Practice Quiz 5 (no calculators allowed) For problems 1 and 2, assume 032tjezjω−=+, 1) The magnitude of , | , is equal to z |z a) 15 b) 113 c) 05jteω− d) 013jteω− e) none of these 2) The complex conjugate of z, , is equal to *z a) 0*32tjezjω−=− b) 0*32tjezjω+=+ c) 0*32tjezjω+=− d) none of these For problems 3 and 4, assume we know 010 45z =∠ 3) The magnitude of the conjugate of , , is equal to z*|z | a) 10 b) -10 c) 5 d) -5 e) none of these 4) The phase of the conjugate of , z*z∠, is equal to a) b) c) d) none of these 045045−00 5) Are the functions and 1() 1vt=2()vt t= orthogonal over the interval [0 ? ,1] a) Yes b) No 6) Are the functions and 1() 1vt=2()vt t= orthogonal over the interval [1,1]−? a) Yes b) No 7) Are the functions 1()ojk tvt eω= and 2()ojm tvt eω= where km≠, and are integers, and k m2ooTωπ= , orthogonal over the interval [0 ? , ]oT a) Yes b) No8) Using Euler’s identity, we can write cos( )tω as a) 2jt jteeωω−+ b) 2jt jteeωω−− c) 2jt jteejωω−+ d) 2jt jteejωω−− 9) Using Euler’s identity, we can write sin( )tω as a) 2jt jteeωω−+ b) 2jt jteeωω−− c) 2jt jteejωω−+ d) 2jt jteejωω−− For problems 10 and 11, assume we have an LTI system with impulse response (())1tht uet−+= 10) Is the system causal? a) yes b) no 11) Is the system BIBO stable? a) yes b) no 12) Assume () 2cos(3)xt= t unction )(2jHej is the input to an LTI system with transfer fω−=. In steady state the output of this system will be ω a) 3() 4cos(3)jytte−= b) c) () 4cos(3 3)yt t=− () 4cos(3 1)yt t=− d) none of these Problems 13-15 refer to a system with transfer function 10()3Hss=+. Assume the input to this system is ( ) 2cos(3 30 )oxt t=+ 13) In steady state, the magnitude of the output will be a) 203 b) 2018 c) 208 d) 206 14) In steady state, the phase of the output will be a) 30 b) c) d) o45o15o− 45o− 15) The bandwidth (-3 dB point) of the system is a) 10 Hz b) 10 radians/sec c) 3 radians/sec d) 3 Hz16) Assume () 2 3cos() 3cos(4) 2cos(6)xttt=+ + + tis the input to an LTI system with the transfer function shown graphically (magnitude and phase) below: -10 -8 -6 -4 -2 0 2 4 6 8 1000.511.522.533.54Frequency (rad/sec)|H(jω)|-10 -8 -6 -4 -2 0 2 4 6 8 10-50-40-30-20-1001020304050Frequency (rad/sec)∠ H(jω) (degrees) The steady state output of the system will be a) 0 b) () 2 3cos() 3cos(4) 2cos(6)yttt=+ + + t c) ( ) 4 6cos( ) 6cos(6 )yttt=++ d) e) 00( ) 4 6cos( 30 ) 6cos(6 45 )yt t t=+ + + +00( ) 2 6cos( 30 ) 6cos(6 45 )yt t t=+ + + +f) 000( ) 4 3cos( 30 ) 2cos(6 45 ) 3cos( 30 ) 2cos(6 45 )yt t t t t=+ + + + + − + −00g) 000( ) 4 6cos( 30 ) 6cos(6 45 ) 6cos( 30 ) 6cos(6 45 )yt t t t t=+ + + + + − + −h) none of these17) Assume is the input to a system with transfer function () 3cos(2 5)xt t=−23||()2jeHjelseωωω−⎧3<=⎨⎩ the output in steady state will be ()yt a) b) () 6cos(2 5)yt t=− () 9cos(2 5)yt t=− c) 4() 9cos(2 5)jyt t e−=− d) ( ) 9cos(2 9)yt t=− 18) Assume () 2cos(3)xt= t is the input to system with transfer function ()2jHj eωω−=. In steady state the output of the system will be a) () 4cos(3)jyt teω−= b) 3() 4cos(3)jyt te−= c) () 4cos(3 3)yt t=− d) e) none of these () 4cos(3 3)yt t=+ 19) Assume () 2cos() 5sin(2) 6sin(3)xtt t=+ +tunction is the input to a system with transfer f()Hj 35ωωdy state the output of the system will be ⎛⎞=Π⎜⎟⎝⎠. In steaa) []( ) 2cos( ) 5sin(2 ) 6sin(3 ) 3rect5yt t t tω⎡⎤⎛⎞=++⎜⎟⎢⎥⎝⎠⎣⎦ b) ( ) 6cos( ) 15sin(2 ) 18sin(3 )yt t t t=+ +c) () 6cos() 15sin(2)yt t t=+d) none of these 20) Assume () 2cos(3) 4cos(5)xtt=+t is the input to a system with transfer function given by 24||6()0Hjelseωω<<⎧=⎨⎩ The output of the system in steady state will be a) () 4cos(3) 8cos(5)yt t t=+b) () 8cos(5)yt t=c) () 4cos(3)yt t=d) none of these21) Assume ( ) cos( ) cos(5 ) cos(9 )xtt t=+ +t is the input to a system with transfer function given below: -10 -8 -6 -4 -2 0 2 4 6 8 1000.511.522.533.54ωH(jω) The output of this system in steady state will be a) b) ( ) 4cos( ) 4cos(5 )yt t t=+ ( ) 4cos( ) 2cos(5 ) cos(9 )yt t t t=++ c) d) none of these ( ) 4cos( ) 2cos(5 )yt t t=+ Answers: 1-b, 2-c, 3-a, 4-b, 5-b, 6-a, 7-a, 8-a, 9-d, 10-b, 11-a, 12-b, 13-b, 14-c, 15-c, 16-d, 17-d, 18-c, 19-c, 20-b,
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