ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 9. EE 350Continuous-Time Linear SystemsRecitation 91 Jeffrey Schiano 2014. All rights reserved. Rec 9. Recitation 9 Topics2• Solved Problems – Periodic and aperiodic signals, fundamental period– Trigonometric Fourier series– Compact Trigonometric Fourier series– Sinusoidal-steady-state response of LTI systems to periodic inputs• MATLAB Exercises– Magnitude and phase of the frequency response function– Sinusoidal steady-state response of LTI systems driven by a periodic input Jeffrey Schiano 2014. All rights reserved. Rec 9. Periodic Signals• A signal f(t) is said to be periodic if for some positive constant Tofor al values of To• The smallest value of Tothat satisfies equation (1) is called the fundamental period of f(t)3 (1)oft ft T Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 1• Determine if the following signals are periodic or aperiodic. If a signal is periodic, determine its fundamental period To.4 (1) ( ) cos sin 2(2) ( ) cos 2 cos 3 sin 4ft t tft t t t Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 2• Consider a LTI system that has the input f(t), zero-state response y(t), and impulse response representation h(t)• If the input f(t) is periodic with fundamental period Toand h(t) is aperiodic, determine if the zero-state response y(t)is periodic, and if it is, the fundamental period of y(t)7y(t)LTI()htf(t) Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 9. Fourier Series Representation• A periodic signal f(t) with fundamental frequency ωoperiod has a trigonometric Fourier Series representationand a compact trigonometric Fourier Series representation10 011f( ) cos sinno nonnta a nt b nt 01f( ) cosnonntC C nt Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 3• Determine the compact trigonometric Fourier series coefficients Co, Cnand ϴn in terms of the trigonometric Fourier series coefficients ao, an, and bn• Determine the trigonometric Fourier series coefficients ao, an, and bn in terms of the compact trigonometric Fourier series coefficients Co, Cnand ϴn 11 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 3 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 9. • Determine the – fundamental period– trigonometric Fourier series representation and– compact trigonometric Fourier series representationof the periodic signal • Sketch the Fourier amplitude (Cn) and Phase spectra (ϴn) of f(t) as a function of frequency ω14Problem 4 ( ) 3 cos 3 cos 7 cos 5 45ft t t t Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 4 Solution15 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 4 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 4 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 4 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 5• Determine the trigonometric and compact trigonometric Fourier series representations of the triangle waveform• Use the fact that the signal f(t) is an even function of time to simplify the calculations 19[ms]tf( )t121211 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 5 Solution20 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 5 Solution21 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 5 Solution22 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 5 Solution23 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 6• The triangle waveform in Problem 5 is applied to the input of the active filter shown below241Cy(t)1R2Rf(t) Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 6• Using the phasor analysis method, determine the frequency response representation of the filter and place your answer in the standard form• Find an expression for the magnitude and phase of the frequency response function in terms of the parameters R1, R2, and C• Determine the DC and high frequency gain of the active filter2511101110() () ()()() () ()mmmmnnnbb bbHaaa Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 6 Solution26 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 6 Solution27 Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 6 Solution28 Jeffrey Schiano 2014. All rights reserved. Rec 9. Sinusoidal Steady-State Response• In Recitation 8 Problem 7 it was shown that the sinusoidal steady-state response of the systemis • The frequency response of a system is characterized graphically by plotting– The magnitude of H(jω) versus frequency– The phase of H(jω) versus frequency29()yt0() cos( )ftA tLTI()Hj y( ) ( ) cos ( )ooo otAHj t Hj Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 7• For the network in Problem 6 let• Using the MATLAB functions polyval, abs, angle, and unwrap, determine the magnitude and phase for 1000 frequency points uniformly spaced between DC and 100 kHz• Plot the magnitude and phase functions in the upper and lower subplots, respectively, of a single figure• Display phase in units of degrees and frequency in units of kHz3012500 , 1k , 0.1 FRRC Jeffrey Schiano 2014. All rights reserved. Rec 9. Problem 7 MATLAB Functions31 Jeffrey Schiano 2014. All rights reserved. Rec 9.
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