FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT
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Estimating the Frequency Response of a Receiver
Blind Identification
Yu Deyue
September 2012
Master’s Thesis in Electronics
Abstract
Acknowledgment
I would like to thank the supervisor, Ph.D. Mr Efrain Zenteno for his constructive comments, careful reviews and scientific supervision.
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
4.4.3 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
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].
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
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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.
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
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
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,
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
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]
[ ]
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 haveThe 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.
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 ‖ ‖√ ∑‖ ‖ [∑ ]
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
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
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.
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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, 4th, 6th, and 8th 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.
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.
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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.
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
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.
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
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-six-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
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.
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.
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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
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-five-multichannel outputs FIR system.
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2 4 6 8 10 -30 -20 -10 0In 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
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-five-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
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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
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.
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
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
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
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.
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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
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.
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
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.
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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
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.
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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.
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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.
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.
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.
References
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