Communications Noise Models
The Shannon-Weaver noise model
Noise Models
•
•
•
•
•
Lect 06
Overview
Channel capacity
Noise sources
Shot and flicker noise
Solar radiation
•
•
•
•
•
Noise spectrum
Thermal noise power
Noise temperature
Noise models
Noise Factor
© 2012 Raymond P. Jefferis III
1
Overview
• Noise is present in all communication
systems. It degrades transmitted data,
causing a lowering of data rates
• Every system design meets a maximum
specified Bit Error Rate (BER).
• System design practices are used to reduce
electrical noise and its effects to attrain the
specified Bit Error Rate goal
Lect 06
© 2012 Raymond P. Jefferis III
2
Well-Known References
• Shannon, Claude E. (1948): A Mathematical
Theory of Communication, Part I, Bell Systems
Technical Journal, 27, pp. 379-423.
• Shannon, Claude E. & Warren Weaver (1949): A
Mathematical Model of Communication. Urbana,
IL: University of Illinois Press.
Lect 06
© 2012 Raymond P. Jefferis III
3
Shannon - Capacity of Channel
• The information capacity of a communications
channel for a given S/N power ratio is
S

C  B log 2 1  
 N
where,
C = information capacity of channel [bits/s]
B = Bandwidth [Hz]
S/N = Signal-to-Noise power ratio [-]
Lect 06
© 2012 Raymond P. Jefferis III
4
Shannon - Capacity of Channel
• The information capacity of a communications
channel for a given (S/N)dB is
 S / N dB 



 10 

C  B log 2 1  10




where,
C = information capacity of channel [bits/s]
B = Bandwidth [Hz]
S/N = Signal-to-Noise power ratio [-]
(S/N)dB = Signal-to-Noise ratio [dB]
Lect 06
© 2012 Raymond P. Jefferis III
5
Example: Satellite Downlink
•
•
•
•
F = 12 GHz (Ku band)
B = 36 MHz (useful bandwidth)
(S/N)dB = 18 dB
C = (36*106)log2[1+1018/10] = 216 Mb/s
Note:
(S/N)dB = 10 log10[S/N]
Lect 06
© 2012 Raymond P. Jefferis III
6
Dynamic Computation
Run sndB model
Lect 06
© 2012 Raymond P. Jefferis III
Lect 00 - 7
Computation of Channel Capacity
Print["Channel capacity [Mb/s] for S/N in dB"];
Manipulate[
bw = 36.0*10^6;
pwr = sndB/10.0;
bw*Log[2, 10^pwr]/10^6,
{sndB, 1, 30}
]
Lect 06
© 2012 Raymond P. Jefferis III
Lect 00 - 8
Noise Sources
• Thermal noise (Johnson noise)
– Is a function of temperature
– Affected by Bandwidth
• Shot noise
– Is a property of solid state amplifier devices
• Flicker noise (1/f noise)
– Is a property of solid state amplifier devices
• Solar radiation noise
– Can cause significant interference at  = 10.7 cm
Lect 06
© 2012 Raymond P. Jefferis III
9
Blackbody (Johnson) Noise
vRMS 
Lect 06
4hfBR
ehf / kT  1
vRMS = RMS voltage noise (Volts)
h = Planck constant
(6.626069E-34 J sec)
f = frequency (1/sec)
B = Bandwidth (1/sec)
R = Resistance (Ohms)
k = Boltzmann constant
(1.380640E-23 J/K)
T = Temperature (Kelvin)
© 2012 Raymond P. Jefferis III
10
Blackbody Noise Example
•
•
•
•
•
•
•
h = 6.626069*10-34
k = 1.390640*10-23
R = 377.0 [Ohms]
f = 13 [GHz]
B = 40 [MHz]
T = 293.156 [K]
VRMS = 15.7 [uV]
Lect 06
© 2012 Raymond P. Jefferis III
11
Blackbody Noise Calculation
Run BBnoise
Lect 06
© 2012 Raymond P. Jefferis III
Lect 00 - 12
Blackbody Noise Calculation
h = 6.626069*10^-34;(*Planck constant*)
k = 1.390640*10^-23;(*Boltzmann constant*)
R = 377.0;
(*Free space-Ohms*)
freq = 1.3*10^10;
(*Hz*)
B = 40*10^6;
(*Hz*)
Manipulate[
Sqrt[4.*h*freq*B*R/(Exp[h*freq/(k*T)] - 1)]*10^6,
{T, 10, 300}]
Lect 06
© 2012 Raymond P. Jefferis III
13
Thermal Noise - Microwave Frequencies
At microwave frequencies the thermal noise is
virtually independent of frequency, and the
equation simplifies to:
vRMS  4kTBR
Lect 06
vRMS = RMS voltage noise
(Volts)
B = Bandwidth (Hz)
R = Resistance (Ohms)
k = Boltzmann constant
(1.3806404E-23 J/K)
T = Temperature (Kelvin)
© 2012 Raymond P. Jefferis III
14
Goldstone Antenna
This deep space
radiotelescope
system is outfitted
with a
cryogenically
cooled receiver to
lower the noise
level for sensitive
reception
Wikipedia
Lect 06
© 2012 Raymond P. Jefferis III
15
Note
•
The 500 kW CW X-band Goldstone Solar System Radar Freiley, A.; Quinn,
R.; Tesarek, T.; Choate, D.; Rose, R.; Hills, D.; Petty, S. Microwave Symposium
Digest, 1992., IEEE MTT-S International Volume , Issue , 1-5 Jun 1992
Page(s):125 - 128 vol.1 Digital Object Identifier ﾊ
10.1109/MWSYM.1992.187924 Summary:In recent years the Goldstone Solar
System Radar (GSSR) has undergone significant improvements in performance in
the areas of increased transmitter power and increased receiver sensitivity. An
overview of the radar system and each of these improvements are discussed. The
transmitter was upgraded with two new state-of-the-art 250 kW X-band klystrons
which increased the radiated power from 360 kW to 460 kW (1.1 dB). The
microwave receiver system was improved by cryogenically cooling a
major portion of the receive feed components, reducing the receiver noise
temperature from 18.0 K to 14.7 K (0.9 dB).
Lect 06
© 2012 Raymond P. Jefferis III
16
Shot Noise
• Statistical noise due to the current carriers
• The shot noise power in a resistor is,
P = 2qIBR
where,
q = electronic charge (1.602176E-19 Coul)
I = average current [Amperes]
B = Bandwidth [Hz]
R = Resistance [Ohms]
• Shot noise arises in semi-conducting detectors
Lect 06
© 2012 Raymond P. Jefferis III
17
Flicker (1/f) Noise
• Usually found at low frequencies
• Can be ignored for microwaves
Lect 06
© 2012 Raymond P. Jefferis III
18
Solar Blackbody Radiation
• The sun is a HOT source
(Blackbody temperature = 5778 K)
(Microwave temperature = 136,000 K)
• Radiation is affected by sunspot cycles
• Radiation can cause significant interference
at  = 10.7 cm ( 1.07*108 nm ) or a
frequency of ~28 GHz.
Lect 06
© 2012 Raymond P. Jefferis III
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Solar Blackbody Radiation
The Columbus Optical SETI Observatory
Lect 06
© 2012 Raymond P. Jefferis III
20
Planck’s Radiation Law ( ,T)
2h 3 
1

I( ,T )  2  (h / kT ) 
c e
 1
where,
I(ν,T) = Power Density (Watts · m-2 · ster-1 · Hz-1)
h = Planck’s constant (6.62606896*10-34 J/s)
c = velocity of light (2.99792458*108 m/s)
 = frequency (Hz)
k = Boltzmann constant (1.3806504*10-23 J/K)
T = temperature (e.g. 5778 K)
Lect 06
© 2012 Raymond P. Jefferis III
21
Spectral Energy Density ( ,T)
Lect 06
© 2012 Raymond P. Jefferis III
22
Spectral Energy Density Calculation
h = 6.62606896*10^-34;
k = 1.3806504*10^-23;
T = 5778;
c = 2.99792458*10^8;
numin = 0.1*10^9;
numax = 1000*10^9;
Ilam = (2*h*nu^3/c^2)*(1/(Exp[(h*nu)/(k*T)] - 1));
LogLogPlot[Ilam, {nu, numin, numax},
PlotStyle -> {Black, Thick},
Frame -> True,
FrameLabel -> {"Frequency [GHz])",
"Spectral Energy Density"},
LabelStyle -> Directive[Bold, Italic]]
Lect 06
© 2012 Raymond P. Jefferis III
23
Planck’s Radiation Law ( ,T)
2 *10 24 hc2 
1

I(,T ) 
 (106 hc /  kT ) 
5

e
1
where,
I(n,T) = Power Density (Watts · m-2 · ster-1 ·  m-1)
h = Planck’s constant (6.62606896*10-34 J/s)
c = velocity of light (2.99792458*108 m/s)
 = wavelength ( m)
k = Boltzmann constant (1.3806504*10-23 J/K)
T = temperature (e.g. 5778 K)
Lect 06
© 2012 Raymond P. Jefferis III
24
Spectral Energy Density ( ,T)
Lect 06
© 2012 Raymond P. Jefferis III
25
Solar Noise Power Density
2 h 3r 2 
1

N planck ( ,T ) 
 (h / kT ) 
2 2
c R
e
1
where,
N = Noise power density (Watts · m-2 · Hz-1)
h = Planck’s constant (6.62606896*10-34 J/s)
c = velocity of light (2.99792458*108 m/s)
r = radius of Sun (6.955*108 m)
 = frequency (Hz)
k = Boltzmann constant (1.3806504*10-23 J/K)
T = temperature (Visible: 5778 K, Microwave: 27, 000)
R = distance of receiver from Sun (1.49597870691*1011 m)
Lect 06
© 2012 Raymond P. Jefferis III
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Solar Noise Spectral Energy Density
Lect 06
© 2012 Raymond P. Jefferis III
27
Solar Noise Spectral Energy Density
h = 6.62606896*10^-34;
r = 6.955*10^8;
k = 1.3806504*10^-23;
R = 149597870691;
T = 27000;
c = 299792458;
nu = c/(10^(lam/10));
NN = (2*π*h*nu^3*r^2)/(c^2*(Exp[(h*nu)/(k*T)] 1)*R^2);
LogLogPlot[NN, {lam, 0.00000001, 1},
PlotStyle -> {Black, Thick},
Frame -> True,
FrameLabel -> {"Wavelength [m])", "Spectral
Energy Density"},
LabelStyle -> Directive[Medium, Italic]]
Lect 06
© 2012 Raymond P. Jefferis III
28
Received Noise Power Formula
Pn = NPlanck( ) · Ar
NPlanck = Noise Power Density
integrated over bandwidth 36 MHz
= 6.26742*10-13 [Watts/m2]
Ar = Area of receiving antenna = 0.049 [m2]
(Diam = 10 at 12 GHz)
B = Receiving input bandwidth [36 MHz]
Pn = 3.07226*10-14 [Watts] = -135.125 [dBW]
Lect 06
© 2012 Raymond P. Jefferis III
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Received Noise Power Calculation
h = 6.62606896*10^-34;
r = 6.955*10^8;
k = 1.3806504*10^-23;
R = 149597870691;
T = 5778;
c = 2.99792458*10^8;
B = 36.0*10^6;
numin = 12.0*10^9;
numax = numin + B;
lam0 = c/numin;
ra = 10*lam0/2;
Ar = *ra^2
NP = (2**h*nu^3*r^2)/(c^2*(Exp[(h*nu)/(k*T)] - 1)*R^2);
NPD = NIntegrate[NP, {nu, numin, numax}]
pn = NPD*Ar
Lect 06
© 2012 Raymond P. Jefferis III
30
Noise Model
• The Shannon - Weaver noise model treats noise as
an additive effect on an otherwise noise-free
communications channel for the purpose of
calculating its effects
Lect 06
© 2012 Raymond P. Jefferis III
31
Noise Factors
• The thermal noise calculated at the receiving antenna
output is:
N0 = kTa [W/Hz]
• Input noise arises from a number of sources:
–
–
–
–
–
Blackbody temperature of space
Blackbody temperature of Sun
Atmospheric noise
Antenna blackbody noise
Receiver system noise calculated at the input
• These contributions can each be converted to equivalent
noise temperatures
Lect 06
© 2012 Raymond P. Jefferis III
32
Noise Power
A black body at a temperature of T [Kelvins]
generates electrical noise according to the relation,
Pn  kTBn
where,
k = Boltzmann constant, 1.3806503*10-23 [J/K]
or -228.6 [dBW/K/Hz]
T = source temperature [Kelvins]
Bn = noise bandwidth [Hz]
Lect 06
© 2012 Raymond P. Jefferis III
33
Boltzmann - Conversion to dBW
k = 1.3806503*10^-23;
kdBW = 10*Log[10, k];
Print["k [dBW] = ", kdBW]
k [dBW] = -228.599
Lect 06
© 2012 Raymond P. Jefferis III
34
Noise Power Conversion to dBm
• Noise power is frequently stated in dBm, or
dB compared to 1 milliwatt.
• The dBm conversion for noise power is:
N dBm
Lect 06
 kTB 
 10 log 
 0.001 
© 2012 Raymond P. Jefferis III
35
Signal Power in Digital Transmission
• Carrier power is the average energy per bit,
in a digital transmission
• Frequently stated in dBm
• The conversion is:
CdBm
Lect 06
 CWatts 
 10 log 
 0.001 
© 2012 Raymond P. Jefferis III
36
Carrier-to-Noise Ratio [dB]
• The ratio of Carrier power to Noise power is a
measure of communication system performance
• Expressed as dB,
(C/N)dB = 10 log10[C/N]
where,
N = kTsBn
k = Boltzmann constant (1.3806503E-23 J/K)
Ts = System noise temperature [Kelvins]
Bn = Noise bandwidth of system [Hz]
Lect 06
© 2012 Raymond P. Jefferis III
37
Carrier-to-Noise Power Ratio
• Relates average carrier energy per bit, in
digital transmission, to noise power density
• In dB (or dBm) units,
 C
C 
   10 log    CdB  N dB
N dB
N 
 C
 C / 0.001 
 10 log 
 CdBm  N dBm
 

N dBm
 N / 0.001 
Lect 06
© 2012 Raymond P. Jefferis III
38
C/N Ratio and Noise Temperature
C Pr

N Pn
where, at the input,
Pn  kTs Bn
in which (to be discussed later in more detail)
Ts  Tin  TRF  TMix (1 / GRF )  TIF (1 / GRF GMix )
Lect 06
Where,
C = Carrier power [W]
N = Noise power [W]
Pr = Received signal power
Pn = Received noise power
Ts = Equiv. input temp. [K]
Bn = Bandwidth of noise [Hz]
Tx = Equiv. temperature at x
Gx = Gain of stage x
k = Boltzmann constant
(1.3806404E-23 J/K)
or, -228.6 [dBW/HzK]
© 2012 Raymond P. Jefferis III
39
Thermal Noise Power Model
The noise power Pn [Watts] delivered to the
matched external resistor, R, is:
2
v 
Pn   RMS  R  kTB
 2R 
Lect 06
[Watts]
© 2012 Raymond P. Jefferis III
40
Energy per Bit
Where,
Eb = energy per bit [Joules/bit]
fb = bit rate [bits/second]
Tb = time of bit [seconds]
C = Carrier power [Watts]
the energy per bit is:
Eb  C / fb  CTb
Lect 06
© 2012 Raymond P. Jefferis III
41
Noise Power Density
Where,
N0 = noise power density [Watts/Hz]
N = thermal noise power [Watts]
B = Bandwidth [Hz]
the noise power density is:
N0 
Lect 06
N kTB

 kT
B
B
© 2012 Raymond P. Jefferis III
42
A Figure of Merit
Eb C / fb  C   B 

  
N 0 N / B  N   fb 
 Eb 
 N 
0
dB
 B
C
 10 log    10 log  
 N
 fb 
where,
Eb/N0 = bit energy/noise power density ratio
C/N= carrier/noise power ratio
B/fb = noise bandwidth/bit rate ratio
Lect 06
© 2012 Raymond P. Jefferis III
43
Example: Earth Station Input
•
•
•
•
•
•
•
•
C = 20 [Watts]
B = 36*106 [Hz]
T = 200 [K]
k = 1.3806404*10-23 [J/K]
Pn = (1.3806404*10-23)(200)(36*106) = 9.94*10-14 [Watts]
CdBm = 10log[20/0.001] = 43.0103 [dBm]
NdBm = 10log[9.94*10-14/0.001] = -100.026 [dBm]
(C/N) dBm = CdBm – NdBm = 143.036 [dBm]
Lect 06
© 2012 Raymond P. Jefferis III
44
Example: Earth Station Input
•
•
•
•
•
q = 1.602176E-19 [Coul]
I = 0.1E-3 [A] = 100 [ A]
B = 36E6 [Hz]
R = 50 [Ohms]
P = (2)(1.602176E-19 )(0.1E-3 )(36E6)(50) = 5.77*
10-14 [Watts]
Lect 06
© 2012 Raymond P. Jefferis III
45
Signal-to-Noise Ratio
• Ratio of signal power to noise power
SNR = Ps/Pn
• The dB form is frequently used
SNRdB = 10 log10(Ps/Pn)
• Is used as a performance measure
Lect 06
© 2012 Raymond P. Jefferis III
46
Noise Figure
• Measures what the system noise contributes to the
input
• Ratio of output noise to POWER gain-multiplied
by input noise
NF = Pno/G*Pni
• Note:
NF = (Ps/SNRo)/(Ps/SNRi) = SNRi/SNRo
• Frequently expressed in dB
Lect 06
© 2012 Raymond P. Jefferis III
47
Noise Computations
• Noise Temperature (T) =
290 * (10^(Noise Figure/10)-1) [K]
• Noise Figure (NF) =
10 * log10 (Noise Factor) [dB]
Lect 06
© 2012 Raymond P. Jefferis III
48
Noise Conversion Table
NF(dB)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
T (K)
7
14
21
28
35
43
51
59
67
75
NF(dB)
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
T (K)
84
92
101
110
120
129
139
149
159
170
NF(dB)
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
T (K)
180
191
202
214
226
238
250
263
275
289
NF(dB)
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
T (K)
302
316
330
344
359
374
390
406
422
438
www.satsig.net/noise.htm
Lect 06
© 2012 Raymond P. Jefferis III
49
Another Figure of Merit
• SPNN = (Ps + Pn) / Pn = 1+ SNR
• Channel Capacity (Shannon) as
calculated using SPNN
C = B log2(SPNN) [bits/sec]
Lect 06
© 2012 Raymond P. Jefferis III
50
Eb/N0 Ratio - Revisited
• Eb/N0 = (Signal energy per bit)/(Noise power
density per Hertz)
Eb PS / R
PS


N0
N0
kTR
Lect 06
where,
PS = Signal power [Watts = J/s]
R = Data rate [bits/sec]
b = Time to send one bit = 1/R
[sec]
Eb = Psb = Energy per bit [J]
T = Temperature [K]
k = Boltzmann constant
(1.3806404E-23 J/K)
© 2012 Raymond P. Jefferis III
51
Summary
 PS   B   S   B 
Eb PS / R
PS


     
N0
N0
kTR  PN   R   N   R 
N  N0 B
Lect 06
© 2012 Raymond P. Jefferis III
52
References
• Stallings, W., Data and Computer
Communications,
Prentice-Hall, 2004.
• Tomasi, W., Advanced Electronic Communications
Systems,
Prentice-Hall, 2001.
Lect 06
© 2012 Raymond P. Jefferis III
53
Component Noise Model
Pn
kB
Po  Pi  Pn Gn
Tn 
Gn  Pi Pn 
To  Pi  Pn   
  Gn  Ti  Tn Gn

kB
kB kB 
Lect 06
© 2012 Raymond P. Jefferis III
54
Meaning of Noise Model
• Noise temperatures can be treated additively
• The input noise plus the input-referred
amplifier noise multiplied by the amplifier
gain tields the effective noise temperature.
Lect 06
© 2012 Raymond P. Jefferis III
55
Noise Factor (Noise Figure)
• Another figure of merit for system components
• Is defined at room temperature (290 K)
• Noise balance
Output = G*(Input + Device)
FGkT0 = Gk(T0+Td)
The noise temperature of a device is:
Td = (NF-1)T0
The noise figure of a device is
NF = 1+ Td / T0
Lect 06
© 2012 Raymond P. Jefferis III
56
Noise Figure [dB]
• The noise figure of a device in dB is,
NF = 10 log10[1+ Td / T0]
(See graph on next slide)
• T0 is typically assumed to be 290 K.
Lect 06
© 2012 Raymond P. Jefferis III
57
NF(Td) – Low Temp. Range
Lect 06
© 2012 Raymond P. Jefferis III
58
Noise Factor Calculation
T0 = 270;
NF = 10 Log[10, 1 + Td/T0];
Plot[NF, {Td, 0, 400},
AxesLabel -> {"Td ( K )", " NF (dB) "},
PlotStyle -> {Black, Thick}]
Lect 06
© 2012 Raymond P. Jefferis III
59
NFdB - Upper Temp. Range
Lect 06
© 2012 Raymond P. Jefferis III
60
Calculation of Noise Factor in dB
T0 = 270;
NF = 10 Log[10, 1 + Td/T0];
Plot[NF, {Td, 0, 10000},
AxesLabel -> {"Td ( K )", " NF (dB) "},
PlotStyle -> {Black, Thick}]
Lect 06
© 2012 Raymond P. Jefferis III
61
Cascaded System Components
Po2  G2 kTn2 B  G1G2 kB(Tn1  Ti1 )
Lect 06
© 2012 Raymond P. Jefferis III
62
Reference to Input Temperature
Let an input noise temperature, TS be defined.
Then,
Po2  G1G2 kTS B
And thus,
Tn2
TS  (Tn1  Ti1 ) 
G1
Note that the first amplifier gain reduces
the noise temperature of the subsequent stage.
Lect 06
© 2012 Raymond P. Jefferis III
63
Noise Temperature Cascade Model
Pno3  GIF kTIF B  GIFGm kTm B  GIFGmGRF kB(TRF  Tr )
Lect 06
© 2012 Raymond P. Jefferis III
64
Noise Temperature Model
Referring all noise to input,
TSource
Lect 06

Tm
TIF 
 Tr  TRF 


G
G
G

IF
m RF 
© 2012 Raymond P. Jefferis III
65
Carrier-to-Noise Ratio, C/N
• Similar to SNR, but more useful for FM
transmission
C 
 N   [Pr ]dB  [Pn ]dB
dB
• Or, substituting the path loss results:
C 
 N   EIRP dB  Gr dB  LOSSi dB  k dB  BdB  TS dB
i
dB
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Typical Antenna Noise Temperatures
3.6m diameter antenna Model 8136 from ViaSat, C + Ku bands (Offset geometry)
Elevation angle (deg)
10
20
30
20
Noise temp (C band) (K)
24
16
15
14
Noise temp (Ku band) (K)
31
23
21
20
From www.satsig.net/antnoise.htm
4.7m diameter antenna Model Vertex, C + Ku bands
Elevation angle (deg)
5
10
20
40
Lect 06
Noise temp (C band) (K)
56
40
45
42
© 2012 Raymond P. Jefferis III
Noise temp (Ku band) (K)
69
62
57
52
67
Receiver Noise Figures
• Noise figures of 0.7 - 2.3 dB and gains of 22 - 27
dB can be achieved in Ku-band amplifiers.
• Noise Temperatures:
– C-Band:
– Ku_Band:
30 - 45 [K]
75 - 85 [K]
• Si/Ge and Ga/As technologies are typically used
• Cooling (thermoelectric, LN2, etc.) can reduce
noise temperatures
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Amplifier Example: NEC NE32584
Noise Figure:
NF = 0.45 dB Typ.,
Gain = 12.5 dB Typ. at f = 12 GHz
Application:
C through Ku Band
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Typical Component: NEC NE325501 Transistor
NF: 0.45 dB at 12 GHz
Gain: 12.5 dB at 12 GHz
From NE325501 Data sheet, NEC
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End
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