ECE 300 Signals and Systems Homework 8 Due Date: Thursday October 29 at the beginning of class Problems 1) Show that if ()xt is real and even, then )(Xωis real and even, and if ()xt is real and odd, then )(Xω is imaginary and odd. 2) By evaluating the Fourier transform integral directly, and using Euler’s identiy, show that the Fourier transform of () cos() ( ) ( )22xt t ut utππ⎡⎤=+−−⎢⎥⎣⎦ is 22cos2()1Xπωωω⎛⎞⎜⎟⎝⎠=− 3) By evaluating the integral by hand, show that the Fourier transform of () ()txteut−= is given by 1211 18() tan()11Xjωω0ωπω−⎡⎤== −⎢⎥+⎣⎦+(degrees 4) By evaluating the integral by hand, show that the Fourier transform of ||()txte−= is 22()1Xωω=+ 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 below is a PWM signal, shown for about one and a half periods. The signal has period , amplitude V (usually fixed at 5 or 3.3 volts), pulse duration ()vt0Tτ, and duty cycle 0DTτ=. Fall 2009The Fourier series representation is 000() sincojkkTjkVkvt e eTTπτktωττ=∞−=−∞⎛⎞ ⎛⎞=⎜⎟ ⎜⎟⎝⎠ ⎝⎠∑ a) For the periodic signalvt , determine an expression for the average power in the periodic signal in terms of T,()0τ, and V. Your answer must contain no sums or integrals. b) Determine an expression for the average value of in terms of T,()vt0τ, and . V c) It is the average value of 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 ()vt 011()11HjjRCjωωαωω==++ where we have set 0RCαω= for convenience. Determine the value of α 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 is0100fHz=, the duty cycle is 00.8Tτ=, and Vvolts. 5.0= d) For your value of αdetermined 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: ( ) 4 0.566cos 100 3(2 .77)tytπ−+≈ ) ()vt V t τ 0T Fall 2009e) For your value of αdetermined 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 () ()txteut−= over the time interval [-4,4]. Hence the period 8T= in this case. You should be sure to look at the Fourier series representation to verify everything is correct. b) Construct a vector 0[:]WNNω=− 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 versus k|kTc|oω and the phase (in degrees) versus kkc(oω. 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 . 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. |kTc|f) Add plots of the magnitude and phase of ()Xω on the existing plots. It might be easiest if you define an anonymous function for this (just as you did for (()xt ). 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 type c_k , and to get kc0ω type \omega_0) Fall 2009-8 -6 -4 -2 0 2 4 6 800.20.40.60.8k ω0T|ck|Fourier SeriesFourier Transform-8 -6 -4 -2 0 2 4 6 8-50050k ω0Phase (deg)Fourier SeriesFourier Transform 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 100N=. Do not change()xt . Here we are increasing the period of the function ()xt to demonstrate that the Fourier transform is just a Fourier series in the limit as , T →∞02kkTπωω⎛⎞=→⎜⎟⎝⎠, and ()kTc Xω→ h) Redo the above for ||()txte−=. Turn in 3 plots for this part (4 plots if you can get T=64 to go…). Fall 20097) (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 () cos(4) cos(5)fttt=+ . b) Now assume we look at ()ft/ )T for a finite time, say T seconds. What we see is actually . Determine an expression for the Fourier transform of , and write your answers in terms of sinc functions. () () (yt f trectt=()ytc) Plot, using Matlab, ()Yωfor ω between 0 and 10 when T=1, T=6, T=10, T20, and T= 0. Can you clearly tell there are two cosines present when
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