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CELLULAR COMMUNICATIONS 3. DSP: A crash course

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Signals

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DC Signal

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Unit Step Signal

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Sinusoidal Signal

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Stochastic Signal

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Some Signal Arithmetic

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Operational Symbols

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Time Delay Operator

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Vector Space of All Possible Signals

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

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

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Fourier Series

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Another version Fourier Series

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Complex Representation

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Parseval Relationship

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

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Discrete Fourier Transform (DFT) FT for discrete periodic signals

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Frequency vs. Time Domain Representation

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Power Spectral Density (PSD)

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Linear Time-Invariant(LTI) Systems

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Example of LTI

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Unit Response of LTI

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24 Convolution sum representation of LTI system Mathematically

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

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

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Frequency Response eigenvalue eigenfunction

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Example: Bandpass filter

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Nyquist Limit on Bandwidth Find the highest data rate possible for a given bandwidth, B Binary data (two states) Zero noise on channel 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

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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 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

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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

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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

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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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