Noise in Cascaded Amplifiers

1 2 313 F2,G2 F1+2,GF1,G1 ≡

1213 33 11 Swhere

NS NS

F =∆+

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ ==

22 11

12 NS NS

N

( ) o2223 FGN −+=

( )( )( )o221133 FGNS −+=∴

( ) ⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ − +=

1 2

1o1 G 1F

FkTS

1

2 1

33

11 G

1FF

NS NS

F −

+== +∴

L1

1+2

2 1 S G G

1 1 o F Recall G F kT

1 1 o kT 1 G F kT G

1 2 1 2 kT 1 F G G S G G

amplified from “2” excess from “2”

2 1

Noise in multiple cascade amplifiers

In general, 1F

G 1F

FF 21

3

1

2 1 … … +

− +

− +=

A B B Avs.

Note:

The better choice is not obvious. F1,2

and gain of the first amplifier, not just on F1

21

3 1FFF

− += +By extension:

1

2 1 G

1FFF

− += +

L2

formula noise cascade G G ,2,1

also depends on interstage impedance mismatches

2 1 3,2,1 G G 2 1

Noise in superheterodyne receivers

×

0 f f0

S1,N1 S2,N2

L.O. Gc,Fc Gi.f.,Fi.f.

“i.f.” “intermediate frequency”∆

( ) ( )S

iS G 1L

1

2

c c =∆∆

( )r 2 ot N i∆ =

3) ( ) orc1

o1

22

11 mixer LS

kTS NS NS

F ==∆

L3

fo fi.f.

0 S2 S1

“upper sideband”

“lower (rf) sideband”

i.f. passband fL.O.

f

1) “Conversion loss of mixer” SSB ., f.r

. f.

2) “Noise temperature ratio of mixer” .f. kT

r c t L kT t

Noise in superheterodyne receivers

3) ( ) orc1

o1

22

11 mixer LS

kTS NS NS

F ==∆

mixer

i mixerimixer G

1FFF

− += +

( ) ( )1tFL1FL r.f.ic.f.icrc −+=−+=

4)

( )2imixer −−≅+

L4

r c t L kT t

. f.amp. . f.

t L

dB 9 3 ~ 8 F e.g. amp. . f.

Basic receiver types-Amplification

vo(t)( )2 f

B

OR

OR

×

N1

h(t)1) Simple detector

2) RF Amplifier

3) Superheterodyne

Basic receiver types-Combinors

1) vo(t) Total power radiometer

2) vo(t)

3) × vo(t)

Multiplication (correlation) receiver

4) N M N-way combiner, N M<>

L I N E A R

N2

Dicke radiometer

Passive Multiport Networks

beamsplitter (4-port)

1)

RCVR+

adding interferometer

2)

3)

white light

(N-port)

diffraction grating

N detectors

N3

4) × 3-dB hybrid

L.O. Zo

λ/4

cancel add in phase

input

output

output

Passive Multiport Networks

5) “Magic tee”

2

1

3 4

Port 1 orthogonal to Port 2 Port 3 orthogonal to Port 4 All 4 ports can be matched

N4

LSB, (lower sideband)

USB, (upper sideband)

I.F.

noise

6) Frequency converter

LSB

f

I.F.

Nonlinear

USB

Linear Passive N-port Networks

Define exchangeable power = ( )⎪⎭⎪⎬⎫

⎪⎩

⎪⎨⎧

2 i

2 i

ib

ia

iia

ib j

Net power entering port “i” = 2 i

2 i ba −

Scattering matrix equation: b =

is defined only when the n-port is imbedded in a network

i and bi can be defined as some linear combinations of those for voltage and current

S Note:

( ) ( ) ooioooi VV2YV −=+== + P1

1­ WHz or W port from

port toward

aS

1) The scattering matrix

2) The phases of a

Z 8 I Z b , Z 8 I Z a .g.e

Gain definition for N-port networks “Transducer gain”

AjDk 2

kj 2

j 2

kTkj PPSabG ==∆

Therefore available power from port K =

But FPA = fractional power emitted, if reciprocity applies.

i.e. fractional power absorbed (FPA) = 2 kk2

k

2 k

2 k S1

a

ba −=

−

( )2 kk

2 k S1b −

Therefore [ ] ( ) =−= 2

j 2

kk 2

kE aS1bG kj kjE2

kk

2 kj G S1

S =

−

“Exchangeable gain” (~”available gain”)

Note: available power out due to possible port-k mismatch

2 jE

E E

a

? P

P G

j

k kj

=∆ 2 kb≥

P2

Constraints on N-port networks

Lossless passive networks ∑∑ ==

=⇒ N

1i

2 i

N

1i

2 i ba

Reciprocity tSS =⇒

P3

Example of constrained N-port networks

bbaa tt ∗∗ =•⇒Losslessness

( ) ( ) SaaSbb tttt ∗∗∗∗ =••=•

Therefore ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ ==

∗

100 010 001

ISSt ; if the combiner is lossless

Does an ideal power combiner exist?

? 2 3

1kT1

kT2 symmetry

k(T1 +T2) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

βαα

αα

= 00 00

SWe want

P4

aS aS

Example of constrained N-port networks

⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢

⎣

⎡

βαα

α

α

= 00 00

S

Therefore ⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢

⎣

⎡ ==

∗

100 010 001

ISSt ; if the container is lossless

Test:

⎥ ⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢ ⎢

⎣

⎡

α

αα

αα

= ∗∗

∗

∗ ∗

22

22

22

t

2 SS

1If 222 ≠α=α

Therefore constraints (8-14) and (8-15) cannot be satisfied simultaneously for this system

P5

β + β α β α

β α

β α

2 then ,1 β +

Ideal matched 2-input-port combiners are impossible

Another N-port network example Can we match all 3 ports simultaneously?

2

3

1

(Note: is defined only when imbedded in network)S

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γβ

γα

βα

=

0 0

0 SFor 3 matched ports:

SSt =

∗ Does (lossless passive constraint)

P6

? I

⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢

⎣

⎡

γβ

γα

βα

=

0 0

0 SFor 3 matched ports:

SSt =

∗ Does (lossless passive constraint)

Matched 3-port example

ISS ?

22

22

22

t =

⎥ ⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢ ⎢

⎣

⎡

β

α

α

= ∗∗

∗∗

∗∗ ∗

Therefore not possible to match all 3 ports simultaneously P7

0∗∗∗ If , then 2 of (α ,β ,γ ) and at least one diagonal element of 0SSt

= ∗

? I

γ + β α γ α

β α γ + γ β

γ α γ β β +

= β α = γ α = γ β

Lossless passive reciprocal symmetric 4-port network

31

42

symmetry axis, all ports matched

Ports 1, 2 are isolated; also 3, 4

We can show ∆φ for 31 41versus paths is unique

using losslessness, reciprocity, and symmetry

P8

R1

vTh+ -

+ = Th) (t) V

signal

local oscillator

i(t) Rs

+

-RL

vo(t)

0 VTh

diode:

diode i,vTh(t)

( )= +o L Si V R R i(t)

2 dvi ≅

load line i = −

+ Th d

L S

V v R R

i

vd

tsinvtsinv)t(v ssod ω+ω≅

termsorder-+

( )= ∝ 2 o L L) iR v R

= + + + +2 3 o 2 3 Thk

dominant

Mixer

S(t

>> Th

diode for

high small

Th v (t

1 Th Th k v k v k v ...

R2

( )= = + + + + ∝ 2 3 2

o L o 1 Th 2 3 Th d Lv (t) iR k k v k v

dominant Spectral content of detector output:

ofsfiff −=

)f(oV image

of sf o ssfof +

o sf0

Pure product (frequency component for pure s(t) • (t))

i4 dv ⇒(Note: , can be in i.f. passband)

1

2 3

4 output

internal noise

signal

image

Normally: a 4-port model suffices to find F,T RSSB

Mixer Th k v ... v R

f 2 f 2 f 3 f 3

. f.f 2

: " T " ,

1

2 3

4 output

internal noise

signal

image

R3

( )1TT)1F(TT rcooSSBR −=−==

for single sideband (SSB) operation

⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡

−

++

⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡

−

==

2 44

2 433

2 422

2 41o

2 44

2 411

o1

44 11

SBB

S1S1

kTS NS NSF

[Note: “S”, signifies signal in port 1, Skj is a scattering matrix element]

o

SSB 2

41

2 43

o

3 2

41

2 42

o 2

SSB T T1

S

S T T

S

S T T1F +=++=Simplifying,

2 41

2 43

32 41

2 42

2SSB S

ST

S

STT +=Therefore

Mixer Noise Figure Using 4-port Model

t L

S kT S kT S kT S S

2 41

2 43

32 41

2 42

2SSB S

ST

S

STT +=Therefore

Both ports 1 and 2 are signal, so

( ) ( )2 44

2 42

2 41o4 S1SSkTS −+=

1

2 3

4 output

internal noise

signal

image

Often suggested: DSBSSBF ≅

R4

It follows that [ ]SST 2

42 2

41 2

433DSB +=

Double-sideband Receiver

F2

S T

DSB2 41

2 43

o

3

o

SSBSSB

S

S T T2

T T1F =+=+=

then;TTS 21o 2

42 2

41 ===

2 41

2 43

o

3

o

DSBDSB

S

S T T

2 11

T T1F +=+=

Often suggested: DSBSSBF ≅

2 41

2 43

32 41

2 42

2SSB S

ST

S

STT +=

1

2 3

4 output

internal noise

signal

image Therefore

R5

Double-sideband Receiver

F2

T and S Let

F2