Transcript Noise

Noise and SNR
Noise
unwanted signals
inserted between
transmitter and
receiver
is the major limiting
factor in
communications
system performance
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• Noise can broadly be defined as any
unknown signal that affects the recovery of
the desired signal.
• The received signal is modeled as
r (t )  s(t )  w(t ) (9.1)
s(t) is the transmitted signal
w(t) is the additive noise
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Categories of Noise
Thermal noise
• due to thermal agitation of
electrons
• uniformly distributed
across bandwidths
• referred to as white noise
Intermodulation noise
• produced by nonlinearities in the
transmitter, receiver, and/or
intervening transmission medium
• effect is to produce signals at a
frequency that is the sum or
difference of the two original
frequencies
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Categories of Noise
Crosstalk:
Impulse Noise:
– caused by external
electromagnetic interferences
– noncontinuous, consisting of
irregular pulses or spikes
– short duration and high
amplitude
– minor annoyance for analog
signals but a major source of
error in digital data
– a signal from one line is
picked up by another
– can occur by electrical
coupling between nearby
twisted pairs or when
microwave antennas pick
up unwanted signals
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Thermal Noise
• Thermal noise known as white noise. Noise is
assumed to be independent of frequency, uniformly
distributed spectrally from 0 to about 1013 Hz.
• Thermal noise, its energy increase with temperature.
• The noise voltage varies in time with a Gaussian
probability distribution function and mean value of zero.
Power spectral density (PSD) of thermal noise
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Thermal Noise (Cont)
• The noise power density (amount of thermal
noise to be found in a bandwidth of 1Hz in
any device or conductor) is:
N0  kT W/Hz
N0 = noise power density in watts per 1 Hz of
bandwidth
k = Boltzmann's constant = 1.3803  10-23 J/K
T = temperature, in kelvins (absolute temperature)
0oC = 273 Kelvin
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Thermal Noise (cont)
• Because of the weakness of the signal received by
satellite earth stations, thermal noise is particularly
significant for satellite communication.
• Thermal noise power present in a bandwidth of B
Hertz (in watts):
N  kTB
or, in decibel-watts (dBW),
N  10log k  10 log T  10log B
 228.6 dBW  10 log T  10log B
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Other noises
• Intermodulation noise – occurs if signals with
different frequencies share the same medium
– Interference caused by a signal produced at a
frequency that is the sum or difference of
original frequencies
• Crosstalk – unwanted coupling between signal
paths
• Impulse noise – irregular pulses or noise spikes
– Short duration and of relatively high amplitude
– Caused by external electromagnetic
disturbances, or faults and flaws in the
communications system
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Signal-to-Noise Ratios
• The desired signal, s(t), a narrowband noise signal, n(t)
x(t )  s(t )  n(t )
• Signal-to-noise ratio is defined by
E[ s 2 (t )]
SNR 
E[n 2 (t )]
• The signal-to-noise ratio is often considered to be a
ratio of the average signal power to the average noise
power.
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Noise in Digital Communications
• Two strong external reasons for the increased
dominance of digital communication
 The rapid growth of machine-to-machine
communications.
 Digital communications gave a greater
noise tolerance than analogue.
• Broadly speaking, the purpose of detection is
to establish the presence of an informationbearing signal in noise.
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Bit Error Rate (BER)
• Let n denote the number of bit errors observed in a
sequence of bits of length N; then the relative
frequency definition of BER is
n
BER  lim  
N  N
 
• BER and Packet error rate (PER)
speech, a BER of 10-2 to 10-3 is sufficient.
data transmission over wireless channels, a bit
error rate of 10-5 to 10-6 is often the objective.
video transmission, a BER of 10-7 to 10-12 is often
the objective.
financial data, a BER of 10-11or better is often the
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requirement.
SNR in digital systems
– The ratio of the modulated energy per information bit to
the one-sided noise spectral density; namely,
SNR
digital
ref
Modulated energy per bit Eb


Noise spectral density
N0
1. The analogue definition was a ratio of powers. The
digital definition is a ratio of energies.
2. The definition uses the one-sided noise spectral
density; that is, it assumes all of the noise occurs on
positive frequencies. This assumption is simply a
matter of convenience.
3. The reference SNR is independent of transmission
rate. Since it is a ratio of energies, it has essentially
been normalized by the bit rate.
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Nyquist Bandwidth
In the case of a channel that is noise free:
• if rate of signal transmission is 2B then can carry
signal with frequencies no greater than B
– given bandwidth B, highest signal rate is 2B
•
•
•
•
for binary signals, 2B bps needs bandwidth B Hz
can increase rate by using M signal levels
Nyquist Formula is: C = 2B log2M
data rate can be increased by increasing signals
– however this increases burden on receiver
– noise & other impairments limit the value of M
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Channel Capacity
Maximum rate at which data can be transmitted over a
given communications channel under given conditions
bandwidth
data rate
noise
in cycles
average
in bits per
per
noise level
second
second or
over path
Hertz
error rate
rate of
corrupted
bits
main
limitations constraint
due to
on
physical
achieving
properties efficiency
is noise
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Shannon Capacity Formula
• considering the relation of data rate, noise and
error rate:
– faster data rate shortens each bit so bursts of noise
corrupts more bits
– given noise level, higher rates mean higher errors
• Shannon developed formula relating these to
signal to noise ratio (in decibels)
• SNRdb=10 log10 (signal/noise)
• capacity C = B log2(1+SNR)
– theoretical maximum capacity
– get much lower rates in practice
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Signal to Noise Ratio – SNR (1)
• Ratio of the power in a signal to the power contained in
the noise present at a particular point in the transmission.
• Normally measured at the receiver with the attempt to
eliminate/suppressed the unwanted noise.
• In decibel unit,
 PS 
SNR dB  10log10  
 PN 
where PS = Signal Power, PN = Noise Power
• Higher SNR means better quality of signal.
Signal to Noise Ratio – SNR (2)
• SNR is vital in digital transmission because it can be
used to sets the upper bound on the achievable data rate.
• Shannon’s formula states the maximum channel capacity
(error-free capacity) as:
C  B log2 1  SNR 
– Given the knowledge of the receiver’s SNR and the signal
bandwidth, B. C is expressed in bits/sec.
• In practice, however, lower data rate are achieved.
• For a fixed level of noise, data rate can be increased by
increasing the signal strength or bandwidth.
Expression of Eb/N0 (1)
• Another parameter that related to SNR for determine data rates
and error rates is the ratio of signal energy per bit, Eb to noise
power density per Hertz, N0; → Eb/N0.
• The energy per bit in a signal is given by: Eb  PS  Tb
– PS = signal power & Tb = time required to send one bit which can be
related to the transmission bit rate, R, as Tb = 1/ R.
• Thus,
• In decibels:
Eb PS / R
PS


N0
N0
kTR
– 228.6 dBW
 Eb   P
S ( dB )  10log10 R  10log10 k  10log10T


 N 0  dB
Expression of Eb/N0 (2)
• As the bit rate R increases, the
signal power PS relative to the
noise must also be increased to
maintain the required Eb/N0.
BER versus Eb/N0 plot
• The bit error rate (BER) for the
data sent is a function of Eb/N0
(see the BER versus Eb/N0 plot).
• Eb/N0 is related to SNR as:
 PS  B
Eb
B
     SNR 
N0
R
 PN  R
where B = Bandwidth, R = Bit rate
Higher Eb/N0,
lower BER
Definition of Q(x)
1 
2
Q( x) 
exp( s / 2)ds

x
2
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Performance comparison
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This example is printed on your tutorial sheet.
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𝑆𝑁𝑅 =
𝐸[𝑆 2 𝑡 ]
𝐸[𝑛 2 𝑡 ]
(9.6)
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