Estimating Influence Parameters in Asynchronous Single Path and Multiple Path Systems, Study Guides, Projects, Research of Electrical and Electronics Engineering

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EE596 Wavelet Final Project

Influence Parameter Estimation of

Asynchronous MIMO system

ID: 4193421378

Name: Kun-Han Lee

Problem Description

A. Asynchronous Single Path System Consider a system having multi-inputs and multi-outputs (MIMO). When inputs have a signal

changing with time, all outputs will be influenced by the signal with two parameters: δ and τ.

Parameter δ denotes how many percentage of the input signal will become the output signal, and

τ means the time delay of the signal from the input to the output. So when all inputs have signals

changing with time, every output is affected by all inputs with two parameters δ and τ that

belong to the input-output pair. Given that we can have some control of input signals, we want to estimate the parameter δ of all input-output pairs by using some wavelet analysis of the output signals. The single path problem is formulated as follows:

Suppose we have M inputs denote as Ii ( ) t , i =1, 2,..., M N outputs denote as P tj ( ) , j =1, 2,..., N Every output signal is influenced by inputs as the following formula:

1

M j ij i i

P t δ I t τ ij

=

= (^) ∑ − for j =1, 2,..., N

where

δ ij denotes the percentage of influence from input i to output j

τ (^) ij denotes the time delay of the signal from input i to output j The system is modeled as:

…… …… …… ……

I 1 (^) ( ) t I 2 (^) ( ) t

I (^) M ( ) t

P 1 (^) ( ) t

P 2 (^) ( ) t

PN ( ) t

M inputs

N outputs

δ 11 ,τ 11 δ 21 ,τ 21

δ (^) MN ,τ MN

δ (^2) N ,τ 2 N

δ 1 N ,τ 1 N

δ (^) M 1 ,τ (^) M (^1) δ (^) M 2 ,τ M 2

δ 22 ,τ 22

System

…… …… …… ……

I 1 (^) ( ) t I 2 (^) ( ) t

I (^) M ( ) t

P 1 (^) ( ) t

P 2 (^) ( ) t

PN ( ) t

M inputs

N outputs

δ 11 ,τ 11 δ 21 ,τ 21

δ (^) MN ,τ MN

δ (^2) N ,τ 2 N

δ 1 N ,τ 1 N

δ (^) M 1 ,τ (^) M (^1) δ (^) M 2 ,τ M 2

δ 22 ,τ 22

System

Fig.1 Asynchronous Single Path System

Given that we can have some simple control of the input signals, we want to estimate the δ ij

for each pair to know the correlation relationship of signals between each input-output pair.

B. Asynchronous Multiple Paths System Now we extended our problem to a more general case: Consider each input-output pair having more than one influence path, that is, there are multiple paths that from an input to an output, each

Given that we can have some simple control of the input signals, we want to estimate the sum

of δ ijk , that is, we want to estimate

1

K ij ijk k

=

∑ for all^ i^ =^ 1, 2,..., M and^ j^ =1, 2,..., N

For the value denoted the correlation relationship of signals between each input-output pair.

Related Works

“Good” Correlation Properties of Sequences In system engineering, a large number of problems required sets of signals (usually required to be periodic) which have one or both of the following two properties:

1. Each signal in the set is easy to distinguish from a time-shifted version of itself;
2. Each signal in the set is easy to distinguish from (a possibly time-shifted version of) every other signal in the set.

Among the applications, Radar systems and CDMA communication system are very important ones. In DS-CDMA multi-user detection, in order to recognize the data transmitting to each specific user, they want to design “good” binary codes which have “good” auto-correlation or/and “good” cross-correlation properties. The “good” auto-correlation property means the code has a low autocorrelation value when the shift is not zero or a multiple of period. The “good” cross-correlation property means the set of codes have low cross-correlation values to each other. “m-sequence” and “Gold sequence” are design of binary codes having some/all of these properties. These codes, with some modifications, are used in many different real world cases.

Compared to DS-CDMA Multi-user Detection Our work, in many aspects, is very similar to the work of DS-CDMA multi-user detection. First of all, all these two systems have multiple inputs and multiple outputs. In CDMA system, the multiple inputs are corresponding to transmitters, and multiple outputs are corresponding to different receivers. More important, when each receiver received the transmitting signal, it is the combination of all transmitting signals from all transmitters. When the channel is modeled as asynchronous multi-path channel, that is, the channel response is:

1

L k k kl kl l

h t A g δ t τ

=

= (^) ∑ −

which

A k is the amplitude; τ kl is the path delay; gkl is the path gain.

It is almost the same model as we suppose on each multiple paths channel. The theory in CDMA system has developed for many years, some ideas may be very helpful to our work, like the design of “good” codes, and this will leave as a future work for us. Although so many aspects are similar in CDMA system and our work, there are still some major differences between this two works. First of all, in CDMA system, each receiver wants to get the data that transmitted to it, which just came from one specific transmitter, so the data transmitted by other transmitters are took as interference, so there is an important issue in CDMA system which is interference cancellation. But in our work, we need to estimate the influence from ALL inputs. So some approaches in CDMA system are not useful in our work. Secondly, in real world, our system may add some “Location” information. All inputs and all

outputs may have the information of geographical locations, and inputs and outputs that are adjacent are more likely to have higher influence. CDMA system does not take this into account. Our work, as so far, also does not consider this, but we can make use of this information to improve our estimation if the location information is available. Thirdly, CDMA system often uses very long sequences as the data bases for transmitting. But it is usually forbidden on the system point of view. To have long sequences means the input signals need to be very long. But many in real world systems, we may want to identity the system by just some short sequences, because long sequences means long time, and it does not satisfy the economic benefits.

Approach and Simulation Results

A. Asynchronous Single Path System Because the system maybe be nonlinear (and usually is the case), we can not set just one input a non-zero value and all other inputs set to zero, to see the influence of this input on all outputs. Instead we need to operate our input signals near some constant operation point, to see the behavior of the system near that operation point. Besides, if it is needed, we want to change our input signals in a very simple way, so that it can apply to many different real applications, such as the oilfield case which inspired me. So we choose our input signals as a constant value plus a periodic signal, which is a Haar wavelet basis from a specific subband of wavelet packet. This periodic signal must has a very small magnitude compared to the constant DC value, to guarantee the parameters we got can approximate the behavior of system under this constant value operation point. For example, the full decomposition of Haar wavelet packet with level 3 was (Fig. 3):

Basis

Subband 1

Subband 2

Subband 3

Subband 4

Subband 5

Subband 6

Subband 7

Subband 8

Basis

Subband 1

Subband 2

Subband 3

Subband 4

Subband 5

Subband 6

Subband 7

Subband 8

Fig.3 Basis of Haar wavelet packet with decomposition level 3

Now observe the bases of Haar wavelet packet with level 3, we found that with periodic expression, subband 2 and subband 4 will be the almost same, only different with a time shift by 2, and this is the same case for subband 3 and subband 7, subband 6 and subband 8. And if we rearrange the subband from the value of center frequency, from low frequency to high frequency, we will find the almost the same bases appear as No.2 and No.3, No.4 and No.5, No.6 and No.7. That is, except for subband 1 and the subband with highest center frequency (Here is subband 5), other subbands will almost the same to one of its adjacent subband in this arrange. And we found it is also true even from composition level higher than 3. So for almost the same subbands, we only use one subband basis from them (Actually, we can not divide them apart if path has a time shift.). Now we observe the autocorrelation and periodic cross-correlation function of subband 2, subband 3, subband 5 and subband 6 of bases (Table 1):

x n yn θ x y , (0) θ x y , (1) θ x y , (2) θ x y , (3) θ x y , (4) θ x y , (5) θ x y , (6) θ x y , (7)

Subband 2 1 0.5 0 -0.5 -1 -0.5 0 0. Subband 3 0 0 0 0 0 0 0 0 Subband 5 0 0 0 0 0 0 0 0

Subband 2

Subband 6 0 0.5 0 0.5 0 -0.5 0 -0. Subband 2 0 0 0 0 0 0 0 0 Subband 3 1 0 -1 0 1 0 -1 0 Subband 5 0 0 0 0 0 0 0 0

Subband 3

Subband 6 0 0 0 0 0 0 0 0 Subband 2 0 0 0 0 0 0 0 0 Subband 3 0 0 0 0 0 0 0 0 Subband 5 1 -1 1 -1 1 -1 1 -

Subband 5

Subband 6 0 0 0 0 0 0 0 0 Subband 2 0 -0.5 0 -0.5 0 0.5 0 0. Subband 3 0 0 0 0 0 0 0 0 Subband 5 0 0 0 0 0 0 0 0

Subband 6

Subband 6 1 -0.5 0 0.5 -1 0.5 0 -0. Table 1. Autocorrelation and cross-correlation function of subband 2, 3, 5, 6 (Autocorrelation are yellow ones.)

From Table 1, we found some subband bases own very “good” periodic cross-correlation properties, although their autocorrelation functions are not so “good”. This motivates us to define a metric to measure how “good” of the periodic cross-correlation properties, and we can use subband bases that are better according to the metric we define. So we define the following metric: Before we start to define the metric, we want to rename the subband basis as “pattern”, because it likes we input some specific pattern in the input signals, and observe the pattern on the output to estimate the influence between this input-output pair. Subband bases only different in time shift when they become periodic are set to the same pattern, so in Haar wavelet packet with full decomposition to level 3, we have four patterns, that is subband bases 2, 3, 5 and 6. Now we define the Pattern Energy Ratio (PER) for a pattern i as:

1 2

0 1 2

0

i i

i j j

N x x l i (^) N x x x Pattern l

l PER l

−

= −

∈ =

= ⎣^ ⎦

∑

∑ ∑

The above definition represents the energy ratio that will be detected for a specific pattern if we use Haar wavelet packet full decomposition and look at the subband the same as the pattern. The higher value of PER means we can have better estimation on the input having the pattern, so for a fixed full decomposition level, we first generate all patterns from subband bases then sort all the patterns from the PER value, and choose patterns as part of input signals from Highest PER, second PER, and so on. And the number of patterns we need to choose depends on the number of inputs, which is M^ in our model. After we have chosen the pattern and set them as input signals, all we need to do is to decompose the output signals into Haar wavelet packet with full decomposition, and repeat doing this with a time shift from 0 to of the output signals, then sum up the energy for the specific subband for all time shifts. That is, for a output , we calculate the value:

N − 1

j

( )

1 2 0

N Subband Energy (^) ij Coeffij τ τ

−

=

= (^) ∑∑

which Coeffij τ means wavelet coefficients of subband corresponding to pattern for output with a time shift

i j τ , that is, Coeff (^) ij τ= W.T. { P tj ( −τ)} subband corresponds Pattern i

So we finally estimate the influence parameters δ ij using the following formula:

( )

( ) 1

ij ij (^) N

j ij

Subband Energy

Subband Energy

=

∑

B. Simulation Results for Asynchronous Single Path System We test our approach in different number of inputs M , different number of outputs , and different level of full decomposition. For all the plots, x-axis means the outputs, from 1 ~. Here the numbers assigned to every output having no any other meanings, just for convenient. The y-axis means the simulation (estimation) ratio of the specific input to outputs. The red solid lines in the plot means the simulation ratio

N

L N

δ ij (which is generated randomly) we used in the simulation,

and the dashed blue lines the estimated ratio δ ij in our method. Another important is we set the DC

value to 100 for all input signals, and the magnitude of the variation is 10, which is 10% of the DC value. Of course we need to adjust these parameters according to the real situations when we apply to real world cases. Fig. 5 is the case for M = 10 , N = 10 , L = 6. Fig. 6 is the case for M = 10 , N = 10 , L = 8. Fig. 7 is the case for M = 20 , N = 20 , L = 6. Fig. 8 is the case for M = 20 , N = 20 , L = 8.

From Fig.5 and Fig.6, the decomposition level equals to 6 or 8 all get good estimation results. And we expect as the input No. increase, we should get a worse result because the PER value of the pattern become smaller, but it is not always the case. In Fig.5, plots for input 7 ~ input 10 all fit the

… …

L / 2 L / 2

…

L / 4

… …

…

...

… …

L / 2 L / 2

…

L / 4

… …

…

...

We found that it is because for these wavelet bases, their autocorrelation function values are not change to much for similar time shift. So when the difference of time shifts between every path is small (Roughly 20 in our simulation), for a decomposition with large level (8 in our simulation),

L

Period = 2 L >> time shift difference

We can still get good result in the method.

Conclusions and Future Works

Here we use Haar bases from wavelet packet with full decomposition to solve the asynchronous single path problem. From simulation results, we can estimate the influence parameters δ (^) ij very well. When the Gaussian noise is added, higher level decomposition performs

better, as the expense of longer period signal needed and more calculations. Of course when more inputs are involved, we also need higher level decomposition to get better result. And there still some effects needed to modify the PER metrics, for some patterns with lower PER values performs better than patterns with higher PER values. But for multiple paths case, there still much need to be done. Multiple paths asynchronous system is much more similar to the systems we deal with in real world, so it is also more important. Maybe some sequences developed from communication system like m-sequences or Gold sequences are suitable for this case, and it is worth to try. Haar wavelet, although very simple, its frequency selectivity is very bad. And if we can use some other kinds of wavelets that frequency selectivity is better, but still simple for real world implementation, we believe we will get a better result than Haar case.

Fig. 5 Case with 10 inputs, 10 outputs, and decomposition level is 6.

Fig. 6 Case with 10 inputs, 10 outputs, and decomposition level is 8.

Fig. 9 Noisy case with 10 inputs, 10 outputs, and decomposition level is 6.

Fig. 10 Noisy case with 10 inputs, 10 outputs, and decomposition level is 8.

Fig. 11 Multiple paths case with 20 inputs, 20 outputs, and decomposition level is 8.

Reference

[1] Dilip V. Sarwate, Michael B. Pursley, “Cross-correlation Properties of Pseudorandom and Related Sequence”, Proceedings of IEEE, Vol. 68, No. 5, May 1980. [2] Shimon Moshavi, Bellcore, “Multi-User Detection for DS-CDMA Communications”, IEEE communication Magazine, Oct. 1996

[3] Vasu Chakravarthy, Abel S. Nunez, and James P. Stephens, “TDCS, OFDM, and MC-CDMA:

A Brief Tutorial”, IEEE Radio Communications, September 2005.

[4] Amol Shah, “Code Division Multiple Access: A Tutorial”, Wireless Communications, Nov.

1999.

[5] Book: Wavelets and Subband Coding, by Martin Vetterli and Jelena Kovacevic, [6] Book: Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar, by Solomon Golomb and Guang Gong