Noise Figure
Signal - to - noise ratio : The ratio of desired signal power to undesired
noise power .
Noise Figure : a measure of the signal - to - noise ratio between input
and output of the component .
Noise Figure F
Si
F
So
Ni
1
No
Si : input signal power
N i : input noise power
S o : output signal power
N 0 : output noise power
N i  kTO B where TO =290K
Consider the following noise network
R
Pi=Si+Ni
To=290K
Noisy
network
G , B , Te
Po=So+No
R
G : gain the network
B: bandwidth
Te
: equivalent noise temperature
Si
F
Ni
So
No

Si
kGB(TO  Te )
T

 1 e  1
kTO B
GSi
TO
Noise Figure is defined for a matched input source and for a noise
source that consists of a resistor at temperature TO =290K .
Te  (T1  1)TO
Noise figure and noise temperature are interchangable .
Consider the following loss line or attenuator with loss L and
temperature T . Since the entire system is in thermal equivalent N i  N o
R
KtB
L , T , Zo=R
Po=Ktb=No
Po  GkTB  GN addid  kTB
N addid 
1 G
kTB  ( L  1)kTB
G
N addid is the noise generated by the line and L=1/G
Te 
N addis 1  G

T  ( L  1)T
kB
G
F  1
Te
T
 1  ( L  1)
To
To
If the line is at temperature To , F=L . This states that a 10 dB
attenuator at room temperature has a noise figure of 10 dB .
Noise Figure of a cascaded system
Consider the cascade of two components as shown below
Ni
To
G1
F1
Te1
N1
G2
F2
Te2
No
G1 ;G2 : gains
F1 ;F2 : noise figures
Te1 ; Te 2 : equivalent noise temperature
N1  G1kTo B  G1kTe1 B
The noise power at the output of the second stage is
N 0  G2 N1  G2 kTe2 B
 G2G1kTo B  G2G1kTe1 B  G2 kTe 2 B
 G2 G1kB(To  Te1 
1
Te 2 )
G1
We define the equivalent network as
Ni
To
G1 G2
F cascaded
Te,cascaded
No
N o  G1G2 kTo B  G1G2 kBTe,cascaded  G1G2 kB(To  Te,cascaded )
Therefore , we obtain
Te,cascaded  Te1 
F  1
Te
T
1 Te 2
 1  e1 
To
To G1 To
 1  ( F1  1) 
 F1 
1
Te 2
G1
1
( F2  1)
G1
1
( F2  1)
G1
For an arbitrary number of stages , we obtain
Tcascaded  Te1 
Te 2
T
 e 3  ...
G1 G1G2
Fcascaded  F1 
F2  1 F3  1

 ...
G1
G1G2
Example :
Consider the following wireless local area network (WLAN) receiver ,
where the bandwidth of the bandpass filter is 100MHz centered at
2.4GHz . If the system is t room temperature .
IL = 1.5 dB
G = 10 dB
F = 2 dB
G = 20 dB
F = 2 dB
(a) Find the noise figure of the overall system .
(b) What is the resulting signal - to - noise ratio at the output , if the
input power level is -90dBm ?
(c) Can the components be rearranged to give a better noise figure ?
Solution :
The noise figure of the cascade is
Fcas  F1 
F2  1 F3  1

 1.41  (1.58  1)(1.41)  (1.41) / 10
G1
G1G2
 2.31  3.64dB
If Pin  90dBm , than we get
Piout  90dBm  1.5dB  10dB  20dB  61.5dBm
The noise power output is
Pn  Gcas kTe,cas B  k ( Fcas  1)To BG cas
 (1.38 10 23 )( 2.31  1)( 290)(108 )(10
 64.3dBm
Thus
SO
NO
 61.5  64.3  2.8dB
2.85
10
)  3.71 10 10W
The best noise figure would be achieved with the arrangement shown
below
G = 20 dB
F = 2 dB
G = 10 dB
F = 2 dB
IL = 1.5 dB
BW = 100 MHz
Then the noise figure is
Fcas  1.58 
(1.58  1) (1.41  1)

 1.586  2.0dB
100
1000
In practice , however , the essential filter may serve to present overload
of the amplifier and may not be allowed to be moved .
Low Noise Amplifier
The noise figure of a two - port amplifier can be expressed as
F  Fmin 
2
RN
YS  Yopt
GS
where
YS  GS  jBS : source admittance presented to transistor
Yopt : optimum source admittance that results in minimum noise figure .
Fmin : minimum noise figure of transistor , attained when YS  Yopt
RN : equivalent noise resistance of transistor
Also we have
YS 
1 1  S
Z o 1  S
Yo p t
1 1  o
Z o 1  o
p t
p t
The quantities Fmin , opt and RN are the characteristics of the
particular transistor being used and are called the noise parameters of the
device .
2
YS  Yopt 
S  opt
2
4
Z O2 1   2 1  
S
opt
2
1 1  S 1  S*
1 1  S
G

Re{
Y
}

(

)

and S
S
2Z O 1  S 1  S*
Z O 1  S
Therefore , we obtain
2
F  Fmin
S  opt
4 RN

Z O (1   2 )(1   2 )
S
opt
2
2
Constant Noise Figure Circles
For a given noise figure Fi , we define a noise figure parameter , called
N i , as
Ni 
S  O
1  S
2
2

Fi  Fmin
1  O
4rn
2
This equation can be written as ( S  O )( *  * )  N i  N i S
S
2
O
*
or S (1  N i )  O  2 Re( S  )  N i
2
2
O
If we now multiply both sides by 1  N i . we obtain
S (1  N i ) 2  O  2(1  N i ) Re( S O* )  N i2  N i (1  O )
2
O
or S  1  N
i
2
2
2
N i2  N i (1  O )
2

(1  N i ) 2
This is a family of circles with N i as a parameter . The circles are

O
centered at C Fi  1  N
i
with radii RFi 
1
1 Ni
N i2  N i (1  O )
2
When Fi  Fmin , then N i =0 , C F min  O , and RF min  0 . The centers
of other noise figure circles are located along the O vector .
Example : Noise Figure Circles
A certain GaAs MESFET has the following noise - figure parameters
measured at Vds  5V , I ds  20mA , with a 50-  resistance for a
frequency of 9 GHz.
Fmin  2dB
o  0.4851550
Rn  4
Plot the noise - figure circles for given values of
Fi
at 2.5 , 3.0 , 3.5 ,
4.0 , and 5.0dB.
Solution :
1. From values of N i , cFi and rFi for
Fi at 2.5dB are computed as
follows :
Ni 
2
1.78  1.59
1  0.4851550  0.21
4(4 / 50)
0.4851550
c Fi 
 0.401550
1  0.21
1
1
2
[( 0.21) 2  0.21(1  0.485 )] 2
1  0.21
 0.37
rFi 
2. Similarly , the values of N i ,
cFi and rFi for Fi at 5dB are also
computed .
3. All values are tabulated in Table .
Table : VALUES OF NOISE - FIGURE CIRCLES
Fi (dB)
2.5
3
3.5
4
5
fi
1.78
2
2.24
2.5
3.16
Ni
0.21
0.45
0.71
1
1.72
cFi
0.40 1550 0.33 1550 0.28 1550 0.24 1550 0.18 1550
rFi
0.37
0.51
0.55
0.66
0.76
4. The noise - figure circles are plotted in the Figure .
Example :
A AaAs is biased for minimum noise figure and has the following S
0
parameters at 4GHz ( Z 0  50 ) , S11  0.6  60 , S21  0.621000 ,
RN  20 . Since S12 is relatively small , we assume the device is
unilateral . Then design an amplifier having 2.0dB noise figure with
the maximum gain that is compatible with this noise figure .
Solution :
We first compute the center and radius of the 2.0dB noise figure
circle :
Ni 
c Fi 
2
2
Fi  Fmin
1.58  1.445
1  opt 
1  0.62100 0  0.0986
4 RN / Z o
4( 20 / 50)
opt
Ni 1
 0.561000
2
RFi 
N i ( N i  1  opt )
Ni 1
 0.24
Next we calculate data for several input section constant gain circles .
GS (dB)
gs
Cs
Rs
1.0
0.805
0.52 600
0.300
1.5
0.904
0.56 60 0
0.205
1.7
0.946
0.58 600
0.15
(a)
(b)
We see that the GS  1.7dB gain circle just intersects the FC  2dB noise
figure circle and that any higher gain will result in a worse noise figure .
0
From the Smith Chart , the optimum solution is then S  0.5375
which yields GS  1.7dB and FC  2dB .
*
0
For the output section , we choose L  S 22  0.560 for a maximum
G L of
GL 
1
1  S 22
2
 1.33  1.25dB
The transistor gain is
Go  S 21  3.61  5.58dB
2
The overall transducer gain is
GTU  GS  GO  GL  1.7  5.58  1.25  8.53dB
A complete AC circuit for the amplifier , using open - circuited shunt
stubs in the matching sections , is shown in the figure .
Example :
The scattering and noise parameters of a GaAs FET measured at three
different optimum bias settings at f=6GHz are :
Minimum Noise Figure ( VDS  3.5V , I DS  15% I DSS ) :
S11  0.674  152 0
Fmin  2.2dB
S12  0.0756.20
O  0.575 1380
S 21  1.7436.4 0
RN  6.64
S 22  0.6  92.60
Linear Power Output( VDS  4V , I DS  50% I DSS )
S11  0.641  171.30
S12  0.05716.30
S 21  2.05828.50
Fmin  2.9dB
O  0.542 1410
RN  9.42
S 22  0.572  95.70
Maximum Gain ( VDS  4V , I DS  100% I DSS )
S11  0.614  167.4 0
S12  0.046650
S 21  2.18732.4 0
S 22  0.716  830
Design a microwave transistor amplifier to have good ac performance .
Solution : There are four ac performances that must be considered :
noise figure , power gain , power output , and input and output VSWR .
The linear power- output bias point ( VDS  4V , I DS  50% I DSS ) provides a
good compromise between the minimum noise figure and maximum
gain . At this bias point , the Table gives the noise , gain , and power
parameters . The output power performance , measured ant the 1-dB
compression point , was experimentally measured and it is given in the
figure .
The data for the output power were taken with an input power drive of
8.3dBm
Noise Parameters
Gain Parameters
Power Parameters
O  0.542 1410
Ms  0.762177.30
Ps  0.7291660
L  0.575104.50
ML  0.718103.90
PL  0.4891010
F  4.44dB
F  3.69dB
GA  9.33dB
G A,max  11.38dB
GP  8.2dB
P1dB  9.3dBm
P1dB  13.4dBm
P1dB  15.5dBm
Fmin  2.9dB
The input VSWR with S  Ms is 1 , and the VSWR =3.82 with
S  O . In order to calculate the VSWR , we obtained a (in the next
1  a
VSWR

page) and used
1  a
Trade - offs between noise figure , power gain , and VSWR
Last Figure shows the noise figure , G A and input and output VSWR as
the reflection coefficient is varied from O to Ms , along a straight
line , in the Smith Chart . The table shows that a good compromise
0
between noise figure , G A , and VSWR is to use S  0.614160 and
L  0.6271060 . The noise figure is increased by 0.24dB from the
minimum noise , but G A is increased by 1.22dB and the input VSWR
is improved by 40% (i.e. , VSWR =2.28) . The ac schematic of the
amplifier for the selected values of S and L is shown in next
Figure and the microstrip board layout is also shown . The board
material is Duroik (  r =2.23 , h=0.031 in.) . The measured
characteristics of the amplifier are shown in next page .
Figure : (a) The ac schematic of the amplifier with  ff =1 ; (b)
microstrip layout with two different dc bias networks.
(c)
(d)
Figure : Measured characteristics of the amplifier : (a) gain performance ;
(b) noise performance ; ( c) input - output VSWR performance ; (d)
wideband gain performance .
Reference : “ A 6GHz amplifier using the HFET -1101 GaAs FET “ HP
Application Note 970 .
Balanced Amplifiers
Figure 4.4.5 Balanced amplifier configuration .
Why use balanced ?
In broadband amplifiers , the design of compensated matching networks
to obtain gain flatness results in impedance mismatching that can
significantly degrade the input and output VSWR . The balanced
configuration can be used to improve the I/O VSWR (Return Loss) .
0
The I/O couplers are 3dB hybrids (usually 90 hybrids) (i.e.
hybrids) .
S11 
1
S11a  S11b
2
S 22 
1
S 22a  S 22b
2
S12 
1
S12a  S12b
2
2
S12 : reverse power loss

4
S 21 
1
S 21a  S 21b
2
S 21
2
: forward power gain
Where a and b indicate the two amplifiers and 1 and 2 refer to the input
and output ports of the balanced amplifiers .
If the two amplifiers are identical , then S11 =0 and S 22 =0 and the gain
S 21 (and also S12 ) is equal to the gain of one side amplifier .
Ref . K.Kurokawa , “Design theory of balanced transistor amplifiers ,
“ pp . 1675-1698 . BSTJ , OCT . 1965 . BSTJ : Bell System Technical
Journal .