CHAPTER THREEDescription of Linear Time Invariant System theoryCommonly known as LTI system theory, it emanated from applied mathematics and has applications in Nuclear Magnetic Resonance (NMR) spectroscopy, seismology, circuit theory and other technical areas. It investigates the response of a linear and time-invariant system to a random input signal. Such systems normally have their trajectories through time however in applications like field theory and image processing, the trajectories are in space. Hence these systems are known as linear translation-invariant to offer the theory the most general reach [2,3]. There are three requirements for an LTI system: Scalability, Additivity, and time invariant. Scalability implies that a change in the input signals amplitude results in a corresponding change in the output signals amplitude. This imply an input of x(t)= 2z produces an output of y(t)= 4z, an input of x(t)= 4z will produce an output of y(t)= 8z. Additivity indicates that the correlation between the input and the output of the system is a linear map: If input x1(t) produces response y1(t), and input x2(t) produces response y2(t), then the scaled and summed input a1(t) x1(t)+ a2(t) x2(t) produces the scaled and summed response a1(t) y1(t)+ a2(t)y2(t) where a1 and a2 are real scalars [4,5]. Time invariance implies that whether an input is applied to the system now or T seconds from now, the output would be identical except for the a time delay of the T seconds.

FLOWCHART OF PROGRAMStartInput Model parametersAre the model parametersvalid?Simulate ModelPlot ModelStopDisplay error messageNo

Yes

SOFTWARE IMPLEMENTATION

The use of software application in the analysis and design of an LTI system is introduced here.The MATLAB software is employed in the development of the lines of code for the analysis and design of the LTI systems. The input parameters are employed in the analysis and the mathematical manipulations are done using MATLABs embedded codes (toolboxes) [6,7,8,9].The program interface is shown in figure 3.0.

Figure 3.0: Program interface

The Program InterfaceOn the mainmenu interface, there are two columns. The Help and Hints column and System analysis column. The System Analysis column covers Second Order Single Input Single Output systems (SISO) in both time and frequency domains. The Help and Hints column provides links to webpages describing the corresponding system analysis in summary. A Graphic User Interface (GUI) is developed for each system domain (time and frequency).

Second Order System (Time Domain)As shown on the interface for a second order system in time domain in figure 4, the required parameters are damping ratio (D), natural frequency (Wn) and response time. The step response is plotted for any given stable system. The corresponding system characteristics are computed and shown as well.

Figure 4: Program Interface for a second order system in time domain

Application interface tabs and their functions "Main Menu" Button Return to the Main Menu Screen. "Help" Button Explanation of the purpose of this screen and how to work with it. "Usage" Button Help about how to manipulate the screen. "Exit" Button Exit from the Signal processing systems Tools. Edit box for the damping ratio. Only non-negative values for the damping ratio are allowed. Edit box for the undamped natural frequency. Only non-negative values for the natural frequency are allowed. Edit box for the "Response time" Maximum time for the time axe of the graph in the right-side window. Replot" Button: After setting the respective values for the system description, by pressing the "Replot" button, the time response curve of the system is displayed on the right side. The horizontal axes scale can be set by changing the value in the "Response Time" edit box. If a plot exists already, "Replot" is performed after clearing the previous shown plot. "Hold plot" Button: This has same usage as "Replot" button, except that if a plot exists already, it is not cleared before a new one is drawn. "Analysis Results": In this part of the window, the results from the analysis of the system are shown as a short text string inside a button. Selecting the button of interest invokes a help screen with more detailed explanation. "Print" Button": Allows printing of the whole screen.

Second Order System (Frequency Domain)

Figure 5: Program Interface for a second order system in frequency domain

With same system description inputs as in figure 4 (damping ratio and natural frequency), this interface gives the bode diagram (Magnitude and phase plots) of the system. Bode diagrams of two systems can be compared by using the hold plot tab to obtain the plots for each system (with different system descriptions) on same graph. These plots can also be printed for clarity and legibility by using the print tab. The system characteristics of the most recent plot is computed and displayed. Information on the gain and phase margins is given as well as the peak gain, resonant peak frequency and bandwidth. The analysis of the most recent plot is also given under the Analysis of result tab.

The SISO System (time domain) InterfaceThe SISO system (time domain) interface is likewise, shown in figure 6.

Figure 6: Program Interface for a SISO system in time domainAs shown in figure 6, the system matrix A, input matrix B, output matrix C and input-output matrix D are the systems parameters required to determine the Impulse and step responses plot on the right hand side. The System characteristics is determined also and displayed within the program interface. In a situation where the transfer function of the system is given, the transfer function input tab [TF (T)] is used to input the numerator and denominator polynomials. As soon as the polynomial parameters are accepted, the equivalent state space parameters (A, B, C and D) are computed and displayed. The makes the system characteristics computed irrespective of how the system dynamics is given.

The SISO System (frequency domain) InterfaceThis shows the frequency analysis of a system/plant.

Figure 7: Program Interface for a SISO system in frequency domainFigure 7 shows the frequency responses of a system/plant when its system dynamic parameters are inputted as in figure 6. The Nyquist, Nichols and Bode plots are obtained on the right side of the interface. The system characteristics (Gain margin and phase margin) are also computed and displayed.

System Analysis InterfaceThis interface encompasses all analytical domains (time and frequency). This is shown in figure 8. For the time domain, the impulse or step response could be computed /plotted while for the frequency domain, bode diagram, nyquist plot, nichols chart or single value plot could be obtained.

Figure 8: Program interface for general System analysis

DISCUSSIONThe graphical representations above are the results (output responses) generated from the simulated MATLAB codes (with arbitrary parameters). The input parameters are modified to system specification in order to achieve the required response. The system characteristics could be in transfer function or state space representation. Irrespective of the form in which it is given, a conversion tool is available on the application interface to convert parameters (transfer function to state space and verse versa, as appropriate). The stability of the system is also determined from the application interface.The design engineer has the option of testing out various system parameters to ascertain which yields a stable system and also observing the impulse and step response. Modifications could be made if results are not satisfactory. Various output responses could be analyzed simultaneously on same graph and their stabilities compared for a better performance. It should also be noted that this program could also be used in the labs to illustrate the practical implementation of LTI systems to students. Students would be delighted to observe in real-time, the effect of various system characteristic parameters on the output responses (step and impulse response) of the system.ConclusionThis project so far, accommodates only Single Input Single Output Systems. The goal of the project has been achieved. The program is simplified owing to the fact that MATLAB has got assigned command codes for every system analysis and design control theories. One may not be vast in the theoretical deductions to develop such a program since MATLAB has codes available for every operation.The program was developed to enable the teaching of signal processing in Communication Engineering easy and effective. The students being taught also get to see a simulated real-time procedure and its varying responses due to varied input parameters. The program can also be employed by Lecturers in Communication Engineering, Control Engineering, Computer Engineering, Mechanical Engineering, Chemical Engineering and in fact, all Engineering fields. The program is built to run on any Windows platform (Operating System) and work is being done to develop same for the LINUX platform.

DeploymentThis program has been deployed in high level languages such as JAVA, C, C++, and C# and program modules can be imported as external function parameters in the use of such high level languages in the development of signal processing devices. Giving to the fact that MATLAB codes are readily incorporated into these high level languages, the execution and compilation delay is very minimal.A signal processing operating system could be developed without great energy in running signal processors for function-specific signal processors. These devices can then be deployed in engineering labs for the training of staff and students. This program was developed using MATLAB 7.0 with NETBEAN 5.0. The use of a lower version to modify any part of this program could result to errors. Also, the presence of a strong graphic user interface platform possessed by Windows Vista also affects the kernel function application in a way that the processing is delayed for a varied time. This actually, does not affect the final result of the program. If for any reason, there happen to be a bug generated in the program procedure, a self-debugging event is generated.

RecommendationI highly recommend an improvement on this work since the work only covers Linear Time-Invariant systems. Same could be done for non-Linear Time-Invariant, non-Linear Time Variant and Linear time Variant systems. The program could also be expanded to accommodate filter designs, modulation and demodulation schemes and other form of signal processing schemes. Aside the simulation generated with MATLAB, an Application Package Interface could be developed to run on the same platform as the simulator but with a programmable hardware interface.This interface could be the Universal Serial Bus (USB) interface, or the RS232 (Serial) interface. This could be interfaced with an oscilloscope or wave analyzer. Such hardware could replace the visual display. The input parameters to the environment could be applied to some similar programs like MultiSIM but a circuit design would have to be present so that the various elements making up the circuit will only need to be moderated in their properties. This should yield same result if well applied. The use of MultiSIM will involve a good knowledge of circuit analysis.References

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