exam_2013_06_26
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Exam of DIGITAL COMMUNICATIONS 26/06/2013 Use separate numbered sheets for each question. Write your name on every sheet. Fasten the sheets by a STAPLER. The use of communication devices (mobile phones, laptops, etc.) is prohibited. Books, notes, pocket calculators are allowed. Available time: 90 minutes. 1. Find the analytic signal and the in-phase and quadrature components corresponding to the signal x(t) = cos(12πt) + 2 sin(16πt) assuming a carrier frequency f c = 7. 2. Consider a binary modulation scheme represented by the following signals: s 1 (t)= u(t)+ u(t - 0.3) - 2u(t - 1), s 2 (t)= u(t) - 2u(t - 0.6) + u(t - 1). Here, u(t)= 1 t>0 , i.e., u(t) = 1 for t> 0 and 0 otherwise, and the symbol time is normalized to T = 1. The transmitted signals are equiprobable. (a) Plot the two signals. (b) Calculate the average bit energy. (c) Calculate the squared distance of the signals. (d) Calculate the error probability over the AWGN channel as a function of E b /N 0 as- suming equiprobable signals. 3. Consider the modulated signal x(t)= ∑ n a n ψ(t - nT ) where the transmitted symbols a n are stationary with zero mean and correlated as follows: E[a n+m a n ]= 1 m =0 0.3 m = ±1 0 otherwise and T is the symbol time. Moreover, assume Ψ(f )= T 1 |f |<1/T (Fourier transform of the modulation pulse ψ(t)). (a) Calculate the power density spectrum of x(t). (b) Calculate the fraction of power contained in the Shannon bandwidth with respect to the total signal power.
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Exam of digital communications
Transcript of exam_2013_06_26
Exam of DIGITAL COMMUNICATIONS
26/06/2013
Use separate numbered sheets for each question.Write your name on every sheet. Fasten the sheets by a STAPLER.The use of communication devices (mobile phones, laptops, etc.) is
prohibited. Books, notes, pocket calculators are allowed.Available time: 90 minutes.
1. Find the analytic signal and the in-phase and quadrature components corresponding tothe signal x(t) = cos(12πt) + 2 sin(16πt) assuming a carrier frequency fc = 7.
2. Consider a binary modulation scheme represented by the following signals:
s1(t) = u(t) + u(t− 0.3) − 2u(t− 1), s2(t) = u(t) − 2u(t− 0.6) + u(t− 1).
Here, u(t) = 1t>0, i.e., u(t) = 1 for t > 0 and 0 otherwise, and the symbol time isnormalized to T = 1. The transmitted signals are equiprobable.
(a) Plot the two signals.
(b) Calculate the average bit energy.
(c) Calculate the squared distance of the signals.
(d) Calculate the error probability over the AWGN channel as a function of Eb/N0 as-suming equiprobable signals.
3. Consider the modulated signal x(t) =∑
n anψ(t− nT ) where the transmitted symbols anare stationary with zero mean and correlated as follows:
E[an+man] =
1 m = 00.3 m = ±10 otherwise
and T is the symbol time. Moreover, assume Ψ(f) = T1|f |<1/T (Fourier transform of themodulation pulse ψ(t)).
(a) Calculate the power density spectrum of x(t).
(b) Calculate the fraction of power contained in the Shannon bandwidth with respect tothe total signal power.
SOLUTION
1. The analytic signal isx̌(t) = ej12πt − j2ej16πt.
The complex envelope isx̃(t) = e−j2πt − j2ej2πt.
The in-phase and quadrature components are:{xc(t) = cos(2πt) + 2 sin(2πt)xs(t) = − sin(2πt) − 2 cos(2πt)
2. (a)
(b) The energies of the two signals are:
E1 = 0.3 × 1 + 0.7 × 4 = 3.1, E2 = 1.
Thus, Eb = 2.05.
(c) The squared minimum distance is
d2min = 0.3 × 1 + 0.4 × 9 = 3.9.
(d) Since d2min/(2Eb) = 0.9512, we have P (e) = Q(√
0.9512Eb/N0).
3. (a) We haveSa(f) = 1 + 0.6 cos(2πfT ).
Hence, Gx(f) = T [1 + 0.6 cos(2πfT )]1|f |<1/T .
(b) The power contained in the Shannon bandwidth is:∫ +1/(2T )
−1/(2T )Gx(f)df = 1.
The total power is: ∫ +1/T
−1/TGx(f)df = 2.
Thus, the fraction of power contained in the Shannon bandwidth is 50%.
