ECE 590 Microwave Transmission for Telecommunications

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ECE 590Microwave Transmission for
Telecommunications

• Noise and Distortion in Microwave Systems
• March 18, 25, 2004

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Random Processes
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Random Processes
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Expected Values
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Expected Values
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Expected Values
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Autocorrelation and Power Spectral Density
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Autocorrelation and Power Spectral Density
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Noise in Microwave Circuits

• Result of random motions of charges or charge
  carriers in devices and materials
• Thermal noise (most basic type)
• thermal vibration of bound charges (also called
  Johnson or Nyquist noise)
• Shot noise
• random fluctuations of charge carriers
• Flicker noise
• occurs in solid-state components and varies
  inversely with frequency (1/f -noise)

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Noise in Microwave Circuits

• Plasma noise
• random motion of charges in ionized gas such as a
  plasma, the ionosphere, or sparking electrical
  contacts
• Quantum noise
• results from the quantized nature of charge
  carriers and photons often insignificant
  relative to other noise sources

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Noise power and Equivalent Noise Temperature
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Noise power and Equivalent Noise Temperature
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Noise in Linear Systems
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Noise in Linear Systems
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Gaussian white noise through an ideal low-pass
filter
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Gaussian white noise through an ideal integrator
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Mixing of noise frequency conversion
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Mixing of noise frequency conversion
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Basic Threshold Detection
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Basic Threshold Detection
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Graphical Representation of Probability of Error
for Basic Threshold Detection
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Noise Temperature and Noise Figure
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Noise Figure
Noisy Rf and microwave components can be
characterized by an equivalent noise temperature.
An alternative is the noise figure which is the
degradation of the signal to noise ratio between
the input and the output of the component, or F
(Si/Ni)/ (S0/N0) ? 1. The input noise power,
Ni k T0 B Pi Si Ni P0 S0 N0 S0 G Si
N0 kGB(T0 Te)
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Noise Figure
So F (Si/ k T0 B)/ (G Si / k G B (T0 Te)
(T0 Te)/ T0 1 Te/ T0 ? 1. Or the
temperature of the noisy network Te (F - 1) T0
. Let Nadded noise power added by the network,
the output noise power, N0 G (Ni Nadded) So F
(Si/ Ni)/ (G Si / G (Ni Nadded) 1
Nadded/ Ni
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Noise Figure of a Lossy Line
Lossy transmission line (attenuator) held at a
physical temperature, T. Power Gain, Glt1 so power
loss factor L 1/Ggt1 If the line input is
terminated with a matched load at temperature T,
then the output will appear as a resistor of
value R and temperature T. Output Noise power is
the sum of the input noise power attenuated
through the lossy line plus the noise power
added by the lossy line itself .
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Noise Figure of a Lossy Line
So the output Noise power, No kTB G(kTB
Nadded), where Nadded is the noise generated by
the line. Therefore, Nadded (1/G) - 1 kTB
(L-1) kTB The equivalent noise temperature Te of
the lossy line becomes Te Nadded / KB (L -
1) T and the noise figure is F 1 Te / T0
1 (L - 1) T / T0
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Noise Figure of Cascaded Components
Consider a cascade of two components having power
gains G1 and G2, noise figures F1 and F2 and
noise temperatures T1 and T2. Find overall
noise figure, T and noise temperature T of the
cascade as if it were the single component with
Ni k T0 B. Using noise temperatures, the noise
power at the output of the first stage is N1 G1
k B T0 G1 k B Te1 and the output at the
second is N0 G2 N1 G2 k B Te2 G1 G2 k B (T0
Te1 Te2 / G1)
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Noise Figure of Cascaded Components
For the equivalent single system N0 G1 G2 k B
(T0 Te) So the noise of the cascade system is
Te Te1 Te2 / G1 Recall F 1 Te/ T0 so
the cascade system F 1 Te1/ T0 Te2 / (G1
T0) F1 ( F2 - 1) / G1 more generally Te
Te1 Te2 / G1 Te3 / (G1G1) F F1 ( F2 - 1)
/ G1 ( F3 - 1) / G1 G2
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Noise Figure of a Passive Two-Port Network
Impedance mismatches may be defined at each port
in terms of the reflection coefficients, ? as
shown in the diagram. Assume the network is
at temperature, T and the input noise power is N1
k T B is applied to the input of the network.
The available output noise at port 2 is N2
G21 k T B G21 Nadded the noise generated
internally by the network (referenced at port 1).
G21 is the available gain of the network from
port 1 to port 2.
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Noise Figure of a Passive Two-Port Network
The available gain can be expressed in terms of
the S-parameters of the network and the port
mismatches as G21 power available from
network divided by power available from source
S212 (1- ?s 2)/ 1S11?s 2(1- ?out
2) and the output mismatch is ?out S22
S12S21?s /(1- S11?s ) From N2k T B, find Nadded
(1/G21-1)k T B, and the equivalent noise
temperature is Te Nadded /kB T(1- G21)/ G21,
and F (1/G21-1)T/T0 Can apply to examples
mismatched lossy line and Wilkinson power divider.
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Gain Compression
General non-linear network with an input voltage
vi and and output voltage v0 can be expressed in
a Taylor series expansion v0 a0 a1vi
a2vi2 a3vi3 where the Taylor coefficients
are given by a0 v0 (0) (DC output)
rectifier converting ac to dc a1 dv0 / dvi
vi 0 (linear output) linear attenuator or
amplifier a2 d2v0 / dvi2 vi 0 (squared
output) mixing and other frequency conversion
functions
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Gain Compression
Let vi V0 cos ?0t then evaluate v0 a0 a1vi
a2vi2 a3vi3 v0 a0 a1 V0 cos ?0t
a2 V0 2 cos 2 ?0t a3 V0 3 cos 3 ?0t ( a0
½ a2 V0 2 ) (a1 V0 ¾ a3 V0 3 ) cos ?0t ½
a2 V0 2 cos 2?0t ¼ a3 V0 3 cos 3?0t This
result leads to the voltage gain of the signal
component at frequency ?0 Gv v0 (?0 ) / vi (?0
) (a1 V0 ¾ a3 V0 3 ) / V0 a1 ¾ a3 V0 2
(retaining only terms through the third order)
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Gain Compression
Gv v0 (?0) / vi (?0) (a1 V0 ¾ a3 V0 3 ) /
V0 a1 ¾ a3 V0 2 here we see the a1 term plus
a term proportional to the square of
the magnitude of the amplitude of the input
voltage. The coefficient a3 is typically
negative so the gain of the amplifier tens to
decrease for large values of V0. This is gain
compression or saturation.
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Intermodulation Distortion
For a single input frequency, or tone, ?0, the
output will consist of harmonics of the input
signal of the form, n ?0, for n 0, 1, 2,
. Usually these harmonics are out of the
passband of the amplifier, but that is not true
when the input consists of two closely
spaced frequencies. Let vi V0(cos ?1t cos ?2t
) where ?1 ?2. Recall v0 a0 a1vi a2vi2
a3vi3 hence
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Intermodulation Distortion
The output spectrum consists of harmonics of the
form, m?1n?2 with m, n 0, ?1, ?2, ?3, These
combinations of the two input frequencies are
call intermodulation products, with order m
n. Generally, they are undesirable however, in
cases, for example a mixer, the the sum or
difference frequencies form the desired outputs.
Note that they are both far from ?1 and ?2. But
the terms 2?1 - ?2 and 2?2 - ?1 are close to ?1
and ?2. Which causes third-order intermodulation
distortion.
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Third-Order Intercept Point
Plot of first and third-order products of the
output versus input power on a log-log plot hence
the slopes represent the powers.
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Dynamic Range