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Basics of Data Transmission
Our Objective is to understand …
Signals, bandwidth, data rate concepts
Transmission impairmentsChannel capacityData Transmission
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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
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Periodic vs. Aperiodic Signals
Periodic signal Pattern repeated over
time s(t+T) = s(t)
Aperiodic signal Pattern not repeated
over time
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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
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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
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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
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Time and Frequency Domains: Example 2
Time domain s(t)
Frequency domain S(f)
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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
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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,
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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Analog Signals Carrying Analog and Digital Data
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Digital Signals Carrying Analog and Digital Data
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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!
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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
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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
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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
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