(a) State the Nyquist sampling theorem. Given a sampling time T, what is the shift in frequency between two aliased harmonics 1    and 2  ?                               [5]

(b)  Consider  a  continuous  signal  R(z)  with  the  following  qualitative  spectral content:

Sketch the magnitude of the same signal after sampling,  TR*   , highlighting aliasing.                     [5]

(c)       Describe a workaround to prevent aliasing to occur, assuming that you cannot alter the sampling frequency s  .           [4]

(d)       Sketch the spectral content of the sampled signal again, when the workaround is in place, showing and explaining how aliasing is prevented.                       [6]

Q2       (a)        …                                                                                                                       []

SECTION B

Q3                   The following transfer function is a lag network designed to increase the steady- state gain by a factor of 10 and have negligible phase lag at 1  = 3 rad/s :

H (s) = 10 

(a)        Find the gain (in dB) at 1  . Assuming a sample time T = 0.25 s, calculate the Nyquist frequency n  . [0.0048 dB; 12.5664 rad/s]                                          [2]

(b)       Design the discrete equivalent of H(s) using the backward rectangular rule. [

H (z) = 1.0224z(z) 0(0).(.)9975(9756) ]                                                      [4]

(c)       Compute the discrete equivalent of H(s) using the pole-zero matching technique

(match the steady-state gain). [ H (z) = 1.011 z(z)  0(0).(.)9975(9756) ]                             [10]

(d)       Find the gain (in dB) at 1    of the discrete equivalents, and compare with that of H(s). [0. 1013 dB; 0.0048 dB]       [4]

Q4                   The following transfer function is a lead network:

H (s) = 

(a)       Find the discrete equivalent of it, when preceded by a zero-order hold (ZOH), for sample time T = 0.25 s. Use 4 significant digits for all numbers in the solution. [ H (z) = 10  ]           [8]

(b)       Using the  inverse z-transform, find the corresponding difference equation.  [ uk  = 0.08208uk−1 +10ek  − 9.082ek−1]                                                                 [4]

(c)       State  a necessary  and  sufficient  condition for BIBO stability and determine whether the difference equation is BIBO stable. [ … , BIBO stable]                 [8]

Section C

Q5       (a)        …                                                                                                                       []

Q6       (a)       ...                                                                                                                        []