6 003 Homework 1 Due at the beginning of recitation on Wednesday February 10 2010 Problems 1 Independent and Dependent Variables Assume that the height of a water wave is given by g x vt where x is distance v is velocity and t is time Assume that the height of the wave is a sinusoidal function of distance at each instant of time Also assume that the positive peaks have a height of 1 meter relative to the average water level and that they occur at integer multiples of 2 meters when the time t 3 seconds a Determine an expression for the height of the wave h x t as a function of distance x and time t if the wave is traveling in the positive x direction at 5 meters second What is the function g for this case b Determine an expression for the height of the wave h x t as a function of distance x and time t if the wave is traveling in the negative x direction at 4 meters second What is the function g for this case c Determine the speed of the wave if successive positive peaks at x 1 3 meters are separated by 0 75 seconds 2 Even and Odd The even and odd parts of a signal x n are defined by the following xe n xe n i e xe is an even function of n xo n xo n i e xo is an odd function of n x n xe n xo n Let x represent the signal whose samples are given by n 1 n 0 2 x n 0 otherwise a Determine the even and odd parts of the signal x b Show that your answer in part a is unique c Plot the results of part a 3 Geometric sums a Expand 1 in a power series For what range of a does your answer converge 1 a 6 003 Homework 1 Spring 2010 2 b Find a closed form expression for N 1 X an For what range of a does your answer n 0 converge c Expand 1 in a power series For what range of a does your answer converge 1 a 2 4 Reconstructing CT Signals from Samples Let a t b t and c t represent the following functions of time a t 1 b t 1 c t 1 t 0 t 1 0 1 t 0 1 Let xc t represent a continuous time signal derived from the discrete time signal xd n using a zero order hold as illustrated below where consecutive samples of xd are separated by T seconds in xc xc t xd 0 xd n xd 1 n 0 2 4 6 8 t 0 10 2T 4T 6T 8T 10T a Determine an expression for xc t in terms of the samples xd n and the functions a t b t and c t Your expression should match xc t at all points in time except possibly at integer multiples of T where xc t is discontinuous Extra credit Briefly discuss whether it is possible to develop an expression that matches xc t at all points including those where xc t is discontinuous Let yc t represent a continuous time signal derived from the discrete time signal yd n using a piecewise linear interpolator so that sucessive samples of yd are connected by straight line segments yc t yd 0 yd n yd 1 n 0 2 4 6 8 10 t 0 2T 4T 6T 8T 10T b Determine an expression for yc t in terms of the samples yd n and the functions a t b t and c t Your expression need not match x t at integer multiples of T dyc t in terms of the samples yd n and the functions dt a t b t and c t Your expression need not match x t at integer multiples of T c Determine an expression for 6 003 Homework 1 Spring 2010 3 5 Missing Parameters Consider the following system X R 1 23 R Y R Assume that X is the unit sample signal x n n Determine the values of and for which y n is the following sequence i e y 0 y 1 y 2 3 7 15 31 0 1 2 4 8 16 Engineering Design Problems 6 Choosing a bank Consider two banks Bank 1 offers a 3 annual interest rate but charges a 1 service charge each year including the year when the account was opened Bank 2 offers a 2 annual interest rate and has no annual service charge Let yi n represent the balance in bank i at the beginning of year n and xi n represent the amount of money you deposit in bank i during year n Assume that deposits during year n are credited to the balance at the end of that year but earn no interest until the following year a Use difference equations to express the relation between deposits and balances for each bank b Assume that you deposit 100 in each bank and make no further deposits Solve your difference equations in part a numerically using Matlab Octave or Python to determine your balance in each bank for the next 25 years Make a plot of these balances Which account has the larger balance 5 years after the initial investment one year without interest and 4 years with interest Which account has the larger balance after 25 years i e at the beginning of the 26th year Hint See the Appendix for help with programming 7 Drug dosing When drugs are used to treat a medical condition doctors often recommend starting with a higher dose on the first day than on subsequent days In this problem we consider a simple model to understand why Assume that the human body is a tank of blood and that drugs instantly dissolve in the blood when ingested Further assume that drug vanishes from the blood either because it is broken down or because it is flushed by the kidneys at a rate that is proportional to drug concentration Let x n represent the amount of drug taken on day n and let y n represent the total amount of drug in the blood on day n just after the dose x n has dissolved in the blood so that 6 003 Homework 1 Spring 2010 4 y n x n y n 1 a Determine the amount of drug in the blood that would result after taking one unit of drug each day for many consecutive days i e determine limn y n when x n 1 b Assume that there is initially no drug in the blood Then starting on day 0 one unit of drug is taken each day Determine the first day when the amount of drug in the blood will equal or exceed half of its final value c Consider the following table of doses and resulting amount of drug in the blood n 1 0 1 2 3 4 5 6 7 x n 0 1 1 …
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