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Homework 12

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Reading

  • Chapter 3 in the Anderson text

Problems

  1. Consider the 4-QASK Y-constellation described
    bits symbol
    0 0 $ \mathbf{s}_1 = (0,0)$
    1 0 $ \mathbf{s}_2 = \left(A\cos\frac{\pi}{6},A\sin\frac{\pi}{6}\right)$
    0 1 $ \mathbf{s}_3 = \left(A\cos\frac{5\pi}{6},A\sin\frac{5\pi}{6}\right)$
    1 1 $ \mathbf{s}_4 = \left(A\cos\frac{9\pi}{6},A\sin\frac{9\pi}{6}\right)$
    1. Sketch the constellation.
    2. Compute the average symbol energy in terms of $ A$ .
    3. Determine the value of $ A$ so that the average symbol energy is $ E_s = 3$ .
    4. Sketch the decision region boundaries for this constellation.
    5. Assuming a unit-energy NRZ pulse, the basis functions are

      $\displaystyle \phi_1(t)$ $\displaystyle = \sqrt{\frac{2}{T}} \cos(\omega_0t)$    
      $\displaystyle \phi_2(t)$ $\displaystyle = -\sqrt{\frac{2}{T}} \sin(\omega_0t)$    

      Sketch the modulated signal corresponding to the bit sequence 1 1 0 1 0 0 1 1. Use $ \omega_0 = 4 \pi/T$ .
    6. Determine the estimated bit sequence for the following sampled matched filter outputs:

      $\displaystyle (r_1,r_2) = \{(-0.03,+1.97),(-0.01,+1.97),(+0.01,+1.97),(+0.03,+1.97)\}$    

    7. Suppose the ML receiver uses a bank of filters matched to the signal pulses $ s_1(t), s_2(t), s_3(t),$ and $ s_4(t)$ and that the matched filter outputs are:

      $\displaystyle \langle \mathbf{r},\mathbf{s}_1 \rangle$ $\displaystyle = 0$    
      $\displaystyle \langle \mathbf{r},\mathbf{s}_2 \rangle$ $\displaystyle = A^2/4$    
      $\displaystyle \langle \mathbf{r},\mathbf{s}_3 \rangle$ $\displaystyle = A^2/5$    
      $\displaystyle \langle \mathbf{r},\mathbf{s}_4 \rangle$ $\displaystyle = A^2/6$    

      What symbol does the ML receiver choose in this case?
    8. What is the probability of making a symbol error? Express your answer in terms of $ E_b/N_0$ .
    9. How does this constellation compare with QPSK in terms of probability of symbol error and bandwidth?
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 29). Homework 12. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Communication_Systems_I_1/hw12.html. This work is licensed under a Creative Commons License Creative Commons License