Homework 6

Reading

• Chapter 2 in the Anderson text.

Problems

1. Prove Nyquist's First Criterion (Theorem 2.2-1 on page 19 of the Anderson text). Note that if you follow the proof given in the book, the you must first prove the Poisson sum formula. Here is a hint.
   Multiply a function by an impulse train as follows.
   
   
   
   Now compute the continuous-time Fourier transform of both sides. Note that the left hand side is a product of two functions (use the modulation property-multiplication in time corresponds to the convolution in the frequency domain). The on the right hand side are the only functions of t. If all goes well, then you end up with the Poisson formula.
   
   
   
   Now you can follow the complete proof.
2. Compute the Fourier transform of the follwing
   
   1. unit-energy NRZ pulse
   2. unit-energy RZ pulse
   3. unit-energy MAN pulse
   4. unit-energy HS pulse which is given by
      
      
      
      
3. Let be the Fourier transform of give by
   
   
   
   
   1. For what symbol rate is a Nyquist pulse?
   2. At this symbol rate, what is the excess bandwith of the pulse?
   3. Sketch terms of the sum in Equation 2.2-2 (page 19 of the Anderson text) and show the Nyquist criterion is satisfied.
4. Let be a pulse with odd symmetry, . Can be a Nyquist pulse?
5. Consider a digital modulator that transmits 3 bits per symbol and uses the unit-energy HS pulse for transmission.
   
   1. Construct a look up table with three columns showing the correspondence between: bits, symbols and the modulated pulses.
   2. Sketch the modulator, the channel, and a sampling receiver.
   3. For the bit sequence , sketch the signal or sequence at each point in the modulator/channel/receiver diagram.
6. Intersymbol interference (ISI)
   
   1. Let be the following pulse,
      
      
      
      What is the ISI?
   2. Let be the following pulse,
      
      
      
      What is the ISI?