[2.1]
NOISE
U
Noise may be defined as any unwanted signal that interferes with the
communication, measurement or processing of an information-bearing signal.
Noise is present in various degrees in almost all environments. For example, in a
digital cellular mobile telephone system, there may be several variety of noise
that could degrade the quality of communication, such as acoustic background
noise, thermal noise, electromagnetic radio-frequency noise, co-channel
interference, radio-channel distortion, echo and processing noise. Noise can
cause transmission errors and may even disrupt a communication process.
1. TYPES OF NOISE
U
Noise in a communication system can be classified into two broad categories,
depending on its source.
1.1 External Noise: Results from sources outside a communication system,
including atmospheric, man-made, and extraterrestrial sources.
U
U
1.1.1 Atmospheric noise: result primarily from spurious radio waves generated
by the natural electrical discharges within the atmosphere associated with
thunderstorms. It is commonly referred to a static or aspheric. Below about l00
MHz, the field strength of such radio waves is inversely proportional to
frequency. Atmospheric noise is characterized in the time domain by largeamplitude, short-duration bursts and is one of the prime examples of noise
referred to as impulsive. Because of its inverse dependence on frequency,
atmospheric noise affects commercial AM broadcast radio, which occupies the
frequency range from 540 kHz to 1.6 MHz, more than it affects television and
FM radio, which operate in frequency bands above 50 MHz.
U
U
1.1.2 Man-made noise: It is include high-voltage power line corona discharge,
commutator-generated noise in electrical motors, automobile and aircraft
ignition noise, and switching-gear noise. Ignition noise and switching noise, like
atmospheric noise, are impulsive in character. Impulse noise is the predominant
type of noise in switched wire line channels, such as telephone channels. For
applications such as voice transmission, impulse noise is only an irritation
factor; however, it can be a serious source of error in applications involving
transmission of digital data. Yet another important source of man-made noise is
radio-frequency transmitters other than the one of interest. Noise due to
interfering transmitters is commonly referred to as radio-frequency interference
U
U
Dr. Ahmed A. Alrekaby
[2.2]
(RFI). RFI is particularly troublesome in situation in which a receiving antenna
is subject to a high-density transmitter environment, as in mobile
communications in a large city.
1.1.3 Extraterrestrial noise: The source of this noise includes our sun and other
hot heavenly bodies, such as stars. Owing to its high temperature (6000: C) and
relatively close proximity to the earth, the sun is an intense, but fortunately
localized source of radio energy that extends over a broad frequency spectrum.
Similarly, the stars are sources of wideband radio energy. Although much more
distant and hence less intense than the sun, nevertheless they are collectively an
important source of noise because of their vast numbers. Radio stars such as
quasars and pulsars are also intense sources of radio energy. The frequency
range of solar and cosmic noise extends from a few megahertz to a few
gigahertz.
U
U
1.1.4 Fading: Another source of interference in communication systems is
multiple transmission paths. These can result from reflection off buildings, the
earth, airplanes, and ships or from refraction by stratifications in the
transmission medium. If the scattering mechanism results in numerous reflected
components, the received multi path signal is noise like and is termed diffuse. If
the multipath signal component is composed of only one or two strong reflected
rays, it is termed specular. Finally, signal degradation in a communication
system can occur because of random changes in attenuation within the
transmission medium. Such signal perturbations are referred to as fading,
although it should be noted that specular multi path also results in fading due to
the constructive and destructive interference of the received multiple signals.
U
U
1.2 Internal Noise: It is the noise generated by components within a
communication system, such as resistors, electron tubes, and solid-state active
devices.
U
U
1.2.1 Thermal Noise: It is caused by the random motion of free electrons in a
conductor or semiconductor excited by thermal agitation.
U
U
1.2.2 shot noise: and is caused by the random arrival of discrete charge carriers
in such devices as thermionic tubes or semiconductor junction devices.
U
U
1.2.3 Flicker Noise: Flicker noise is due to various causes. It is characterized by
a spectral density that increases with decreasing frequency. The dependence of
the spectral density on frequency is often found to be proportional to the inverse
first power of the frequency. Therefore, flicker noise is sometimes referred to as
U
U
Dr. Ahmed A. Alrekaby
[2.3]
one-over-f noise. More generally, flicker noise phenomena are characterized by
power spectra that are of the form constant/fα, where α is close to unity. The
physical mechanism that gives rise to flicker noise is not well understood.
2. THERMAL NOISE
U
Thermal noise is the noise arising from the random motion of charge carriers in
a conducting or semiconducting medium. Such random agitation at the atomic
level is a universal characteristic of matter at temperatures other than absolute
zero. Nyquist was one of the first to have studied thermal noise. Nyquist's
theorem states that the mean-square noise voltage appearing across the terminals
of a resistor of R ohms at temperature T kelvin in a frequency band B hertz is
given by
2
𝑣𝑣𝑟𝑟𝑟𝑟𝑟𝑟
= ⟨𝑣𝑣𝑛𝑛2 (𝑡𝑡)⟩ = 4𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘
𝑉𝑉 2
(2.1)
where k = Boltzmann's constant = 1.38 x 10-23 J/K
Thus a noisy resistor can be represented by an equivalent circuit consisting of a
noiseless resistor in series with a noise generator of rms voltage υrms as shown in
Fig.(2.1a). Short-circuiting the terminals of Fig.(2.1a) results in a short-circuit
noise current of mean-square value
2
𝑖𝑖𝑟𝑟𝑟𝑟𝑟𝑟
= ⟨𝑖𝑖𝑛𝑛2 (𝑡𝑡)⟩ =
⟨𝑣𝑣𝑛𝑛2 (𝑡𝑡)⟩ 4𝑘𝑘𝑘𝑘𝑘𝑘
=
= 4𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘
𝑅𝑅
𝑅𝑅2
𝐴𝐴2
(2.2)
where G = 1/R is the conductance of the resistor. The Thevenin equivalent of
Fig.(2.1a) can therefore be transformed to the Norton equivalent shown in
Fig.(2.1b).
irms=(4kTGB)1/2
Fig.(2.1)
Example 2.1
U
Consider the resistor network shown in Fig.(2.2a) . Assuming room temperature
of T = 290 K, find the rms noise voltage appearing at the output terminals in a
100 kHz bandwidth.
Dr. Ahmed A. Alrekaby
[2.4]
Sol. We use voltage division to find the noise voltage due to each resistor across
the output terminals. Then, since powers due to independent sources add, we
find the rms output voltage vo by summing the square of the voltages due to each
resistor, which gives the total mean-square voltage and take the square root to
give the rms voltage. The calculation yields
U
U
Fig.(2.2)
In the preceding expressions, �4𝑘𝑘𝑘𝑘𝑅𝑅𝑖𝑖 𝐵𝐵 represents the rms voltage across
resistor Ri. Thus
For more complex circuits Nyquist's formula is utilized to simplify such
computations considerably. It states: The mean square noise voltage produced at
Dr. Ahmed A. Alrekaby
[2.5]
the output terminals of any one-port network containing only resistors,
capacitors, and inductors is given by
∞
⟨𝜐𝜐𝑛𝑛2 (𝑡𝑡)⟩ = 2𝑘𝑘𝑘𝑘 � 𝑅𝑅(𝑓𝑓) 𝑑𝑑𝑑𝑑
(2.3)
𝑉𝑉 2
(2.4)
−∞
where R (f) is the real part of the complex impedance seen looking back into the
terminals. If the network contains only resistors, the mean-square noise voltage
in a bandwidth B is
⟨𝜐𝜐𝑛𝑛2 (𝑡𝑡)⟩ = 4𝑘𝑘𝑘𝑘𝑅𝑅𝑒𝑒𝑒𝑒 𝐵𝐵
where Req is the Thevenin equivalent resistance of the network. If we look back
in to the terminals of the network shown in Fig.(2.2), the equivalent resistance is
Thus
which can be shown to be equivalent to the result obtained previously.
3. AVAILABLE POWER
U
Because calculations involving noise involve transfer of power, the concept of
maximum power available from a source of fixed internal resistance is useful.
Figure (3.1) illustrates the familiar theorem regarding maximum power transfer,
which states that a source of internal resistance R delivers maximum power to a
resistive load RL if R = RL and that, under these conditions, the power P
produced by the source is evenly split between source and load resistances. If R
= RL, the load is said to be matched to the source, and the power delivered to the
load is referred to as the available power Pa= P/2. Consulting Fig.(3.1a), in
which υrms is the rms voltage of the source, we see that the voltage across RL = R
υ
is 𝑟𝑟𝑟𝑟𝑟𝑟 , This gives
2
2
1 𝜐𝜐𝑟𝑟𝑟𝑟𝑟𝑟 2 𝜐𝜐𝑟𝑟𝑟𝑟𝑟𝑟
𝑃𝑃𝑎𝑎 = �
� =
𝑅𝑅 2
4𝑅𝑅
(3.1)
Similarly, when dealing with a Norton equivalent circuit as shown in
Fig.(3.1b),we can write the available power as
Dr. Ahmed A. Alrekaby
[2.6]
2
𝑖𝑖𝑟𝑟𝑟𝑟𝑟𝑟
𝑖𝑖𝑟𝑟𝑟𝑟𝑟𝑟 2
𝑃𝑃𝑎𝑎 = �
� 𝑅𝑅 =
2
4𝐺𝐺
(3.2)
where irms = υrms / R is the rms noise current. Returning to Eq.(2.1) and (2.2) and
using Eq.(3.1) and (3.2), we see that a noisy resistor produces the available
power
𝑃𝑃𝑎𝑎,𝑅𝑅 =
4𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘
= 𝑘𝑘𝑘𝑘𝑘𝑘
4𝑅𝑅
𝑊𝑊
(3.3)
Fig.(3.1)
Example 3.1: Calculate the available power per hertz of bandwidth for a
resistance at room temperature, taken to be To = 290 K. Express in decibels
referenced to 1 watt (dBW) and decibels referenced to 1 mill watt (dBm).
U
U
Sol.
U
Power/hertz = Pa,R/ B = (1.38 x 10-23) (290) = 4.002 × 10-21 W/Hz
Power/hertz in dBW = 10 logl0 (4.002 × 10-21 /1) ≅ −204 dBW
Power/hertz in dBm = 10 logl0 (4.002 × 10-21/10-3) ≅ −174 dBm
4. NOISE FIGURE OF A SYSTEM
U
Figure (4.1) illustrates a cascade of N stages or subsystems that make up a
system. For example, if this block diagram represents a superheterodyne
receiver, subsystem 1 would be the RF amplifier, subsystem 2 the mixer,
subsystem 3 the IF amplifier, and subsystem 4 the detector. At the output of
each stage, we wish to be able to relate the signal-to-noise power ratio to that at
the input. This will allow us to pinpoint those subsystems that contribute
significantly to the output noise of the overall system.
Dr. Ahmed A. Alrekaby
[2.7]
Fig.(4.1)
One useful measure of system noisiness is the so-called noise figure F, defined
as the ratio of the signal-to-noise power ratio (SNR) at the system input to the
SNR at the system output.
1 𝑆𝑆
𝑆𝑆
� � = � �
𝑁𝑁 𝑙𝑙 𝐹𝐹𝑙𝑙 𝑁𝑁 𝑙𝑙−1
(4.1)
𝐹𝐹𝑑𝑑𝑑𝑑 = 10 𝑙𝑙𝑙𝑙𝑙𝑙10 𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟
(4.2)
2
𝑒𝑒𝑠𝑠,𝑙𝑙−1
=
4𝑅𝑅𝑙𝑙−1
(4.3)
𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙−1 = 𝑘𝑘𝑇𝑇𝑠𝑠 𝐵𝐵
(4.4)
2
𝑒𝑒𝑠𝑠,𝑙𝑙−1
𝑆𝑆
=
� �
𝑁𝑁 𝑙𝑙−1 4𝑘𝑘𝑇𝑇𝑠𝑠 𝑅𝑅𝑙𝑙−1 𝐵𝐵
(4.5)
For an ideal, noiseless subsystem, Fl = 1; that is, the subsystem introduces no
additional noise. For physical devices, Fl > 1. Noise figures for devices and
systems are often stated in terms of decibels (dB). Specifically
Consider the lth subsystem in the cascade of the system shown in Fig.(4.1a). If
we represent its input by a Thevenin equivalent circuit as shown in Fig.(4.1b),
with rms signal voltage es,l-1 and equivalent resistance Rl-1, the available signal
power is
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙−1
If we assume that only thermal noise is present, the available noise power for a
source temperature of Ts is
giving an input SNR of
Dr. Ahmed A. Alrekaby
[2.8]
The available output signal power, from Fig.(4.1b), is
2
𝑒𝑒𝑠𝑠,𝑙𝑙
=
4𝑅𝑅𝑙𝑙
(4.6)
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙 = 𝐺𝐺𝑎𝑎 𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙−1
(4.7)
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙
We can relate Psa,l to Psa,l-1 by the available power gain Ga of subsystem l.
defined to be
which is obtained if all resistances are matched. The output SNR is
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙
1 𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙−1
𝑆𝑆
=
� � =
𝑁𝑁 𝑙𝑙 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙 𝐹𝐹𝑙𝑙 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙−1
or
𝐹𝐹𝑙𝑙 =
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙−1 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙−1 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙
𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙
=
=
𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙−1 𝐺𝐺𝑎𝑎 𝑃𝑃𝑠𝑠𝑠𝑠 ,𝑙𝑙−1 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙−1 𝐺𝐺𝑎𝑎 𝑃𝑃𝑛𝑛𝑛𝑛 ,𝑙𝑙−1
(4.8)
(4.9)
Noting that Pna,l = GaPna,l-1 + Pint,l, where Pint,l is the available internally
generated noise power of subsystem l, and that Pna,l-l = kTsB, we may write
Eq.(4.9) as
𝐹𝐹𝑙𝑙 = 1 +
𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖 ,𝑙𝑙
𝐺𝐺𝑎𝑎 𝑘𝑘𝑇𝑇𝑠𝑠 𝐵𝐵
(4.10)
Thus, for 𝐺𝐺𝑎𝑎 ≫ 1, F ≅ 1, which shows that the internally generated noise
becomes negligible for systems with large gains. For cascade of subsystems the
overall noise figure is
Example 4.1
𝐹𝐹 = 𝐹𝐹1 +
U
𝐹𝐹2 − 1 𝐹𝐹3 − 1
+
+⋯
𝐺𝐺𝑎𝑎1
𝐺𝐺𝑎𝑎1 𝐺𝐺𝑎𝑎2
(4.11)
A preamplifier, with power gain to be found, having a noise figure of 2.5 dB is
cascaded with a mixer with a gain of 5 dB and a noise figure of 8 dB. Find the
preamplifier gain such that the overall noise figure of the cascade is at most 4
dB.
Sol. From Friis's formula
U
U
𝐹𝐹 = 𝐹𝐹1 +
𝐹𝐹2 − 1
𝐺𝐺1
Dr. Ahmed A. Alrekaby
[2.9]
Solving for G1,
𝐹𝐹2 − 1
108⁄10 − 1
=
= 7.24 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟) = 8.6 𝑑𝑑𝑑𝑑
𝐺𝐺1 =
𝐹𝐹 − 𝐹𝐹1 104⁄10 − 102.5⁄10
Note that the gain of the mixer is immaterial.
5. WHITE NOISE
U
A random process X(t) is called White noise if (see Fig.(5.1a))
𝑆𝑆𝑋𝑋𝑋𝑋 (𝜔𝜔) =
𝜂𝜂
2
Taking the inverse Fourier transform of Eq.(5.1), we have
𝜂𝜂
𝑅𝑅𝑋𝑋𝑋𝑋 (𝜏𝜏) = 𝛿𝛿(𝜏𝜏)
2
(5.1)
(5.2)
Which illustrated in Fig.(5.1b). It is assumed that the mean of white noise is
zero.
τ
Fig.(5.1)
Example 5.1: The input to the RC filter show below is a white noise process.
U
U
(a) Determine the power spectrum of the output process Y(t).
(b) Determine the autocorrelation and the mean-square value of Y(t).
Sol. the frequency response of the RC filter is
1
𝐻𝐻(𝜔𝜔) =
1 + 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗
U
U
Dr. Ahmed A. Alrekaby
[2.10]
(a) From Eq.(5.1), 𝑆𝑆𝑋𝑋𝑋𝑋 (𝜔𝜔) =
𝜂𝜂
2
𝑆𝑆𝑌𝑌𝑌𝑌 (𝜔𝜔) = |𝐻𝐻(𝜔𝜔)|2 𝑆𝑆𝑥𝑥𝑥𝑥 (𝜔𝜔) =
(b) Rewriting Eq.(5.3) as
1
𝜂𝜂
1 + (𝜔𝜔𝜔𝜔𝜔𝜔)2 2
𝜂𝜂 1
2[1⁄(𝑅𝑅𝑅𝑅)]
2 2𝑅𝑅𝑅𝑅 𝜔𝜔 2 + [1⁄(𝑅𝑅𝑅𝑅)]2
and
𝑆𝑆𝑌𝑌𝑌𝑌 (𝜔𝜔) =
Finally
𝑅𝑅𝑌𝑌𝑌𝑌 (𝜏𝜏) = ℱ −1 [𝑆𝑆𝑌𝑌𝑌𝑌 (𝜔𝜔)] =
(5.3)
𝜂𝜂 1 −|𝜏𝜏|/(𝑅𝑅𝑅𝑅)
𝑒𝑒
2 2𝑅𝑅𝑅𝑅
𝐸𝐸[𝑌𝑌 2 (𝑡𝑡)] = 𝑅𝑅𝑌𝑌𝑌𝑌 (0) =
𝜂𝜂
4𝑅𝑅𝑅𝑅
6. BAND-LIMITED WHITE NOISE
U
A random process X(t) is called band-limited white noise if
Then
𝜂𝜂
𝑆𝑆𝑋𝑋𝑋𝑋 (𝜔𝜔) = �2
0
|𝜔𝜔| ≤ 𝜔𝜔𝐵𝐵
|𝜔𝜔| > 𝜔𝜔𝐵𝐵
1 𝜔𝜔 𝐵𝐵 𝜂𝜂 𝑗𝑗𝑗𝑗𝑗𝑗
𝜂𝜂𝜔𝜔𝐵𝐵 𝑠𝑠𝑠𝑠𝑠𝑠𝜔𝜔𝐵𝐵 𝜏𝜏
�
𝑒𝑒 𝑑𝑑𝑑𝑑 =
𝑅𝑅𝑋𝑋𝑋𝑋 (𝜏𝜏) =
2𝜋𝜋 −𝜔𝜔 𝐵𝐵 2
2𝜋𝜋 𝜔𝜔𝐵𝐵 𝜏𝜏
(6.1)
(6.2)
And 𝑆𝑆𝑋𝑋𝑋𝑋 (𝜔𝜔) and 𝑅𝑅𝑋𝑋𝑋𝑋 (𝜏𝜏) of band-limited white noise are shown in Fig.(6.1).
τ
Fig.(6.1)
Note that the term white or band-limited white refers to the spectral shape of the
process X(t) only, and these terms do not imply that the distribution associated
with X(t) is Gaussian.
Dr. Ahmed A. Alrekaby
[2.11]
Example 6.1: The input X(t) to an ideal bandpass filter having the frequency
response characteristic shown below is a white noise process. Determine the
total noise power at the output of the filter.
U
U
𝑆𝑆𝑋𝑋𝑋𝑋 (𝜔𝜔) =
𝜂𝜂
2
𝜂𝜂
𝑆𝑆𝑌𝑌𝑌𝑌 (𝜔𝜔) = |𝐻𝐻(𝜔𝜔)|2 𝑆𝑆𝑥𝑥𝑥𝑥 (𝜔𝜔) = |𝐻𝐻(𝜔𝜔)|2
2
The total noise power at the output of the filter is
1 ∞
1 𝜂𝜂 ∞
� 𝑆𝑆𝑌𝑌𝑌𝑌 (𝜔𝜔)𝑑𝑑𝑑𝑑 =
� |𝐻𝐻(𝜔𝜔)|2 𝑑𝑑𝑑𝑑
𝐸𝐸[𝑌𝑌 (𝑡𝑡)] =
2𝜋𝜋 −∞
2𝜋𝜋 2 −∞
2
where B = WB/(2π) (in hertz).
=
𝜂𝜂 1
(2𝑊𝑊𝐵𝐵 ) = 𝜂𝜂𝜂𝜂
2 2𝜋𝜋
Dr. Ahmed A. Alrekaby