# Cellular Communications

## Presentation on theme: "Cellular Communications"— Presentation transcript:

Cellular Communications
3. DSP: A crash course

Signals

DC Signal

Unit Step Signal

Sinusoidal Signal

Stochastic Signal

Some Signal Arithmetic

Operational Symbols

Time Delay Operator

Vector Space of All Possible Signals

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

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

Fourier Series

Another version Fourier Series

Complex Representation

Parseval Relationship

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

Discrete Fourier Transform (DFT)
FT for discrete periodic signals

Frequency vs. Time Domain Representation

Power Spectral Density (PSD)

Linear Time-Invariant(LTI) Systems

Example of LTI

Unit Response of LTI

Convolution sum representation of LTI system
Mathematically

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

Sinusoidal and Complex Exponential Sequences
LTI h(n)

Frequency Response eigenvalue eigenfunction

Example: Bandpass filter

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

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

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

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!

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