# Transmitted Signal Arrives At Receiver From Different Directions Biology Essay

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A multi-path channel is one in which the transmitted signal arrives at receiver from different directions after undergoing multiple reflections. The reflections may occur because of the surrounding walls, stationary objects and moving people. In order to establish a radio communication channel in such an environment, a channel impulse response will be useful in identifying the location of the transmitter to be fixed so that there will minimum losses. Saleh and Valenzuela have proposed a model to measure the impulse response of such a multi-path propagation channel.

They have made an experimental setup of a trans-receiver system to find out the signal strengths that are received in particular building at various locations. They collected this experimental data and proposed a statistical model for the channel. [1,2].

## 2.2 Statistical Model for indoor communication:

Saleh and Valenzuela have conducted their experiments on a two-storeyed building whose first floor plan is given in Fig.1 of [1]. The external walls of this building were made of steel beams and glass, while the internal walls were almost entirely made of wood studs covered with plasterboard. The rooms contained typical metal office furniture and/or laboratory equipment. They made the measurements by using low-power 1.5 GHz radar-like pulses to obtain a large ensemble of the channel's impulse responses with a time resolution of about 5ns.

The transmitter consists of a sweep oscillator and a pulse wave generator. The pulse generator produces a pulse train of 10ns width and a pulse repetition period of 600ns. These pulses are modulated using a 1.5GHz sinusoidal signal which is generated by the sweep oscillator. The modulated signal is transmitted through the air medium using a vertically polarized discone antenna.[1,16]

Figure - 1

IEEE Journal on selected areas in communications, FEB 1987. [1]

The receiver consists of another vertically polarized discone antenna which collects the transmitted signal and feeds it to a square law envelope detector. The output of the envelope detector is then feeded to a computer controlled digital storage oscilloscope.

This trans-receiver system is used to conduct the experiment. The experiments are conducted by fixing the transmitter at the center of a hallway and the receiver is moved to various positions in the building. Saleh and Valenzuela have conducted their experiments in 8 different rooms and calculated the signal power at the receiver. [2]

## Measured Signal Responses:

The measured signal strengths are used in determining the power delay profiles of the signals. The power delay profile plots are taken for the measurements done in different rooms. These plots show that the signal arrives in clusters. The plots show the original signal and the echoes that arrive after some delay. The delay of the echoes depends on the location where the measurements are taken.

For example, when the experiment is conducted with the transmitter and receiver located in a hallway of the building at a distance of 1m, only the direct signal is received as the signal takes only the Line of Sight path. When the distance was 60m, a small echo has been observed along with the direct signal. Similarly, multiple echoes are obtained for measurements in room at various locations. These observations are shown in [1].

From the observations, Saleh and Valenzuela have proposed that the signals (also called rays) arrive in clusters. With this assumption they tried to find out a distribution model for the arriving delays of the signals. [3,4]

## 2.4 S-V Model assumptions and results:

it is assumed the inter arrival times of the clusters as well as the rays within each of these clusters as statistically independent random variables. The distributions of these random variables take a Poisson distribution. Similarly the probability distribution of the path gains also takes a Poisson distribution.

Once these distributions are considered, the model parameters are estimated. The model parameters are cluster arrival rate, ray arrival rate, power decay constants of the clusters and the rays. In the measured observations, the cluster arrival rate is found to be approximately around 200 to 300 ns in the test building. The arrival rate of the rays within the clusters was found to be in the range of 1 to 5ns.The power decay constant for the rays was approximately 20 ns and the power decay constant of the clusters was 60ns.

An indoor multipath channel impulse response is simulated using these estimated parameters. The simulation procedure is explained in [1]. From these observations, it was concluded that the channel was quasi-static i.e. slowly time varying, the channel impulse response was virtually independent of the polarization of the antennas used, the maximum delay spread in the test building was observed to be between 200 to 300 ns, the rms delay spread being around 25 ns and the signal attenuation of the signals other than the direct signal ranges around 60dB.

## 2.5 Probability Distribution of a Random Variable:

The probability distribution of a random variable gives the probability of occurrence of a particular value of the random variable. If the random variable is continuous then the pdf gives probability of occurrence of an interval of values.

Consider the arriving times of the first ray within a cluster. It is difficult to predict at what time a new cluster can arrive, even if the arriving time of the current cluster is known. The new cluster may arrive with a delay of 1 to hundreds of nanoseconds from the previously arrived cluster. This difference (inter arrival times) can take any random value. Thus the inter arrival times can be considered a statistically independent random variable. Similarly, the inter arrival times of the rays within a cluster can be considered as another random variable [5].

## 2.6 Estimation of the Probability Density function:

The probability density function of a random variable can be estimated by a simple histogram based approach. In the histogram based method, the values of a random variable can be first divided into equal number of bins. For example, consider that the inter arrival times of the clusters range from 0 to 200ns.

In order to plot the histogram having 20 bins, the range of the inter arrival times is divided into 10 equal parts of 0, 10, 20...90. Now the arrival times that fall in the range of 0 to 10ns are considered as the times arriving at 0ns and that in range of 10 to 20 ns as that of 10ns and so on. Once all the values of the random variable are identified, the count of the values at each of these bins is plotted as a vertical bar on a plot with the bin value on the x-axis and count on y-axis. This plot is the histogram of the random variable.[1,14,16]

The following is one of the histograms for the experimental data considered where the bin width of the histogram is 20ns with number of bins being 10. The figure taken from matlab .

Fig 2. Example for Histogram of a random variable [18]

From the histogram, the probability distribution function of the random variable can be determined. The probability of a particular value of the random variable can be found by dividing the count of the bin value with the total number of possible outcomes of the random variable. The following is the probability distribution function of the random variable.

Fig3.Example for Probability Density Function of a random variable [18]

The probability distributions functions are of many types such as Gaussian distribution, exponential distribution, uniform distribution, lognormal distribution, Rayleigh distribution, Poisson distribution and so on.

In the current project, some of these probability distributions must be explored which can give a best fit to the experimental data. From the previous researches conducted in this area it can be understood that the signals transmitted in a multi path propagation channel can take the distributions like Poisson, Rayleigh, exponential and log normal distributions. [6]

## 2.7 Exponential Probability Distribution:

The exponential distribution of a random variable X is given as

(2.7) [7]

Here µ is the model parameter for the exponential distribution whose reciprocal gives the rate of occurrence of the event. The plot of an exponential distribution curve of a random variable x is shown in the figure below

Fig3. Probability Density function of an exponential distribution [18]

This figure is generated using a matlab code snippet. The x-axis represents the random variable x and y-axis gives the distribution of the random variable x.

In general, exponential distribution can be observed, for example, in the decay times of a radioactive element. If the decay time of the radioactive element is chosen to be a random variable, the radio activity of the element takes the above shape. [7,8]

## 2.8 Parameter Estimation of Exponential Distribution:

The model parameter for an exponential distribution is µ. This parameter can be estimated from the given data using a maximum likelihood estimate method. In order to estimate the parameter, first a likelihood function of the exponential distribution is established. The likelihood function is given as

(2.8.1)[18]

Once the Likelihood estimate function is calculated then maximum likelihood is estimated by finding the maximum of the logarithm of the above equation. The natural logarithm is given as [6]

(2.8.2)

The derivative of this logarithm is given as

(2.8.3)

In order to find the maximum estimate of the model parameter µ, the above equation is equated to zero

(2.8.4)

(2.8.5)

## 2.9 Curve Fitting of Exponential Distribution:

For the exponential distribution, the model parameter is estimated as described above. Once the parameter is known for the given data, the model fit has to be done to the experimental data. In order to fit a curve, first the distribution function values at each value of the random variable need to be calculated. These calculated values are plotted over the given data. The curve that can best fit to this data can be approximated by changing the model parameter. For example, the following figure shows a curve that is fitted to the given experimental data.

The experimentally determined values are represented by the red colored asterisks and fitted curve is shown in the blue color.

Fig4.Curve fitting of exponentially distributed data [18]

## 2.10 Log-normal Distribution:

If a random variable has its probability distribution as a normal distribution then the logarithm of such a variable has log-normal distribution. The probability density function of a log-normal distribution is as given below [9, 10, and 18]

(2.10.1)

The parameters mu and sigma are called the model parameters which can be used for curve fitting of the given data as well as finding out the mean and variance of the log-normal distribution [9, 10]. The mean or expectation of the distribution in terms of mu is given as

(2.10.2)

The variance of the log-normally distributed random variable is given as

(2.10.3)

The probability density function of a log-normally distributed data will as shown below

Fig5. Probability Density Function of a log-normally distributed Random Variable

[18]

## 2.11 Maximum Likelihood Estimate of Model Parameters:

The model parameters for the log-normal distribution can be estimated in similar method as done for the exponential distribution. First the likelihood function for the distribution is evaluated. The likelihood function for the log-normal distribution is [10]

(2.11.1)

The logarithm of this function is applied and the first order derivative of this log-likelihood function is equated to find the maximum estimate of the μ and σ. The logarithm to the above function is taken and differentiated. The differentiated equation is equated to zero and the parameters are obtained as

(2.11.2)

The constant k in the above equation denotes the summation term of the ln(1/x) terms in the likelihood function. Differentiating and equating the above equation, the parameters are obtained as

(2.11.3)

## 2.12 Curve Fitting of Log-Normal Distribution:

The model parameters are first estimated using the above described method. With these parameters log-normally distributed data is generated and fitted to the experimental data. For example, consider the following figure, it shows a fitted curve (blue) to the experimental data (shown in red) collected.

Fig6. Curve fitted to a log-normally distributed data

[18]

## 2.13 Poisson distribution:

Poisson distribution of a random variable is given as given below [11]

(2.13.1)

In this distribution, the random variable x denotes the number of occurrences of an event that happen in a time-interval. The parameter λ denotes the expected number of occurrences that can happen in the given time interval. For example, if the events occur on an average for every 2 minutes, the in order to find the number of events that occur in 5 minutes time interval, a Poisson distribution model has to be used with a the parameter λ = 2.5.

The mean and the variance of the Poisson distribution are equal and they are equal to the rate parameter lambda.

E(X) = Var(X) = λ. (2.13.2)

The shape of a Poisson distribution curve is as given below figure Fig7. Probability Density function of a Poisson distribution

The red curve is obtained for λ = 1, blue curve for λ = 5 and green curve for λ = 10. It can be observed that this distribution takes the form of a uniform distribution as value of lambda increases. The above figure is generated using the matlab code snippet given in the help documentation of Poisson distribution. [11]

## 2.14 Parameter Estimation of Poisson distribution:

Poisson distribution is a single parameter model. The only model parameter it has is the rate parameter λ. The likelihood function of the Poisson distribution is given below [11,18]

(2.14.2)

A logarithm is applied to the above equation and differentiated to and equated to zero to obtain the rate parameter λ.

(2.14.3)

## 2.15 Curve fitting of Poisson distribution:

Once the parameter lambda is estimated a curve has to be fitted to the experimental data. In this case, the experimental data is generated using the matlab function poissrnd which gives the Poisson distributed random data for a given rate parameter of lambda = 3 over the inter Val of 1 to 50. Using this random generated data maximum likelihood estimate of the parameter lambda is found. With this estimated parameter, a Poisson distribution curve is fitted to the random data. [11]

## 2.16 Summary

In order to propose the model, it is assumed that the inter arrival times of the clusters as well as the rays within each of these clusters are statistically independent random variables. The distributions of these random variables take a Poisson distribution. Similarly the probability distribution of the path gains also takes a Poisson distribution.

Once these distributions are considered, the model parameters are estimated. The model parameters are cluster arrival rate, ray arrival rate, power decay constants of the clusters and the rays. In the measured observations, the cluster arrival rate is found to be approximately around 200 to 300 ns in the test building. The arrival rate of the rays within the clusters was found to be in the range of 1 to 5ns.The power decay constant for the rays was approximately 20 ns and the power decay constant of the clusters was 60ns.

An indoor multipath channel impulse response is simulated using these estimated parameters. The simulation procedure is explained in [1]. From these observations, it was concluded that the channel was quasi-static i.e. slowly time varying. The channel impulse response was virtually independent of the polarization of the antennas used. The maximum delay spread in the test building was observed to be between 200 to 300 ns. The rms delay spread being around 25 ns, and the signal attenuation of the signals other than the direct signal ranges around 60dB.

In this project, the information about the indoor multi path channel is used in finding out the best fit of the experimental data. In this model it is shown that the experimental data has best fitted to a Poisson distribution. For the current observation, data which obtains a best fit has to be estimated. In order to do this, various types of distribution functions have to be analyzed. Distribution functions such as Exponential, Log-Normal are taken into account.