9.1. Probability and Statistics

Transcription

1 9. Probability and Statistics Measured signals exhibit deterministic (predictable) and random (unpredictable) behavior. The deterministic behavior is often governed by a differential equation, while the random behavior is modeled in terms of probability distributions and described with statistics. Example: Additive White oise Electrons in wires, resistors, and semiconductor devices are always in motion. When no potential is applied across the device, electrons move in all directions at different rates (Brownian motion). The measured current and voltage fluctuations are often referred as noise. The noise signal is unpredictable from one time moment to the next; however, noise signals can sometimes be described by their average power. Use Matlab to plot a 6 Hertz sinusoidal voltage (deterministic) with an average power of Watt measured from a circuit with a noise power of.25 Watts. Assume the probabilistic model for the noise is a ormal distribution. Show the magnitude spectrum of signal plus noise and histogram of noise. Increase noise power to 6 Watts and repeat. Explain what you see.

2 Matlab Code: t= [:23]/496; % time axis, sampling rate 496 Hz f= 496*[:23]/24; % frequency axis for FFT noise25 = sqrt(.25)*randn(,24); % Create noise array with average power.25 watts. sig = sqrt(2)*cos(2*pi*6*t); % Create signal with average power watt. figure(); plot(t,noise25); xlabel(''); ylabel(''); title('oise') % Create a histogram of the noise samples with 2 bins [n, x] = hist(noise25,2); figure(2); bar(x,n); xlabel(''); ylabel('frequency of Occurrence'); title('histogram of oise Voltages'); % Create and plot signal plus noise signo = sig + noise25; figure(3); plot(t, signo); xlabel(''); ylabel(''); title('signal Plus oise') % Plot signal in the frequency domain spec = fft(signo)/(496*.25); % Scale by sampling rate and segment duration figure(4); plot(f(:52), abs(spec(:52))); xlabel('hertz'); ylabel(''); title('signal Plus oise Magnitude Spectrum') % Increase noise power to 6 watts and plot; noise6 = sqrt(6)*randn(,24); % Create noise array with average power 4 watts. figure(); plot(t,noise6); xlabel(''); ylabel(''); title('oise') % Create a histogram of the noise samples with 2 bins [n, x] = hist(noise6,2); figure(2); bar(x,n); xlabel(''); ylabel('frequency of Occurrence'); title('histogram of oise Voltages'); % Create and plot signal plus noise signo = sig + noise6; figure(3); plot(t, signo); xlabel(''); ylabel(''); title('signal Plus oise') % Plot signal in the frequency domain spec = fft(signo)/(496*.25); % Scale by sampling rate and segment duration figure(4); plot(f(:52), abs(spec(:52))); xlabel('hertz'); ylabel(''); title('signal Plus oise Magnitude Spectrum') 9.2

3 9.3.5 oise 2 Histogram of oise Voltages Frequency of Occurrence Signal Plus oise.8 Signal Plus oise Magnitude Spectrum Hertz

4 9.4 5 oise 6 Histogram of oise Voltages Frequency of Occurrence Signal Plus oise.8 Signal Plus oise Magnitude Spectrum Hertz

5 9.5 Some Important Statistics: Sample mean of data points x i : m = x i (in Maltab: m = mean(x) % x is a vector of data points) n= Sample variance of data points x i : 2 2 s = ( x m) (in Maltab: s2 = std(x)^2 % x is a vector of data points) i n = Sample standard deviation of data points x i : 2 s = s = 2 ( x m) (in Maltab: s = std(x) % x is a vector of data points) i n = ote the average power of a signal with zero mean is close to the sample variance, for large, they are almost equal: lim

6 9.6 The histogram of data points x i : Datum values are compared to a set of cells (or bins), which are a subintervals covering the full range of the datum values. The number of data that fall within each subinterval are counted and plotted. The x-axis corresponds to the cell and the y-axis is the frequency (number of datum values within in the cell interval). In Matlab: [n,c] = hist(x,cnumb); % x is vector of data points, cnumb is the number of equal interval cells over which the range of data points is divided, n is a vector corresponding to the number of data in each cell, and c is a vector containing the midpoints in each cell. To plot: bar(c,n) Pseudorandom umber Generators: Complicated and nonlinear deterministic expressions can be used to generate a sequence of numbers that appear random:. uniformly distributed values over the interval of interest 2. no periodicities or repeated patterns in the sequence 3. small changes in the expression's parameter (seed) result in significant changes in the sequences

7 9.7 Two random number generators exist in Matlab: >> rand(rows, columns); Generates a matrix of uniformly distributed random numbers between and. These numbers can be scaled and translated to any 25 interval: 2 Example >>ur = *(rand(,2) -.5) + 5; % uniform numbers between 45 and 55. >>[n,x] = hist(ur,); bar(x,n); xlabel('bin Range'); ylabel('frequency'); frequency 5 5 >>randn(rows, columns); Generates a matrix of normal distributed numbers with zero mean and standard deviation. These numbers can be scaled and translated to any mean and standard deviation: Example >>nr =.5*randn(,2) + ; % ormal numbers with mean and std.5 >>[n,x] = hist(nr,); bar(x,n); xlabel('bin Range'); ylabel('frequency'); frequency Bin Range Bin Range