIR system
Applying the blind identification algorithm by using two-multichannel outputs FIR system through the
Matlab simulation the performance of blind identification is shown as follows.
Figure 3.3 Frequency response of identified channel using two-multichannel outputs FIR system method
compare with Frequency response of the exact channel. Both are in magnitude and phase. Comparison is shown
between exact channel and identified channel in noise free situation.
In order to establish the performances of the blind identification technique, the frequency response
function of the exact and identified channel using two-multichannel outputs FIR system are plotted
together in Figure.3.3. In magnitude plot, both the two filters behave the low-pass response of the
channel in frequency domain. The frequency response of identified channel follows the shape of the
exact channel. However, there is about dB magnitude deviation (magnitude error) between exact
and identified channel frequency response. The identified channel is not flat as the exact channel. In
phase plot, a big phase error occurred when doing the blind identification for the identified channel
and it appears to be a nonlinear phase. The constant arbitrary factor is found . The RMSE is
calculated dB.
0 0.2 0.4 0.6 0.8-1000
-500
0
Normalized Frequency ( rad/sample)
Ph
ase
(d
eg
ree
s)
0.2 0.4 0.6 0.8-15
-10
-5
0
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
Exact
Identified
Yu Deyue Estimating the Frequency Response of a Receiver
15
Figure 3.4 Channel coefficients of the identified channel using two-multichannel outputs FIR system method
compare with channel coefficients of the exact channel in time domain in noise free situation. Comparison is
taken in real part. Channel order is .
The comparison of channel coefficients between the exact channel and the identified channel that is
using two-multichannel outputs FIR system blind identification method is shown in Figure.3.4.
Comparison is taken only in magnitude of the real part. The imaginary part presents the computer
noise in Matlab simulation that is too much small closes to zero, and can be ignored. It is observed that
the real parts of identified channel coefficients have larger errors compared to the exact channel
coefficients at the 2th, 4
th, 6
th, and 8
th number of elements.
3.1.1.2 Six-multichannel outputs FIR system
Applying the blind identification algorithm by using six-multichannel outputs FIR system through the
Matlab simulation the performance of blind identification is shown as follows.
2 4 6 8 10 12
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Index of elements
Ma
gn
itu
de
(re
al)
Exact
Identified
Yu Deyue Estimating the Frequency Response of a Receiver
16
Figure 3.5 Frequency response of identified channel using six-multichannel outputs FIR system method compare
with Frequency response of the exact channel. Both are in magnitude and phase. Comparison is shown between
exact channel and identified channel in noise free situation.
In order to establish the performances of the blind identification technique, the frequency response
function of the exact and identified channel using six-multichannel outputs FIR system are plotted
together. See Figure.3.5, in magnitude plot, both the two filters behave the low-pass response of the
channels in frequency domain. The frequency response of identified channel follows the shape of the
exact channel. There is about dB magnitude deviation (magnitude error) between exact and
identified channel frequency response. The identified channel is as flat as the exact channel, they are
very much close. In phase plot, a phase error occurred when doing the blind identification for the
identified channel at the high frequency and it appears to be a linear phase. The constant arbitrary
factor is found . The RMSE is calculated dB that indicates a better channel
estimation than previous efforts.
0.2 0.4 0.6 0.8
-1000
-500
0
Normalized Frequency ( rad/sample)
Ph
ase
(d
eg
ree
s)
0.2 0.4 0.6 0.8-8
-6
-4
-2
0
2
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
Exact
Identified
Yu Deyue Estimating the Frequency Response of a Receiver
17
Figure 3.6 Channel coefficients of the identified channel using six-multichannel outputs FIR system method
compare with channel coefficients of the exact channel in time domain in noise free situation. Comparison is
taken in real part. Channel order is .
The comparison of channel coefficients between the exact channel and the identified channel that is
using six-multichannel outputs FIR system blind identification method is shown in Figure.3.6. The
real parts of identified channel coefficients are very close to exact channel coefficients.
3.1.1.3 Comparison
When the order of the identified channel is known, in noiseless situation, Several SIMO FIR systems
are tested through Matlab simulation. They have different performance measure in minimum
eigenvalue and RMSE value. See Table 3.1.
The number of
multichannel
output FIR
Two-
multichannel
Three-
multichannel
Four-
multichannel
Five-
multichannel
Six-
multichannel
Minimum
eigenvalue
RMSE
(in dB)
Table 3.1 Comparison of using different SIMO FIR systems in minimum eigenvalue, RMSEs.
2 4 6 8 10 12-0.2
0
0.2
0.4
0.6
0.8
Index of elements
Ma
gn
itud
e (
rea
l)
Exact
Identified
Yu Deyue Estimating the Frequency Response of a Receiver
18
See the Table 3.1, the minimum eigenvalue for every specific SIMO FIR system is close to zero. By
using two-multichannel FIR outputs system method for the blind identification, but it has the largest
RMSE value, which means the largest error of the identification. By adding the numbers of the
multichannel outputs FIR system up to six, the minimum eigenvalue is increased, but still close to zero
and the RMSE is reduced to dB with the range that becomes more and more smaller, but it
yields about dB improvement in RMSE. There is no regular pattern to follow for minimum
eigenvalue in different cases. But the RMSE value is decreased when the number of the multichannel
outputs FIR system is increased. The six-multichannel outputs FIR system gives the best estimation of
the unknown channel compare to the others. There is no necessary to add more numbers of the
multichannel outputs FIR system for blind identification, because the reduced RMSE value compare
with using the six-multichannel method may smaller than dB.
3.1.2 Noisy channel
In the real world, the communication channel will not be noiseless. When the SIMO FIR systems are
constructed from the signal Equation (3.1), with equal amplitude ( = , constant and zero
phases ( ). The oversampled output signal is going through a noisy channel, by adding the white
Gaussian noise. SNR level is increased from dB up to dB with steps dB. The RMSE is
calculated in a Monte-Carlo simulation with times at each specific SNR level.
3.1.2.1 Eigenvalues
The solution of the blind identification algorithm is the eigenvector that corresponds to minimum
eigenvalue of correlation matrix. In noisy situation, the level of SNR compensates for the bias in the
eigenvalues due to the different noise levels at the multichannel outputs. For this reason, the minimum
eigenvalues at different SNR levels in six-multichannel outputs FIR system is shown followed. We
will analyze how the SNR affects the minimum eigenvalue. Furthermore, the minimum eigenvalue
affects the performance of the blind identification.
Yu Deyue Estimating the Frequency Response of a Receiver
19
Figure 3.7 Comparison of eigenvalues formed by using six-multichannel outputs FIR system in noisy situation at
different SNR values. Channel order is .
All the eigenvalues of correlation matrix formed by six-multichannel outputs FIR system are indicated
in Figure.3.7. It is shown the comparison of all eigenvalues at different SNR levels in six-multichannel
FIR outputs system. This Figure is plotted as magnitude versus eigenvalues. It is easily to see that the
first number of eigenvalues has the minimum error value that is reduced by increasing the SNR level
of the output data to high values with steps dB, and the reduced range becomes more and more
smaller for each step. The rest of eigenvalues’ error values are not influenced by the high SNR level
very much. The blind algorithm for channel estimation search for the minimum eigenvalue, thus the
high SNR level is required for the accuracy blind identification. These smallest eigenvalues are
relative to the noise power in the same criterion as described in some eigenvalue algorithms for
frequency estimation [9]. The smallest eigenvectors is treated as the noise eigenvectors that, ideally,
have eigenvalues will only approximately equal noise power.
0 2 4 6 8 10 1210
-3
10-2
10-1
100
101
Index of eigenvalues
Ma
gn
itu
de
SNR (0 dB)
SNR (5 dB)
SNR (10 dB)
SNR (15 dB)
SNR (20 dB)
SNR (25 dB)
Yu Deyue Estimating the Frequency Response of a Receiver
20
3.1.2.2 SIMO systems
Applying the blind identification algorithm to the output by using several different SIMO FIR systems
in noisy situation at different SNR levels, the RMSEs are calculated by applying the Equation (3.2),
and the results are plotted together in one figure for comparison, see the Figure.3.8 below.
Figure 3.8 The comparison of RMSEs that are calculated by using several different SIMO FIR systems in noisy
situation.
The RMSE is the performance measure of the identified channel. The comparison of RMSE values
that are calculated by using different SIMO output FIR systems is shown in Figure.3.8. The figure is
plotted as the RMSEs versus different SNR levels. It is observed that for every specific SIMO FIR
system, the RMSE value could be reduced by increasing the SNR from a low level to a threshold. For
the blind identification using several different SIMO output FIR systems, the RMSE is reduced by an
increased SNR level from dB up to a certain SNR level (around 23 dB), an further increase of the
SNR level will not yield any improvement in the RMSE. At the SNR level of the output signal dB,
the RMSE can be reduced approximately dB by using the six-multichannel outputs FIR system
instead of the two-multichannel outputs FIR system. Once you start to increase the SNR from a low
0 10 20 30 40 50 60-30
-25
-20
-15
-10
-5
0
5
SNR (dB)
RM
SE
(d
B)
2 multichannel
3 multichannel
4 multichannel
5 multichannel
6 multichannel
Yu Deyue Estimating the Frequency Response of a Receiver
21
level to high, the RMSE is reduced very fast at the low SNR level ( up to dB) for using the six-
multichannel outputs FIR system compare with the two-multichannel outputs FIR system. At the high
SNR level dB, the RMSE is reduced approximately dB by using the six-multichannel outputs
FIR system instead of the two-multichannel outputs FIR system. At the high SNR level (higher than
dB), the range of RMSE reduction becomes more and more smaller if you increasing the numbers
of the multichannel outputs FIR system from two to six.
3.1.2.3 Comparison
According to the Figure.3.8, a table is created to show the RMSEs that are calculated from the
Equation (3.2) by using the different SIMO FIR systems at different SNR level of output, which could
be easily used to compare the results. See Table 3.2.
The number of
multichannel
output
Two-
multichannel
Three-
multichannel
Four-
multichannel
Five-
multichannel
Six-
multichannel
RMSE (in dB) at
𝑆 dB
RMSE (in dB) at
𝑆 dB
Table 3.2 Comparison of RMSEs by using several different SIMO FIR systems in noisy situation.
3.1.2.4 Oversampling influence
The blind identification algorithm is based only on the output of the unknown channel. The output
signal must be oversampled for a good identification. So the oversampling of the output may affect the
performance of the blind identification. The RMSEs of identified channel using only the output signal
with different numbers of samples in six-multichannel outputs FIR system is shown as follows.
Yu Deyue Estimating the Frequency Response of a Receiver
22
Figure 3.9 RMSEs of six- multichannel outputs FIR system using the output with different sampling rate in noisy
channel.
Oversampling of the observed output signal is important for blind identification. See Figure.3.9, for
the larger number of samples in the output, results a smaller value of RMSE, when the SNR level is
lower than dB. The largest improvement of the reduction of RMSE value that using the highest
oversampling rate compare to the smallest is about dB at the SNR level dB. The RMSE reduction
range is decreasing when the SNR level is increased from dB up to dB. It is no need to have a
larger oversampling rate of the output when the SNR level is higher than dB, the RMSE will not be
reduced any more.
3.1.2.5 Identified channel (𝑆 dB)
From the Figure.3.8, it is easily to see, when using the six-multichannel outputs FIR system, the blind
identification has the best performance compare to the others. For a specific SNR level of the output
signal at dB, the RMSE is dB that results a very good identification of the unknown channel.
For this reason, a figure is plotted to show how the performance of the identified channel is at the SNR
level dB. The figure is shown as follows.
0 10 20 30 40 50 60-30
-25
-20
-15
-10
-5
0
SNR (dB)
RM
SE
(d
B)
300 samples
600 samples
1200 samples
1500 samples
3000 samples
Yu Deyue Estimating the Frequency Response of a Receiver
23
Figure 3.10 Frequency response of identified channel using six-multichannel FIR outputs system method at
𝑆 dB compare with frequency response of the exact channel. Both are in magnitude and phase.
Comparison is shown between exact channel and identified channel in noisy situation. The number of samples is
.
In order to establish the performances of the blind identification technique, the frequency response
function of the exact channel and identified channel using six-multichannel outputs FIR system at
𝑆 dB in noisy channel are plotted together. See Figure.3.10, in magnitude plot, both the two
filters behave the low-pass response of the channels in frequency domain. The frequency response of
identified channel follows the shape of the response of the exact channel. However, there is about
dB (approximately) magnitude deviation between exact and identified channel frequency response.
The identified channel is as flat as the exact channel. In phase plot, a phase error occurred when doing
the blind identification for the identified channel at the high frequency and it appears to be a linear
phase. The constant arbitrary factor is found . The RMSE is calculated dB that results a
good estimation of the channel.
0 0.2 0.4 0.6 0.8
-1000
-500
0
Normalized Frequency ( rad/sample)
Ph
ase
(d
eg
ree
s)
0.2 0.4 0.6 0.8 1
-3
-2
-1
0
1
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
Exact
Identified (SNR=10 dB)
Yu Deyue Estimating the Frequency Response of a Receiver
24
Figure 3.11 Channel coefficients of the identified channel using six-multichannel outputs FIR system method
compare with channel coefficients of the exact channel in time domain in noise free situation. Comparison is
between real part and imaginary part. The number of samples is . Channel order is .
The comparison of channel coefficients between the exact channel and the identified channel that is
using six-multichannel outputs FIR system blind identification method at 𝑆 dB in noisy
situation is shown in Figure.3.11. The real parts of identified channel coefficients are very close to
exact channel coefficients.
3.1.3 Consider the different feature input
For all the simulations above, the SIMO FIR systems are constructed from the signal Equation
(3.1), with equal amplitude ( = , constant and zero phases ( ), and the results were
shown above. If we consider the input signal is non-periodical, the performance of blind identification
still maintain good. But the blind identification failed when the input signal has a random phase
behavior.
0 2 4 6 8 10 12-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Index of elements
Ma
gn
itu
de
(re
al)
Exact
Identified (SNR=10 dB)
Yu Deyue Estimating the Frequency Response of a Receiver
25
3.2 Channel order unknown
The blind identification algorithm is based only the oversampled output of the channel. When the
order of the identified channel is known, the blind identification becomes straight forward. On the
other hand, if the order of the identified channel is unknown, then the blind identification becomes
very much difficult. First of all, the unknown channel order must be properly selected. The idea of
determining the unknown channel order is to repeat doing the blind identification by increasing the
number of elements in every sub-multichannel from to infinity until you find the smallest RMSE
value. RMSE is calculated using Equation (3.2). There is no limitation for how big the number of the
elements in every sub-multichannel should go. The RMSE values that are calculated for different
number of elements in every sub-multichannel using different SIMO FIR systems are shown as
follows.
Figure 3.12 RMSEs calculation for unknown channel order selection in noiseless situation. (a)Using
oversampled output into four-multichannel outputs FIR system. (b) Using oversampled output into five-
multichannel outputs FIR system. (c) Using oversampled output into six-multichannel outputs FIR system.
2 4 6 8 10-30
-20
-10
0
Number of elements in each sub-channel(a)
RM
SE
(dB
)
2 4 6 8 10-30
-20
-10
0
Number of elements in each sub-channel(b)
RM
SE
(dB
)
2 4 6 8 10-30
-20
-10
0
Number of elements in each sub-channel(c)
RM
SE
(dB
)
6 multichannel (noiseless)
5 multichannel (noiseless)
4 multichannel (noiseless)
Yu Deyue Estimating the Frequency Response of a Receiver
26
In noiseless situation, the RMSEs are calculated for unknown channel order selection that is shown in
Figure.3.12. The smallest RMSE results a proper channel order selection, of course it may give
reasonable blind identification. From Figure.3.12.a, it is easily to see, at the 3th element, the RMSE
reaches its deepest level, it is calculated as the smallest dB, so the identified channel order is
selected as ( ). From Figure.3.12.b, it is easily to
see, at the 2th element, the RMSE reaches its deepest level, it is calculated as the smallest dB, so
the identified channel order is selected as . From Figure.3.12.c, it is easily to see, at the 2th element,
the RMSE reaches its deepest level, it is calculated as the smallest dB, so the identified channel
order is selected as . By the comparison, using the six-multichannel outputs FIR system gives the
best performance of blind identification with proper channel order selection. It is noticed that the exact
channel length has elements, with zero or much closed to zeros for last two elements, so when
using the five-multichannel outputs FIR system, the last two elements are ignored, which may not
affect the blind identification.
Yu Deyue Estimating the Frequency Response of a Receiver
27
Figure 3.13 RMSEs calculation for unknown channel order selection in noisy situation with a 𝑆 dB.
(a)Using oversampled output into four-multichannel outputs FIR system. (b) Using oversampled output into five-
multichannel outputs FIR system. (c) Using oversampled output into six-multichannel outputs FIR system.
In noisy situation, it is known from the previous simulation that the blind identification performed well
at dB SNR level. So at the SNR level of the output signal dB, the RMSEs are calculated for
unknown channel order selection that is shown in Figure.3.13. It is observed that Figure.3.13 is very
similar to the Figure.3.12. As we known the RMSE is the performance measure of the blind
identification, which also could be used for unknown channel order selection. The smallest RMSE
results a proper channel order selection, of course it may give reasonable blind identification. The
order selection and blind identification are done in the same way as the noiseless situation
(Figure.3.12). From Figure.3.13.a, it is easily to see, at the 3th element, the RMSE reaches its deepest
2 4 6 8 10-30
-20
-10
0
Number of elements in each sub-channel(a)
RM
SE
(d
B)
2 4 6 8 10-30
-20
-10
0
Number of elements in each sub-channel(b)
RM
SE
(d
B)
2 4 6 8 10-30
-20
-10
0
Number of elements in each sub-channel(c)
RM
SE
(d
B)
4 multichannel (SNR=20 dB)
5 multichannel (SNR=20 dB)
6 multichannel (SNR=20 dB)
Yu Deyue Estimating the Frequency Response of a Receiver
28
level, it is calculated as the smallest dB, so the identified channel order is selected as . From
Figure.3.13.b, it is easily to see, at the 2th element, the RMSE reaches its deepest level, it is calculated
as the smallest dB, so the identified channel order is selected as . From Figure.3.13.c, it is
easily to see, at the 2th element, the RMSE reaches its deepest level, it is calculated as the smallest
dB, so the identified channel order is selected as . By the comparison, using the six-
multichannel outputs FIR system gives the best performance of blind identification with proper
channel order selection. It is noticed that the exact channel used for calculating the RMSEs here is the
same as the one used in noiseless situation (Figure.3.12).
Yu Deyue Estimating the Frequency Response of a Receiver
29
4 Measurements
In the previous Matlab simulations, the blind identification algorithm performed very well in both
noiseless and noisy situation (at high SNR level). So the blind algorithm shows promise in measured
data experiments. In this chapter, the performance of the blind identification algorithm will be verified
in the experimental measurements.
4.1 Hardware device introduction
The ADQ 214 data acquisition card is used as a receiver in this thesis. It has two channels for data
acquisition card featuring two -bits, MSPS capture rate ADC converters, and a high speed USB
2.0 interface. It is suitable to be used in RF sampling of RF signals or high speed data recording for the
high input bandwidth GHz, and MSamples of memory buffer per channel. The ADQ 214 is
equipped with two advanced Xilinx Virtex5 LX50T FPGA that are available for customized
applications [10].
4.2 Measurement set-up
In this section, the measurement set-up is built to collect the output data for blind identification. The
ADQ 214 data acquisition card is used as a receiver to collect the signal, which is oversampled at
MHz. The signal is modulated by the SMU 200A (vector signal generator with good quality)
equipment with a carrier frequency MHz. The baud rate is MHz. The experiment setting up
is shown in the figure as follows.
Yu Deyue Estimating the Frequency Response of a Receiver
30
Figure 4.1 The experimental measurement setting up of collecting the data. AWG is known as standard
American wire gauge.
4.3 Noise reduction
Noise is present in RF experiment as in this case. A noise averaging technique is employed to improve
the level of SNR in the output signal for an accurate identification. This averaging technique will
reduce the noise level in the collecting data without affecting the original signal. It should be
considered that the signal and noise are uncorrelated, the signal is periodic, and noise is assumed to
have zero mean with some unknown variance.
The noise level can be expressed by
∑
Where is the measurement data in frequency domain, and is a frequency location that does
not contain any input signal. Hence, it contains noise and distortion, is excluded.
Clock
PC
AWG Modulator
(SMU 200A)
Receiver Device
(ADQ 214) Signal
Generator
Yu Deyue Estimating the Frequency Response of a Receiver
31
Figure 4.2 Noise level versus number of averaging.
The noise averaging technique is employed to the measured data. The result of noise level reduction is
shown in Figure.4.2. It is easily to see that the noise level is reduced by increasing the number of
averaging. The level of noise is reduced very fast at the beginning of increasing the number of
averaging. Without employing the noise averaging technique, the noise level is at around dB.
When employing the noise averaging technique, the noise level could be reduced around dB, which
is down to dB at a specfic number of averaging, and there is no much improvement of noise
reduction if you continue to increase the number of averaging to an even high value (big number of
averaging requires long computing time). In this experiment, the number of averaging is chosen
for a tradeoff of low noise level and short computing time.
4.4 Receiver identification
When the blind identification algorithm is implemented in the measured data experiment, the real data
are collected from the RF experiment. The measured data may suffer from unknown phase shift. For
the unknown phase shift in the RF equipment, a phase correction method is employed to reduce or
even remove the phase error. An input signal with zero phase elements passes directly through the RF
0 200 400 600 800 1000-60
-55
-50
-45
-40
-35
Number of averaging
No
ise
le
ve
l (d
B)
Yu Deyue Estimating the Frequency Response of a Receiver
32
equipment to the output with some phase information, those phase information are used for phase
correction. For the rest of data measurement, the input signal with a minus phase elements of the
measured phase information passing through the RF equipment may remove the phase error in the
output. A good quality of the measured output data may give a good performance of blind
identification.
For the blind identification of the experimental measurement, the collected data in the receiver is the
only one that is used for the receiver identification. The Figure.4.3 followed shows the spectrum of the
collected data in the receiver. This data is used to form a correlation matrix for the blind identification.
The blind identification algorithm described in Section 2.2 is employed for the receiver identification.
The solution of the algorithm is the eigenvector that corresponds to the minimum eigenvalue, which is
the identified receiver impulse response. Since the order of the receiver is unknown, the method that
described in Section 3.2 is employed to select the proper order of the receiver.
4.4.1 Received data
The data is measured by using the ADQ 214 data acquisition card. The SNR level of the measured
data is the most important that we are interested in. A high level of SNR, may give a good blind
identification. The collected data is oversampled that is shown as follows.
Yu Deyue Estimating the Frequency Response of a Receiver
33
Figure 4.3 The spectrum of collected data in the receiver.
The spectrum of collected data in the receiver that is used for blind identification is shown in
Figure.4.3. The signal level is at about dB. The noise level is at a low level, this results a sufficient
SNR level of the collected data. According to the previous simulation results, we have a preliminary
judgment that the blind identification may give a good estimation by using this collected data.
4.4.2 Existing channel
Consider the existing time-invariant linear system is modeled by the FIR system, which could be
obtained by using the Equation (2.28). This time-invariant linear system is the exact filter in the RF
measurement system, which is used to calculate the RMSEs that could measure the performance of the
identified system .
0 100 200 300 400 500 600 700-70
-60
-50
-40
-30
-20
-10
0
10
Frequency (MHz)
Ma
gn
itu
de
(d
B)
Yu Deyue Estimating the Frequency Response of a Receiver
34
Figure 4.4 The existing time-invariant linear system in frequency domain that is obtained from the input and
output.
The existing time-invariant linear system in frequency domain is shown in Figure.4.4. The
calculated linear system is the complete combination of the SMU 200A and the ADC receiver. The
frequency response of the linear system seems to be nearly flat in the pass band. For all of input
signal pass directly through the system to the output, result an increasing gain in the pass band that
is nearly up to dB. The phase angle of the system appears to be a nonlinear phase with a
maximum out of phase.
0 0.2 0.4 0.6 0.8 1-2
0
2
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
0 0.2 0.4 0.6 0.8 1-20
-10
0
10
Normalized Frequency ( rad/sample)
Ph
ase
(d
eg
ree
)
Yu Deyue Estimating the Frequency Response of a Receiver
35
Figure 4.5 The impulse response of the existing time-invariant linear system that is obtained from the input
and output. This Figure shows the positive part of the impulse response of the system .
The impulse response of the existing time-invariant linear system is shown in Figure.4.5. It is
observed that this existing time-invariant linear system is very close to impulse response. See the
Figure.4.5, the magnitude is dB at element. Since the noise level of measured data
(Figure.4.3) is at about dB, so the number of the elements is limited to . For this reason, the
number of the coefficients of the identified receiver is limited up to .
4.4.3 Order selection
The collected data in the receiver is only used for blind identification. It should be noticed that the
identified receiver’s frequency response is the complete combination of the SMU 200A and the ADC
receiver. Since the order of the identified receiver is unknown that results a difficulty for blind
identification. As the previous simulations, the idea of determining the unknown receiver order is to
repeat doing the blind identification by increasing the number of elements in every sub-multichannel
from to infinity until you find the smallest RMSE value. From Figure.4.5, we know that the
1 2 3 4 5 6-40
-35
-30
-25
-20
-15
-10
-5
0
5
Index of elements
Ma
gn
itu
de
of im
pu
lse
ele
me
nts
(d
B)
Yu Deyue Estimating the Frequency Response of a Receiver
36
limitation of the elements is . The RMSE values for different number of elements in each sub-
multichannel using different SIMO FIR systems are shown as follows.
Figure 4.6 RMSEs calculation for unknown receiver order selection using different SIMO FIR systems in
measured data experiment.
The performance of blind identification for unknown order of receiver is shown in Figure.4.6. It is
easily to see, when the number of elements in each sub-multichannel is , the blind identification has
good performance for all different SIMO FIR systems. When the number of elements in each sub-
multichannel is increased, the RMSE values become larger that results worse identification, but these
values seem to be concentrated at the big number of elements in each sub-multichannel. Using the
four-multichannel outputs FIR system, the receiver order is selected as , the RMSE value is the
smallest dB. Using the five-multichannel outputs FIR system, the receiver order is selected as
, the RMSE value is dB. Using the six-multichannel outputs FIR system, the receiver order is
selected as , the RMSE value is dB.
2 4 6 8 10-40
-38
-36
-34
-32
-30
-28
-26
-24
-22
Number of elements
RM
SE
(d
B)
6 multichannel
5 multichannel
4 multichannel
Yu Deyue Estimating the Frequency Response of a Receiver
37
4.4.4 Identified receiver models
From the Figure.4.6, it is easily to see the blind identification has a good performance for all SIMO
FIR systems when the number of elements in each sub-multichannel is . So there are three options of
systems for the identified receiver. The three options of the finite impulse response of identified
receiver using different SIMO FIR systems that have only one element in each sub-multichannel are
shown in Table 4.1.
Blind identification algorithm Finite impulse of identified receiver
Four-multichannel FIR system
Five-multichannel FIR system
Six-multichannel FIR system
Table 4.1 The finite impulse response of the identified receiver in three options. Three different SIMO FIR
systems are used for the receiver identification.
All the three identified receivers’ finite impulse response are similar and they are all close to the
impulse response.
4.4.4.1 Frequency response
The frequency responses of the three identified receiver systems are shown followed. They are
compared to the existing time-invariant linear system .
Yu Deyue Estimating the Frequency Response of a Receiver
38
Figure 4.7 The frequency response of identified receiver using different SIMO FIR systems compare with the
frequency response of linear system . Both are in magnitude and phase. The comparison is shown between the
linear system and identified receivers.
The comparison of the three options of identified receiver systems and the existing time-invariant
linear system in frequency domain is shown in Figure.4.7. It is easily to see, the shapes of three
identified receiver follow the existing linear system , and they are nearly flat in the pass band. The
maximum gain of each identified receiver is increased when the number of the multichannel outputs
FIR system is increased. The normalized frequency of each identified receiver that corresponding to
maximum gain is increased as well when the number of the multichannel outputs FIR system is
increased. The phase angle of each identified receiver is nonlinear and with an increased maximum out
of phase at 0.9 (normalized frequency). Each maximum gain at the specific normalized frequency and
the maximum out of phase are shown in Table 4.2.
0 0.2 0.4 0.6 0.8-40
-20
0
20
Normalized Frequency ( rad/sample)
Ph
ase
(d
eg
ree
s)
0 0.2 0.4 0.6 0.8-6
-4
-2
0
2
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
Linear system
Identified (4 multichannel)
Identified (5 multichannel)
Identified (6 multichannel)
Yu Deyue Estimating the Frequency Response of a Receiver
39
Maximum gain (dB) normalized frequency at
Maximum gain (𝜋 rad/sample)
Maximum out of phase
(degree)
Four-multichannel FIR
system
Five-multichannel FIR
system
Six-multichannel FIR system
existing time-invariant linear
system
Table 4.2 Comparison of different parameters in Figure.4.7.
4.4.5 Validation of the identified receiver models
Another method is employed to check the performance of the three options of identified receiver
systems, which is called system identification. See the scheme in Figure.4.8 below. An input signal
passed directly through the unknown linear system to the output signal , the blind
identification algorithm is employed to identify the system from the output . Another input
signal passed through both unknown system and the identified system , the two outputs
and are compared for the validation of the identified system .
Figure 4.8 The scheme of the system identification. is used for system blind identification to find the
identified system . and are used for the validation of identified system .
Applying the method of system identification, an input signal passed respectively through the three
systems of identified receiver and the unknown linear system , the three outputs that corresponding
to the three identified systems respectively are compared to the measured data in the receiver that
through the unknown system . The figure is shown as follows.
Identification
Validation
nnn
Yu Deyue Estimating the Frequency Response of a Receiver
40
Figure 4.9 The spectrum of output signals through different systems. The comparison of output signals is shown
between the unknown system output and the three identified systems outputs.
The comparison of output signals is shown between the measured data in the receiver and the three
identified systems outputs. See the Figure.4.9, the output signals that contain energy are at the level
about . The noise level is below dB results a larger SNR value. Since the outputs
comparisons that contain energy are difficult to see in Figure.4.9, an enlarged view is shown in the
figure as follows.
0.2 0.4 0.6 0.8 1
-60
-50
-40
-30
-20
-10
0
10
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
Output of existing linear system
Output of identified (4 multichannel)
Output of identified (5 multichannel)
Output of identified (6 multichannel)
Yu Deyue Estimating the Frequency Response of a Receiver
41
Figure 4.10 The comparison of output signals is shown between the measured data in the receiver and the three
identified systems outputs.
The outputs through the three identified systems and measured data in the receiver are shown in
Figure.4.10. The shapes of all outputs through the three identified systems are tried to follow the shape
of measured data in the receiver. The variant of the output through identified system is increased as the
number of the multichannel outputs FIR system is increased. This variant may result a negative in
accuracy, however the blind identification is not as accurate for the hardware devices influence (noise).
The output through the identified channel using six-multichannel outputs FIR system gives the highest
maximum gain compare to the other models’ outputs.
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
8
Normalized Frequency ( rad/sample)
Ma
gn
itu
de
(d
B)
Output of existing linear system
Output of identified (4 multichannel)
Output of identified (5 multichannel)
Output of identified (6 multichannel)
Yu Deyue Estimating the Frequency Response of a Receiver
42
5 Discussion
The advantage of the blind identification in this thesis is less consumption of receivers or sensors, and
the algorithm provides exact identification of a possibly non-minimum phase as well as non-causal
channel, whenever the large number of SIMO FIR system the more identification accuracy. However,
the tested blind identification algorithm is computationally intensive that suffers from the fact that the
identification of higher order statistics usually converge slower than those lower order statistics.
Moreover, the received signal may be sensitive to the uncertainties that associated with timing
recovery, unknown phase jitter, and frequency offset. Finally, the effect of non-Gaussian noise may
affect the performance of blind identification [11].
In order to have a reasonably and accurate blind identification, the received signal has to be
oversampled, it is shown in [12] that the oversampling provides good immunity to noise, interference
and frequency selective fading. Hence, at low SNR level, large numbers of samples of received signal
are required for multichannel FIR system blind identification.
In the experimental measurement, the blind algorithm leads to a robust solution to either receiver order
or receiver’s frequency response. The algorithm can be used with any persistently exciting input signal
and can be easily generalized to an arbitrary number of FIR channels. The cost of enhanced accuracy
will probably be a significant increase in the computational requirements compared to many of other
algorithms.
It should be noticed that the identified receiver’s impulse response is the complete combination of the
SMU 200A (signal generator) and the ADC receiver. There is no knowledge to separate the
combination at present so that we can obtain only the impulse response of the identified receiver. If we
assume that the SMU 200A is the exact impulse response, then the estimated receiver’s frequency
response is exactly close to the real. Fortunately, the identified FIR system of the receiver device in
the measured data experiment is a minimum phase system, which could definitely restore the input
signal.
Yu Deyue Estimating the Frequency Response of a Receiver
43
6 Conclusions
In this thesis, the blind identification algorithm is presented and tested for identifying a single output
channel (the output of signal channel is transformed into multichannel outputs FIR systems) with
unknown input knowledge. Therefore, the input is no need for the system identification in this thesis.
The solution of blind identification algorithm is the eigenvector that corresponds to minimum
eigenvalue of correlation matrix, which is formed by the observed outputs that are oversampled.
Hence, the blind identification algorithm is employed in Matlab simulations and RF experiment.
Several transformed SIMO FIR systems are tested and compared to find a good performance of blind
identification. With known channel order, the blind identification is straight forward. But with
unknown channel order, the blind identification becomes more intricate. However, applying the
RMSE method and with proper channel order selection, the algorithm could accomplish the blind
identification reasonably. The channels that need be identified are allowed to be non-minimum phase
as well as non-causal, which make the blind identification algorithm particularly suitable for
applications such as echo problem.
In Matlab simulation, several useful results characterize that the large number of SIMO FIR system
give the perfect performance of blind identification. In noisy situation, the blind identification
algorithm performs well when SNR level is high. However, the performance of blind identification is
limited when the SNR level is below a threshold. For the blind identification algorithm to be effective
at low SNR level, a larger number of samples at outputs are necessary, which may limit the
effectiveness of the algorithm for rapidly varying channels.
It is observed through experimental measurement results that the eigenvector based algorithm tested in
this thesis seems to be a useful algorithm that could give a reasonable identification of the frequency
response of the receiver device. The identified system is close enough to the exact system, which is
close to the impulse response as well.
The useful Matlab simulations and RF experimental results demonstrate the potential of the blind
algorithm.
Yu Deyue Estimating the Frequency Response of a Receiver
44
7 Future work
Although the blind algorithm provides good channel identification, the performance of the algorithm
when a small number of samples (limitation of samples) is used needs to be further examined from
both theoretical and experimental points.
The choice of the parameters (channel order or receiver order) involved by the blind algorithm is still
need a further research.
In the measured data experiment, it seems we identified the whole system that is the convolution of the
SMU 200A and ADQ 214. The further work could be finding a method to separate the combination of
the convolution so that we can obtain the exact frequency response of the receiver
Yu Deyue Estimating the Frequency Response of a Receiver
45
References
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[2] A. Benveniste and M. Goursat, ‘‘Robust identification of a nonminimum phase system: Blind
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[3] T. Öberg, Modulation, Detection and Coding. Baffins Lane, Chichester, West Sussex, PO19 1UD,
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[8] J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and
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[9] R. Schmidt, ‘‘Multiple emitter location and signal parameter estimation,’’ Proc. RADC Spectrum
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[10] ‘‘ADQ 214 High-speed data acquisition card’’. Productsheet. Linköping, Sweden: Signal
Processing Devices Sweden AB. 2009.
[11] L. Tong, G. Xu, T. Kailath, ‘‘Blind identification and equalization based on second-order
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[12] W. A. Gardner, W. A. Brown, ‘‘Frequency-shift filtering theory for adaptive co-channel
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Asilomar Conf. Signals, Systs., and Comput., Pacific Grove,
CA, Oct. 1989, pp. 562-567.
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