Data

1
Data Signals

• Chapter 3

2
Analog Digital Data

• Data can be analog or digital
• Analog data is continuous information.
• Digital data is information with discrete states.

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Signals

• Signals are used to transport data typically in
  the form of electromagnetic waves.
• Signals can be either analog or digital
• AM and FM Radio transmissions are analog.
• Digital signals take on discrete values.

4
Figure 3.1 Comparison of analog and digital
signals
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Analog Digital Signals

• An analog signal can be a combination of one or
  more continuous frequencies.
• A pure digital signal is made up of constant
  output levels over a fixed amount of time.

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Periodic Signals

• Can be either analog or digital
• Periodic signals repeats a pattern over identical
  time periods.

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Periodic Analog Signals

• Are
• simple a single sine wave that cannot be
  decomposed into simpler signals.
• composite a combination of sine waves that can
  be decomposed into individual sine waves.

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Periodic Signals

• Any composite signal is a combination of simple
  sine waves with varying frequencies, amplitudes,
  and phases.
• A composite periodic signal is made up of
  discrete frequencies.
• A composite non-periodic signal is made up of
  continuous frequencies.

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Figure 3.9 A composite periodic signal
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Non-Periodic Signals

• Can be analog or digital
• Changes over time but does not repeat a pattern.

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Figure 3.1 non-periodic analog and digital
signals
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Sine Wave Properties

• Wavelength the distance covered by one period
  of the waveform. Note ? c / f
• Phase position of the waveform relative to time
  0.
• Period the amount of time required to complete
  one full waveform cycle.
• Frequency is the number of periods in one second.
• Amplitude distance from the midline to peak or
  trough.
• Phase a shift in time.

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Figure 3.2 A sine wave
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sin function

• g(t) A sin( Bt P )
• The magnitude of A is the amplitude.
• B angular frequency (assume B gt 0)
• P phase shift, P 2 p f h, where h
  horizontal shift
• T 2 p/B, where T period
• f frequency, f B/(2 p), B 2p f
• T 1/f

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Figure 3.4 Two signals with the same amplitude
and phase, but different
frequencies
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Figure 3.5 Three sine waves with the same
amplitude and frequency,
but different phases
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Frequency vs time

• A sine wave in time vs Amplitutde, can be
  represented using a single spike using a
  frequency vs Amplitude plot

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Time Frequency

• A periodic sine wave in the time domain is a
  single spike in the frequency domain sharing the
  same amplitude.

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Composite Signals

• Any composite signal is a combination of simple
  sine waves with varying frequencies, amplitudes,
  and phases.
• A composite periodic signal is made up of a
  finite number of discrete frequencies.
• A composite non-periodic signal is made up of an
  infinite number of continuous frequencies.

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Figure 3.9 A composite periodic signal
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Figure 3.10 Decomposition of a composite
periodic signal in the time and
frequency domains
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Figure 3.11 The time and frequency domains of a
nonperiodic signal
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Bandwidth

• Bandwidth difference between highest and lowest
  frequencies.

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Figure 3.12 The bandwidth of periodic and
nonperiodic composite signals
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Digital Signals

• Let L be the number of signal levels.
• Let n be the number of bits

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Figure 3.16 Two digital signals one with two
signal levels and the other
with four signal levels
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Digital Signals

• L 2n
• n log2(L)

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Bit Rate Bit Length

• Bits per second or bps.
• Similar to the analog frequency
• Bit length speed bit duration
• Similar to analog wavelength
• The number of samples required for a signal is
  twice the highest frequency of the signal.
• The worst case for digital data is alternating
  ones and zeros. This can be thought of as a peak
  and trough in a sin wave.

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Figure 3.16 Two digital signals one with two
signal levels and the other
with four signal levels
30
Figure 3.17 The time and frequency domains of
periodic and nonperiodic
digital signals
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Digital Signal Bandwidth

• A pure digital signal has bandwidth from 0Hz to
  infinity Hz.
• 0 Hz frequency is constant output (signal never
  changes).
• Sudden vertical change has infinite bandwidth

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Baseband

• Sending a digital signal without changing it to
  an analog signal
• Must include 0 Hz
• Perfect transmission of a digital signal requires
  a bandwidth from 0 to infinity.
• By including more frequencies, the signal will
  resemble the square wave more accurately.

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Figure 3.9 contains the 1, 3, and 5 harmonics
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Figure 3.19 Bandwidths of two low-pass channels
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Figure 3.20 Baseband transmission using a
dedicated medium
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Figure 3.21 Rough approximation of a digital
signal using the first harmonic
for worst case
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Fig 3.21

• Let N bit rate.
• Note that f N/2 repeats twice as often as f
  N/4.
• N/2 represents the worst case, alternating ones
  and zeros.
• A better approximation includes the odd
  harmonics 3N/2 5N/2 7N/2,

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Figure 3.22 Simulating a digital signal with
first three harmonics
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Bandwidth for Baseband Transmission

• The highest frequency determines the bps
  bandwidth.
• If B bandwidth, and B 500Hz, then the bps is
  about 1000bps
• B N/2
• Adding harmonics increases the required bandwidth.

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Table 3.2 Bandwidth requirements
41
Broadband Transmission

• Most transmission media have limited bandwidth
  and cannot send a baseband transmission.
• Broadband changes the digital signal to an analog
  signal for transmission.
• Example Amplitude modulation of a frequency.

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Figure 3.24 Modulation of a digital signal for
transmission on a bandpass
channel
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Transmission Impairment

• Attenuation loss of energy
• Distortion undesired signal changes
• Noise unwanted signal energy

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Figure 3.25 Causes of impairment
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Attenuation

• Measure attenuation dB 10 log10( p2/p1 )
• Pn is measured in power units watts. It can also
  be measured in volts.
• DB 20log10(v2/v1)

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Figure 3.26 Attenuation
47
Figure 3.27 Decibels for Example 3.28
48
Distortion

• Each signal component travels with its own
  propagation speed.
• The components will arrive at their destination
  at different times.
• The phase differences caused by the different
  arrival times induce distortion.

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Figure 3.28 Distortion
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Noise

• Thermal noise caused by excited electrons in
  the transmission media.
• Crosstalk wires will act as an antenna and
  receiver. Each wire transmits a signal and each
  wire receives signals from the other wires.
• Impulse noise caused by power surges
• Induced noise generated by magnetic fields near
  your transmission media.

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Noise

• Signal noise ratio SNR.
• SNR avg signal power / avg noise power
• SNR(db) 10log10(SNR)

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Figure 3.29 Noise
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Data Rate Limits

• Bandwidth available
• Level of signal
• Quality of the channel due to the amount of noise.

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Data Rate Limits

• Nyquist Bit Rate defines the theoretical
  maximum bit rate.
• Shannon Capacity the theoretical highest data
  rate for a noisy channel.

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Nyquist Bit Rate

• Theoretical Bit Rate 2 bandwidth log2(L) ,
  Where L is the number of signal levels.
• In practice, increasing the number of levels
  reduces reliability. The receiver must be able to
  distinguish the levels to work correctly.

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Shannon Capacity

• C bandwidth log2( 1 SNR )
• Where C capacity in bps.

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Data Rate Limits

• Given the SNR, the Shannon capacity will give an
  upper bound on the transmission rate.
• Use the Nyquist bit rate to determine the number
  of levels that will not exceed the Shannon
  capacity.
• The number of levels should be a power of 2.

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Example

• A telephone line has a bandwidth from 100 to 4100
  Hz and a SNR(db) of 40 db. What is the capacity
  in bps of the line?

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Example

• SNR(db) 10log10(SNR)
• SNR10(SNR(db)/10)
• C bandwidthlog2(1SNR)
• about 53kbps

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Performance

• Bandwidth
• Throughput
• Latency aka delay
• Bandwidth delay product
• Jitter

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Performance Bandwidth

• Range of frequencies measured in hertz.
• Bits per second

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Performance Throughput

• Throughput lt Bandwidth
• The actual amount of data transferred.

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Performance Latency

• The amount of time required for a complete
  message to arrive at its destination.
• Latency is the sum of
• Propagation time
• Transmission time
• Queuing time
• Processing delay

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Latency

• Propagation time
• Propagation time Distance / Propagation speed
• Transmission time
• Transmission time Message size / bandwidth
• Queuing time A router will queue messages as
  they arrive before forwarding them.
• Processing delay The compute time required by
  the hardware to process the message.

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Bandwidth Delay Product

• The product of the bandwidth and the delay is the
  number of bits that can fill the link.

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Figure 3.31 Filling the link with bits for case 1
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Figure 3.32 Filling the link with bits in case 2
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Example

• How much data should be sent using a full duplex
  channel?
• 2bandwidthdelay
• When the first bit arrives, an acknowledgement
  is sent back. By the time the sender gets the
  acknowledgement, the channel is ready for the
  next burst.

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Jitter

• When packets of information do not arrive at
  uniform time intervals.
• This is an issue for real-time connections like
  video and sound.
• E-mail is not impacted by jitter.