EECE 340 Homework 1 Solution, Spring 2016

1 | P a g e AMERICAN UNIVERSITY OF BEIRUT ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT EECE 340 Homework 1 – Solution Problem 1 Determine whether the following signal is periodic. If it is periodic, find its period.      t sin x(t ) 2 13 5  x(t) is periodic with a period T=26/5π seconds Problem 2 Determine whether or not the following signal f(t)=4u(t)+2sin(3t) is periodic. If the signal is periodic, determine its fundamental period. The signal f(t) is not periodic Reason: u(t) is not a periodic signal Problem 3 Determine whether or not each of the following signals is periodic. If the signal is periodic, determine its fundamental period. a) )]3 [cos(3 t x )t(     X(t) is periodic with a period T=6 seconds b) ) 4 sin(3 cos(3 ) ( )       t t x t The cosine function is periodic with a period T=6 seconds. The sine function is also periodic with a period T=8/3 seconds. X(t) is also periodic with a period of 24 seconds Problem 4 Consider the periodic signal x(t) given by the expression t3j 2j t 2j t t3j 2 e)j (2 3je 5 3je 2 e)j (2 x )t(          Determine the period of x(t) and its fundamental frequency. / 1 seconds, 2 0 rad s T     2 | P a g e Problem 5 a. Consider the everlasting signal e at X t( )   . Is X(t) an energy signal?        )t( dt x E 2 , therefore X(t) is not an energy signal. b. For which values of “a” X(t) is a power signal? Determine its average power. X(t) is a power signal if a is a complex quantity. In this case Pav=1Watt. Problem 6 Classify these signals into energy-type signals, power-type signals, and signals that are neither energy type nor power type signals. For energy-type and power-type signals, find the energy or the power content of the signals a. 𝑓(𝑡) = 4𝑒𝑗2𝜋𝑓0 + 3𝑒𝑗(2𝜋𝑓1+𝜃) Solution: Each of the above signal is periodic. Each is a power signal, therefore f(t) is a power signal if the sum is periodic. Average power=16+9=25 Watts b. 𝑓(𝑥) = 𝑒−2|𝑡| Solution: The above signal is an energy signal as most of the energy is concentrated within a finite period of time. 𝐸 = 2 ∫ 𝑒−𝑡𝑑𝑡 ∞ 0 = 2 𝐽𝑜𝑢𝑙𝑒𝑠 Problem 7 Categorize each of the following signals as an energy signal or a power signal. Sate the reason for your answer. (a) The continuous-time signal x(t) , defined by E is finite. Energy Signal 3 | P a g e (b) The continuous-time signal z(t) , defined for - < t < by z(t) = 3 sin(πt) + 2 cos(3πt) Periodic. Power Signal (c) None Problem 8 Consider the signal shown below a. What is the total energy of the signal x(t) The total energy is given by:           1 1 T T 1 2 2 2 2A T A dt dt )t( f E Joules b. What is the time-averaged power of the signal x(t) The time-averaged power is given by: 0 dt )t( f T 1 P / 2 T / 2 T 2 T 1 1 lim       4 | P a g e Problem 9 Consider the signal shown below          1 t 0 1 t t 1 x )t( a. Determine the total energy of this signal. Is x(t) an Energy Signal?      1 t dt 1 E X(t) is not an energy signal b. Determine the average power of this signal. Is x(t) a power signal. does not exist tdt 1 T lim 1 P / 2 T /t 2 T      . X(t) is not a power signal. Problem 10 A continuous-time signal g(t) is defined as: 2 12cos (2 ), 8 31 ( ) 0, elsewhere t t g t         a. Is g(t) an energy signal? Show your work. g(t) is an energy signal as it is a finite duration signal. b. Is g(t) a power signal? Show your work. g(t) is not a power signal as it cannot be both energy and power.