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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
Recommended