8 - Chapman MATLAB Programming for Engineers

For example:

>>mean(data1) ans =

2.9940

>>sum(data1)/length(data1) ans =

2.9940

>>mean(data2)

ans = 2.9768

>>median(data1) ans =

3.0285

>>median(data2) ans =

3.0574

These results can be confirmed by observing the plots of the data in Figure 7.1, in which the data values can be seen to be centered on 3.0. For the two data sets considered, the values of the mean and median are very close. This is not necessarily the case for other data sets. Also note that the results above show that mean(data1) = sum(data1)/length(data1).

Measures of variation

Measures of variation: indicate the degree of deviation of random samples from the measure of central tendency.

Referring again to the plots of our random data sets data1 and data2 in Figure 7.1, observe that data2 has greater variation from the mean.

The sample standard deviation of vector x having N elements is

The sample variance s2 is the square of the standard deviation.

std(x) Returns the sample standard deviation of the elements of the vector x. Returns a row vector of the sample standard deviations of the columns of matrix x.

For example:

>> std(data1)

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ans = 0.5989

>> std(data2) ans =

0.9408

Thus, the variation of data2 is greater than that of data1, as we concluded from observation of the plotted data values.

7.4Random Number Generation

Many engineering problems require the use of random numbers in the development of a solution. In some cases, the random numbers are used to develop a simulation of a complex problem. The simulation can be tested over and over to analyze the results, with each test representing a repetition of the experiment. Random numbers also used to represent noise sequences, such as those heard on a radio.

Uniform Random Numbers

Random numbers are characterized by their frequency distributions. Uniform random numbers have a constant or uniform distribution over their range between minimum and maximum values.

The rand function in Matlab generates uniform random numbers distributed over the interval [0,1]. A state vector is used to generate a random sequence of values. A given state vector always generates the same random sequence. Thus, the sequences are known more formally as pseudorandom sequences. This generator can generate all the floating point numbers in the closed interval [2−53, 1 − 2−53]. Theoretically, it can generate over 21492 values before repeating itself.

The following commands generate and display two sets of ten random numbers uniformly distributed between 0 and 1; the di erence between the two sets is caused by the di erent states.

rand(’state’,0)

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set1 = rand(10,1); rand(’state’,123) set2 = rand(10,1); [set1 set2]

The results displayed by these commands are the following, where the first column gives values of set1 and the second column gives values of set2:

0.9501 0.0697

0.2311 0.2332

0.6068 0.7374

0.4860 0.7585

0.8913 0.6368

0.7621 0.6129

0.4565 0.3081

0.0185 0.2856

0.8214 0.0781

0.4447 0.9532

Note that as desired, the two sequences are di erent.

To generate sequences having values in the interval [xmin, xmax], first generate a random number r in the interval [0, 1], multiply by (xmax −xmin), producing a number in the interval [0, (xmax −xmin], then add xmin. To generate a row vector of 500 elements, the command needed is

x = (xmax - xmin)*rand(1,500) + xmin

The sequence data1, plotted in Figure 7.1, was generated with the command:

data1 = 2*rand(1,500) + 2;

Thus, xmax − xmin = 2 and xmin = 2, so xmax = 4 and the range of the data sequence is (2,4), which can be confirmed from Figure 7.1.

Example 7.4 Flipping a coin

When a fair coin is flipped, the probability of getting heads or tails is 0.5 (50%). If we want to represent tails by 0 and heads by 1, we can first generate a uniform random number in the range [0,2). The probability of a result in the range [0,1) is 0.5 (50%), which can be mapped to 0 by truncation of this result. Similarly, the probability of a result in the range [1,2) is 0.5 (50%), which can be mapped to 1 by truncation. The trunction function in Matlab is floor.

A script to simulate an experiment to flip a coin 50 times and display the results is the following:

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%Coin flipping simulation

%Coin flip:

The histogram plot is shown in Figure 7.6. Note that in this trial, there were 26 tails and 24 heads, close to the expected result. Since this is a random process, repeated trials will give di erent results.

Example 7.5 Rolling dice

When a fair die (singular of dice) is rolled, the number uppermost is equally likely to be any integer from 1 to 6. The following statement will generate a vector of 10 random numbers in the range 1-6:

die = floor( 6 * rand(1,10) + 1 )

The following are the results of two such simulations:

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Figure 7.6: Histogram of 50 Coin Flips

Gaussian Random Numbers

A random distribution that is an appropriate model for data sets from many engineering problems is the Gaussian or Normal distribution. This is the distribution having a bell shape, with a high relative frequency at the mean, decreasing away from the mean, but extending indefinitely in both directions. The Gaussian distribution is specified by two parameters: (1) the population mean µ, and (2) the population standard deviation σ, with the distribution function given by

This distribution is plotted in Figure 7.7, where it is observed to be symmetrical about the mean, dropping o rapidly away from the mean. The distribution can be normalized by performing a change of scale that converts the units of measurement into standard units

z = x − µ

σ

as shown in Figure 7.7. It can be shown that approximately 68% of the values will fall within one standard deviation of the mean (−1 < z < 1), 95% will fall within two standard deviations of the mean (−2 < z < 2), and 99% will fall within three standard deviations of the mean (−3 < z < 3).

The Matlab functions for generating Gaussian random numbers are:

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To generate a row vector x of 500 Gaussian random variables with mean m and standard deviation s:

x = m + s*randn(1,500);

The sequence data2, plotted in Figure 7.1, was generated with the command:

data2 = rand(1,500) + 3;

Thus, m = 3 and s = 1. The mean value can be confirmed from the plot in Figure 7.1 and the mean value computed in Section 7.3. The standard deviation can be confirmed by the value computed in Section 7.3. The computed values are not identical to the desired values because the computed values are due to a finite number of random samples. The computed and desired values become closer as the number of samples increases.

Example 7.6 Noisy signal simulation

Many sensors, which convert physical quantities into electrical signals, produce measurement errors, resulting in what is known as noisy signals. Additive Gaussian random numbers are often an accurate model of the errors. The signal model is

x = s + n

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where s is the noise-free signal of interest, n is the Gaussian measurement error, and x is the noisy signal. A measure of the signal quality is the signal to noise ratio (SNR), defined by

σ2

SNR = 10 log10 s

σn2

σs

= 20 log10 σn

where σs is the standard deviation of the signal and σn is the standard deviation of the noise.

Consider simulating a noisy signal consisting of a sine wave with additive Gaussian measurement noise. The variance of a sine wave can be shown to be

σs2 = A2

where A is the sinusoidal amplitude. For A = 1 and Gaussian noise standard deviation σn = 0.1, the SNR is

0.5

SNR = 10 log10 0.12 = 10 log10 50 = 17.0

The following script simulates this noisy signal. In the next section, methods to reduce the a ect of the noise will be considered.

% Noisy signal generation and analysis

The computed SNR is

Signal to Noise Ratio (SNR), dB

SNR =

16.9456

This is very close to the value computed above. The reason for the di erence is the finite number of samples used to estimate the noise standard deviation. The resulting noise signal is shown in Figure 7.8.

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Noisy signal

Figure 7.8: Simulated noisy sinusoidal signal

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Section 7

Data Analysis

Outline

•Maximum and minimum

•Sums and products

•Statistical analysis

•Random number generation

The purpose of data analysis is to determine some properties and characteristics of the data, which may contain measurement errors or other random influences.

For the examples to be considered in this section, generate, save, and plot the two data vectors, data1 and data2 as shown below. The commands used to generate this data will be explained later. The two data vectors, data_1 and data_2 are plotted in Figure 7.1.

data1 = 2*rand(1,500) + 2; data2 = randn(1,500)+3;

save data

subplot(2,1,1),plot(data1),...

axis([0 500 0 6]),...

title(’Random Numbers - data1’),...

subplot(2,1,2),plot(data2),...

title(’Random Numbers - data2’),...

xlabel(’Index, k’)

Other data analysis functions available in Matlab are described with the command help datafun.

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Random Numbers − data1

Figure 7.1: Random sequences

7.1Maximum and Minimum

Determining the maximum and minimum of a vector:

>>v = [3 -2 4 -1 5 0];

>>max(v)

ans =

5

>> [vmin, kmin] = min(v) vmin =

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