Homework 8 - 7 Problems on Signals and Systems with Solution | ECE 300, Assignments of Electrical and Electronics Engineering

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ECE 300 Signals and Systems Homework 8 Due Date: Tuesday May 7 at the beginning of class Problems 1) Show that if x(¢) is real and even, then X(w)is real and even, and if x(s) 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(f) = cosy] D -ut-| is 2eos{ 0) X(@) =— 4 ow 3) By evaluating the integral by hand, show that the Fourier transform of x(t) = e“u(t) is given by 1 X(@)= = l+jo Vi+e° 4~tan (wv) [=] degrees oe 4) By evaluating the integral by hand, show that the Fourier transform of x(t)=e" is X@)= 5 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 signais. In this problem we will use what we have learned in the course to investigate how to do this. The signal v(r) below is a PWM signal, shown for about one and a half periods. The signal has period 7,, amplitude V (usually fixed at 5 or 3.3 volts), pulse duration 7, and v duty cycle D=— q, Spring 2009