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