1-1

Basics of Data Transmission

Our Objective is to understand …

Signals, bandwidth, data rate concepts

Transmission impairmentsChannel capacityData Transmission

1-2

Signals A signal is

generated by a transmitter and transmitted over a medium

function of time function of frequency, i.e.,

composed of components of different frequencies

Analog signal varies smoothly with time E.g., speech

Digital signal maintains a constant level for

some period of time, then changes to another level

E.g., binary 1s and 0s

1-3

Periodic vs. Aperiodic Signals

Periodic signal Pattern repeated over

time s(t+T) = s(t)

Aperiodic signal Pattern not repeated

over time

1-4

Sine Wave The fundamental periodic

signal Peak Amplitude (A)

maximum strength of signal

volts Frequency (f)

Rate of change of signal Hertz (Hz) or cycles per

second Period = time for one

repetition (T) T = 1/f

Phase () Relative position in time

1-5

Signals in Frequency Domain Signal is made up of many components

Components are sine waves with different frequencies

In early 19th century, Fourier proved that Any periodic function can be constructed as the

sum of a (possibly infinite) number of sines and cosines

11

2cos2sin2

1)(

nn

nn nftbnftacts

dttsT

cdtnfttsT

bdtnfttsT

aTT

n

T

n 000

)(2

,)2cos()(2

,)2sin()(2

This decomposition is called Fourier series f is called the fundamental frequency an, bn are amplitude of nth harmonic c is a constant

1-6

Frequency Domain (cont’d) Fourier Theorem

enables us to represent signal in Frequency Domain i.e., to show

constituent frequencies and amplitude of signal at these frequencies

Example 1: sine wave:

s(t) = sin(2πft) Frequency, f

S(f)

1 f

1-7

Time and Frequency Domains: Example 2

Time domain s(t)

Frequency domain S(f)

1-8

Frequency Domain (cont’d) So, we can use Fourier theorem to represent a

signal as function of its constituent frequencies, and we know the amplitude of each constituent

frequency. So what?

We know the spectrum of a signal, which is the range of frequencies it contains, and

Absolute bandwidth = width of the spectrum

Q: What is the bandwidth of the signal in the previous example? [sin(2πft) + sin(2π3ft)]

A: 2f Hz

1-9

Frequency Domain (cont’d) Q. What is the absolute bandwidth of square wave?

Hint: Fourier tells you that Absolute BW = ∞ (ooops!!) But, most of the energy is contained within a

narrow band (why?) we refer to this band as effective bandwidth, or just bandwidth

kftk

tskoddk

2sin14

)(1,

1-10

A. BW = 6*f Hz

Approximation of Square Wave

Using the first 3 harmonics, k=1, 3, 5

Using the first 4 harmonics, k=1, 3, 5, 7

Q. What is BW in each case?

A. BW = 4*f Hz

Cool applet on Fourier Series

1-11

Signals and Channels

Signal can be decomposed to components (frequencies) spectrum: range of frequencies contained in signal (effective) bandwidth: band of frequencies

containing most of the energy

Communications channel (link) has finite bandwidth determined by the physical

properties (e.g., thickness of the wire) truncates (or filters out) frequencies higher than its

BW• i.e., it may distort signals

can carry signals with bandwidth ≤ channel bandwidth

1-12

Bandwidth and Data Rate Data rate: number of bits per second (bps) Bandwidth: signal rate of change, cycles per sec (Hz) Well, are they related? Ex.: Consider square wave with high = 1 and low = 0

We can send two bits every cycle (i.e., during T = 1/f sec) Assume f =1 MHz (fundamental frequency) T = 1 usec

Now, if we use the first approximation (3 harmonics) BW of signal = (5 f – 1 f) = 4 f = 4 MHz Data rate = 2 / T = 2 Mbps

So we need a channel with bandwidth 4 MHz to send at date rate 2 Mbps

1-13

Bandwidth and Data Rate (cont’d)

But, if we use the second approx. (4 harmonics) BW of signal = (7 f – 1 f) = 6 f = 6 MHz Data rate = 2 / T = 2 Mbps

Which one to choose? Can we use only 2 harmonics (BW = 2 MHz)?

It depends on the ability of the receiver to discern the difference between 0 and 1

Tradeoff: cost of medium vs. distortion of signal and complexity of receiver

1-14

Bandwidth and Data Rate (cont’d)

Now, let us agree that the first appox. (3 harmonics) is good enough Data rate of 2 Mbps requires BW of 4 MHz

To achieve 4 Mbps, what is the required BW? data rate = 2 (bits) / T (period) = 4 Mbps T = 1 /2

usec f (fundamental freq) = 1 /T = 2 MHz BW = 4 f = 8 MHz

Bottom line: there is a direct relationship between data rate and bandwidth Higher data rates require more bandwidth More bandwidth allows higher data rates to be sent

1-15

Bandwidth and Data Rate (cont’d)

Nyquist Theorem: (Assume noise-free channel) If rate of signal transmission is 2B then signal with

frequencies no greater than B is sufficient to carry signal rate, OR alternatively

Given bandwidth B, highest signal rate is 2B

For binary signals Two levels we can send one bit (0 or 1) during each

period data rate (C) = 1 x signal rate = 2 B That is, data rate supported by B Hz is 2B bps

For M-level signals M levels we can send log2M bits during each period

C= 2B log2M

1-16

Bandwidth and Data Rate (cont’d)

Shannon Capacity: Considers data rate, (thermal) noise and error rate Faster data rate shortens each bit so burst of noise

affects more bits At given noise level, high data rate means higher error

rate

SNR ≡ Signal to noise ration SNR = signal power / noise power Usually given in decibels (dB): SNRdB= 10 log10 (SNR)

Shannon proved that: C = B log2(1 + SNR) This is theoretical capacity, in practice capacity is

much lower (due to other types of noise)

1-17

Bandwidth and Data Rate (cont’d)

Ex.: A channel has B = 1 MHz and SNRdB = 24 dB, what is the channel capacity limit? SNRdB = 10 log10 (SNR) SNR = 251

C = B log2(1 + SNR) = 8 Mbps

Assume we can achieve the theatrical C, how many signal levels are required? C = 2 B log2M M = 16 levels

1-18

Transmission Impairments

Signal received may differ from signal transmitted

Analog - degradation of signal quality Digital - bit errors Caused by

Attenuation and attenuation distortion Delay distortion Noise

1-19

Attenuation

Signal strength falls off with distance Depends on medium Received signal strength:

must be enough to be detected must be sufficiently higher than noise to be

received without error Attenuation is an increasing function of

frequency attenuation distortion

1-20

Delay Distortion

Only in guided media Propagation velocity varies with

frequency Critical for digital data

A sequence of bits is being transmitted Delay distortion can cause some of signal

components of one bit to spill over into other bit positions

intersymbol interference, which is the major limitation to max bit rate

1-21

Noise (1) Additional signals inserted between

transmitter and receiver Thermal

Due to thermal agitation of electrons Uniformly distributed across frequencies White noise

Intermodulation Signals that are the sum and difference of

original frequencies sharing a medium

1-22

Noise (2)

Crosstalk A signal from one line is picked up by

another Impulse

Irregular pulses or spikes, e.g. external electromagnetic interference

Short duration High amplitude

1-23

Data and Signals

Data Entities that convey meaning Analog: speech Digital: text (character strings)

Signals electromagnetic representations of data Analog: continuous Digital: discrete (pulses)

Transmission Communication of data by propagation and

processing of signals

1-24

Analog Signals Carrying Analog and Digital Data

1-25

Digital Signals Carrying Analog and Digital Data

1-26

Analog Transmission

Analog signal transmitted without regard to content

May be analog or digital data Attenuated over distance Use amplifiers to boost signal But, it also amplifies noise!

1-27

Digital Transmission

Concerned with content Integrity endangered by noise,

attenuation Repeaters used

Repeater receives signal Extracts bit pattern Retransmits

Attenuation is overcome Noise is not amplified

1-28

Advantages of Digital Transmission

Digital technology Low cost LSI/VLSI technology

Data integrity Longer distances over lower quality lines

Capacity utilization High bandwidth links economical High degree of multiplexing easier with digital

techniques Security & Privacy

Encryption Integration

Can treat analog and digital data similarly

1-29

Summary

Signal: composed of components (Fourier Series) Spectrum, bandwidth, data rate

Shannon channel capacity Transmission impairments

Attenuation, delay distortion, noise Data vs. signals Digital vs. Analog Transmission