##### Personal tools
•
You are here: Home Digital Modulation, Linear Receiver, Square-Root Nyquist Pulses

# Digital Modulation, Linear Receiver, Square-Root Nyquist Pulses

## Objective

The objective of this lab is to deepen students understanding of principles of digital modulation.

## Building the basic digital modulator and linear receiver

2. Open up your final model from Lab 5.  The essential elements in this model are:
• Binary source
• Look up table for converting bits to symbols
• Impulse generator (upsample block)
• Pulse generator (transmitter pulse filter)
• Additive white Gaussian noise channel
• Delay block (implemented by a filter)
• The sampling receiver (downsample block)

3. In this lab, we want to replace the sampling receiver by a linear receiver.
4. Use the "Save as..." option in the file menu to create a new copy of your linear receiver model.  Use the new copy as the starting point in this lab to create your linear receiver.
5. At the receiver end, between the delay filter and the downsampler, insert another Discrete Filter block.  This filter will serve as our matched filter whos impulse response will be a time reversed version of the transmit filter impulse response.
6. Add another Discrete-Time Eye Diagram Scope to the diagram.  You should now have two eye diagram blocks.
7. Connect one of the eye diagram blocks to the matched filter input and one to the matched filter output.
8. Your model should now look like this.

9. Let's try the NRZ pulse first.  In class we discussed how the NRZ pulse is both a Nyquist pulse and a square-root Nyquist pulse. We will verify this via simulation now.
10. At the Matlab prompt type
11. nrzpulse = ones(1,100)/10;

12. This is a unit energy NRZ pulse.
13. Set the impulse response of the transmitter filter to nrzpulse .  Do this by setting the Numerator to nrzpulse and the Denominator to 1 in the Discrete Filter parameter dialog.
14. The linear receiver uses a filter matched to the NRZ pulse.  The matched filter impulse response is a time reversed version of the NRZ pulse.  Because the NRZ pulse is symmetric, the impulse response of the matched filter is again an NRZ pulse.
15. Set the impulse response of the receiver filter (this is the new filter you just added to your model), to nrzpulse .
16. Set the impulse response of the delay filter to 1 (numerator) and 1 (denominator).  This sets the delay to zero.
17. Set the Gaussian channel noise to have a variance of 10^(-7) so that the channel noise will be very small.
18. Run the simulation.
19. The eye diagrams and scatter plots should look like this.
20.

21. Using the information contained in these diagrams, explain why the NRZ pulse is both a Nyquist pulse and a square-root Nyquist pulse.
22. Now let's repeat this experiment using the sinc pulse.  In a homework assignment, we proved that the sinc pulse is both a Nyquist pulse and a square-root Nyquist pulse.  Let's verify this via simulation now.
23. Define the sinc pulse by typing the following at the Matlab prompt.  It will define a unit energy sinc pulse.
24. sincpulse = sinc([-12:0.01:12]);
sincpulse = sincpulse/norm(sincpulse);

25. Because this is a symmetric pulse, we can use it for the impulse response of the transmitter pulse filter and the matched filter too.
26. Note that the since pulse is infinite in length.  Here, we have truncated the response to 24 symbol periods.  So, our simulation is only an approximation.
27. Change the impulse response of the transmitter filter and the matched filter in the receiver to sincpulse .
28. Run the simulation.
29. The eye diagrams and scatter plot should look something like this.
30.

31. Using the information contained in these diagrams, explain why the sinc pulse is both a Nyquist pulse and a square-root Nyquist pulse.
32. Next let's experiment with the raised-cosine (RC) pulse.  It is a Nyquist pulse but not a square-root Nyquist pulse.
33. In the Matlab workspace, define a 30% rasied-cosine pulse.  We have done this in previous labs.  There's Matlab code in the Anderson text that shows how to do this.  We want a unit-energy RC pulse.
34. rcpulse = rcpuls(0.3,[-6:0.01:6]);
rcpulse = rcpulse/norm(rcpulse);

35. Set the impulse responses of the transmitter filter and receiver matched filter to rcpulse .
36. Run the simulation.
37. The eye diagrams and scatter plot should look something like this.
38.

39. Using the information contained in these diagrams, explain why the RC pulse pulse is a Nyquist pulse but not a square-root Nyquist pulse.
40. Next let's experiment with the square-root RC pulse.  This pulse is a square-root Nyquist pulse but not a Nyquist pulse.
41. Define the unit-energy sqrare-root RC pulse in the Matlab workspace by typing the following at the Matlab prompt.
42. rtrcpulse = rtrcpuls(0.3,[-6:0.01:6]);
rtrcpulse = rtrcpulse/norm(rtrcpulse);

43. The function rtrcpuls is given on page 27 of the Anderson text.
44. Change the impulse response of the transmit and receive matched filters to rtrcpulse .
45. Run the simulation.
46. The eye diagrams and sactter plot should look something like this.
47.

48. Using the information contained in these diagrams, explain why the square-root RC pulse pulse is not a Nyquist pulse but is a square-root Nyquist pulse.
49. To complete this lab exercise, write a description about what you learned about the following topics.  Use one or two paragraphs to describe each topic.  Include figures that you produced to help explain each topic.  However, please be paper conscious.  Save the forests.  See GreenPeace .
• Nyquist pulses versus square-root Nyquist pulses
• Linear versus the sampling receiver