1. Cellular Communications
3. DSP: A crash course

2. Signals

3. DC Signal

4. Unit Step Signal

5. Sinusoidal Signal

6. Stochastic Signal

7. Some Signal Arithmetic

8. Operational Symbols

9. Time Delay Operator

10. Vector Space of All Possible Signals

11. Shifted Unit Impulse (SUI) signals are basis for the signal vector space

12. Periodic Signals Periodic Signals have another basis signal: sinusoids
Example: Building square wave from sinusoids

13. Fourier Series

14. Another version Fourier Series

15. Complex Representation

16. Parseval Relationship

17. Fourier Transform
Works for all analog signals (not necessary periodic)
Some properties

18. Discrete Fourier Transform (DFT)
FT for discrete periodic signals

19. Frequency vs. Time Domain Representation

20. Power Spectral Density (PSD)

21. Linear Time-Invariant(LTI) Systems

22. Example of LTI

23. Unit Response of LTI

24. Convolution sum representation of LTI system
Mathematically

25. Sum up all the responses for all K’s
Graphically
Sum up all the responses for all K’s

26. Sinusoidal and Complex Exponential Sequences
LTI
h(n)

27. Frequency Response
eigenvalue
eigenfunction

28. Example: Bandpass filter

29. Nyquist Limit on Bandwidth
Find the highest data rate possible for a given bandwidth, B
Binary data (two states)
Zero noise on channel
Example shown with band from 0 Hz to B Hz (Bandwidth B) Maximum frequency is B Hz
Period = 1/B 1
Nyquist: Max data rate is 2B (assuming two signal levels)
Two signal events per cycle

30. Nyquist Limit on Bandwidth (general)
If each signal point can be more than two states, we can have a higher data rate
M states gives log2M bits per signal point
Period = 1/B
4 signal levels:
2 bits/signal 10 00 11 01
General Nyquist: Max data rate is 2B log2M
M signal levels, 2 signals per cycle

31. Practical Limits
Nyquist: Limit based on the number of signal levels and bandwidth
Clever engineer: Use a huge number of signal levels and transmit at an arbitrarily large data rate
The enemy: Noise
As the number of signal levels grows, the differences between levels becomes very small
Noise has an easier time corrupting bits
2 levels - better margins
4 levels - noise corrupts data

32. Characterizing Noise Noise is only a problem when it corrupts data
Important characteristic is its size relative to the minimum signal information
Signal-to-Noise Ratio
SNR = signal power / noise power
SNR(dB) = 10 log10(S/N)
Shannon’s Formula for maximum capacity in bps
C = B log2(1 + SNR)
Capacity can be increased by:
Increasing Bandwidth
Increasing SNR (capacity is linear in SNR(dB) )
SNR in linear form
Warning: Assumes uniform (white) noise!

33. Shannon meets Nyquist From Nyquist: From Shannon: Equating: or
M is the number of levels needed to meet Shannon Limit
SNR is the S/N ratio needed to support the M signal levels
Example: To support 16 levels (4 bits), we need a SNR of 255 (24 dB)
Example: To achieve Shannon limit with SNR of 30dB, we need 32 levels