rf signal - polimi.it
TRANSCRIPT
RF Signal
0
1Bf<<
Analytic Formulation:
( ) ( ) ( )( ) ( ) ( ) ( ) ( )0 0 0cos cos sinRF M I Qs t V t t t V t t V t tω ω ω= +Φ = +
f0 f
B
VM(t), Φ(t): Magnitude and phase (instantaneous)VI(t), VQ(t): In phase and Quadrature signal components
Information associated to RF Signal is carried by these signals only (carrier does not convey any information)
Complex envelop representation
( ) ( ) ( ) ( ) ( ) ( ) ( )( )0 0 0cos sin cosRF I Q Ms t V t t V t t V t t tω ω ω= + = +Φ
Phasor notation:( ) ( ) ( ) ( ) 0 0Re Rej t j t j t
RF M Cs t V t e e t eω ωΦ= = V
Vc(t) is a complex function of time called complex envelope. In case of RF signals the spectrum of Vc(t) is around 0 (base band) and contains all the information carried by the modulated signal.The spectrum of Vc(t) is related to the spectrum SRF(f) of the real signal (sRF(t)) as follows:
( ) ( ) ( ) ( )0 *0 0Re j t
RF C C CS f t e f f f fω= = − + +V S S
Magnitude of real spectrum (bandpass)
0f
Magnitude of complex envelope spectrum (lowpass)
-f0 f0 f
Sc(f)
SRF(f)
RF Signal across a linear 2-port:no distortion condition
f0
Bf0
B H(ω)Sin Sout
Given the input RF signal in the frequency domain (Sin), the output signal is given by Sout=H(ω).Sin. To avoid a degradation of the output signal (=loss of information), the transfer function must satisfy the following conditions:
( ) cost, ( )H H
These conditions are requested in the signal band (bandpass characteristic).
Baseband equivalent 2-portThe response of a 2-port to a RF signal excitation can be obtained from the complex envelope signal. Defining the baseband as Ω=ω−ω0 and Cin(Ω) the spectrum of the complex signal, we have:
( ) ( ) ( )0out inC H Cω ωΩ = − Ω
Note that H(ω−ω0) is in this case a lowpass function (it is centered at zero frequency).Note also that the use of complex envelope equivalence is meaningful when the transient response is exhausted (steady-state condition)
Advantage of using the complex envelope representation
• In a RF signal all the information is associated to the complex envelope while the carrier don’t carry information.
• Moreover the minimum time step associated to the carrier is several order of magnitude smaller than the one associated to the envelope.
• Using CAD techniques for analyzing RF systems, a huge computation time is requested if the analysis is performed directly in the bandpass domain
• Using equivalent lowpass networks and complex envelope equivalent signals the computation time is strongly reduced without affecting the accuracy of the steady-state response
0RFout RFf f f= ±
Principle of Frequency Conversion
RFs
( )0cos 2 fπ
RFouts
fRF
BfRF
BfRFout
BfRFout
B
f
Intermediate Frequency (fixed)
Frequency Down Conversion (from fRF to fRFout (< fRF)
fOL fRF
fRFout
fRFout
How can we get the frequency conversion?Ideally: By multiplying the RF signal by a sinusoidal tone
ωOL
ωRF + ωOL ωRF - ωOL
Filter
ωRF
( ) ( )cos ( )RF M RFS V t t tω= +Φ
( )cosOL o OLV V tω=
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )( )
0
0 0
cos ( ) cos
cos ( ) cos ( )2 2
OUT RF OL M RF OL U D
M MRF OL M RF OL
S S V V t V t t t S t S t
V V t V V tt t V t t t
ω φ ω
ω ω φ ω ω φ
= ⋅ = + = +
⋅ ⋅− + + + +
SD(t) SU(t)
( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
20 1 0
2 220 2 0
cos cos2
cos 2 cos 2 2 cos cos ...2
out M OL M RF
OL M RF M OL RF
aV V V a V t V t
a V t V t a V V t t
ω ω
ω ω ω ω
= + + + +
+ + + ⋅ +
Non-lineardevice
Vin=SRF+VOL Vout 21 2 ...out in inV a V a V= + +
In RF practice: exploiting the non-linearity of a device
( ) ( )cosRF M RFS V t tω=
( )cosOL o OLV V tω=
ωRF + ωOLωRF - ωOL
Filter
ωOL ωRF 2ωOL 2ωRF
ωOL ωRF( ) ( )cos cosin M RF o OLV V t V tω ω= +
Image Frequency
Filter
ωIF =ωOL - ωRF ωOLωRF
ωIF ωIF
ωIM
The image frequency is separated 2ωIF from ωRF
The passband filter at ωRF is needed for eliminating the image band (signal and noise)
Power Transfer from Source to Load (Amplifier)
zS
zL
i1i2
vin v22-port
Power from Source (PS)
v1
Power to Load (PL)
Problem: there are several ways to define PS and PL!
Definition of Power Gain GP
( )*1 1
1 Re Power Entering the 2-port2S inP P v i= = ⋅ =
outP
in
PGain GP
= =
BUT: Pin may be very small, making Gp unreasonable large. In fact:zS i1
vin v1 zin
1 1 goes to 0 for inin in S
in S
Zv v v Z ZZ Z
= ⇒ <<+
Gp tends to infinite when the 2-port presents a very small (or very large) input impedance (with respect to Zs)!
1 1 goes to 0 for inin S
in S
vi i Z ZZ Z
= ⇒ >>+
( )*2 2
1=Power delivered to Load = Re2L outP P v i= − ⋅
Available power from a source and conjugate matching
The Available power of a source is the maximum power the source can deliver to a load. This happens when the load impedance is equal to the conjugate of the source impedance (conjugate matching)
( )
218 Re
Sav
S
VP
Z=
zSVS
I
V zL
zSVS
I
V zL
* L S L avZ Z P P= ⇒ =
( )
2
,1 Available Power from Source = 8 Re
inS AV in
S
vP P
Z= =
( )( )
2
2
,
Re 4
ReSout
TAV in L in
ZP vTransducer Power Gain GP Z v
= = =
Meaning of GT: It represents the power delivered to the load normalized to the maximum power deliverable from the source.
GT is the power gain definition more suitable for describing the attitude of a device to transfer power to a load (with respect to the use of a passive matching network connecting the source to the load )
Definition of Transducer Power Gain GT
( )
221 Power delivered to Load =
2 ReL outL
vP P
Z= =
PS=Available power from source = PAV, in
zS
zL
i1 i2
vin v22-portv1
( )( )
2,
,
ReRe
AV out S outA
AV in out in
P Z vAvailable Power Gain GP Z v
= = =
Another Power Gain definition: GA
zSvin
i1
v1( )
2
,18 Re
inAV in
S
vP
Z=
PL = Available power at output =PAV, out
( )
2
,18 Re
outAV out
out
vP
Z=
zoutvout
i2
v2
Specific features of Power Gain definitions
1. The value of GT is always smaller than or equal to the value of GP and GA(for a given 2-port with assigned ZL e ZS)
2. There is a particular condition where all the three gains assume the same value: it happens when both input and output are conjugate matched:
zS
zLvin2-port
zin zout
Conjugate Match at input:
Conjugate Match at output:
*in SZ Z=
*out LZ Z= GT=GA
GT=GP
T P AG G G= =If both these conditions are verified:
Cascade of 2-ports
The overall GP and GA is simply the product of the 2-ports gains:
1 2 3 1 2 3, P P P P A A A AG G G G G G G G= =
This does not apply, in general, for the overall transducer GT (it depends on the in/out matching of the 2-ports). However, if all the 2-ports are matched, also the total GT is given by the product of the 2-ports GTk.
GP1, GA1 , GT1
GP2, GA2 , GT2
GP3, GA3 , GT3
Z1,out Z2,in
*1 2 3 1,in , for T T T T k k outG G G G Z Z+= =
Noise generated by a resistor at T (°K)
The Available Noise Power produced by a resistor is given by:
K= Boltzman’s Constant, T=Temperature (°K), ∆F: Bandwidth
NP K T F= ⋅ ⋅∆
R
vn
R4nv K T R F= ⋅ ⋅ ⋅∆
214
nN
vP K T FR
= = ⋅ ⋅∆
RMS noise voltage
Available Noise Power:Maximum power delivered to a load
Characterization of Noisy 2-ports
Available noise power at the output of a noiseless two-port:
PAV,outGAVPN
PAV,out= GAVPN + PD=K.T.∆F.GAV+PD
PAV,out=GAV.PAV,in=GAV
.PN=GAV.K.T.∆F
GAV=Available Power Gain
If the 2-port produces noise (PD), the output power becomes
PD
GAV.PN PAV,out++
Teq: Equivalent Noise Temperature at the input of the 2-port
PAV,out = K.T.∆F.GAV+PD=K.(T+Teq) .∆F.GAV
Equivalent Noise Temperature at input
PD=K.Teq.∆F.GAV
Noisy2-portGAV
=PN
PAV,out Noiseless2-portGAVTeq
+TPAV,out
Teq of a dissipative Attenuator
R RThe 2-port is matched at input and output:GAV=GT<1.
Assuming T0 the physical temperature (°K), the noise power in the resistor at the output port must be Pout=KT0∆F. This power must be equal to the available power at the output port obtained with the equivalent model:
Noiseless2-portGAVTeq
+TPAV,outNoiseless
2-portGAVTeqTeq
+TTPAV,out PAV,out = K.(T0+Teq) .∆F.(GT) = Pout =KT0∆F
The equivalent temperature Teq is then given by:
Teq=(1/GT-1)T0 =(A-1)T0
In the practice is more used the transducer attenuation A=1/GT>1
T
1A =G
Noise Figure NF of a 2-port
NF=Noise Power at output
Noise Power at output with noiseless 2-port
0
0 0 0
1 1 eqAV D D
AV AV
TG KT F P PNFG KT F G KT F T
∆ += = + = +
∆ ∆
PAV,outGAVK.T0.∆F
Source of noise at T0
Note that NF depends on the temperature T0, which is conventionally assumed 293 °K (20 °C)
Added noise at output: PD
D AV eqP G K T F= ⋅ ∆
Noise Figure for cascaded 2-ports
The noise figure is mainly determined by the first stage
Ga1, NF1 Ga2, NF2 Ga3, NF3
( ) 321
1 1 2
11TOT
a a a
NFNFNF NFG G G
−−= + + + ⋅⋅⋅⋅
Relationship between Teq and Noise Figure NFFrom D AV eqP G K T F= ⋅ ∆ it is obtained:
0
1 eqTNF
T= + T0 = 290 °K
Note that NF is a multiplicative factor while Teq is additive. The generation of noise is however an additive phenomenon, so the model based on the equivalent temperature is more close to the physical reality. The Noise Figure has been introduced because it is more easy to be measured than Teq. It has in fact:
( )( )
in
out
S NNF
S N=
NF is the degradation of the S/N ratio from the input to the output of a 2-port(input noise KT0∆F)
( ) 01eqT NF T= −
Noisy RF Amplifier
GTNF
GAV+T
Teq
Noisy
Noiseless
( ) 01eqT NF T= −
NF is in general dependent on the source impedance ZS
ZS ZL
Noise in Frequency Conversion (DBS)
The mixer can be considered as a 3-port attenuator:
=>
T’0 is slightly larger than T0 to take into account the noise generated by the mixer directly at intermediate frequency
AU≈AL=AC
f
Noise at RF
fOLIntermediate Frequency
∆F
fRFfIM
Noise at IF
∆F∆F
RF Band
0 12C
DSBA
T T ′= −
AL
AU
RF Channel
IM Channel
T
T
+
TDSB
TDSB
T
T +
RF Channel
IM Channel
+
TDSB
TDSB
T
T +
+
TDSB
TDSB
T
T +
RF Channel
IM Channel
( )00
2 DSBAV ,out
C
K T TPKT
F A+
= =∆
SSB Noise model
( )2SSBAV ,out
C
K T TPKT
F A+
= =∆
+
TSSB
T
T
RF Channel
IM Channel
In this case we have only one equivalent noise source at input:
=>AU≈AL=AC
AL
AU
RF Channel
IM Channel
T
T
( )0 2SSB CT T A′= −
Noise Figure of a Mixer
The NF of a mixer can be expressed as the degradation of the signal-to-noise ratio from input to output.We assume PS the power of the signal (RF channel); TSSB the equivalent noise temperature of the mixer (SSB); KT0∆F the noise power in the RF channel at the mixer input. It has:
( )0 0
2
S S c
in out SSB c
P P AS S,N KT F N K T T F A
= = ′ ′∆ + ∆
0
2in SSB
out
STNNF
S TN
= = +
′
IEEE Noise Figure definitionThe previous expression says that a noiseless mixer (TSSB=0) would present NF=2 (3dB), which sounds strange (it should be NF=1).This result is due to the inclusion of the noise coming from the image channel in the mixer noise at output. If this contribute is not included (because for instance a sharp filter is added in that channel), the expression of the noise figure become the following:
( )0 0
S S c
in out SSB c
P P AS S,N KT F N K T T F A
= = ∆ + ∆
0
1 SSBIEEE
in out
TS SNFN N T
= = + ′ IEEE Definition
+
TSSB
T
T
RF Channel
IM Channel
Bandpass Filter (RF Channel)
RF Filter
LocalOscillator
Mixer
RF + IM
Noise from local oscillatorThe spectrum produced by a real oscillator is different from the ideal one (a single tone at fOL). The tone actually spreads out in in a spectrum with not zero width called “phase noise spectrum”
fOL fOL
Ideal LOReal LO
During the mixing process, the phase noise spectrum is converted to IF as it would be located in the RF band. It then contributes to worsen the overall noise behavior of the receiver.From an equivalent point of view, the contribution of the phase noise can represented by an equivalent temperature (TPN) at the mixer output
+
TDSB
TDSB
T
T +
+
TPN+
TDSB
TDSB
T
T +
+
TPN
General configuration of a RF Link
Information source(Baseband)
Baseband processing
RF Processing
RF PowerAmplifier
TransmittingAntenna
Information extraction
Baseband Translation
RF Processing
Low noiseAmplifier
ReceivingAntenna