RF Circuit Design Hexagon

Linear Time-Invariant systems

• Linear

• Time-Invariant

• Memoryless

)()()()(),()(),()(

2121

2211

tbytaytbxtaxthentytxtytxIf

+→+→→

　　

　

)()(),()(

ττ −→−→

tytxthentytxIf

　　

　

"."　　　

)),(()(dynamiccalledelse

txfty =

Nonlinearity Issues

Static model

Dynamic model

3rd order polynomialapproximation adequatefor weak nonlinearities,

eg. LNALinear part

Also prefer polynomial approximation

Nonlinearity issues

• Harmonic distortions• Gain compression• Desensitization and Blocking• Cross Modulation• Intermodulation distortions• Volterra series-based model• Effect of feedback on nonlinearity

Harmonic distortions

DC component(offset) Fundamental

(amplitudeinfluenced bynonlinearity)

Harmonics(not of major concernin a receiver – as out

of band)

β1β0 β2 β3

3cos4

2cos2

cos)4

3(2

)3coscos3(4

)2cos1(2

cos

coscoscos)(

33

22

33

1

22

33

22

1

333

2221

tAtAtAAA

ttAtAtA

tAtAtAty

ωαωαωααα

ωωαωαωα

ωαωαωα

++++=

++++=

++=

Atx cos)( =

Advantages of differential circuits

Even harmonics can besuppressed by usingdifferential mode(as in Differ. Amplifier):

In practice, only partial suppression, mismatchcorrupts symmetry between the two signal paths

tanh 2 T

inVv

EEout RIv =

Balanced or Differential system

Gain compression• As A ↑, β1/α1A ↓• When β1/α1A = - 1 dB, A = “1-dB compression

point”

3

11

112

31

33

311

145.0

1log2043log20

0,43

αα

ααα

αααβ

=

−=+

<+=

−

−

dB

dB

A

dBA

AA

1 dB

20 log Ain

20 log Aout

A1-dB

12

2313

112

231

21

12

2133

1311

2211

23 ,0

........cos)23()(

),ceInterferen(　For

..,cos)23

43()(

,coscosx(t)

αααα

ωαα

ωααα

ωω

<+<

++=

<<

+++=

+=

Aif

tAAty

AA

tAAAAty

tAtA

Desensitization and Blocking

Gain of desired signal is reduced by interferer signal.

Nonlinearity issues

Blocking: 3

1

32

αα

=blockA

SNR

Pin

Pout

-20

-20

-70

-70

-45-70

-100

-110

blocker

signal

noise

3dbdesensitization

point

As block power become excessive, signal gain is reduced, equivalent noise is increase.

GSM blocking test:

Wanted signal is -99 dBm

At 3dB desensitization point, SNR=9dB

Cross ModulationIf the blocker’s amplitude varies in time the amplitude modulation occurs at the output:

Nonlinearity issues

Extra harmonics are produced.Similar effect for large noise imposed.

Typical of multi-channel transmitterswith a common amplifier

If the interferer is amplitude modulated:A2cos(w2t) = A2(1+m(t))cos(w2t)

Where m(t) is the message being transmitted.The cross modulation component becomes:

( ) )cos())()(21( 122

2323

1 tAtmtmA ωαα +++

The component that can be heard in our channel is:

( )

)cos()(31)(

)cos()(31)(

)cos())(21(

111

2232

2323

1

112232

31

2232

2323

1

112232

31

tAtmAA

tAtmA

AA

tAtmA

ωα

ααα

ωαα

ααα

ωαα

⎟⎟⎠

⎞⎜⎜⎝

⎛++≈

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++=

++

The modulation index that of int.ch. When: A2 =(α1/3α3)^0.5

32213

22212

22111

2211

)cos(cos)cos(cos

)coscos()(,coscos)(

tAttAt

tAtAtytAtAtx

ωωα

ωωα

ωωαωω

++

++

+=+=

221312

212321

2123

323212

2213

313111

43:2

43:2

23

43:

23

43:

AA

AA

AAAA

AAAA

αωω

αωω

αααω

αααω

−

−

++

++• Intermodulation

ω1 ω2ω ω1 ω2

ω

2ω1-ω2 2ω2-ω1

DUT

Intermodulation distortions

Corruption of a signal due to intermodulationbetween two interferers (of big concern)

Two-tone test for nonlinearity: A1=A2 and ω1≈ω2

Intermodulation distortionsThird intercept point IP3

For small amplitudes the fundamental rises linearlywith A, and the 3rd order intermodulation terms with A3

At IP3, |α1A| = |¾ α3A3|

Two-tone Distortion• 3rd Order Intermodulation Intercept

Point (IP3)

...)2cos(43

)2cos(43

cos)49(

cos)49()(

,

123

3

213

3

22

31

12

31

21

+−+

−+

++

+=

=

tA

tA

tAA

tAAty

AALet

ωωα

ωωα

ωαα

ωαα

　

33

3

13

33331

log2034

43

IPIP

IPIPIP

AP

AAA

=

=⇒=αααα

If α1>>α3

Let Aω1ω2 be the amplitude of ω1 and ω2Let Ain be the input level at each frequencyLet AΙΜ3 be the amplitude of the IM3 products

...)2cos(43

)2cos(43

cos)49(

cos)49()(

123

3

213

3

22

31

12

31

+−+

−+

++

+=

tA

tA

tAA

tAAty

ωωα

ωωα

ωαα

ωαα

23

1

33

1

3

2,1

1||3||4

4/||3||

in

in

in

IM

A

AA

AA

αα

ααωω

=

≈

2

23

3

2,1

in

IP

IM AA

AA ≈ωω

||3

134

3 αα=IPA

2

23

3

2,1

in

IP

IM AA

AA ≈ωω

log20log20log20log20 22332,1 inIPIM AAAA −=−ωω

log20)log20log20(log20 32,121

3 inIMIP AAAA +−= ωω

The IP3 can be measured with single input.

log20log20log20log20 22332,1 inIPIM AAAA −=−ωω

log20)log20log20(log20 32,121

3 inIMIP AAAA +−= ωω

3rd Order IM Intercept Point (IP3)

32

23

2)()(

2

3

3

XIIPin

Xin

Xinin

YoutinIIP

PPP

PP

GPGPP

PPPP

+=

−=

+−++=

−+=

Pin

Pout

Pin PIIP3

Pout

PY

3rd IM power

fundamental

PX

If PX is the minimum receivable input power, and if the corresponding output PY is not to be exceeded by IM3, then Pin must stay below the above expression

IM2• 2nd order intermodulation (IM2) can be

similarly defined and computed• Main cause:

– Coupling from ant to LO (self mixing)– Distortion in amp and mixer– Mismatch in differential topologies

• Main effect is on DC• Can be minimized with fully differential

circuit

IP3 of Cascade Networks• IP3 of Cascade Networks with odd nonlinearity

G1IP31

G2IP32

GtotIP3output

)(

31

1322

22

1

mW

GGGIP

IP

i niii

output

∑++

=

L

2 stage: 2212

2212

2221

3333

31

31

13GIPIPGIPIP

IPGIP

IP output +⋅⋅

=+

⋅

=

223GAIP

)()()()(

)()()()(313

212112

33

2211

tytytyty

txtxtxty

βββ

ααα

++=

++=

333

2213

233

2212

33

22112

])()()([

])()()([])()()([)(

txtxtx

txtxtxtxtxtxty

αααβ

αααβαααβ

+++

+++++=

........)()2()()( 33

322113112 1

++++= txtxty βαβααβαβα

||3

3122113

11

234

3 βαβααβαβα

++=IPA

G1=α1G2=β1

22,3

21

1

222

1,3

11

33122113

23

231

|||||2|||

431

IPIP

IP

AA

A

αββα

βαβαβααβα

++=

= ++

22,3

21

21,3

23

11IPIPIP AAAα+=

...... 23,3

21

21

22,3

21

21,3

23

11 +++=IPIPIPIP AAAA

βαα

In general:

L

L

L

++++=

++++=

=

++++=

ttttAtAtAty

tAtxtxtxtxty

ωβωβωββωαωαωαα

ωαααα

3cos2coscoscoscoscos)(

cos)()()()()(

3210

333

22210

32

2210

　

Harmonics Distortion

4,

2,

43,

2

33

3

22

2

33

11

22

00AAAAA αβαβααβααβ ==+=+=

2

1

3

1

33

1

2

1

22

41

,21

AHD

AHD

αα

ββ

αα

ββ

≈≡

≈≡

1

23

22

βββ L++

=THD

Third Order Harmonic Intercept• As A ↑, HD’s increase faster than basic• At one point, HD3 becomes 1.

Pin

Pout

P0 PHIP3

P1

P3

3rd harmonic

fundamental

2/2 3

313 HDPPPPP sigsigHIP +=

−+=

33

3

13

2331

33

311

log20

4

41

41

HIPHIP

HIP

HIP

AP

A

A

AA

=

=

=

===

αα

αα

βααβ

Comparison of these points

AAIP3 AHIP3ABlockA1-dB

3

1

32

αα

=blockA

3

11 145.0

αα

=− dBCA3

13 3

4αα

=IPA

3

13 4

αα

=HIPA

PIIP3= 10*log(3) + Psig+HD3/2

3

1.. 3

1αα

=mcA

Ac.mA1-dB

3

13 195.0

αα

=− dBDA

Dynamic Range• Could be measured at either input or

output• Could be based on various nonlinearity

measures

min,,11 inindBdB PPDR −=

3

)(23

2 min,3min,

,3

min,,

inIIPin

inIMIIP

ininsfsf

PPP

PP

PPDR

−=−

+=

−=

Examples:

Dynamic Range

Pin

Pout

Psf,in PIIP3

Psf,out

PIM,out

DR1dB

1dB

P1dB,in

DRsf

∆P ∆P ∆P

Pin,min

Nonlinearity issues

Volterra series-based modelH1(ω)H2(ω1, ω2)H3(ω1, ω2 , ω3)….

Frequency domaintransfer functions(esp. useful in analysis of mixers)

•Can be calculated from nonlinear model equations by “harmonic balance” approach provided the nonlinearity is “weak”• Represent gains for harmonic distortions when all ωk = ω0, eg. for the second harmonic (assuming symmetry):

•Represent gains for intermodulation distortions for arguments with different ωk respectively

Example results (details skipped)

),,()(

32,

),,()(2

)(),,(

43,

)(),(

)(),,(

4,

)(),(

2

2213

113

1113

113

11

2213412

11

21212

11

11132

13

11

11212

ωωωω

ωωωω

ωωωω

ωωω

ωωωω

ωωω

jjjHjHIP

jjjHjHHIP

jHjjjHAIM

jHjjHAIM

jHjjjHAHD

jHjjHAHD

−==

−==

==

Noise Issues

• Randomness of noise• Signal to Noise ratio• Time/frequency domain description• Circuit noise• Noise figure• Sensitivity• Spurious Free Dynamic Range

Noise Issues

Additive White Gaussian Noise (AWGN) – random processes

Classical model:Additive: v(t) + x(t) (signal + noise) typical of LNA’s and Mixers

Gaussian Probability Distribution Function (PDF)

Randomness of noise

Randomness of noiseTime dependence: x = x(t)

For stationary noise the mean (and “mean square”) values over the probability domain, and over the time domain are same:

)()()( tntstx +=

srmsT

s PSrmsSdttsT

P === ∫ )(,)(10

2

nrmsT

n PNrmsNdttnT

P === ∫ )(,)(10

2

Signal and noise power

Physical interpretation

power

If we apply a signal (or noise) as a voltage source across a one Ohm resistor, the power delivered by the source is equal to the signal power.

Signal power can be viewer as a measure of normalized power.

Signal to noise ratio

)(log20)(log10 1010rms

rms

n

s

NS

PPSNR ==

SNR = 0 dB when signal power = noise power

Absolute noise level in dB:w.r.t. 1 mW of signal power

)log(10dB30mW1

log10din

n

nmn

P

PP

+=

=B

SNR in bits• A sine wave with magnitude 1 has power

= 1/2.• Quantize it into N=2n equal levels between

-1 and 1 (with step size = 2/2n)• Quantization error uniformly distributed

between +–1/2n

• Noise (quantization error) power=1/3 (1/2n)2

• Signal to noise ratio = 1/2 ÷ 1/3 (1/2n)2 =1.5(1/2n)2

= 1.76 + 6.02n dB or n bits

Adding Noises

Frequency domain description of noise

∫−∞→+=

TTT

n dttntnT

R )()(21lim)( ττ

))(()()( τnnn RFfSfPSD ==

dffPSDP nn )(∫+∞

∞−=

dffPSDP nn )(0∫+∞

=

Given n(t) stationary, its autocorrelation is:

The power spectral density of n(t) is:

For real signals, PSD is even. can use single sided spectrum: 2x positive side

↑ single sided PSD

Time-frequency domain relation

)()( fXtx ⇔

⇓dffXdttx

22)()( ∫∫

∞+

∞−

∞+

∞−=

)()( fPSDR xx ⇔τ⇓

dffPSDRdttx xx

T

TT ∫∫+∞

∞−

+

−∞→== )()0()(lim

2

Parseval’s Theorem:

If

If x(t) stationary,

Interpretation of PDS

PSDx(f)

Pxf1 = PSDx(f1)

Filtering of noise

H(s)x(t) y(t)

|H(f )|2 = H(s)|s=j2πf H(s)|s=-j2πf

Total output noise = transferred noise + circuit generated noise

Circuit noise

Example:R = 1kΩ, B = 1MHz, 4µV rms or 4nA rms

Example:ID= 1mA, B = 1MHz, 17nA rms

MOS Noise Model

BJT Noise

Sampling Noise• Commonly called “kT/C” noise• Applications: ADC, SC circuits, …

von

R

C

Used:

NF: Noise Figure (dB) Noise Factor (value)

Might be for 1-Hzbandwidth at agiven frequency

For a system with no inherent noise NF =1 (0 dB)Noisy system degrades SNR and NF > 1

GF

Nout = F*G*Nin

Ninh = (F – 1)*Nin1

F–1G

Noise Figure Example

Input referred noise

NIN = 2SRV ( )2

, nSninh

riout IRVN +=−

( )2

2, 11

SR

nSn

in

inhriout

VIRV

NN

NF ++=+= −

Noise Figure of Cascade NetworksG1F1

G2F2

G3F3

G4F4

G1 G2 G3 G41F1-1 F2-1 F3-1 F4-1

G1G2 G3 G41F1-1 (F2-1)/G1 F3-1 F4-1

Noise Figure of Cascade Networks

G1G2 G3 G41F1-1 (F2-1)/G1 F3-1 F4-1

G1G2G3 G41F1-1 (F2-1)/G1 (F3-1)/G1G2 F4-1

G1G2G3G41F1-1 (F2-1)/G1 (F3-1)/G1G2

(F4-1)G1G2G3

Noise Figure of Cascade NetworksG1F1

G2F2

GtotFtot

L+−

+−

+−

+−

+=321

4

21

3

1

21 1111

11GGG

FGG

FG

FFFtot

G3F3

G4F4

Gtot1Ftot-1

Note: here G is power gain. If Ai is voltage gain, Gi= Ai^2

EXAMPLE

If the NF and gain of the blocks are:LNA: 20dB gain, 3 dB NF (A1, F1)Mixer: 10dB gain, 10 dB NF (A2, F2)VGA + Filter: 80dB gain, 20 dB NF (A3, F3)

dB

dbdBdB

dBdBdB

GGF

GFFFtot

39.3184.210*100

99100

9995.1

10*20120

20110311

21

3

1

21

==++=

−+

−+=

−+

−+=

Sensitivity

out

in

SNRSNR

NF =

inN

insigin P

PSNR −=

Psig: Input signal power

PIN: Input noise power

BWSNRNFPP

SNRNFBWPSNRNFPP

dBdBHzdBmNdBminsig

outHzNoutNinsig

in

inin

log10min/min.

/

+++=

⋅⋅⋅=⋅⋅=

−

−

The minimum required signal power at the input terminal in order for the circuit to achieve a specific SNR at the output, which determines the Bit Error Rate (BER)

Input Noise Power

System MatchedCircuit

Noise voltage: (4kTR)^0.5 /HZ^0.5Power delivered to matched load:

P = ¼ V^2 / RInput noise power: PN-in = kT /HzAt room temp: PN-in = –204 dBWatt /Hz

= –174 dBm /HzNoise power over a bandwidth: = –174 dBm + 10log(BW)

BW

dBGGGG

FGGG

FGG

FG

FFFtot 8.342.24

1111

321

5

321

4

21

3

1

21 ==

−+

−+

−+

−+=

dBmdBdBdBm

BWSNRNFPP dBdBHzdBmNdBminsig in

9.106)10*3log(105.848.2174

log105

min/min.

−=+++−=

+++=−

If the received signal power is below this, it cannot be detected correctly

Example: GSM receiver• Baseband required SNR

– 6 dB for static conditions– 9 dB with fading

• Channel bandwidth typically 170kHz– 10log(170000)=52.3 dB

• GSM specification required sensitivity:– <= – 102 dBm