Estimating the Frequency Response of a Receiverhig.diva-portal.org/smash/get/diva2:557446/FULLTEXT01.pdf · Yu Deyue Estimating the Frequency Response of a Receiver 1 1 Introduction

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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

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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.


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.


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


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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


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1

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


2

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


3

(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


4

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.


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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].


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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,

[ ]

[ ]


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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

‖ ‖


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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

,

, ...,


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


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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].


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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.


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)


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


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


Ma

gn

itu

de

(d

B)

Exact

Identified


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


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.


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


Ph

ase

(d

eg

ree

s)

0.2 0.4 0.6 0.8-8

-6

-4

-2

0

2


Ma

gn

itu

de

(d

B)

Exact

Identified


17

Figure 3.6 Channel coefficients of the identified channel using six-multichannel outputs FIR system method


taken in real part. Channel order 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


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.


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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)


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


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.


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


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

.


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


Ph

ase

(d

eg

ree

s)

0.2 0.4 0.6 0.8 1

-3

-2

-1

0

1


Ma

gn

itu

de

(d

B)

Exact

Identified (SNR=10 dB)


24

Figure 3.11 Channel coefficients of the identified channel using six-multichannel outputs FIR system method


between real part and imaginary part. The number of samples is . Channel order 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)


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)




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.


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)




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).


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.


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


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)


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.


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)


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


Ma

gn

itu

de

(d

B)

0 0.2 0.4 0.6 0.8 1-20

-10

0

10


Ph

ase

(d

eg

ree

)


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)


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


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 .


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


Ph

ase

(d

eg

ree

s)

0 0.2 0.4 0.6 0.8-6

-4

-2

0

2


Ma

gn

itu

de

(d

B)

Linear system

Identified (4 multichannel)




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


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


Ma

gn

itu

de

(d

B)

Output of existing linear system

Output of identified (4 multichannel)




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


Ma

gn

itu

de

(d

B)

Output of existing linear system





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.


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.


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


45

References

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May. 2004, vol. 5, pp. V-673-6 vol. 5.

[2] A. Benveniste and M. Goursat, ‘‘Robust identification of a nonminimum phase system: Blind

adjustment of a linear equalizer in data communications’’, IEEE Trans. Automat. Contr., pp. 385-

399, June 1980.

[3] T. Öberg, Modulation, Detection and Coding. Baffins Lane, Chichester, West Sussex, PO19 1UD,

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[4] L. Ahlin, J. Zander, B. Slimane, Principles of Wireless Communications. Lund, Sweden:

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[5] D. C. Lay, Linear Algebra and Its Applications. Boston: Pearson Education, Inc. 2006, ch. 5.

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IEEE Trans. Signal Processing, vol. 43, pp. 2982-2993, Dec. 1995.

[7] M. I. Gurelli and C. L. Nikias, ‘‘EVAM: An eigenvector-based algorithm for multichannel blind

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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

statistics: A time domain approach’’, IEEE Trans. Inform. Theory, vol. 40, PP. 340-349, Mar.

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[12] W. A. Gardner, W. A. Brown, ‘‘Frequency-shift filtering theory for adaptive co-channel

interference removal’’, in Proc. 23rd

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Estimating the Frequency Response of a Receiverhig.diva-portal.org/smash/get/diva2:557446/FULLTEXT01.pdf · Yu Deyue Estimating the Frequency Response of a Receiver 1 1 Introduction