ECE 300 Signals and Systems Homework 7 Due Date Tuesday October 21 2008 at the beginning of class Exam 2 Thursday October 23 2008 Problems 1 Assume x t which has a fundamental period of 2 seconds has the following spectrum all phases are multiples of 45 degrees Amplitude 3 2 1 0 4 3 2 1 0 Harmonic 1 2 3 4 3 2 1 0 Harmonic 1 2 3 4 Phase degrees 200 100 0 100 4 a What is x t Your expression must be real b What is the average value of x t c What is the average power in x t Fall 2008 2 Assume x t has the spectrum shown below the phase is shown in radians and a fundamental frequency o 2 rad sec 4 Magnitude 3 2 1 0 4 3 2 1 0 Harmonic 1 2 3 4 3 2 1 0 Harmonic 1 2 3 4 3 Phase radians 2 1 0 1 2 3 4 Assume x t is the input to a system with the transfer function e j 1 3 j 2 H 2e 3 5 0 else Determine an expression for the steady state output y t Be as specific as possible simplifying all values and using actual numbers wherever possible Fall 2008 3 A periodic signal x t is the input to an LTI system with output y t The signal x t has period 2 seconds and is given over one period as x t e t 0 t 2 x t has the Fourier series representation 0 4323 jk t x t e k 1 jk The system is an ideal highpass filter that eliminates all signals with frequency content less than 0 75 Hz a Find the average power in x t b Determine an expression for the output y t Your expression for y t must be real Answer y t e t 0 4323 0 2622 cos t 1 2626 c Determine the average power in y t d What fraction of the average power in x t is contained in the DC and fundamental frequency components 4 Assume x t t 2 t with Fourier Series representation x t k X k e jkt where 2 3 Xk k 2 1 k 2 k 0 k 0 a Assume x t is the input to a system that eliminates all signals with frequencies outside the range 0 5 to 0 7 Hz What is the output of the system y t and what fraction of the average power in x t is in y t Note your answers must be real no e ja terms b Assume x t is the input to a system that eliminates all signals with frequencies in the range 0 5 to 0 7 Hz What is the output of the system y t and what fraction of the average power in x t is in y t Fall 2008 5 Assume two periodic signals have the Fourier series representations x t X k e jk ot y t Yk e jk ot For the following system input output relationships a y t bx t a b y t bx t a c y t bx t cos o t Answer Yn b X n 1 X n 1 2 d y t 2 n y t n2 y t K n2 x t i ii iii write Yk in terms of the X k If possible determine the system transfer function H j A system must be both linear and time invariant to have a transfer function If you cannot determine the transfer function indicate which system property is not satisfied L or TI 6 Consider an LTI system with impulse response h t and input x t A cos 0t a Using the convolution integral show that y t A j 0t A e H 0 e j 0t H 0 2 2 Where H 0 h e j 0 d This is the Fourier transform of the impulse response evaluated at the input frequency b Show that for a real impulse response H 0 H 0 the magnitude is even and H 0 H 0 the phase is odd c Using the results from parts a and b show that y t A H 0 cos 0t H 0 Cultural Information This problem shows that e j 0t is an eigenfunction of LTI systems Fall 2008 7 Most microcontrollers are capable of generating pulse width modulation PWM signals on one or more output pins These signals are square waves where both the period and the duty cycle can be programmed in the microcontroller by the use of timers and different reference clocks These PWM output signals can then be used in conjunction with lowpass filters to produce reasonable approximations to analog output signals In this problem we will use what we have learned in the course to investigate how to do this The signal v t below is a PWM signal shown for about one and a half periods The signal has period T0 amplitude V usually fixed at 5 or 3 3 volts pulse duration and duty cycle D T0 v t V T0 t jk To V k jk 0t sinc e k T0 T0 a For the periodic signal v t determine an expression for the average power in the periodic signal in terms of T0 and V Your answer must contain no sums or integrals The Fourier series representation is v t k e b Determine an expression for the average value of v t in terms of T0 and V c It is the average value of v t that we want to use as our analog output Hence we need to design a lowpass filter that allows us to keep our DC term and ideally remove all of the other harmonics Let s assume we want to use a simple first order RC lowpass filter with transfer function H j where we have set RC Fall 2008 1 j RC 1 for convenience 0 1 j 1 0 Determine the value of so that the average power in the first harmonic of the output signal is 20 dB lower than the average power in the DC component the output signal Assume here that the fundamental frequency is f 0 100 Hz the duty cycle is 0 8 T0 and V 5 0 volts d For your value of determined in part c and the parameter values given in part c determine an expression for the first two terms the DC and first harmonic in the Fourier series representation of the output signal Answer y t 4 0 566 cos 2 100t 3 77 e For your value of determined in part c and the parameter values given in part c determine the bandwidth of the filter you designed Be sure to include units Fall 2008
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