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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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**Convolution sum representation of LTI system**

Mathematically

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**Sum up all the responses for all K’s**

Graphically Sum up all the responses for all K’s

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**Sinusoidal and Complex Exponential Sequences**

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 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

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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 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

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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 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!

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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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