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Unformatted text preview: Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 3: Frequency Analysis October 6, 2010 1 Introduction In this experiment, we will use Fourier series and Fourier transforms to analyze continuous- time and discrete-time signals and systems. The Fourier representations of signals involve the decomposition of the signal in terms of complex exponential functions. These decompositions are very important in the analysis of linear time-invariant (LTI) systems, due to the property that the response of an LTI system to a complex exponential input is a complex exponential of the same frequency! Only the amplitude and phase of the input signal are changed. Therefore, studying the frequency response of an LTI system gives complete insight into its behavior. In this experiment and others to follow, we will use the Simulink extension to Matlab. Simulink is an icon-driven dynamic simulation package that allows the user to represent a system or a process by a block diagram. Once the representation is completed, Simulink may be used to digitally simulate the behavior of the continuous or discrete-time system. Simulink inputs can be Matlab variables from the workspace, or waveforms or sequences generated by Simulink itself. These Simulink-generated inputs can represent continuous- time or discrete-time sources. The behavior of the simulated system can be monitored using Simulinks version of common lab instruments, such as scopes, spectrum analyzers and network analyzers. Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494- 0340; firstname.lastname@example.org Purdue University: ECE438 - Digital Signal Processing with Applications 2 2 Background Exercises INLAB REPORT: Submit these background exercises with the lab report. 2.1 Synthesis of Periodic Signals Each signal given below represents one period of a periodic signal with period T . 1. Period T = 2. For t [0 , 2]: s ( t ) = rect( t- 1 2 ) 2. Period T = 1. For t [- 1 2 , 1 2 ]: s ( t ) = rect (2 t )- 1 2 For each of these two signals, do the following: i. Compute the Fourier series expansion in the form s ( t ) = a + summationdisplay k =1 A k sin(2 kf t + k ) where f = 1 /T . Hint: You may want to use one of the following references: Sec. 4.1 of Digital Signal Processing, by Proakis and Manolakis, 1996; Sec. 4.2 of Signals and Systems, by A. Oppenheim and A. Willsky, 1983; Sec. 3.3 of Signals and Systems, A. Oppenheim and A. Willsky, 1997. Note that in the expression above, the function in the summation is sin(2 kf t + k ), rather than a complex sinusoid. The formulas in the above references must be modified to accommodate this. You can compute the cos/sin version of the Fourier series, then convert the coefficients....
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