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CELLULAR COMMUNICATIONS 3. DSP: A crash course. Signals.

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Presentation on theme: "CELLULAR COMMUNICATIONS 3. DSP: A crash course. Signals."— Presentation transcript:


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 24 Convolution sum representation of LTI system Mathematically

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

26 Sinusoidal and Complex Exponential Sequences LTI h(n) 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 101000101101000 Period = 1/B Nyquist: Max data rate is 2B (assuming two signal levels) Two signal events per cycle Example shown with band from 0 Hz to B Hz (Bandwidth B) Maximum frequency is B Hz

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 log 2 M bits per signal point 10001100 110110 0100 11 Period = 1/B General Nyquist: Max data rate is 2B log 2 M M signal levels, 2 signals per cycle 4 signal levels: 2 bits/signal

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 log 10 (S/N) Shannon’s Formula for maximum capacity in bps C = B log 2 (1 + SNR) Capacity can be increased by: Increasing Bandwidth Increasing SNR (capacity is linear in SNR(dB) ) Warning: Assumes uniform (white) noise! SNR in linear form

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

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