Basic Concepts in RF Design
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Overview
System Theory
Effects of Nonlinearities
Gain Compression
Desensitization and Blocking
Crossmodulation
Intermodulation
Adjacent Channel Power Rejection
Random Signals
Noise and Noise Figure
Input Referred Noise
Noise Figure of Lossy Circuits
Sensitivity
Dynamic Range
References
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System Theory (1)
Linear Systems
If inputs x1 and x2 generate outputs y1, y2
x1 → y1
x2 → y 2
For a linear system output can be expressed as a linear combination of inputs
ax1 + bx2 → ay1 + by2
for all values of the constants a and b
Any system that does not satisfy this condition is nonlinear
Obs. A system is nonlinear if it has nonzero initial conditions
Time Invariant Systems
For a time invariant system time shift in input results in the same time shift in output
x(t ) → y (t )
then
x(t − τ ) → y (t − τ )
for all values of τ
A system is time variant if it does not satisfy this condition
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System Theory (2)
Example
vin1 (t ) = A1 cos ω1t
vin 2 (t ) = A2 cos ω 2t
and the switch is on if vin1 > 0
Is the system (a) nonlinear or time variant?
• Case (b)
Nonlinear - the control is sensitive to the polarity
of vin1
Time variant - because vout also depends on vin2
• Case (c)
Linear
Time variant
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System Theory (3)
Remark: a linear system can generate frequency components that do not exist in
the input signal
In example (c) vout can be considered as the product of vin2 and a square wave
toggling between 0 and 1, the output spectrum is given by:

n  +∞ sin (nπ / 2)
n
sin (nπ / 2) 
Vout ( f ) = Vin 2 ( f ) ∗ ∑
Vin 2  f − 
δ  f −  = ∑
nπ
T1  n = −∞
nπ
T1 
n = −∞


+∞
Where * denotes convolution, δ(#) is the Dirac delta function, T1 = 2π/ω1
The output consists of vertically scaled replicas of Vin2(f) shifted by n/T1
• A system is memoryless if its output does not depend on the past values of its input
• A memoryless linear system:
• A memoryless nonlinear system:
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y (t ) = αx(t )
y (t ) = α 0 + α1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t ) + ...
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System Theory (4)
A system with „odd“ symmetry: its response to -x(t) is the negative of that to x(t)
This occurs if αj = 0 for even j
A circuit with odd symmetry is called differential or „balanced“ - has no even order
harmonics
Example: bipolar differential pair
vout = RI EE tanh
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vin
2VT
6
Effects of Nonlinearity (1)
Consider a nonlinear, memoryless, time - variant systems y(t) = f(x(t)) which can
be modeled by Taylor’s series (consider only the first three terms). Then:
y (t ) ≈ α1 x(t ) + α 2 x (t ) + α 3 x (t )
2
3
1  ∂k y 
 k 
=
k!  ∂ x 
αk
x=0
Harmonics
If a sinusoid is applied to our system, the output exhibits frequency components that
are integer multiples of the input
If x(t ) = A cos ωt
y (t ) = α1 A cos ωt + α 2 A2 cos 2 ωt + α 3 A3 cos 3 ωt
α 3 A3
α 2 A2
= α1 A cos ωt +
(1 + cos 2ωt ) +
(3 cos ωt + cos 3ωt )
2
4
3α 3 A3 
α 2 A2 
α 2 A2
α 3 A3
 cos ωt +
cos 2ωt +
cos 3ωt (1)
=
+ α1 A +
2

4

2
4
The term with the input frequency - the fundamental, the higher order terms - the
harmonics
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Effects of Nonlinearity (2)
1 Even-order harmonics result from αj with even j and vanish if the system has
odd symmetry - in reality, however mismatches corrupt the symmetry yielding
finite even-order harmonics
1 The amplitude of the nth harmonic consists of a term proportional to An and other
terms proportional to higher powers of A. Neglecting the latter for small A, we can
assume the nth harmonic grows approximately in proportion to An
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Gain Compression (1)
Neglecting the harmonics in eq. (1), the output is:
3


y (t ) = α1 A + α 3 A3  cos ωt
4


The gain is:
3
av = α 1 A + α 3 A3
4
and usually α3 < 0!
• For A small, the small signal gain is:
av ≈ α1
• When A increases the gain begin to decrease,
and the second term cannot be anymore neglected
1 Nonlinearity = variation of the small-signal gain with the input level
1 The output is a compressive (saturating) function of the input: the gain
approaches zero for sufficiently high input levels
1 The gain is a decreasing function of A
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Gain Compression (2)
This effect is quantified by the “1-dB
compression point” (1-dBCP):
the input signal level that causes the smallsignal gain to drop by 1dB.
20log α 1 − 1dB = 20log α 1 +
A 1- dB = 0.145
α1
α3
3
α 3 A 21-dB
4
1 Measure of maximum applicable input signal
1 In typical RF amplifiers, 1-dBCP occurs around –20 to –25dBm (63.2 to 35.6mVpp
in a 50Ω system)
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Gain Compression (3)
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Desensitization and Blocking
Problem: when process a weak desired signal in the presence of a strong interferer
the small signal gain is reduced by the interferer - the receiver is desensitized by the
interferer!
x(t ) = A1 cos ω1t + A2 cos ω 2t
y (t ) = α1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t )
3
3

3
2
y (t ) = α1 A1 + α 3 A1 + α 3 A1 A2  cos ω1t + ...
4
2


3

2
y (t ) = α1 + α 3 A2  A1 cos ω1t + ...
2


for A1 << A2:
1 The gain is a decreasing function of A2; for sufficiently large A2, the gain drops to
zero, and we say the signal is „blocked“
1 In RF design „blocking signals“ usually refers to interferers that desensitize a
circuit even if the gain does not fall to zero
1 RF receivers require to withstand blocking signals 60 to 70dB greater than the
wanted signal
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Cross Modulation
Phenomenon that occurs when a weak signal and a strong interferer pass through
a nonlinear system: the transfer of the amplitude modulation (or noise) of the
interferer to the amplitude of the weak signal
x(t ) = A1 cos ω1t + A2 (1 + m cos ω mt ) cos ω 2t
m – the modulation index


m2 m2
3
2

y (t ) = α1 + α 3 A2 1 +
+
cos 2ω mt + 2m cos ω m t   A1 cos ω1t + ...
2
2
2



1 The desired signal at the output contains amplitude modulation at ωm and 2ωm
1 Common case of crossmodulation: amplifiers that must simultaneously
process many independent signal channels, e.g. in cable television transmitters
1 Nonlinerities in the amplifier corrupt each signal with the amplitude variations
in other channels
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Intermodulation (1)
1 When two signals with different frequencies are applied to a nonlinear system,
the output in general exhibits some components that are not harmonics of the input
frequencies - intermodulation
1 Intermodulation arises from mixing of the two signals when their sum is raised to
a power greater than unity
x(t ) = A1 cos ω1t + A2 cos ω 2t
y (t ) = α1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t )
y (t ) = α 1(A1 cos 11t + A2 cos 1 2 t) + α 2(A1 cos 11t + A2 cos 1 2 t) 2
+ α 3(A1 cos 11t + A2 cos 1 2 t) 3
After expanding and collecting the terms, at fundamental frequencies we have:
At ω1:
At ω2:
3
3

2
2
+
+
2
2
A
2
A
 1
3 2
3 1  A1 cos 11t
2
4


3
3

2
2
2
2
A
2
A
+
+
 1
3 1
3 2  A2 cos 1 2 t
2
4


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Intermodulation (2)
Second-order terms:
At 2ω1, 2ω2:
At ω1 ± ω2 :
1
2
 2 2 A1  cos 2 11t
2

1
2
2
A

2 2  cos 2 1 2 t

2
2 2 A1 A2 (cos (11 + 1 2 ) t + cos (11-1 2 ) t )
Third-order terms:
1
3
 2 3 A2  cos 31 2 t
4

At 3ω1, 3ω2 :
1
3
 2 3 A1  cos 311t
4

At 2ω1 ±ω2 :
3

2
2
A
A

3 1
2  (cos (2 11 + 1 2 ) t + cos (2 11-1 2 ) t )
4

At ω1 ±2ω2 :
3
2
2
A
A

3 1 2  (cos (11 + 2 1 2 ) t + cos (11- 2 1 2 ) t )

4
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Intermodulation (3)
Very important: the third-order intermodulation (IM3) terms at 2ω1 -ω2 and 2ω2 -ω1
If the difference between ω1 and ω2 is small, the IM3 components (2ω1 - ω2) and
(2ω2 - ω1) are in the vicinity of ω1 and ω2 ((2ω1 - ω2) ~ ω1) ⇒ signal corruption due
to the two strong interferers!
In a typical two tone test: A1 = A2 = A
The ratio of the amplitude of the output
IM3 products to α1A defines the IM
distortion
• Example: if α1A = 1Vpp and 3α3A3/4
= 10mVpp then the IM3 components
are at –40dBc
IM is a critical effect in RF systems!
⇒ IP3 point (third intercept point)
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Intermodulation (4)
1 The fundamental increases in proportion to A (~ α1A) whereas the IM3 products
increase in proportion to A3 (~ 3α3A3/4)
1 Input IP3 is the the signal level at which the amplitude of the IM3 would become
equal to the that of the fundamental in a two-tone test
1 IIP3 (input IP3) & OIP3 (output IP3)
1 Two-tone test: A is small enough to neglect the high-order nonlinear terms and the
gain is relatively constant and equal to α1
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Intermodulation (5)
9


y (t ) =  21 + α 3 A 2  A cos 11t +
4


3
+
2 3 A 3 cos ( 2 11 -1 2 )t +
4
If 21 >>
9
23 A 2
4
21 AIP 3
9

2
α
+
2
A
 1
 A cos 1 2 t
3
4


3
2 3 A 3 cos ( 2 1 2 -11 )t + ...
4
3
= 2 3 A 3 IP 3
4
AIP 3 = IP 3 =
4 21
3 23
• Relationship between AIP3 and A1-dB:
A1− dB
0 .145
=
≈ − 9 .6 dB
AIP 3
4/3
1 IP3 characterizes only third-order nonlinearities; if the input signal is increased, the
condition α1A>> 9α3A3/4 no longer holds, the gain drops, and higher order IM
products become significant
1 Many circuits have IP3 beyond the allowable input range, sometimes even higher
than the supply voltage
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Intermodulation (6)
Method of measuring the IP3
Ain – input level at each frequency
A ω1, ω2 – amplitude of the output components @ω1 and ω2
AIM3 – amplitude of the IM3 products
From the previous equation:
Aω 1,ω 2
Then
3
2 3 A 3 in
= 21 Ain and AIM 3 ≈
4
Aω 1,ω 2
AIP2 3
4 21 1
=
= 2
2
AIM 3
3 α 3 Ain
Ain
Expressed in dB:
20 log AIP 3
20 log Aω 1,ω 2 − 20 log AIM 3 = 20 log AIP2 3 − 20 log Ain2
1
= (20 log Aω 1,ω 2 − 20 log AIM 3 ) + 20 log Ain
2
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Intermodulation (7)
Graphical calculation
L1 – slope equal to 1, L2 – slope equal to 3
An input increment ∆P/2 yields an equal increment in L1 and an increment equal to
3∆P/2 in L2 reducing the difference between the two lines to zero
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Cascaded Nonlinear Systems (1)
y1 (t ) = α1 x(t ) + α 2 x (t ) + α 3 x (t )
2
3
[
]
+ β [α x(t ) + α x (t ) + α x (t )]
+ β [α x(t ) + α x (t ) + α x (t )]
y2 (t ) = β1 α1 x(t ) + α 2 x 2 (t ) + α 3 x 3 (t )
2
y2 (t ) = β1 y1 (t ) + β 2 y1 (t ) + β y (t )
2
3
3 1
2
1
2
1
2
2
3
3
3
2
3
3
3
Considering the first and third-order terms, we have:
(
)
y2 (t ) = α1 β1 x(t ) + α 3 β1 + 2α1α 2 β 2 + α13 β 3 x 3 (t ) + ...
AIP 3 =
4
21 β 1
3 2 3 β 1 + 2α 1α 2 β 2 + α 13 β 3
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Cascaded Nonlinear Systems (2)
A worst case estimate:
3
β
α
α
β
α
2
+
2
+
1
3 3 1
1 2 2
1 β3
=
2
A IP 3 4
21 β 1
1
1
3 α 2β2
α 12
= 2 +
+ 2
2
A IP 3
A IP 3 ,1 2 β 1
A IP 3 , 2
As α1 increases, the overall IP3 decreases: with higher gain in the first stage, the
second stage senses larger input levels, producing much greater IM3 products
Since each stage has a narrow passband, out-of-band signals are heavily
attenuated, and:
1
1
α 12
≈ 2 + 2
For more stages:
2
A IP 3
A IP 3 ,1 A IP 3 , 2
α 12
α 12 β 12
1
1
≈ 2 + 2 + 2 + ...
2
A IP 3
A IP 3 ,1 A IP 3 , 2
A IP 3 , 3
Note that AIP3, AIP3,1, AIP3,2 … are voltage quantities rather than power quantities
and rather real values, not dB
For gain greater than 1, the IP3 of latter stage becomes increasingly critical
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Adjacent Channel Power Rejection – ACPR (1)
Problem
• Intermodulation lead to higher in
band noise power
• Signal power leakage to the
adjacent channel degrade
adjacent channel (used by other
users) SNR
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Adjacent Channel Power Rejection – ACPR (2)
Example: CDMA ACPR
Measure spectral power of the channel (1.23 MHz bandwidth)
Measure upper and lower band edge ( ±885 KHz) of the next adjacent channel
power (30KHz bandwidth)
ACPR [dBc ] =
total power in desired channel
total power in adjacent channel
Alternative method:
ACPR [dBc ] =
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P1 ⋅ 10 log (1230 / 30 )
P2
24
Random Processes (1)
Random process
• we do not know / need to know everything about it
• it can be characterized with only a few parameters and functions
• we can solve most problems without any other information about the process
Example:
The noise voltage across a resistor as a function of time today is different from
that measured tomorrow
To know everything about the noise voltage, infinite number of measurements,
each one for infinite length of time are required
Random processes
• require a collection of measurements - family of time functions
• can be modeled with simple statistical functions that indicates how much and
how fast the amplitude varies with time
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Random Processes (2)
Time average - consider one resistor and measure the noise n(t) over a long time
T, and calculate the average as:
T /2
1
n(t ) = lim
n(t )dt
T →∞ T ∫
−T / 2
Ensemble (statistical) average - consider an ensemble of identical resistors and
average simultaneous samples of the noise waveforms in an ensemble:
∞
n(t ) = ∫ n(t )Pn (n )dn
−∞
where Pn(n) is the probability density function of the process
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Random Processes (3)
Q1: Is the time average measured today equal to that measured tomorrow?
A: Not necessarily. If system is “stationary”, its statistical properties are invariant to
a time shift. Fortunately most of the random phenomena in RF systems can be
considered stationary.
Q2: Is the time average of a stationary process equal to ensemble average?
A: Not always. But in most cases they are equal.
Remark: second order averages are of particular interest because they represent
the power of the signals
T /2
1
2
2
(t )dt
n (t ) = lim
n
T →∞ T ∫
−T / 2
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∞
n 2 (t ) = ∫ n 2 (t )Pn (n )dn
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Probability Density Functions (PDFs)
The PDF - Px(x) - of a random signal x(t)
1 shows how often the amplitude of a random process falls in a given range of
values
1 provides no information as to how fast the random signal varies in the time domain
Px (x )dx = probability of x < X < x + dx
X – is the measured value of x(t)
at some point in time
Example: the Gaussian (normal) distribution
1
− (x − m )
Px (x ) =
exp
2σ 2
σ 2π
x
1
− u2
exp
erf ( x ) =
du
∫
2
2π 0
2
1
− (x − m )
Px (x1 < x < x2 ) = ∫
dx
exp
2
2σ
2π
x1 σ
x2
2
1 Central limit theorem: if many independent random processes with arbitrary
PDFs are added, the PDF of the sum approaches a Gaussian distribution
1 Many natural phenomena exhibit Gaussian statistics (noise of a resistor, etc.)
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Power Spectral Density (PSD) (1)
The PSD - Sx (f) - of a random signal x(t) shows how much power the signal
carries in a unit bandwidth around frequency f
1 Sx(f) estimation: apply the signal to a
bandpass filter with 1-Hz bandwidth
centered at f and measure the average
output power over a sufficiently long
time
1 If this measurement is performed for
each value of f, the overall spectrum of
the signal is obtained
S x ( f ) = lim
T →∞
CMOS RFIC Design
XT ( f )
T
T
2
where
X T ( f ) = ∫ x(t ) exp(− j 2πft )dt
0
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Power Spectral Density (PSD) (2)
Computational algorithm for PSD
1. Truncate x(t) to a relatively long interval [0, T]
2. Calculate the Fourier transform of the result and hence |XT(f)|2
3. Repeat steps 1 and 2 for many sample functions of x(t)
4. Take the average of all |XT(f)|2 functions and normalize the result to T
− f1
Sx (f) is an even function of f for real x(t)
f2
∫ S ( f )df + ∫ S ( f )df = ∫ 2S ( f )df
x
− f2
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x
f1
x
f1
30
Power Spectral Density (PSD) (3)
If a signal of Sx(f) is applied to a linear, time-invariant system with transfer function
H(s), then:
S y ( f ) = Sx ( f ) H ( f )
2
where
H ( f ) = H (s = j 2πf )
• The spectrum of the signal is shaped by the transfer function of the system
• If x(t) is Gaussian, so is y(t)
In general, PDF and PSD have no relationship:
• thermal noise: Gaussian PDF, white PSD
• flicker noise: Gaussian PDF, 1/f PSD
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Noise (1)
Thermal noise
1 Brownian random motion of thermally agitated charge carriers
1 Generated in every physical resistors
1 Modeled by a voltage source in series (Thevenin representation) or current source
in parallel (Norton representation) with the PSD given by:
2
V
4kT∆f
= 4kTG∆f
I n2 = n2 =
R
R
V = 4kTR∆f
2
n
where k = 1.38 ⋅10 −23 J / K Boltzmann constant and K is the absolute temperature
• „white“ noise – contain the same level of power at all frequencies
• Is the mean square noise voltage generated by a resistor R in a bandwidth ∆f
⇒ is measured in V2/Hz or in
[
Vn2 = Vrms / Hz
]
• Is not power! We tacitly assume that this voltage is applied across a 1-Ω resistor
to generate a power of 4kTR in a 1-Hz bandwidth
• Example: 1kΩ ⇒ Vn,rms = 4nV/Hz1/2,
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50Ω ⇒ Vn,rms = 4nV/Hz1/2
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Noise (2)
G
Thermal noise in MOSFET
D
Cgs
• Channel resistance noise
2
I nD
= 4kTγg d 0 ∆f ≈ 4kT (2 g m / 3)
gg
InG
gmvgs
InD
S
where γ is a bias-dependent factor: γ ~ 2/3 in long channel devices; γ ~ 2 - 3 in
short channel devices
gd0 - the zero-bias drain conductance ( = gm for long channel devices)
• Gate noise
2
I nG
= 4 kT δ g g ∆ f
ω 2 C gs2
gg =
5gd 0
δ ~ 4/3 in long devices
Thermal agitation of channel charge cause fluctuation of channel potential ⇒
this couples capacitively with gate terminal, generating gate noise
Gate noise is negligible at low frequency, but it can dominate at RF
• Both drain and gate noise share a common origin ⇒ they are correlated
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Noise (3)
Shot noise (Schottky noise)
• Gaussian white process associated with the transfer of charge across an energy
barrier (e.g., a p-n junction)
I = 2 qI DC ∆ f
2
n
where
q = 1.6 ⋅10 −19 C is the charge of the electron
• For a bipolar transistor the collector and emitter shot noise is modeled as a current
source connected between C and E and another between B and E
• In MOS devices the shot noise of the very small gate leakage current can be
neglected
• „white“ noise, no temperature dependence
• Example: for a IDC = 1mA ⇒ In,rms = 18pA/Hz1/2
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Noise (4)
Flicker noise (1/f noise, „pink“ noise)
• Random trapping of charge at oxide interface modeled as a voltage source in
series with gate
• Modeled by a noise current source with a spectral density given by:
2
K gm
I n2 =
∆f
2
f WLC ox
where K is a technology-dependent parameter
• At high frequency may be neglected; however in
mixers and oscillators the 1/f-shaped spectrum
can be translated to RF range
Small signal noise model of a MOSFET
G
D
2
I nG
= 4 kT δ g g ∆ f
Cgs
InG
gg
gmvgs
S
CMOS RFIC Design
2
InD
I
2
nD
K gm
= 4 kT γ g d 0 ∆ f +
∆f
2
f WLC ox
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Input - Referred Noise (1)
1 The noise of a two-port system can be modeled by two input noise generators: a
series voltage source Vn2 and a parallel current source In2
1 Require both Vn2 and In2 for adequate representation
1 In general, the correlation between Vn and In must be taken into account
1 Vn2 is determined by shorting the input port of the two circuits and equating the
output noise
1 In2 is determined by opening the input port of the two circuits and equating the
output noise
1 Even though a circuit may have no physical input noise current, the representation
using input-referred sources must include In2
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Input - Referred Noise (2)
Example
The circuit should produce the same output noise in both cases!
Short the input port and equate the output noise:
2
g m2 Vn2 = I nD
Open the input port and equate the output noise:
2
g m2 I n2 Z in = I nD
For
I
2
nD
= 4kT (2 g m / 3) we obtain V = 8kT / (3 g m )
2
n
2
(
I = 8kT / 3 g m Z in
2
n
Since Vn and In represent the same mechanism, they are correlated
Z in → ∞ I n2 → 0 and Vn is sufficient to represent the noise
In RF design Z in is low (50Ω) ⇒ we need both Vn and In
If
CMOS RFIC Design
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2
)
Noise Figure (NF) (1)
Noise Figure =
total output noise power
output noise due to input source
Noise Factor =
SNRin
SNRout
(2)
(1)
Noise Figure = 10log10 (Noise Factor )
1 The two definitions are equivalent!
1 Noise figure (or noise factor) measures the SNR degradation as a signal pass
through a system
1 If a system has no noise, then SNRin = SNRout ⇒ NF = 1 = 0dB
1 In reality, the finite noise of a system degrades the SNR, yielding NF > 1
1 If the input signal contains no noise, then SNRin = ∞ and NF = ∞ (even though
the system has a finite internal noise): this does not occur in real world!
1 Take care whether discuss about noise figure or noise factor
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38
Noise Figure (2)
For NF calculations – usually
use the first definition!
VRS2 + (Vn + I n RS )
2
NF =
VRS2
2
(
Vn + I n RS )
NF = 1 +
4kTRS
2
(
Vn + I n RS )
= 1+
VRS2
or
NF =
Vn2,out
2
v ,tot
A
2
RS
V
=
Vn2,out
Av2,tot 4kTRS
1 V2n,out – is the total noise at the output; A v,tot – the total gain from Vin to Vout
1 No correlation between VRS and Vn (or In), but correlation between Vn and In
1 NF is typically specified for a 1-Hz BW
1 Vn and In are also measured in 1-Hz BW
1 NF is function of source impedance RS
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Noise Figure (3)
Example
Determine the NF of RP with respect to a source resistance RS
2
n ,out
V
NF =
= 4kT (RS // RP )
Vn ,out
2
VRS2 ,out
2
RS ,out
V
=AV
2
v
2
RS
= A 4kTRS
2
v
Av =
RP
RS + RP
RS
= 1+
RP
NF is minimized by maximizing RP
The condition for minimum noise figure does not coincide with that for maximum
power transfer RS = RP
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Noise Figure of Cascaded Stages (1)
The overall NF can be obtained in terms of NF and gain of each stage
Av1, Av2 – the unloaded voltage gain
The total noise power at the input of the first stage can be written as:
(
Vn2,in1 = α12 (VRS + I n1 RS + Vn1 ) = α12 VRS2 + (I n1 RS + Vn1 )
2
2
)
The total noise power at the input of the second is:
(
Vn2,in 2 = α 22 Av21Vn2,in1 + (I n 2 Rout1 + Vn 2 )
2
)
The total noise power at the output is:
2
n ,out
V
=α A V
2
3
2
v2
2
n ,in 2
CMOS RFIC Design
where
α1 =
Rin1
RS + Rin1
α2 =
Institute of
Microelectronic
Systems
Rin 2
Rout1 + Rin 2
α3 =
RL
Rout 2 + RL
41
Noise Figure of Cascaded Stages (2)
α 32 Av22 Vn2,in 2
α 22 Av21Vn2,in1 + α 22 (I n 2 Rout1 + Vn 2 )2
= 2 2 2 2 2
=
NFtot = 2
Av ,tot 4kTRS α1 Av1α 2 Av 2α 3 4kTRS
α12 Av21α 22 4kTRS
Vn2,out
=
Vn2,in1
α12 4kTRS
2
(
I n 2 Rout1 + Vn 2 )
+
α12 Av21 4kTRS
1 (I n 2 Rout1 + Vn 2 )
VRS2 + (I n1 RS + Vn1 )
=
+ 2 2
4kTRS
4kTRS
α1 Av1
Where the voltage gain from Vin to Vout is:
2
2
Av ,tot = α1 Av1α 2 Av 2α 3
In the special case where RS = Rin1 = Rout1 = Rin2
NFtot = NF1, RS +
NF2, RS − 1
Av21
NF1,RS, NF2,RS - noise figure of the1st and 2nd stage with respect to a
source impedance RS
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Noise Figure of Cascaded Stages (3)
AP =
Available power gain AP
1 Available power at the output: the
power that the circuit would deliver to a
conjugate-matched load
available power at the output
available source power
α12 Av21Vin2
Pout ,av1 =
4 Rout1
1 Available source power: power that source
would deliver to a conjugate-matched circuit
Vin2
Psource,av1 =
4 RS
So that:
RS
AP1 = α A
Rout1
2
1
2
v1
The total NF is:
(I n 2 Rout1 + Vn 2 )
1 (I R + V )
R
+ 2 2 n 2 out1 n 2 = NF1, RS + 2 out2 1
α1 Av1
4kTRS
α1 Av1 RS
4kTRout1
2
NFtot = NF1, RS
NFtot = NF1, RS +
NF2, Rout1 − 1
CMOS RFIC Design
AP1
where
2
NF2, Rout1
Institute of
Microelectronic
Systems
2
(
I n 2 Rout1 + Vn 2 )
= 1+
4kTRout1
43
Noise Figure of Cascaded Stages (4)
Similarly, for m stages (Friis equation):
NFtot = 1 + ( NF1 − 1) +
NFm − 1
NF2 − 1
+ ... +
AP1
AP1 ⋅ ... ⋅ AP (m −1)
1 NF of each stage is calculated with respect to the output impedance of the
previous stage (the source impedance driving that stage)
1 The noise contributed by each stage decreases as the gain preceding the stage
increases
1 The first few stages in a cascade are the most critical
1 If a stage exhibits attenuation (loss), then NF of the following circuit is amplified
when referred to the input of that stage
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Noise Figure of Lossy Passive Circuits (1)
Lossy Circuit
1 Passive devices attenuate desired
signal, and contribute noise
1 Power Loss (L) – similar to the concept
of available power gain
Pin,av
Vin2 / 4 RS
Vin2 Rout
= 2
= 2
L=
Pout,av VTH / 4 Rout VTH RS
2
n ,out
V
= 4kTRout
NF =
Vn2,out
Av2,tot
RL2
(RL + Rout )2
Av ,tot
VTH
RL
=
Vin RL + Rout
Vin2
1
1
= 4kTRout 2
=L
VTH 4kTRS
4kTRS
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Noise Figure of Lossy Passive Circuits (2)
Cascade of lossy filter and
low noise amplifier (LNA)
NFFilt = L
NFtot = NFFilt +
NFLNA − 1
NFLNA − 1
= L + ( NFLNA − 1)L = L ⋅ NFLNA
= NFFilt +
−1
AP , Filt
L
NFtot [dB ] = L[dB ] + NFLNA [dB ]
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Sensitivity (1)
Sensitivity is the minimum power level of the input signal that the receiver can
detect while providing a required SNR at the output (for a required Bit Error Rate)
NF =
P /P
SNRin
= s,in n,in
SNRout
SNRout
Ps,in - the input signal power per unit bandwidth (/Hz)
Pn,in - the input noise power per unit bandwidth (/Hz)
The power of input signal, distributed across the channel bandwidth BW is:
Ps,in = Pn,in ⋅ NF ⋅ SNR out ⋅ BW
equivalent, in dB:
Ps,in [dBm] = Pn,in [dBm / Hz] + NF[dB] + SNR out [dB] + 10 log (BW)
min
Psmin
, in [dBm] = Pn , in [dBm / Hz] + NF[dB] + SNR out [dB] + 10 log (BW)
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Sensitivity (2)
Pn,in - the available noise power that the source resistance RS delivers to a
matched input is:
Pn,in =
4 kTR S
= kT = − 174 dBm / Hz = 4 × 10 − 21 W
4 Rin
min
Psmin
[dBm]
=
−
174
dBm
/
Hz
+
10
log
(BW)
+
NF[dB]
+
SNR
, in
out [dB]
min
min
Sensitivit y = Psmin
[dBm]
=
Input
Noise
Floor
+
SNR
=
INF
+
SNR
, in
out
out
Sensitivity:
1 is function of bandwidth
1 degrades with higher data
rates (BW increases)
1 Input Noise Floor - INF
Input Noise Floor = Output Noise Floor - Gain
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Dynamic Range (1)
1 Dynamic Range (DR)
DR =
Pin,max [dBm]
Pin,min [dBm]
Pin,max - maximum input level that the circuit can tolerate
Pin,min - minimum input level at which circuit provides a reasonable signal quality
Differently quantified in different applications
1 Spurious Free Dynamic Range (SFDR)
Pin,min - sensitivity
Pin,max - maximum input level in a two-tone test for which the IM3 products are
smaller than the noise floor
1
(20 log Aω 1,ω 2 − 20 log AIM 3 ) + 20 log Ain
2
Pout − PIM 3 ,out
20 log AIP 3 =
PIIP 3 = Pin +
CMOS RFIC Design
2
Institute of
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49
Dynamic Range (2)
Pout = Pin + G and PIM3,out = PIM3,in + G so that:
PIIP 3 = Pin +
Pin − PIM 3,in
2
Pin =
2 PIIP 3 + PIM 3 ,in
3
• Pin,max is obtained when P IM3,in = INF
Pin , max
2 PIIP 3 + INF
=
3
• Pin,min is sensitivity
min
Pin , min = INF + SNR out
2 PIIP 3 + INF
2
min
min
SFDR =
− INF + SNR out = (PIIP 3 − INF ) − SNR out
3
3
(
)
Example: NF =-15dBm, PIIP3 =-15dBm, B =200kHz, SNRmin =12dB ⇒ SFDR =53dB
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References
B. Razavi - RF Microelectronics
http://info.nsu.ac.kr/cwb-data/data/ycra2/Basic_Concepts_in_RF_Design.pdf
(ICE669 Wireless Transceiver Design Lecture / H.J. Yoo Information and Communication University, Seoul, Korea)
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