EE422

Lab #2 (counts as HW#19 and HW#20)

Be sure to check your scores and let me know of any problems.

1.a) Determine the Nyquist sampling frequency for the signal, y=10sin(2p t)

b) Starting at t=0, plot 10 or so points if we sample the signal, y=10sin(2p t), at

i) fs=4xNyquist Frequency ii) fs=2xNyquist Frequency iii) i) fs=Nyquist Frequency

c) From your time domain plot, is there any way that we can recover the sampled signal when we sampled at fs=Nyquist Frequency?

d) Sketch two complete periods of the results of quantizing the signal, y=10sin(2p t), if our A/D converter has:

i) N=1 bit ii) N=2 bits iii) N=4 bits

e) Using the formula derived in class, calculate the MSE due to quantizing your signal in part a) if your A/D converter has:

i) N=12 bits (which we have) ii) N=6 bits iii)N=2 bits

2.(Some basic Z-transform problems)

a) Prove the final-value theorem for Z-transforms (hint: )

b) Recall for Laplace transforms, we had a condition on sX(s) in order to apply the final-value theorem. (i.e., the region of convergence of sX(s) had to contain the jw -axis). What is the corresponding condition for the Z-transform final value theorem?

c) Use the final value theorem (if possible!) on the following to determine the steady-state value of xk if

i) ii) iii)

d) Use the definition of the Z-transform to prove that

e) Finally, Use the results of part d) to find the Z-transform of

i) ii) iii)

1. Our D/A system has a sample and hold circuit such that the value of the sampled signal is held constant until the next sampling instant. Plot 20 or so points of sampling and holding the real signal, y=10*sin(2p t) at:

i) fs=4xNyquist ii) fs=2xNyquist iii) fs=Nyquist

Note: before running on the computer, you must calculate the sampling period, Ts, and how long we need to run the test to obtain 100 points (i.e., 100Ts).

2. Choose Ts=0.005 sec and 200 points to investigate the the effects of quantizing the signal, y=10*sin(2p t) by, using

i) N=12 bits (which we have) ii) N=6 bits iii)N=2 bits

Obtain plots and compare to your theoretical calculations.

3.Recall that the formula, MSE=S²/12, was based upon assumption that the signal was linear in the quantizing band! Is our signal, y=10*sin(2p t), linear in the quantizing band? If not, repeat part 2 using the ramp, y=10t. Are your results closer to your theoretical answers?