Noise in Cascaded Amplifiers F1 G1 F2 G2 1 2 F1 2 3 F1 2 G1 2 1 3 S1 N1 where S3 G1G2S1 S3 N3 S N N2 kToF1G1 Recall F1 1 1 S2 N2 N3 G2kToF1G1 G2 F2 1 kTo amplified from 2 excess from 2 S3 N3 G1G2S1 G1G2F1 G2 F2 1 kTo S1 kTo F 1 F1 2 G1 S1 N1 F2 1 F1 F1 2 S3 N3 G1 L1 Noise in multiple cascade amplifiers F3 1 By extension F1 2 3 F1 2 G1G2 F2 1 F1 2 F1 G1 In general F2 1 F3 1 cascade noise formula F1 2 F1 G1G2 G1 Note A B vs B A The better choice is not obvious F1 2 also depends on interstage impedance mismatches and gain of the first amplifier not just on F1 L2 Noise in superheterodyne receivers S1 N1 S2 N2 0 fo f Gc Fc 0 fi f L O lower rf sideband f Gi f Fi f i f intermediate frequency upper sideband S2 0 S1 i f passband f fL O S2 i f 1 1 Conversion loss of mixer Lc Gc S1 r f SSB 2 Noise temperature ratio of mixer t N i f kT r S1 kTo S1 N1 3 Fmixer Lc tr S2 N2 S1 Lc tr kTo 2 o L3 Noise in superheterodyne receivers S1 kTo S1 N1 3 Fmixer Lc tr S2 N2 S1 Lc tr kTo Fi f 1 4 Fmixer i f amp Fmixer Gmixer L c tr L c Fi f 1 L c Fi f t r 1 e g Fmixer i f amp 2 8 3 9 dB L4 Basic receiver types Amplification 1 Simple detector B 2 h t vo t f OR 2 RF Amplifier OR 3 Superheterodyne N1 Basic receiver types Combinors 1 2 vo t Total power radiometer vo t Dicke radiometer vo t Multiplication correlation receiver 3 4 N L I N E A R M N way combiner N M N2 Passive Multiport Networks 1 2 beamsplitter 4 port RCVR adding interferometer 3 N detectors 3 dB hybrid 4 Zo white light diffraction grating L O cancel add in phase output N port input 4 output N3 Passive Multiport Networks 5 Magic tee 6 Frequency converter USB upper sideband 2 I F 3 4 Nonlinear noise 1 LSB lower sideband LSB Port 1 orthogonal to Port 2 Port 3 orthogonal to Port 4 All 4 ports can be matched USB f I F N4 Linear Passive N port Networks a 2 toward port i Define exchangeable power i 2 1 bi from port i W or WHz ai i bi j 2 Net power entering port i ai bi 2 Scattering matrix equation b Sa Note 1 The scattering matrix S is defined only when the n port is imbedded in a network 2 The phases of ai and bi can be defined as some linear combinations of those for voltage and current e g ai V Yo 2 V Zo I 8Zo bi V Zo I 8Zo P1 Gain definition for N port networks Transducer gain GTkj bk 2 aj 2 Skj 2 PDk PAj Exchangeable gain available gain PE 2 GE k Note available power out bk 2 kj PE aj due to possible port k mismatch j i e fractional power absorbed FPA ak 2 bk ak 2 1 Skk 2 2 But FPA fractional power emitted if reciprocity applies Therefore available power from port K bk 1 S a Therefore GE bk kj 2 kk 2 2 j 2 1 S kk Skj 2 2 1 Skk 2 GE kj P2 Constraints on N port networks N Lossless passive networks ai i 1 Reciprocity S S 2 N bi 2 i 1 t P3 Example of constrained N port networks Does an ideal power combiner exist kT1 0 0 We want S 0 0 k T1 T2 1 2 symmetry kT2 3 t t Losslessness a a b b t S a a b b Sa Therefore t t t S Sa 1 0 0 S S I 0 1 0 if the combiner is lossless 0 0 1 t P4 Example of constrained N port networks 1 0 0 Therefore S S I 0 1 0 if the container is lossless 0 0 1 0 0 0 0 S 2 2 2 t 2 Test S S 2 2 2 t 2 2 2 If 1 then 2 1 Therefore constraints 8 14 and 8 15 cannot be satisfied simultaneously for this system Ideal matched 2 input port combiners are impossible P5 Another N port network example Can we match all 3 ports simultaneously 1 3 2 Note S is defined only when imbedded in network For 3 matched ports t Does S S I 0 S 0 0 lossless passive constraint P6 Matched 3 port example 0 S 0 0 For 3 matched ports t Does S S I lossless passive constraint 2 2 t S S 2 2 I 2 2 If 0 then 2 of and at least t one diagonal element of S S 0 Therefore not possible to match all 3 ports simultaneously P7 Lossless passive reciprocal symmetric 4 port network Ports 1 2 are isolated also 3 4 1 3 2 4 We can show for 1 3 versus symmetry axis all ports matched 1 4 paths is unique using losslessness reciprocity and symmetry P8 Mixer vThsignal S t t VTh vo t Rs i t RL local oscillator diode i io VTh i v 2 for diode d v d t v sin o t v s sin s t small high order terms 2 v o t iRL v Th RL 2 k o k1v Th k 2v Th k 3 v 3Th RL RS i t load line i RL RS vd 0 diode i vTh t VTh v d VTh dominant R1 Mixer 2 v R 3 2 d L v o t iRL k o k1v Th k 2 v Th k 3 v Th dominant Spectral content of detector output Vo f 0 fif fs fo image fo fs 2fo 2fs fo fs 3fo 3fs f Pure product frequency component for pure s t t Note v 4 2fi f can be in i f passband d Normally a 4 port model suffices to find F TSSB TR signal 1 4 image 2 3 output internal noise R2 Mixer Noise Figure Using 4 port Model TR TSSB F 1 To To Lc tr 1 signal 1 4 for single sideband SSB operation image 2 3 FSBB S1 N1 S 4 N4 S S 2 1 41 2 1 S 44 output internal noise S1 kTo kT S 2 kT S 2 kT S 2 2 42 3 43 o 41 2 1 S 44 Note S signifies signal in port 1 Skj is a scattering matrix element 2 2 TSSB T3 S 43 T2 S 42 Simplifying FSSB 1 1 2 2 To To S To S 41 41 Therefore TSSB T2 S 42 2 S 41 2 T3 S 43 2 S 41 2 R3 …