1. EE 495 Modern Navigation Systems Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Slide 1 of 19

2. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems We can NOT describe all of the parameters that contribute to a signal or sensor measurement  Deterministic: Can exactly model the relationship between the input (stimulus) and output (sensor meas)  Random: Can NOT exactly model the relationship o Can characterize attributes of the signal – e.g., mean (  ), standard deviation (  ), probability density function (pdf), power spectral density (PSD), … o i.e., Noise (random signals) Slide 2 of 19

3. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems A random process  A “family” of random variables  A function of both time and event  Can compute statistics across the ensemble or across time If the time statistics and ensemble statistics are equal, then the random process is ergodic. For example: Ensemble mean = time average!! Slide 3 of 19

4. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems An Example: A gyro sitting “still” on a lab bench  Mean =  and Standard Deviation =  Mean  -0.8 deg/s SD  0.22 deg/s Slide 4 of 19

5. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Comparison of two random signals  Green signal has a lower standard deviation Slide 5 of 19

6. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems For a Gaussian (or Normal) distribution Slide 6 of 19

7. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Autocorrelation function: R xx (0)  0.0488  /s ?? Mean of x was removed before computing R xx Slide 7 of 19  x  0.22  /s

8. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Cross-correlation function: Means of x & y were removed before computing R xy Slide 8 of 19

9. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Ideal signals and real-world signals differ!! Infinitely long Continuous in time and magnitude True distribution of power vs frequency Infinitely long Continuous in time and magnitude True distribution of power vs frequency Ideal Signals Finite length Discrete in time and magnitude (digital) Approximate distribution of power vs frequency Finite length Discrete in time and magnitude (digital) Approximate distribution of power vs frequency Real-World Signals Slide 9 of 19

10. Noise & Random Processes Energy Signals vs Power Signals Mon, March 02 EE 495 Modern Navigation Systems Consider the voltage, v(t), across a resistor (R) producing a current, i(t).  Assume that R = 1  Total Energy Average Power Slide 10 of 19

11. Noise & Random Processes Energy Signals vs Power Signals Mon, March 02 EE 495 Modern Navigation Systems Parseval’s Theorem Power Spectral Density: S(f) Power Spectral Density Slide 11 of 19

12. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Input-Output Relationships & the PSD  The PSD of the output is scaled by the transfer function squared. If x(k) was a white noise process then y(k) is now a colored noise process If x(k) was a white noise process then y(k) is now a colored noise process Slide 12 of 19

13. Noise & Random Processes Relationship between Continuous & Discrete Time Mon, March 02 EE 495 Modern Navigation Systems Slide 13 of 19

14. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Key Points:  Must sample more than twice bandwidth to avoid aliasing  FFT represents a periodic version of the time domain signal o Could have time domain aliasing  Number of points in FFT is the same as number of points in time domain signal. [Pxx,f] = pwelch(x, window, noverlap, nfft, Fs, 'range') % x - The time domain signal (samples) of length N % window - Length in samples of a smoothing filter (default N/8) % noverlap - 50% overlap is the default % nfft - Number of points in the discrete freq domain % Fs - Sample frequency (Hz) % range - Two-sided or one-sided frequency range [Pxx,f] = pwelch(x, window, noverlap, nfft, Fs, 'range') % x - The time domain signal (samples) of length N % window - Length in samples of a smoothing filter (default N/8) % noverlap - 50% overlap is the default % nfft - Number of points in the discrete freq domain % Fs - Sample frequency (Hz) % range - Two-sided or one-sided frequency range Slide 14 of 19

15. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems An Example: Fs = 1000; % Sample frequency (Hz) dT = 1/Fs; % Sample interval (sec) t = 0:dT:100-dT;% Discrete sample times N = length(t); % Number of samples sigma = 10; % Standard deviation of the noise x = sigma*randn(1,N); % A noise signal: N[0, 10] [Pxx,f] = pwelch(x,1024,[],[],Fs,'onesided'); % Decrease the window size => Increase averaging [Pxx,f] = pwelch(x,128,[],[],Fs,'onesided'); % Decrease the window size => Increase averaging [Pxx,f] = pwelch(x,128,[],[],Fs,'onesided'); Slide 15 of 19

16. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Two tones + noise Fs = 100; % Sample frequency (Hz) dT = 1/Fs; % Sample interval (sec) t = 0:dT:5-dT; % Discrete sample times N = length(t); % Number of samples sigma = sqrt(0.1*Fs); % Standard deviation of the noise x = cos(2*pi*10*t) + cos(2*pi*11*t)+sigma*randn(1,N); [Pxx,f] = pwelch(x,[],[],[],Fs,'onesided'); Need better freq resolution  increase data length Need better freq resolution  increase data length Slide 16 of 19

17. Noise & Random Processes Mon, March 02 EE 495 Modern Navigation Systems Two tones + noise Fs = 100; % Sample frequency (Hz) dT = 1/Fs; % Sample interval (sec) t = 0:dT:50-dT; % Discrete sample times N = length(t); % Number of samples sigma = sqrt(0.1*Fs); % Standard deviation of the noise x = cos(2*pi*10*t) + cos(2*pi*11*t)+sigma*randn(1,N); [Pxx,f] = pwelch(x,256,[],[],Fs,'onesided'); Now we can see the two tones!! Slide 17 of 19

18. Noise & Random Processes _ PSD of Cobham Sr. Design IMU (Accels) Mon, March 02 EE 495 Modern Navigation Systems Accel_x = N[ -0.004, 0.0062] g Accel_y = N[ -0.0129, 0.0062] g Accel_z = N[ 1.0771, 0.0069] g Mean Pxx = 7.5965e-07 (g)^2   = 0.0062 g Mean Pyy = 7.6005e-07 (g)^2   = 0.0062 g Mean Pzz = 9.4731e-07 (g)^2   = 0.0069 g [Pxx,f] = pwelch(accel(:,1)-mean(accel(:,1)),[],[],[],Fs,'onesided'); Slide 18 of 19

19. Noise & Random Processes _ PSD of Cobham Sr. Design IMU (Gyros) Mon, March 02 EE 495 Modern Navigation Systems Gyro_x = N[ 0.54717, 0.2973] deg/s Gyro_y = N[ -0.80251, 0.2209] deg/s Gyro_z = N[ -0.086707, 0.3432] deg/s Mean Pxx = 0.001769 (  /s)^2   = 0.2974  /s Mean Pyy = 0.00097461 (  /s)^2   = 0.2208  /s Mean Pzz = 0.0023621 (  /s)^2   = 0.3437  /s Slide 19 of 19