3. 6.2 - The power Spectrum of a Digital PAM Signal A digtal PAM signal at the input to a communication channl scale factor (where 2d is the “Euclidean distance”) pulse waveform reciprocal of the symbol rate

4. to determine the spectral characteristics of the random process V(t), we must evaluate the power spectrum.  mean value  autocorrelation function

5.  average autocorrelation function

6. The power spectrum of V(t) is the Fourier transform of the average autocorrelation function. The power spectrum of amplitude sequence a n

7. Problem 6.1 Answer ip_06_01 Determine the power spectrum of V(t) when {a n } is an uncorrelated sequence and g(t) is the following rectangular pulse.

8. The Fourier trasform of g(t) is and

9. This power spectrum is illustrated in the following Figure:

10. Problem 6.2 Answer ip_06_02 Suppose the autocorrelation function of the sequence {a n } is and g(t) is the rectangular pulse.shown at the previous problem. Evaluate S v (f) in this case.

11. SOLUTION The power spectrum of the PAM signal was given by from and using also the data we can replace...

12. Consequently,

13. This power spectrum is illustrated in the following Figure:

14. 6.3 - Characterization of bandlimited channels and channels distortion We can characterize many communication channels as bandlimited linear filters, described by their frequency response Amplitude response phase response

15. : we can also express it as an envelope delay (group delay)

16. nondistorting (ideal) channel Amplitude response -constant phase response - linear function of frequncy ( envelope delay -constant) distorting channel Amplitude distortion - Amplitude response isn’t constant delay distortion - envelope delay isn’t constant)

17. Effect of channel distortion channel input channel output equalizer

18. Average amplitude and delay characteristics of a medium-range telephone channel,

19. Impulse response of the average channel with amplitude and delay shown before

20. Problem 6.3 Answer ip_06_03 T.B.D

21. Impulse response Frequency response of linear phase FIR filter

22. Problem 6.4 Window method Impulse response if the desired channel response is C(f) for |f|<W and C(f)=0 for |f|>W then for example: if channel is ideal, then C(f)=1, |f|<W and hence

23. Answer ip_06_04 Let us design an FIR filter with: W=2000 Hz and Fs =1/Ts=10KHZ. (this digital filter may be implemented by sampling h(t) at t=nTs, where Ts is the sampling interval and n = 0,+/-1,+/-2,...)

24. Samples of h(n)

25. Frequency response (rectangular window)

26. Frequency response (Hanning window)

27. Problem 6.5 The impulse response of a two-path(multipath) radio channel can expressed as: random processes the delay between 2 paths

28. Answer ip_06_05 simulate such a channel on the computer