Nyquist Nyquist FormulaFormula

• Assume a channel is noise free.• Nyquist formulation: Nyquist formulation: if the rate of signal transmission is 2B,

then a signal with frequencies no greater than B is sufficient to carry the signal rate.– Given bandwidth B, highest signal rate is 2B.

• Why is there such a limitation?– due to intersymbol interference, such as is produced by delay

distortion.• Given binary signal (two voltage levels), the maximum data

rate supported by B Hz is 2B bps.– One signal represents one bit

Nyquist Nyquist FormulaFormula

• Signals with more than two levels can be used, i.e., each signal element can represent more than one bit.– E.g., if a signal has 4 different levels, then a signal can be used to

represents two bits: 00, 01, 10, 11• With multilevel signalling, the Nyquist formula becomes:

– C = 2B log2M– M is the number of discrete signal levels, B is the given

bandwidth, C is the channel capacity in bps.– How large can M be?

• The receiver must distinguish one of M possible signal elements. • Noise and other impairments on the transmission line will limit the

practical value of M.• Nyquist’s formula indicates that, if all other things are

equal, doubling the bandwidth doubles the data rate.

Channel CapacityChannel Capacity

• Channel capacity is concerned with the information handling capacity of a

given channel. It is affected by:

– The attenuation of a channel which varies with frequency as well as

channel length.

– The noise induced into the channel which increases with distance.

– Non-linear effects such as clipping on the signal.

Some of the effects may change with time e.g. the frequency response of a

copper cable changes with temperature and age.

Obviously we need a way to model a channel in order to estimate how much

information can be passed through it. Although we can compensate for non

linear effects and attenuation it is extremely difficult to remove noise.

The highest rate of information that can be transmitted through a

channel is called the channel capacity, C.

Channel CapacityChannel Capacity

• Shannon’s Channel Coding Theorem states that if the information

rate, R (bits/s) is equal to or less than the channel capacity, C, (i.e. R < C) then

there is, in principle, a coding technique which enables transmission over the

noisy channel with no errors.

• The inverse of this is that if R > C, then the probability of error is close to 1

for every symbol.

• The channel capacity is defined as: the maximum rate of reliable (error-

free) information transmission through the channel.

Shannon’s Channel Capacity TheoremShannon’s Channel Capacity Theorem

Shannon’s Channel Capacity TheoremShannon’s Channel Capacity Theorem

• Shannon’s Channel Capacity Theorem (or the Shannon-Hartley

Theorem) states that:

where C is the channel capacity, B is the channel bandwidth in hertz, S is

the signal power and N is the noise power ( with being the

two sided noise PSD).

Note: S/N is the ratio watt/watt not dB.

• The channel capacity, C, increases as the available bandwidth increases and

as the signal to noise ratio increases (improves).

• This expression applies to information in any format and to both analogue

and data communications, but its application is most common in data

communications.

• The channel capacity theorem is one of the most important results of

information theory. In a single formula it highlights the interplay between 3

key system parameters:

– channel bandwidth,

– average transmitted or received signal power,

– noise power at the channel output.

Shannon’s Channel Capacity TheoremShannon’s Channel Capacity Theorem

• For a given average transmitted power S and channel bandwidth, B, we

can

transmit information at the rate C bits/s with no error, by employing

sufficiently complex coding systems. It is not possible to transmit at a rate

higher than C bits/s by any coding system without a definite probability of

error. Hence the channel capacity theorem defines the fundamental limit

on the rate of error-free transmission for a power-limited, band-limited

channel.

Shannon’s Channel Capacity TheoremShannon’s Channel Capacity Theorem

Capacity versus BandwidthCapacity versus Bandwidth

• It appears from the expression:

that as the bandwidth increases the capacity should increase proportionately. But this does not happen, because increasing the bandwidth, B, also increases the noise power N = giving:

Capacity versus BandwidthCapacity versus Bandwidth

Transmission ImpairmentsTransmission Impairments

• With any communications system, the signal that is received may differ from the signal that is transmitted, due to various transmission impairments.

• Consequences:– For analog signals: degradation of signal quality– For digital signals: bit errors

• The most significant impairments include– Attenuation and attenuation distortion– Delay distortion– Noise

AttenuationAttenuation• Attenuation: signal strength falls off with distance.• Depends on medium

– For guided media, the attenuation is generally exponential and thus is typically expressed as a constant number of decibels per unit distance.

– For unguided media, attenuation is a more complex function of distance and the makeup of the atmosphere.

• Three considerations for the transmission engineer:1. A received signal must have sufficient strength so that the

electronic circuitry in the receiver can detect the signal. 2. The signal must maintain a level sufficiently higher than noise to be

received without error.

These two problems are dealt with by the use of amplifiers or repeaters.

Attenuation DistortionAttenuation Distortion(Following the previous slide)

Attenuation is often an increasing function of frequency. This

leads to attenuation distortion:

• some frequency components are attenuated more than

other frequency components.

Attenuation distortion is particularly noticeable for analog

signals: the attenuation varies as a function of frequency,

therefore the received signal is distorted, reducing intelligibility.

Delay DistortionDelay Distortion

• Delay distortion occurs because the velocity of propagation

of a signal through a guided medium varies with frequency.

• Various frequency components of a signal will arrive at the

receiver at different times, resulting in phase shifts between

the different frequencies.

• Delay distortion is particularly critical for digital data

– Some of the signal components of one bit position will spill over into

other bit positions, causing intersymbol interference, which is a major

limitation to maximum bit rate over a transmission channel.

Noise (1)Noise (1)

• For any data transmission event, the received signal will consist of the

transmitted signal, modified by the various distortions imposed by

the transmission system, plus additional unwanted signals that are

inserted somewhere between transmission and reception.

• The undesired signals are referred to as noise, which is the major

limiting factor in communications system performance.

• Four categories of noise:

– Thermal noise

– Intermodulation noise

– Crosstalk

– Impulse noise

Noise (2)Noise (2)

• Thermal noise (or white noise)Thermal noise (or white noise)– Due to thermal agitation of electrons– It is present in all electronic devices and transmission media, and

is a function of temperature.– Cannot be eliminated, and therefore places an upper bound on

communications system performance.• Intermodulation noiseIntermodulation noise

– When signals at different frequencies share the same transmission medium, the result may be intermodulation noise.

– Signals at a frequency that is the sum or difference of original frequencies or multiples of those frequencies will be produced.

– E.g., the mixing of signals at f1 and f2 might produce energy at frequency f1 + f2. This derived signal could interfere with an intended signal at the frequency f1 + f2.

Noise (3)Noise (3)

• CrosstalkCrosstalk– It is an unwanted coupling between signal paths. It can occur by

electrical coupling between nearby twisted pairs.– Typically, crosstalk is of the same order of magnitude as, or less

than, thermal noise.• Impulse noiseImpulse noise

– Impulse noise is non-continuous, consisting of irregular pulses or noise spikes of short duration and of relatively high amplitude.

– It is generated from a variety of cause, e.g., external electromagnetic disturbances such as lightning.

– It is generally only a minor annoyance for analog data.– But it is the primary source of error in digital data

communication.

twisted-pair cable twisted-pair wire

Twisted Pair - ApplicationsTwisted Pair - Applications

• Most common medium• Telephone network

– POTS– Between house and local exchange (subscriber loop), also

called the end office. From the end office to Central Office (CO) class 4 CO class 1 via Public Switched Telephone Network (PSTN)

• Within buildings– To private branch exchange (PBX)

• For local area networks (LAN)– 10Mbps or 100Mbps– Possible to rev up to 1Gbps – Gigabit Ethernet

UTP Categories• Cat 1

– Used for audio frequencies, speaker wire, etc. Not for networking.

• Cat 2– Up to 1.5Mhz, used for analog phones, not for networking

• Cat 3 – EIA 568-A Spec from here on up– up to 16MHz– Voice grade found in most offices– Twist length of 7.5 cm to 10 cm

• Cat 4– up to 20 MHz– Not frequently used today, was used for Token Ring

UTP Categories Cont.• Cat 5

– up to 100MHz– Twist length 0.6 cm to 0.85 cm– Commonly pre-installed in new office buildings

• Cat 5e “Enhanced”– Up to 100Mhz– Specifies minimum characteristics for NEXT (Near End Crosstalk) and ELFEXT (Equal

level far end crosstalk)• Coupling of signal from one pair to another• Coupling takes place when transmit signal entering the link couples back to

receiving pair, i.e. near transmitted signal is picked up by near receiving pair• Cat 6

– Proposed standard up to 250Mhz• Cat 7

– Proposed standard up to 600Mhz

plastic outer coating

woven or braided metal

insulating material

copper wire

protective coating

glass cladding

optical fiber core

Optical FiberOptical Fiber

An optical fiber is a thin (2 to 125µm), flexible medium capable of guiding an optical ray.

Preferable because of,

• Greater capacity

• Smaller size and lighter weight

• Lesser attenuation

• Greater repeater spacing

• Electromagnetic isolation

Optical FiberOptical Fiber

Five basic categories of application have become important for

optical fiber:

• Long-haul trunks

• Metropolitan trunks

• Rural exchange trunks

• Subscriber loops

• Local area networks

Fiber Optic TypesFiber Optic Types

• Step-index multimode fiberStep-index multimode fiber– the reflective walls of the fiber move the light pulses to

the receiver• Graded-index multimode fiberGraded-index multimode fiber

– acts to refract the light toward the center of the fiber by variations in the density

• Single mode fiberSingle mode fiber– the light is guided down the center of an extremely

narrow core

Optical Fiber Transmission CharacteristicsOptical Fiber Transmission Characteristics

Optical Fiber Transmission Modes