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

24

Convolution sum representation of LTI system

Mathematically

25

Graphically

Sum up all the responses for all K’s

Sinusoidal and Complex Exponential Sequences

LTI

h(n)

LTI

h(n)

njenx )(

k

knxkhny )()()(

k

knjekh )()(

jn

k

jk eekh )(

jnj eeH )(

Frequency Response

nje jnj eeH )()( jeH

eigenvalueeigenfunction

k

jkj ekheH )()(

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

1 0 1 0 0 0 1 0 1 1 0 1 00 0

Period = 1/B

• Nyquist: Max data rate is 2B (assuming two signal levels)• Two signal events per cycle

Example shown with bandfrom 0 Hz to B Hz (Bandwidth B)Maximum frequency is B Hz

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

10 00 11 00 00 00 11 01 10 10 01 00 0000 11

Period = 1/B

• General Nyquist: Max data rate is 2B log2M • M signal levels, 2 signals per cycle

4 signal levels:2 bits/signal

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)

)

Warning: Assumes uniform (white) noise!

SNR in linear form

Shannon meets Nyquist

From Nyquist: MBC 2log2From Shannon: )1(log2 SNRBC

Equating: )1(loglog2 22 SNRBMB )1(loglog2 22 SNRM

)1(loglog 22

2 22 SNRM SNRM 12

SNRM 1 12 MSNRor

M is the number of levelsneeded to meet Shannon Limit

SNR is the S/N ratio needed tosupport 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

)1(loglog 22

2 SNRM