National Mathematics Day

Presentation on theme: "National Mathematics Day"— Presentation transcript:

1 National Mathematics Day
“An equation means nothing to me unless it expresses a thought of God” 

2 Power Spectrum Density

3 What is Maths?

4 Why maths in engineering?

5 Random process for Engineering Application
Communication systems and computer networks are designed to provide high performance consistently and reliably in the presence of noisy communication channels; equipment faults; a wide range of media applications that combine voice, images and video; and high variability in user demand.  

6 Random process for Engineering Application
Probability models provide the mathematical framework for characterizing random variability and form the basis for tools to design systems that perform predictably in the face of random inputs and environments. 

7 Random process for Engineering Application
The joint distribution function as well as the correlation and the covariance functions are essential tools in achieving these objectives. Random processes are used to describe signals and dynamic behavior encountered in engineering systems.

8 What is Power Spectrum Density?
Power spectral density function (PSD) is the strength of the variations(energy) as a function of frequency. It shows at which frequencies variations are strong and at which frequencies variations are weak.

9 What is Power Spectrum Density?
The unit of PSD is energy per frequency(width) and you can obtain energy within a specific frequency range by integrating PSD within that frequency range.

10 What is Power Spectrum Density?
Computation of PSD is done directly by the method called FFT or computing autocorrelation function and then transforming it. First compute the auto correlation function and then compute its Fourier Transform.

11 Communication System

12 What can you do with power spectral density function?
PSD is a very useful tool if you want to know frequencies and amplitudes of oscillatory signals in your time series data. A chocolate factory example is given below

13 Is power spectral density function always useful?

14 Estimate the Power Spectral Density in Simulink
In DSP System Toolbox™, you can perform real-time spectral analysis of a dynamic signal using the Spectrum Analyzer block. Alternatively, you can use the Spectrum Estimator block from the dspspect3 library to compute the power spectrum, and Array plot  block to view the spectrum.

15 Estimate the Power Spectral Density in Simulink
The model ex_psd_sa feeds a noisy sine wave signal to the Spectrum analyzer block. The sine wave signal is a sum of two sinusoids: one at a frequency of 5000 Hz and the other at a frequency of 10,000 Hz. The noise at the input is Gaussian, with zero mean and a standard deviation of 0.01.

16 Estimate the Power Spectral Density in Simulink
To open the model, enter ex_psd_sa in the MATLAB® command prompt.

17 Estimate the Power Spectral Density in Simulink
lock Parameter Changes Purpose of the block Sine Wave 1 Frequency to 5000 Sample time to 1/44100 Sample per frame to 1024 Sinusoid signal with frequency at 5000 Hz Sine Wave 2 Frequency to 10000 Phase offset (rad) to 10 Sinusoid signal with frequency at Hz Random Source Source type to Gaussian Variance to 1e-4 Random Source block generates a random noise signal with properties specified through the block dialog box Add List of signs to +++. Add block adds random noise to the input signal Spectrum Analyzer Click the Spectrum Settings icon . A pane appears on the right.In the Main options pane, under Type, select Power density. In the Trace options pane, clear the Two-sided spectrum check box. This shows only the real-half of the spectrum. Clear the Max-hold trace and Min-hold trace check boxes if needed. Spectrum Analyzer block shows the Power Spectrum Density of the signal

18 Estimate the Power Spectral Density in Simulink

19 Estimate the Power Spectral Density in Simulink
To distinguish between two frequencies in the display, the distance between the two frequencies must be at least RBW. In this example, the distance between the two peaks is 5000 Hz, which is greater than RBW. Hence, you can see the peaks distinctly. Change the frequency of the second sine wave from Hz to 5015 Hz. The difference between the two frequencies is less than RBW.

20 Estimate the Power Spectral Density in Simulink

21 Estimate the Power Spectral Density in Simulink
To increase the frequency resolution, decrease RBW to 1 Hz.

22 Estimate the Power Spectral Density in Simulink
Change the Spectrum Analyzer Settings on the Main options pane, select Window length. The window length is 1024.

23 Estimate the Power Spectral Density in Simulink
 change the Window length from 1024 to 500  The Samples/update (Nsamples) parameter changes to 500.

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