Q1.

(a)        Out  of the  continuous  Fourier  transform,  the  fast  Fourier  transform,  and  the  convolution

product, explain which of these operators/algorithms is the most useful in current wireless communication systems. Justify your answer (in a maximum of 100 words). [10%]

(b)         Let    y (t )  and    z (t )  be   two    time-domain    signals,    where    z (t )= rect (t − 12)  and

y (t )=〈t(t)herw(0 ≤ t)is(≤)e1 .

(i)       Calculate the convolution between  y (t ) and  z (t ), i.e.  w (t )= y (t )∗ z (t ). Justify your answers by providing all the details of your calculations. [20%]

(ii)      Determine if the resulting signal from the cross-correlation between  y(t ) and  z (t ) ,i.e. y (t )*z (t ), is the time reflection of w(t ). Justify your answer without calculation. [10%]

(c)         Let  s(t) = 7sin (2πt) − 2sin (πt)t − cos(πt  2) be a time -domain signal.

(i)       Calculate  its  discrete  Fourier  Transform  for  a  time  window  of  T=  4/3  second,  a sampling time of Ts= 1/3 second and a number of samples N= 5. [20%]

(ii)      Do the values  of the  parameters  T,  Ts    and N in the previous  questions  meet  the requirements of the Nyquist-Shanon theorem. Justify your answer. [10%]

(d)        Let  u (t )= cos (2απt), v (t )= βsinc (βt ) be two time domain signals, where α and   βare two non-negative real parameters. What are the different types of signals that can be obtained when performing the convolution product of u(t )by v (t )(based on different values of α and   β)? Justify your answer.  [10 %]

(e)        Prove that the Fourier Transform of  g (t )= −  + nrect (2t + 2n − 12) can be expressed as G (f )= δ(f + (2n − 1)) −δ(f − (2n − 1)) , where δ(.) stands for the delta Dirac

function. Justify your answer by providing all the steps of your derivation.    [20 %]

Q2.

(a)        Explain, in your own words, what the maximum a posteriori (MAP) rule is and how it is useful in communication systems (in a maximum of 100 words).  [10%]

(b)         Let    r= [3   10    − 15]T be  a   signal  received  by   a   3x2   MIMO   system.   Knowing  that

「 3    − 1]

H= |−6   − 1 | and that the transmit symbols are binary phase shift keying (BPSK) symbols, |L 7     2 」|

(i)         calculate the value of the detected signal when applying the zero forcing (ZF) detection method. Justify your answer by providing all the details of your calculations;     [15 %]

(ii)        calculate the value of the detected signal when applying the minimum mean square error (MMSE) detection method for a signal-to-noise ratio of 10 dB. Justify your answer by providing all the details of your calculations;   [15 %]

(iii)       calculate the value of the detected signal when applying the maximum-likelihood

(ML) detection method. Justify your answers by providing all the details of your calculations;   [15 %]

(iv)       discuss the performance of these three detection methods, i.e. ZF, MMSE, and ML,

in terms of probability of detection error and computational complexity in the

general case. Which one provides a good trade-off? (in a maximum of 150 words) [15%]

(c)         Let us assume that a symbol  s, s ∈ {−α, α} is transmitted over a wireless channel, where noise

is added to it at the receiver such that  r = s + n .

(i)         Assuming that the noise n follows a uniform distribution such that n →u( −β, β), determine which of the following two settings of the parameters “ andβ  is the best in terms of probability of detection error for the symbol s (when assuming a decision threshold set to 0); setting 1: α = 3 and  β= 1 ; setting 2: α = 1/ 3 and β= 1 . Justify your answer by providing all the details of your reasoning.  [15 %]

(ii)        Assuming that the noise n follows a Gaussian distribution with a mean of 0 and a

variance of β, i.e.  n → N (0, β), determine which of the following two settings of the parameters “ and β  is the best in terms of probability of detection error for the    symbol s (when assuming a decision threshold set to 0); setting 1: α = 1and  β= 3 ;  setting 2: α = 13 and β= 1 . Justify your answer by providing all the details of your  reasoning. [15 %]