Transmission

Physical Layer I - communicating over a single "wire"

bandwidth and signal units
digital vs analog: source, signal, transmission
multiplexing/channels
- TDM
- FDM
- SDM
limits on communication rate - noise & attenuation
- Nyquist's theorem (noiseless)
  - H = bandwidth = high frequency - low frequency, in Hertz
  - L = number of distinct signal elements (voltage levels, e.g.)
  - R = data rate in bits per second
  - lg = log base 2
  - R = 2H lg L
- Shannon's theorem (white noise)
  - S = signal power
  - N = noise power
  - R < H lg (1 + S/N)
- Note that usually only 1/4 to 1/2 the Shannon limit can be attained in practice.
- Note that signal to noise ratio (SNR) is often given in decibels, defined to be
  - SNR (dB) = 10 log (S/N)
  - where log = log base 10.
- EX: Suppose you have a channel from 10,000 Hz to 22,000 Hz, with an SNR of 18 dB. Assuming that you can achieve about 1/3 of the Shannon limit, (a) what is the data rate of your channel and (b) how many signal elements should you use?
Bandwidth = H = 22,000 - 10,000 = 12,000 Hz
SNR (dB) = 18 = 10 log S/N
1.8 = log S/N
63 = S/N (approx)
Shannon Limit = R* = H lg (1 + S/N) = 12,000 lg 64 = 12,000*6
R* = 72,000 bps
Achievable rate R = (1/3) R* = 24,000 bps << answer (a)
Nyquist rate R = 2H lg L = 24,000 = 2 (12,000) lg L
lg L = 24,000/24,000 = 1
L = 2 distinct signal elements << answer (b)

The Spectral Efficiency is defined to be the number of bits of data transmitted per hertz, or 2 lg L.

In practice, Nyquist's formula must be modified according to how the signal is filtered. A useful measurement of performance of this filtering is r, where typically 0 < r < 1. Nyquist's theorem assumes that r = -1/2, and so, like Shannon's formula, is really a limit. The general equation for multilevel signaling relating bandwidth to data rate and number of levels used is:

R = Data Rate desired
L = number of distinct signal elements (levels)
BT = Transmission Bandwidth required
BT = R (1+r)/(lg L),
or R = BT (lg L)/(1+r)
These are used for ASK and PSK. For FSK, the bandwidth is
BT = 2 (f2 - fc) + (1+r)R,
where f2 - fc = fc - f1 is the offset of the modulated frequency from the carrier frequency, fc. Twice that is the width of the transmission spectrum.

While Nyquist's result (sort of a dual of the Sampling Theorem) states that at most two signal elements can be sent per cycle, in practice it is more like one signal element per one or two cycles.

transmission media and characteristics

hardwire
- ribbon cable
- twisted pair (UTP)
- shielded TP (STP)
- coax (75 Ohm CATV, 50 Ohm digital data)
- optical fiber (multimode, single mode)
softwire
- radio
- microwave
- IR
- Ground Wave
- free space laser
- satellites

Transmission -

Balanced vs. unbalanced
synchronous vs. asynchronous

Once we are able to transmit and receive a signal, we wish to be able to encode data in that signal. While we have considered some fundamental limitations of transmission here, the next page considers the problem of encoding data in signals for transmission.

Send comments to nemo@cis.ufl.edu