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Homework 7

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Reading

  • Chapter 2 in the Anderson text

Problems

  1. Suppose you wish to transmit $ 21000$ bits per second over a channel with one-sided bandwidth $ \sqrt{2} \cdot 1000$ Hertz.
    1. What is the maximum symbol rate that avoids ISI?
    2. What is the fewest number of bits per symbol that avoids ISI? Let this number be denoted $ k_$min.
    3. Design a raised-cosine (RC) pulse that occupies the full channel bandwidth. What is the excess bandwidth $ \alpha$ when there are $ k_$min bits per symbol?
    4. Design a raised-cosine (RC) pulse that occupies the full channel bandwidth. What is the excess bandwidth $ \alpha$ when there are $ k_$min$ +1$ bits per symbol?
    5. Design a raised-cosine (RC) pulse that occupies the full channel bandwidth. What is the excess bandwidth $ \alpha$ when there are $ k_$min$ +2$ bits per symbol?
    6. Show graphically (via a sketch in the frequency domain) that the $ k_$min$ +2$ case avoids ISI.

  2. Using a $ \alpha = 1/\sqrt{2}$ RC pulse to transmit $ 2400$ bits/second with $ 3$ bits/symbol, what is the minimum channel bandwidth needed for zero ISI in the samples taken by the sampling receiver?

  3. Let the transmitter pulse be given by

    $\displaystyle v(t) = \left\{ \setlength{\extrarowheight}{6pt}\begin{array}{rc} ... ... -\frac{1}{T}t+1 & 0 \leq t \leq T 0 & \vert t\vert > T \end{array} \right.$ T \end{array} \right.$" align="middle" border="0" height="101" width="237" />    

    Suppose that the symbol set is $ \{ +3, +1, -1, -3\}$. Sketch the eye diagram. Be sure to show all possible transitions between symbols.

  4. Let the transmitter pulse be given by

    $\displaystyle v(t) = \left\{ \setlength{\extrarowheight}{6pt}\begin{array}{rc} ... ...eq t \leq \frac{10T}{9} 0 & \vert t\vert > \frac{10T}{9} \end{array} \right.$ \frac{10T}{9} \end{array} \right.$" align="middle" border="0" height="101" width="264" />    

    Suppose that the symbol set is $ \{ +1, -1\}$. Sketch the eye diagram. Be sure to show all possible transitions.

  5. Prove that the NRZ pulse defined by

    $\displaystyle v(t) = \left\{ \setlength{\extrarowheight}{6pt}\begin{array}{rc} 1 & -\frac{T}{2} \leq t < \frac{T}{2} \\ 0 & \text{otherwise} \end{array} \right.$< \frac{T}{2} \\ 0 & \text{otherwise} \end{array} \right.$" align="middle" border="0" height="73" width="193" />    

    is a Nyquist and a root-Nyquist pulse. Do this in the time domain.
  6. Prove that the sinc pulse defined by

    $\displaystyle v(t) =$   sinc$\displaystyle \left(\frac{t}{T}\right) = \frac{\sin(\pi t/T)}{\pi t/T}$    

    is a Nyquist pulse and a root-Nyquist pulse. Do this in the frequency domain.
Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, June 29). Homework 7. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Communication_Systems_I_1/hw7.html. This work is licensed under a Creative Commons License. Creative Commons License
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