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Periodic Signals: 1. For the following signals, (i) determine analytically which are periodic (if periodic, give the period) and (ii) sketch the signals. (Scale your time axis so that a sufficient amount of the signal is being plotted.). a) b) c) d) e) f) g) h)



x(t) = 4 cos(5πt) x(t) = 4 cos(5πt-π/4) x(t) = 4u(t) + 2sin(3t) x(t) = u(t) - 1/2 x[n] = 4 cos(πn) x[n] = 4cos(πn-2) x[n] = 2sin(3n) x[n] = u[n]+p4[n]



2. Determine if the following signals are periodic; if periodic, give the period. a) x(t) = cos(4t) + 2sin(8t) b) x(t) = 3cos(4t) + sin(πt) c) x(t) = cos(3πt) + 2cos(4πt) 3. Give an expression for the signal: 2 1 0



x(t)



-1 -2 -3 -4 -5 -6 -2



-1



0



1 Time (sec)



2



3



4



4.



3 2.5 2



x(t)



1.5 1 0.5 0 -0.5 -1 -4



-3



-2



-1



0 Time (sec)



1



a) Give an expression for x(t). b) Plot dx/dt. 5. Are the following periodic? If so, give the period. a) x(t) =4cos(3πt+π/4) + u(t) b) x[n] = 4cos(0.5πn + π/4) c) x(t) = 4cos(3πt+π/4) + 2cos(4πt) d) x[n] = 12cos(20n) e) x(t) = cos(2ω1t) + cos(3ω1t) where ω1 is a specific frequency f) x(t) = 4cos(3πt+π/2) + 2cos(8πt+π/2) g) x(t) = 2cos(3πt+π/2) + 4cos(10t-π/2) h) x[n] = 10cos(2π(8)n) i) x[n] = 10cos(8n)



2



3



4



3. Give an expression for x(t).



2 1 0 -1 x(t -2 ) -3



T=2 sec



-4 -5 -6 -2



-1



0



offset is -2 amplitude is 8/2 = 4 frequency is 2 π / 2 = πrad / sec shift is 0.35 sec to the left x(t) = -2 + 4cos(π(t+0.35)) = -2 + 4cos(πt+0.35π)



1 Time



2



3



4



System Response: 1. Sketch the response of each of the systems below to a step input. 10 s+ 2 0.2 b) H (s) = s + 0.2



a) H (s) =



2. Given, the two step responses shown below, the first one is a first order system and the second one is a second order system. Determine the transfer functions for both systems. Step Response



Amplitude



2 1.5 1 0.5 0



0



2



4



6



8



10



12



Time (sec.) Step Response



Amplitude



1.5 1 0.5 0 0



5



10



15



20



25



30



35



40



45



Time (sec.)



3. Plot the pole positions for each of the following systems, determine the values for ζ and ωn for the stable second order systems with complex poles. a) b) c) d)



1 H (s) = s+ 4 1 H (s) = s + 10 1 H (s) = s− 2 1 H (s) = 2 s + 4s + 16



e) H (s) = f) H (s) = g) H (s) =



1 s + 4s + 3 1 2



s 2 + 4s + 2 1 s 2 − 4s + 16



4. Give the general form of the response of the systems in Problem 3 to a step input. 5. Determine the steady-state response of the systems in Problem 3 a), d), and f) to an input of x(t) = 2cos(4t-20o)u(t).



6. Given the following system: H (s) =



10 s + 10s + 100 2



a) Plot the poles. Identify the values of ωn and ζ. b) Sketch the step response. c) What is the steady-state response of the system to the following input? x(t) = cos(10t)u(t)



2. Step Response



Amplitude



2 1.5 1 0.5 0



0



2



4



6



8



10



12



Time (sec.) Step Response



Amplitude



1.5 1



envelope of decay



0.5 0



0



5



10



15



20



25



30



35



40



45



Time (sec.)



k . The time-constant is the time that the s+a response is equal to 2(1-e-1 ) = 63% of 2 = 1.26, so τ ≈ 2 sec. a=1/τ = 0.5. The steady-state value (due to a unit step input) is H(0) = k/a = k/0.5. From the plot 2 = k/0.5 so k = 1.



First order system (top plot) has general form H (s) =



Final answer: H (s) =



1 s + 0.5



Second order system (bottom plot) has general form H (s) = H (s) =



k s + 2ζω n s + ω n 2 2



or



k



where the real part of the pole is at -ζωn is the real part of the pole (it governs (s + ζω n ) 2 + ω d 2 the envelope of decay) and ωd is the imaginary part of the pole (it governs the frequency of the oscillations, ωd = 2π/T). From the plot, T ≈ 12 sec, so ωd = 2π/13. The time constant of the envelope of decay is about τ ≈ 7 sec, k . Solving fo k yields k ≈ so ζωn=1/7. k is found from the steady-state value 1 = H (0) = (ζω n ) 2 + ω d 2 0.254. Final answer: H (s) =



0.254 s + 0.286s + 0.254 2



Plotting Signals: 1. Sketch the following signals: a)   0 if t < −4  x( t ) = t + 2 if − 4 ≤ t < 3  t − 2 if  3≤ t b) y(t) = x(t-1) where x(t) is defined in part a)



c) if  0  x[ n] = 2 n − 4 if  4 − n if 



n