1. Communications Theory
N Amanquah
Ashesi University

2. DATA COMMUNICATION THEORY
FOURIER ANALYSIS
DATA COMMUNICATION THEORY
Harmonics - Signals of any flavor are made up of harmonics.
It carries no real information other than its frequency (musically it's pretty dull too.)
Information-carrying-signals are made up of a number of frequencies.
the fundamental frequency, +other frequencies that are integer multiples of the fundamental.
Higher frequency components are called harmonics.
Fundamental = 1st harmonic
Chap. 2- Physical 2

3. Fourrier Series
Any wave can be formed by the addition of a number of sinusoidal waves.
21 March 1768)-16 May 1830) (aged 62)
Mathematician, Physicist and historian
Institutions: École Normale,
École Polytechnique
Doctoral advisor: Joseph Lagrange

4. DATA COMMUNICATION THEORY
FOURIER ANALYSIS
If the wave g is a function of time t, then
An and Bn =amplitude of the individual component waves.
Chap. 2- Physical 4

5. Example An = 2/T  g(t)sin(2 p n f t) dt
Eg, for a square wave:
g(t) = 1 ( 0 <= t < 1, <= t < 3, )
= 0 ( 1 <= t < 2, <= t < 4, )
then solve for (from 0 to T)
An = 2/T  g(t)sin(2 p n f t) dt
Bn = 2/T  g(t)cos(2 p n f t) dt
C = 2/T  g(t) dt

6. Example Review Example graphically
Review Example graphically

7. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Relevance of Fourier Series (decomposition) to data communications:
Fourier series represents component harmonics (frequencies) and amplitudes
A channel attenuates component frequencies by different amounts.
A channel can be deliberately constrained (by filters) for system efficiency. Eg telephone system uses 3KHz filter -allows multiple channels per medium.
Chap. 2- Physical 7

8. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Compare original signal vs signal received depending on the component frequency present
The greater the number of components, the higher the similarity with the original signal.
Higher frequency components may be absent
Chap. 2- Physical 8

9. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Why Square waves?
Wave Shape - A pure sine wave (the fundamental only) alone does not provide any information.
some degree of "squareness" is necessary, to be able to detect a voltage level/signal  requires some harmonics in the signal to be square
the frequency of the fundamental (also called the first harmonic)
Note: not all frequencies are passed by a medium
Bandwidth
The range of frequencies transmitted without being strongly attenuated.
The 3dB cutoff – the frequency at which half the power gets through.
Channels are usually LOW PASS, (Filters can also be employed to limit bandwidth.)
BW is a physical property of the medium – construction, thickness, length etc.
Decibels
10log10 (ratio of powers)
Chap. 2- Physical 9

10. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Discuss
Components freqs & amplitudes of a “periodic” 8-bit transmission
effect of bandwidth size on output
Next: Analyze max data rate
Chap. 2- Physical 10

11. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Voice-grade Line - is an ordinary telephone line. Its cutoff frequency is near 3,000 Hz- adequate for speech.
Analysing max data rate
If we assume:
1 bit per baud
Assume we send 8bits (in a period as in the example)
then
1) the bit rate is b bits/sec, time to send 8bits = 8/b
2) the frequency of the fundamental (also called the first harmonic) is b/8 Hz.
3) the highest harmonic passed through a voice grade line: 3000/(b/8) = 24,000/b. ie max freq/first harmonic freq
4) The Table shows how this equation works in practice.
Chap. 2- Physical 11

12. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Observations:
Transmissions at 9600 (2 harmonics) will transforms (a) into (c) –accurate reception will be difficult!
Transmissions at 4800 (5 harmoics) more like (d)
At lower speed, more faithful reproduction eg (10 harmoics – just better than (e)
Chap. 2- Physical 12

13. DATA COMMUNICATION THEORY
Definitions
observation from table: at data rate higher than 38.4kbps, no hope for binary signals even if a completely noiseless channel.
Limiting the bandwidth limits the data rate even for a perfect (noiseless) channel.
Modems try to employ multiple symbols to achieve more bits per symbol.
Chap. 2- Physical 13

14. DATA COMMUNICATION THEORY
BANDWIDTH-LIMITED SIGNALS
Baud - The number of changes in the signal per second. (ie symbols per sec)
A b baud line does not necessarily transmit b bits/second - each signal may convey several bits - for example if 8 voltages are possible per signal, then 3 bits are sent on every signal. If the signal is BINARY (only two voltage levels), then the bit rate is equal to the baud rate.
Baud rate == symbol rate.
Chap. 2- Physical 14

15. DATA COMMUNICATION THEORY
Definitions
Henry Nyquist: when an arbitrary signal is passed through a low pass filter of bandwidth H, the filtered signal can be completely reconstructed by making only 2H samples per sec. Sampling a signal faster than 2H is pointless because the higher frequency samples that such sampling can recover have already been filtered out.
NB: According to Nyquist, for 3000Hz line, no use sampling more than 6000Hz. (typical is 2400/sec). Modulation schemes eg QPSK is employed. QPSK has 4 levels per symbol =>bit rate=2*the baud rate.
(February 7, 1889 – April 4, 1976)
Swedish, US Citizen
University of North Dakota in 1912 and received the B.S. and M.S. degrees in electrical engineering in 1914 and 1915, respectively. He received a Ph.D. in physics at Yale University in He worked at AT&T (& Bell Telephone Laboratories) from 1917 until his retirement in 1954.
Chap. 2- Physical 15

16. DATA COMMUNICATION THEORY
MAXIMUM DATA RATE OF A CHANNEL
The Nyquist equation evaluates the amount of data that can be pushed through a (perfect) channel with a given bandwidth, H. If the signal consists of V levels (for example for binary V= 2), assuming no noise (i.e., perfect channel)
maximum data rate = 2 H log2 V ( in bits/sec )
On a noiseless 3KHz channel, max data rate for a binary signal is
6000bps
Chap. 2- Physical 16

17. DATA COMMUNICATION THEORY
MAXIMUM DATA RATE OF A CHANNEL
Claud Shannon
Signal to noise ratio - random (thermal) or interference causes a degradation of the signal. This is measured in terms of the ratio of signal power to noise power. Usually this is measured in decibels, in terms of 10 log10 S/N. So an S/N of 100 = 20 dB.
Shannon's equation is another way of expressing maximum data rate. It's given as
maximum data rate = H log2 ( 1 + S/N )
Typical: SNR=30dB, H=3000Hz,
Maximum data rate =?
~ 30,000bps. (29,901.7kbps)
Claude Elwood Shannon ( )
U Michigan- 2BSc EE & Maths
MIT –postgrad & PHD 1940
"the father of information theory"
possibly the most important, and also the most famous, master's thesis of the century
Also circuit theory, Boolean algebra, genetics, many inventions.
See Wikipedia!
Chap. 2- Physical 17

18. Modems- Higher data rates?
The Telephone System
Modems- Higher data rates?
The 3 kHz phone line can only be sampled at 6 kHz.
So it doesn't do any good to sample more - instead try to get in more bits per sample.
For example, the Figure shows a combination of phase and amplitude modulation leading to multiple bits/baud.
Chap. 2- Physical 18

19. Notes number of signal levels Number of samples
Shannon gives max data rate irrespective of
number of signal levels
Number of samples