ECE 300 Signals and Systems Homework 8 Due Date Thursday October 29 at the beginning of class Problems 1 Show that if x t is real and even then X is real and even and if x t is real and odd then X is imaginary and odd 2 By evaluating the Fourier transform integral directly and using Euler s identiy show that the Fourier transform of x t cos t u t u t is 2 2 2 cos 2 X 1 2 3 By evaluating the integral by hand show that the Fourier transform of x t e t u t is given by X 1 1 180 tan 1 degrees 1 j 1 2 4 By evaluating the integral by hand show that the Fourier transform of x t e t is 2 X 1 2 5 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 Fall 2009 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 1 j RC 1 1 j 1 0 for convenience Determine the value of so that 0 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 and V 5 0 volts where we have set RC T0 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 Fall 2009 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 6 Matlab Problem In this problem you will utilize your Matlab program Complex Fourier Series m to demonstrate that as the period of a periodic function increases the Fourier series approximates the Fourier transform Use your answers for Problems 3 and 4 to plot the Fourier transform results a Use your program to determine the Fourier series for x t e t u t over the time interval 4 4 Hence the period T 8 in this case You should be sure to look at the Fourier series representation to verify everything is correct b Construct a vector W N N 0 just as you did in lab This will make plotting and evaluation functions much easier c Construct the vector C fliplr conj c c0 c This will make plotting easier d Modify your code to plot the amplitude T ck versus k o and the phase ck in degrees versus k o You should use the subplot command and plot both on one page You should use the command orient tall before any plotting to use more of the page Some Matlab commands you might find useful are angle length and abs Instead of using the stem command you should use the plot command and plot discrete points like dots e By using the axis command limit the axes for magnitude plot to the range 8 to 8 radians sec and from 0 to the maximum value of T ck The max command may prove useful here Limit the axes for the phase plot to the range 8 to 8 radians sec and from 90 to 90 degrees f Add plots of the magnitude and phase of X on the existing plots It might be easiest if you define an anonymous function for this just as you did for x t You may need the functions sqrt and or Use a solid line type and be sure to add legends If you have done everything correctly and you use N 100 points in the Fourier series you should get the plot shown in Figure 1 Be sure to modify the title and axes so they look like those in this figure to get ck type c k and to get 0 type omega 0 Fall 2009 T ck 0 8 Fourier Series Fourier Transform 0 6 0 4 0 2 0 8 6 4 2 0 2 4 6 8 Phase deg k 0 Fourier Series Fourier Transform 50 0 50 8 6 4 2 0 2 4 6 8 k 0 Figure 1 Example plots for part d g Change the duration of the periodic signal to 8 8 so the period is T 16 and 16 16 so the period is T 32 and rerun your code Try and run a duration of 32 32 so the period T 64 though this may not work well Turn in your plots Keep the number of points at N 100 Do not change x t Here we are increasing the period of the function x t to demonstrate that the Fourier transform is just a Fourier series in the limit 2 as T k k 0 and Tck X T h Redo the above for x t e t Turn in 3 plots for this part 4 plots if you can get T 64 to go Fall 2009 7 Matlab Problem In this problem we ll look at a real world situation when we have to truncate a signal This actually happens more with digital signal processing but we can get the basic idea using our continuous time abilities a Find an expression for the Fourier transform of f t cos 4t cos 5t b Now assume we look at f t for a finite time say T seconds What we see is actually y t f t rect t T Determine an expression …
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