66
1 Edgar Sánchez-Sinencio Department of Electrical and Computer Engineering Analog and Mixed-Signal Center Texas A&M University http://www.ece.tamu.edu/~sanchez/ N O I S E
1
Edgar Sánchez-Sinencio
Department of Electrical and Computer Engineering
Analog and Mixed-Signal Center
Texas A&M University
http://www.ece.tamu.edu/~sanchez/
N O I S E
2
NOISE
• NOISE limits the minimum signal level that a circuit can process withacceptable quality.
Can you identify the signal buried in the noise?
How does the minimal signal must be with respect to the noise level?
Noise Level
Mr. Signal
3
• Let us consider the street noise, can one predict the (exact) noise at
any time?
No, because it is a random process
• If you decide to blow your car’s horn every 5 minutes, then you can say
this signal is deterministic.
• So, how can we incorporate noise in circuit design?
- Observe the noise for a long time
- Construct a “statistical model”
• The average power noise is predictable
• Most noise sources in circuits exhibit a constant average power
4
• Average power delivered by a periodic voltage v(t), of period T, toa load resistance RL is given by
• For non-periodic signals, T becomes a large quantity
• How much power a signal carries at each frequency is defined by the“power spectral density” (PSD) (Sx(f))
• Sx(f) is defined as the average power carried by x(t) in a one-hertzbandwidth around f.
2
T
2T
2T
2T
L
2
av dt)t(i)t(vT
1dt
R
)t(v
T
1P
t
x(t)
t( )2
f
AMP
2
f )t(x1
BP Filter x2
xf1(t)
t tf1 f2 f3 fn
Sx(f)1Hz
f1
5
• Sx(f) is expressed in V2/Hz rather than VI/Hz = w/Hz, in fact
Sx(f) 1/2 = v/Hz-1/2 is often used. Say a filter at 1MHz is equal to
1.414 nV/Hz-1/2, means an average power in a 1-Hz bandwidth at
1MHz is equal to (1.414x10-9)2V2=2x10-18V2
• Another spectrum is the “White Spectrum” or white noise
In practice is band limited.
• How do you determine the output spectrum of a linear, time-
invariant system with transfer function, H(s)?
f
Sn(f)
6
• How is the shape of the output spectrum?
• A practical example is the telephone bandwidth, where BWof Sxin(f) is between 0-20KHz, BW of H(f) is 4KHz andconsequently BW of Syout(f) is 4 KHz.
H(s)xin yout
inxS 2xy )f(H)f(S)f(S
f
x)f(Sx )f(Sy
2)f(H
7
If there are more than two noise sources, how do you computetheir total effects?
xn1(f), xn2(f), … are uncorrelated noise sources
Superposition is applied to obtain the total output spectrum.
H1(s))f(X 1n
H2(s))f(X 2n
Hn(s))f(Xmn
+...
)f(Xf2jH)f(S2ni
2
iy
8
Assume an input signal, Vni(f)=Vnw (constant)
If this same input signal, Vnw, is applied to the (brick-wall) filter
2
fV
f
ftanfVdf
ff1
Vdf)fj2(H)f(VV onw
oo
1onw
o 2
o
2nw
o
22ni
2)rms(out,n
2
ffthen,VVcesin,fVdfVV o
xbrickout,nx
2nw
f
o
2nw
2)rms(brick
x
100
fo
10
fo
Noise Bandwidth
0dB
-20
)j(H
o/s1
1)s(H
of of10
-12 dB/oct
0
-20
)j(Hbrick
100
fo of
dB
fox f2
f
First - Order LP Brick - Wall Filter
9
Equivalent Noise Bandwidth
df)f(AA
1f
o
2v2
vo
Examples
First-order system:
dB3
o2
dB3dB3
v f571.1
ff1
dff,
ffj1
1)f(A
dBa
o
dBdB
v ffdf
ff
f
ff
jfA 32
3
2
3
6436.022.122.1
1
1,
1
1)(
10
• Amplitude Distribution of Noise
Probability Density Function (PDF), the distribution of x(t) is
• An important example of PDFs is the Gaussian (or normal)Distribution.
Where s and m are the standard deviation and mean of thedistribution, respectively.
dxxXxofyprobabilitdx)x(px
2
2
x2
)mx(exp
2
1)x(p
s
s
x(t)
t
#samples
x
11
• How do you determine the total average power due to two or
more noise sources?
If noise sources are uncorrelated the third term is zero.
Example of uncorrelated and correlated noises is that of spectators
in a sports stadium.
2T
2T
2122
21
T
2T
2T
221
Tav dt)t(x)t(x2dt)t(xdt)t(x(
T
1limdt)]t(x)t(x[
T
1limP
2T
2T
21T
avavav dt)t(x)t(x2T
1limPPP
21
= 0 ? When?
12
• Device Electronics Noise
• Environmental Noise
NoiseTypes
• Thermal Noise
Example22192
N )Hz/nV91.0(Hz/V1028.8VK300T50R
m
2N g
3
2KT4I Thermal (White) Noise
fWLC
KV
ox
dev2N Flicker Noise
R
NoiseResistor
Noiseless
R
- +
2NV
KTR4V2N R
2NI
R
KT4I
2N
4KTR
SV(f)
f
T=Temperature
Noisy
- +2NI
2NV
13
Noise Voltage
* EtRL(noiseless)
RS
Eo Power Supplied to RL is
L
2o
ootR
EEIP
RRR
2
ER
R2
EE,fKT
R4
E
P
LS
tto
2t
t
Then
)V(fkTR4E rmst
rmst Vn4
Hz1f
K1RE
BW
1Hz1k
14
• Flicker or 1/f Noise
PSD is given by
f
1
WLC
K
f
K
f
K)f(V
ox
2V
2V2
n
for aCMOS
1/f noisedominates
LowFrequency
white noisedominates
HighFrequency KT8
3g
WLC
Kf m
oxc
2m
coxm g
f
1
WLC
Kg
3
2KT4
Equating output currents
K~10-25V2F
fcorner
Vn(f)
f0.1 1.0 10 100 103 104
Hz
V10
1
-10 dB/dec
15
Noise Considerations
FlickerNoise
Thermal or JohnsonNoise (White Noise)
fcp f
2nv
PMOSNMOS
fcn
B
2eqV
~
• To reduce Flicker Noise:
Increase W * L (Area)
To keep same bias point keep same ( W/L )
• To reduce White Noise
Increase gm. Since
(a) (W/L) and (ID) will increase gm, power consumption (ID), and area
(b) (W/L) and modify the bias to keep ID same as before increasing gm.
L
W)VV(K
L
WIK2g TgspDpm
A107.2KF 21p
A103.4KF 21n
i.e.,
m
mb
g
g
f
g3
1kT8
WLKfC2
fKFVV
mpox
2eq
2N
FlickerNoise
WhiteNoise
16
How to compute the total output noise due to individual noise sources?
How to compute the input-referred noise?
Example.___
VDD
Vout
Vin
RDVn,out
2
InRD2
In12
RD
VDD
WLfC/Kgg3
2KT4I ox2mm
21n
D
2nRD R/KT4I
2D
D
2m
oxm
2out,n R
R
KT4g
f
1
WLC
Kg
3
2KT4V
2out,n
D2m
2V
2out,n2
in,n VRg
1
A
VV
D2moxm
2in,n
Rg
KT4
f
1
WLC
K
g3
2KT4V
in 1Hz at frequency f
Good performance parameter
17
• Can we always use the input-referred noise by a single voltagesource in series with the input?
A necessary and sufficientrepresentation
• Simplifications
- For zero source impedance, I2n,in not affecting the output
- For infinite source impedance, then V2n, in has no effect
• Note that both sources will not count the noise twice.
NoiselessC
I R
CU
IT
Vn,in2
- +
In,in2
18
To compute the rms output noise:
(i) Compute the frequency response bandwidth f3dB
(ii) Compute the noise bandwidth, f
f = 1.571 * f3dB (assuming a single pole)
(iii)
where is the spectral density
value of the output noise at low
frequency.
rms2
12
out,nnoise,out VfVV
2out,nV
622 (ESS)NOISE CONSIDERATIONS REMARKS
Basic elements and their noise models
Resistor. -
Op Amp
19
RR
KTR4V2NR
RKT4I2NR
A
NAV
2NaI1NaI
20
Noise in an op amp macromodel
Particular Cases:
• Ideal capacitors and inductors do not generate noise, but accumulate.
-
+
In-(f)
Noiseless
In+2Vn
2(f)
-
+
Vno2(f) -
+
Vno2(f) -
+
Vno2(f)
R
R
If Vn2(f) ignored:
Vno2 = 0
Actual:
Vno2 = Vn
2
If In-(f) ignored:
Vno2 = Vn
2
Actual:
Vno2 = Vn
2 + (In-R)2
If In+(f) ignored:
Vno2 = Vn
2
Actual:
Vno2 = Vn
2 + (In+R)2
21
Computation of Total RMS Noise Voltage at Filter Output
1. Determine the noise transfer function k
2k
N
VsT from each equivalent
voltage or current noise element, kth theof Ior Vn 2n
2n
2k kk
to the outputof the filter.
2. Spectral density from the different noise sources are added:
2k2
kk
2no njTV
where jTk is the absolute value of the noise transfer function
from source kth.
22
3. Obtain the total output noise power by integrating the mean squarenoise spectral density .V2
no
That is:
d)(V2
NOo
2rmsno
This is often referred as the noise floor.
Dynamic range of an Active-RC is obtained as:
dBV
log20DRrmsno
maxrmsso
Where rmssoV is the maximum undistorted rms voltage at the output.
23
How do you simulate noise in SPICE?
- Noise is associated with AC analysis
• Noise V(N) VIN
• AC DEC FI FSTEP FFINAL
Reader.___
Simulate an example dealing with first-order active
RC filter, plot frequency responses and noise spectrum, as well
as the total noise at each frequency (total rms noise)
Obtain the signal to noise ratio
)rms(NoiseTotal
)rms(THD%forsignallog20
NS
LTSPICE Noise:
Analysis & Simulation
Contributed by Kyoohyun Noh
24
LM741’s open loop ac response
25
LM741’s DC gain : 106dB
LM741’s -3dB frequency: 5Hz
LM741’s unity gain frequency: 1MHz
LM741’s noise simulation
26
LM741’s input-referred noise PSD is simulated: R4’s noise contribution is small enough to be neglected.
HznVfv inputopenLMn /51.6/2,,741, HzfVfv inputRn /67.0/2
,4,
Filter Schematic for .noise simulation
27
Filter’s midband gain : 50 (34dB) from R2/R1
Expected upper -3dB bandwidth : about 5kHz (from 1/R2C1)
Expected lower -3dB bandwidth : about 0.16Hz (from 1/R3C2)
Coupling cap
Output: V(OUT), Input source:V320 simulation points per decade over 0.1Hz ~ 100kHz
Filter Transfer Function
28
Simulated Midband gain = 34dB
Simulated Upper -3dB = 4kHz
Simulated Lower -3dB = 0.16Hz
Input/Output-referred noise PSD
29
Midband input-referred noise PSD
HznVfv inputn /77.6/2
,
Midband output-referred noise PSDHznVfv outputn /338/2
,
+20dB/dec of filter gain flattens input-referred noise PSD
-20dB/dec of filter gain flattens input-referred noise PSD
Noise contribution to Input-referred
noise
30
Output Resistance R3
FeedbackResistance R2
InputResistance R1
Noise
Component
Theoretical
input-referred noise voltage
Simulation
R1
R2
R3 0
HznVkTR /29.14 1
HzpVkTRRR /1824)/( 221
HznV /29.1
HzpV /182
HzfV /90
Assume T=300K
Input-referred noise contribution in the filter
31
LM741’s noise contribution cannot be obtained directly in the filter simulation. Instead, it should be estimated from the other noise values
HznVfVpVnVnVfv inputfilterLMn /64.6)90()182()29.1()77.6(/ 22222,,741,
HznVfv inputopenLMn /51.6/2,,741,
The input-referred noise voltage of the LM741 was obtained in the open loop simulation.
Theoretical LM741’s input-referred noise contribution agrees well with the simulation result
fvR
Rfv inputopenLMninputfilterLMn /)1(/ 2
,,741,2
12,,741,
HznVHznV /64.6/51.6*02.1
Summary
32
A dominant contributor to the filter noise is LM741
Theoretical noise analysis agrees well with simulation results
Noise
components
Midband input-referred
noise voltage
[nV/sqrt(Hz)]
Midband input-
referred noise PSD
[V2/Hz]
Contributio
n [%]
R1 1.29 1.66x10-18 3.63
R2 0.182 3.31x10-20 0.07
R3 0.00009 8.10x10-27 ~0
LM741 6.64 4.41x10-17 96.3
Total 4.58x10-17 100
Biquad Filter NoiseCourtesy of Mohamed Abuzaid
Biquad Filter
• 𝐻𝐿𝑃𝐹 𝑗𝜔 =1
1−𝜔
𝜔𝑜
2+𝑗
𝜔
𝜔𝑜𝑄
• 𝐻𝐵𝑃𝐹 𝑗𝜔 = 𝑗𝜔
𝜔𝑜𝐻𝐿𝑃𝐹 𝑗𝜔
• R2=R4=R5=R6=R
• C3=C7=C
• 𝜔𝑜 = 1/𝐶𝑅
• 𝑄 = 𝑅3/𝑅
INBPF
LPF
R1
R3
C3
R7=1G
C7
R4
R5
R6
R8
R9R10
R2
Bruton, L.T.; Trofimenkoff, F.N.; Treleaven, D.H., "Noise Performance of low-sensitivityactive filters," Solid-State Circuits, IEEE Journal of , Feb. 1973
Noise Analysis
• Power Spectral Density of output noise:
• 𝜖𝑂𝑅2 𝑓 = 4𝐾𝑇 𝑖=1
6 𝑅𝑖 𝐻𝑖𝑜 𝑗𝑓2 (
𝑉2
𝐻𝑧)
• Power Spectral Density of output noise:
• 𝜖𝑂𝑅2 𝜔 =
2𝐾𝑇
𝜋 𝑖=16 𝑅𝑖 𝐻𝑖𝑜 𝑗𝜔
2 (𝑉2
𝑟𝑎𝑑/𝑠)
• Integrated output noise:
• 𝐸𝑂𝑅 = 𝐾𝑇𝑅 5𝑄 + 1 𝜔𝑜 𝑉2
Divide by:2𝜋
Continue Noise Analysis
• Assumptions for next results:
– In band, 𝐻𝐵𝑃𝐹 𝑗𝜔 = 𝐻𝐿𝑃𝐹 𝑗𝜔
– Assume 𝑄 >> 1
• Power Spectral Density of output noise:
• 𝜖𝑂𝑅2 𝑓 ≈ 20𝐾𝑇.𝐻𝐵𝑃𝐹 𝑗𝜔 (
𝑉2
𝐻𝑧)
• At the center frequency
• 𝜖𝑂𝑅2 𝑓 ≈ 20𝐾𝑇. 𝑄 (
𝑉2
𝐻𝑧)
– “this is not correct in paper (12), they put an extra 𝜔 term”
Simulation Ideal Opamp
• Biquad with the specs:
• 𝑓𝑜 = 1.5 𝑘𝐻𝑧
• 𝑄 = 150
• So, the values:
• 𝑅 = 10 𝐾Ω
• 𝑅3 = 1.5 𝑀Ω
• 𝐶 = 10.6 𝑛𝐹
INBPF
LPF
R1
R3
C3
R7=1G
C7
R4
R5
R6
R8
R9R10
R2
Simulation Ideal Opamp
• Compare the PSD of the output noise and
the theoretical expression.
• Inband, expressions
are identical.
Noise Contribution
• There are three groups in terms of
contribution
Noise ComparisonTheoretical Spice Simulation
Integrated Noise -95.33 dB -95.33 dB
Noise at peak -107 dB/Hz -107 dB/Hz
Using Actual Opamp
• Opamp noise: 6.3 𝑛𝑉/ 𝐻𝑧
• Compare the PSD of the output noise and
the theoretical expression.
• In band, there is
extra contribution
due to opamp
(15 dB higher)
Noise ComparisonTheoretical Spice Simulation
Integrated Noise -95.33 dB -80.17 dB
Noise at peak -107 dB/Hz -92 dB/Hz
Resistors
Opamps
Biquad Tow-Thomas Filter
Noise Analysis & Simulation
Courtesy of Kyoohyun Noh
43
Tow-Thomas(TT) Biquad Filter
44
Simultaneous Biquad Filter Implementations•Low-pass(LP), Band-pass(BP) output
Independent tuning of Q and filter frequency
TT Filter Analysis
45
Key Specifications of TT BPF
46
64273
50
RRRCC
R
4267
5333330
RRRC
RRRCCRQ
20
02
0
1
3)(
sQ
s
sQ
R
RsTBPinv
Output Noise PSD from passive components
47
22
310, )()
1(
11sH
CRSS BPRBPinvR
1)(
)(
0
2
0
0
Q
ss
s
sH BP
22
320, )()
1(
22sH
CRSS BPRBPinvR
22
330, )()
1(
33sH
CRSS BPRBPinvR
2
, )(44
sHSS LPRBPinvR
22
5
4, )()(
55sH
R
RSS LPRBPinvR
22
5
4, )()(
66sH
R
RSS LPRBPinvR
1)(
1)(
0
2
0
Q
sssH LP
L.T. Bruton, et. al., “Noise Performance of Low-Sensitivity Active Filters,” JSSC. pp.85-91, Feb. 1973
Ideal Op-amps are assumed
over R of PSD noise theis ,2kTR
ii
iRS
Integrated output noise components
48
22
310, )()
1(
11sH
CRSS BPRBPinvR
22
320, )()
1(
22sH
CRSS BPRBPinvR
22
330, )()
1(
33sH
CRSS BPRBPinvR
2
, )(44
sHSS LPRBPinvR
22
5
4, )()(
55sH
R
RSS LPRBPinvR
22
5
4, )()(
66sH
R
RSS LPRBPinvR
)2
()1
(0
2
31
2, 11
Q
CRSv RBPinvR
)2
()1
(0
2
32
2, 22
Q
CRSv RBPinvR
)2
()1
(0
2
33
2, 33
Q
CRSv RBPinvR
)2
( 02, 44
QSv RBPinvR
)2
()( 02
5
42, 55
Q
R
RSv RBPinvR
)2
()( 02
5
42, 66
Q
R
RSv RBPinvR
L.T. Bruton, et. al., “Noise Performance of Low-Sensitivity Active Filters,” JSSC. pp.85-91, Feb. 1973
Angular freq. integration
Ideal Op-amps are assumed
over R of PSD noise theis ,2kTR
ii
iRS
Total integrated output noise
49L.T. Bruton, et. al., “Noise Performance of Low-Sensitivity Active Filters,” JSSC. pp.85-91, Feb. 1973
2,
2,
2,
2,
2,
2,
2,, 654321 BPinvRBPinvRBPinvRBPinvRBPinvRBPinvRBPinvtotalnoise vvvvvvv
If R=R1=R2=R4=R5=R6, C=C3=C7
RC
10
R
RCRQ 3
30
02
,, )15( QkTRv BPinvtotalnoise
6
11,
,1,
,,00
iQ
BPinvRQ
peakBPinvtotal iSS
)]//([ 10 22
1,,,
0
sradVQ
kTRSQ
peakBPinvtotal
Ideal Op-amps are assumed
over R of PSD noise theis ,2kTR
ii
iRS
BPF Simulation
50
Ideal Op-amps are assumed
BPF with different Qs are implemented with ideal op-amps
•C=10.6nF is assumed•Each R is listed in the next slide
Simulated total integrated output
noise
51
Ideal Op-amps are assumed
1,2,4,5,6ifor )4
(4 02,
QkTRv BPinvRi
)4
(4 0
3
2,3
Q
R
RkTRv BPinvR
02
,, )15( QkTRv BPinvtotalnoise
Theoretical estimation agrees well with the simulation results Feedback resistor R3 of the lossy integrator makes the least contribution to the High-Q High-gain BPF output noise among passive components
BPF Output Noise PSD
52
Theoretical estimation agrees well with the simulation results
)]//([ 10 22
1,,,
0
sradVQ
kTRSQ
peakBPinvtotal
]/[ 20 1,0
HzVkTRQSQ
v, peaktotal,BPin
Theory
[uV/sqrt(Hz)]
Simulation
[uV/sqrt(Hz)]
Peak
output
noise PSD
[V/sqrt(Hz)]
4.31 4.3
Q=150, R=10kIdeal opamps are assumed
OP Amp Design for BPF
53
BPF Q =15, f0=1.5kHz
Assume GB is large enough•GB=8MHz >>Qf0=22.5kHz
Amplifier gain is set to larger than 3000 from behavioral simulation
Q vs. op amp gain under GB=8MHz
OP Amp example
54
Gain Stage1 Gain Stage2 Buffer Stage
OP Amp details
55
Bias Current Generation
ibias = 20uA
W/L
Diode connected PMOS 14/0.6
Current source 14/0.6
Specification
Process CMOS 180nm
VDD [V] 1.8
1st Stage
VBP=0.85,
VBN=0.7
W/L 2nd Stage W/L Buffer
Stage
Cm=2pF
W/L
PMOS input 10/0.36 PMOS current
source
14/0.6 NMOS 20/0.18
PMOS cascode 10/0.36 NMOS battery 20/0.18 PMOS 20/0.18
NMOS
cascode
2/0.36 PMOS battery 80/0.18
NMOS Mirror 2/0.36 NMOS Gm 40/0.36
OP Amp/BPF AC response
56
Gain = 77.8dB (7.7k)GB=8.15MHz
Op AmpFreq. Response
BPFFreq. Response f0=1.5kHz
Q=14.83
Op Amp input-referred noise PSD
57
Op Amp’s flicker noise is dominant below about 100kHz
•BPF output noise is expected to be dominated by flicker noise of the Op amps
BPF output noise
58
BPF OUTPUT NOISE PSD [V2/Hz]
Noise
Components
Simulation
[V2]
Contribution
[%]
OA1 9.35e-8 45.9
OA2 7.39e-8 36.3
OA3 3.61e-8 17.8
Total 2.04e-7 100
Op Amp’s noise is dominant in this example
•Flicker noise of each amp’s 1st stage current mirror is dominant in this design due to low BPF frequency
BPF OUTPUT Integrated NOISE
Noise from passive components are negligible
Major noise contributors
/I20: lossy Integ., /I22: Inverting amp, /I21: Integ.
ECEN 622 (ESS)
TAMU
NOISE REDUCTION IN ACTIVE-RC FILTERS*
* K.Gharib Doust and M. S. Bakhiar, “A Method for Noise Reduction in Active-RC Circuits”, IEEE Trans. on Circuits and Systems – II,
Vol. 58, pp. 906-910, December 2011.
yy,outinin,outout VTVTV
59
60
Introduce Af to reduce noise without modifying the original transfer function.
2n2
y,x1
2
y,out212
out
y,x1
y,outy,xin,out1in,out
in
out
0Vy
outy,out
in
outin,out
1
in
0yV
VTA1
TAV
TA1
TTTAT
V
V
V
VT
V
VT
Node j must be a ground before inserting Aj
j,outin,xxyin,out
f1
j,outfy,out1
j,vfy,x1
xp
TTTT
responsefrequency original keep To
AA isThat
TkATA
TAkTA
and VkV Make
61
62
Class - AB
Design Example: rmn,outrmson,outo V524 noise integratedV,V34.1f@V,20Q,MHz5.2f
63
025.35.0j025.3A
12
s05.6
kxQ
kTA
Then
s05.6
2
s1RRQ1
2xQ
1T
1k,VV
1f
2o
22
y,outf
oo12
y,out
xp
2
Note that in order to obtain the noise reduction without Af , A1 would have to increase its power
by 150%. The Af power added is about 60% of A1. Also note that a simple short circuit of x to
node j also reduce the output noise by 2.4dB.
64
Noise Reduction Comparison Plots
Conclusions
65
Noise reduction can be done by inserting one or more reduction paths
without affecting original transfer function. These paths can be active
or passive when power is limited.
66
References
1. B. Razavi, “Design of Analog CMOS Integrated Circuits”, Preview
Edition, Mcgraw Hill, 2000.
2. D.A. Johns and K. Martin, Analog Integrated Circuit Design, New
York; Wiley, 1997.
3. M.S. Gupta, “Selected papers on Noise in Circuits and Systems”,
IEEE Press 1998.
