6.776 high speed communication circuits lecture 22 noise
TRANSCRIPT
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6.776High Speed Communication Circuits
Lecture 22
Noise in Integer-N and Fractional-N Frequency Synthesizers
Michael PerrottMassachusetts Institute of Technology
May 3, 2005
Copyright © 2005 by Michael H. Perrott
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2 M.H. Perrott MIT OCW
Outline of PLL Lectures
Integer-N Synthesizers- Basic blocks, modeling, and design- Frequency detection, PLL Type
Noise in Integer-N and Fractional-N Synthesizers- Noise analysis of integer-N structure- Sigma-Delta modulators applied to fractional-N
structures- Noise analysis of fractional-N structure
Design of Fractional-N Frequency Synthesizers and Bandwidth Extension Techniques- PLL Design Assistant Software- Quantization noise reduction for improved bandwidth
and noise
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3M.H. Perrott MIT OCW
Frequency Synthesizer Noise in Wireless Systems
Zin
Zo LNA To Filter
From Antennaand Bandpass
Filter
PC boardtrace
PackageInterface
LO signal
MixerRF in IF out
FrequencySynthesizer
ReferenceFrequency
VCO
f
PhaseNoise
fo
Synthesizer noise has a negative impact on system- Receiver – lower sensitivity, poorer blocking performance- Transmitter – increased spectral emissions (output spectrum must
meet a mask requirement)Noise is characterized in frequency domain
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4M.H. Perrott MIT OCW
Impact of Synthesizer Noise on TransmittersSx(f)
f
Sout(f)
ffLO
Sy(f)
ffRFfIF
x(t) y(t)
out(t)
Synthesizer
close-inphase noise
far-awayphase noise
reductionof SNR
out-of-bandemission
Synthesizer noise can be lumped into two categories- Close-in phase noise: reduces SNR of modulated signal- Far-away phase noise: creates spectral emissions
outside the desired transmit channelThis is the critical issue for transmitters
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5M.H. Perrott MIT OCW
Impact of Remaining Portion of TransmitterSx(f)
f
Sout(f)
ffLO
Sy(f)
ffRFfIF
x(t) y(t)
out(t)
Synthesizer
close-inphase noise
far-awayphase noise
reductionof SNR
out-of-bandemission
To Antenna
Band SelectFilter
PA
Power amplifier- Nonlinearity will increase out-of-band emission and create
harmonic contentBand select filter- Removes harmonic content, but not out-of-band emission
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6M.H. Perrott MIT OCW
Why is Out-of-Band Emission A Problem?
Near-far problem- Interfering transmitter closer to receiver than desired
transmitter- Out-of-emission requirements must be stringent to
prevent complete corruption of desired signal
Transmitter 2 Base
Station
Transmitter 1
Desired Channel( )
Interfering Channel( )
Relative PowerDifference (dB)
Desired Channel
Interfering Channel
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7M.H. Perrott MIT OCW
Specification of Out-of-Band Emissions
ffRF
MaximumRF OutputEmission
(dBm) M0 dBm
M1 dBm
M2 dBm
Channel Spacing= W Hz
IntegrationBandwidth= R Hz
M3 dBm
Maximum radiated power is specified in desired and adjacent channels- Desired channel power: maximum is M0 dBm- Out-of-band emission: maximum power defined as
integration of transmitted spectral density over bandwidth R centered at midpoint of each channel offset
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8M.H. Perrott MIT OCW
Example – DECT Cordless Telephone Standard
Standard for many cordless phones operating at 1.8 GHzTransmitter Specifications- Channel spacing: W = 1.728 MHz- Maximum output power: Mo = 250 mW (24 dBm)- Integration bandwidth: R = 1 MHz- Emission mask requirements
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9M.H. Perrott MIT OCW
Synthesizer Phase Noise Requirements for DECT
Calculations (see Lecture 16 of MIT OCW 6.976 for details)
Graphical display of phase noise mask
-8 dBm X1 = -29.6 dBcX2 = -51.6 dBc
-92 dBc/Hz-114 dBc/Hz
24 dBm
-30 dBm
ChannelOffset
MaskPower
Maximum Synth. NoisePower in Integration BW
Maximum Synth. Phase Noise at Channel Offset
01.728 MHz3.456 MHz
X3 = -65.6 dBc -128 dBc/Hz-44 dBm5.184 MHz
set by required transmit SNR
ffLO
SynthesizerSpectrum
(dBc)-92 dBc/Hz
-114 dBc/Hz
-128 dBc/Hz
Channel Spacing = 1.728 MHz
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10M.H. Perrott MIT OCW
Critical Specification for Phase Noise
Critical specification is defined to be the one that is hardest to meet with an assumed phase noise rolloff- Assume synthesizer phase noise rolls off at -20
dB/decadeCorresponds to VCO phase noise characteristic
For DECT transmitter synthesizer- Critical specification is -128 dBc/Hz at 5.184 MHz offset
ffLO
SynthesizerSpectrum
(dBc)-92 dBc/Hz
0 dBc
-114 dBc/Hz
-128 dBc/Hz
Channel Spacing = 1.728 MHz
Phase NoiseRolloff: -20 dB/dec
CriticalSpec.
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11M.H. Perrott MIT OCW
Receiver Blocking Performance
fRFf
Synthesizer
LNATo
IF ProcessingStage
Band SelectFilter
ChannelFilter
Band Select Filter MustPass All Channels
-73-58-39
RF Input(dBm)
fIF
fLO
f
Channel FilterBandwidth
IF Output(dBm)
f
SynthesizerSpectrum(dBc/Hz)
0 dBc
Radio receivers must operate in the presence of large interferers (called blockers)Channel filter plays critical role in removing blockers
Passes desired signal channel, rejects interferers
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12M.H. Perrott MIT OCW
Impact of Nonidealities on Blocking Performance
fRFf
Synthesizer
LNATo
IF ProcessingStage
Band SelectFilter
ChannelFilter
Band Select Filter MustPass All Channels
-73-58-39
RF Input(dBm)
fIF
fLO
f
Channel FilterBandwidth
IF Output(dBm)
Synthesizer Noise and Mixer/LNA Distortion
Produce Inband Interference
Phase Noise(dBc/Hz)
Spurious Noise(dBc)
f
SynthesizerSpectrum(dBc/Hz)
0 dBc
Blockers leak into desired band due to- Nonlinearity of LNA and mixer (IIP3)- Synthesizer phase and spurious noise
In-band interference cannot be removed by channel filter!
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13M.H. Perrott MIT OCW
Quantifying Tolerable In-Band Interference Levels
fRFf
Synthesizer
LNATo
IF ProcessingStage
Band SelectFilter
ChannelFilter
Band Select Filter MustPass All Channels
-73-58-39
RF Input(dBm)
fIF
fLO
f
Channel FilterBandwidth
IF Output(dBm)
Synthesizer Noise and Mixer/LNA Distortion
Produce Inband Interference
Phase Noise(dBc/Hz)
Spurious Noise(dBc)
f
SynthesizerSpectrum(dBc/Hz)
Min SNR: 15-20 dB
0 dBc
Digital radios quantify performance with bit error rate (BER)- Minimum BER often set at 1e-3 for many radio systems- There is a corresponding minimum SNR that must be achieved
Goal: design so that SNR with interferers is above SNRmin
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14M.H. Perrott MIT OCW
Example – DECT Cordless Telephone Standard
Receiver blocking specifications- Channel spacing: W = 1.728 MHz- Power of desired signal for blocking test: -73 dBm- Minimum bit error rate (BER) with blockers: 1e-3
Sets the value of SNRmin
Perform receiver simulations to determine SNRmin
Assume SNRmin = 15 dB for calculations to follow- Strength of interferers for blocking test
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15M.H. Perrott MIT OCW
Synthesizer Phase Noise Requirements for DECT
ffLO
SynthesizerSpectrum
(dBc)
X0 dBc
0 dBc
X1 dBcX2 dBc
RF
LO
IF
fRFf
RF Input(dBm)
fIFf
Channel FilterBandwidth
In-ChannelIF Output
(dBm)
Inband InterferenceProduced by
Synth. Phase Noise
W = 1.73 MHz
Channel Spacing= 1.73 MHz
Channel Spacing= 1.73 MHz
SNRmin: 15 dB
Y1 = 15 dB X1 = -30 dBcX2 = -49 dBcY2 = 34 dB
ChannelOffset
Relative BlockingPower
Maximum Synth. NoisePower at Channel Offset
1.728 MHz3.456 MHz
X3 = -55 dBcY3 = 40 dB5.184 MHz
0 dB X0 = -15 dBc0-92 dBc/Hz
-111 dBc/Hz
Maximum Synth. PhaseNoise at Channel Offset
-117 dBc/Hz
-77 dBc/Hz
-73 dBm-58 dBm-39 dBm-33 dBm
X3 dBc
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16M.H. Perrott MIT OCW
Graphical Display of Required Phase Noise Performance
Mark phase noise requirements at each offset frequency
Calculate critical specification for receive synthesizer- Critical specification is -117 dBc/Hz at 5.184 MHz offset
Lower performance demanded of receiver synthesizer than transmitter synthesizer in DECT applications!
ffLO
SynthesizerSpectrum
(dBc)
-92 dBc/Hz
0 dBc
-111 dBc/Hz
-117 dBc/Hz
Channel Spacing = 1.728 MHz
Phase NoiseRolloff: -20 dB/dec
CriticalSpec.
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17M.H. Perrott MIT OCW
Noise Modeling for Frequency Synthesizers
vin(t)vc(t)
vn(t)
PLL dynamicsset VCO
carrier frequency f
PhaseNoise
fo
Sout(f)
Extrinsic noise(from PLL)
out(t)
Intrinsicnoise
To PLL
PLL has an impact on VCO noise in two ways- Adds noise from various PLL circuits- Suppresses low frequency VCO noise through PLL feedback
Focus on modeling the above based on phase deviations- Simpler than dealing directly with PLL sine wave output
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18M.H. Perrott MIT OCW
Phase Deviation Model for Noise Analysis
vin(t)vc(t)
vn(t)
PLL dynamicsset VCO
carrier frequency f
PhaseNoise
fo
Frequency-domain viewSout(f)
Extrinsic noise(from PLL)
Intrinsicnoise
2πKvs
Φout
Φvn(t)
2cos(2πfot+Φout(t))out(t)
To PLL
Model the impact of noise on instantaneous phase- Relationship between PLL output and instantaneous phase
- Output spectrum
Note: Kv units are Hz/V
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19M.H. Perrott MIT OCW
Phase Noise Versus Spurious Noise
Phase noise is non-periodic
- Described as a spectral density relative to carrier power
Spurious noise is periodic
- Described as tone power relative to carrier power
SΦout(f)Sout(f)
f-fo fo
1dBc/Hz
Sout(f)
f-fo fo
dBc1
fspur
21 dspur
fspur
2
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20M.H. Perrott MIT OCW
Sources of Noise in Frequency Synthesizers
PFD ChargePump
e(t) v(t)
N
LoopFilter
DividerVCO
ref(t)
div(t)
f
Charge PumpNoise
VCO Noise
f
-20 dB/dec
1/Tf
ReferenceJitter
f
ReferenceFeedthrough
T
f
DividerJitter
Extrinsic noise sources to VCO- Reference/divider jitter and reference feedthrough- Charge pump noise
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21M.H. Perrott MIT OCW
Modeling the Impact of Noise on Output Phase of PLL
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
espur(t)Φjit[k] Φvn(t)Ιcpn(t)
Icp
VCO Noise
f0
SΦjit(f)
f0
SΙcpn(f)
f0
SEspur(f)
Divider/ReferenceJitter
ReferenceFeedthrough
Charge PumpNoise
1/T f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
Determine impact on output phase by deriving transfer function from each noise source to PLL output phase- There are a lot of transfer functions to keep track of!
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22M.H. Perrott MIT OCW
Simplified Noise Model
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
Φvn(t)en(t)
Icp
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
Refer all PLL noise sources (other than the VCO) to the PFD output- PFD-referred noise corresponds to the sum of these
noise sources referred to the PFD output
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23M.H. Perrott MIT OCW
Impact of PFD-referred Noise on Synthesizer Output
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
Φvn(t)en(t)
Icp
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
Transfer function derived using Black’s formula
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24M.H. Perrott MIT OCW
Impact of VCO-referred Noise on Synthesizer Output
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
Φvn(t)en(t)
Icp
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
Transfer function again derived from Black’s formula
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25M.H. Perrott MIT OCW
A Simpler Parameterization for PLL Transfer Functions
Define G(f) as
- A(f) is the open loop transfer function of the PLL
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
Φvn(t)en(t)
Icp
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
Always has a gainof one at DC
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26M.H. Perrott MIT OCW
Parameterize Noise Transfer Functions in Terms of G(f)
PFD-referred noise
VCO-referred noise
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27M.H. Perrott MIT OCW
Parameterized PLL Noise Model
Φvn(t)en(t)
Φout(t)Φc(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
foG(f)�πNα
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
Divider Controlof Frequency Setting
(assume noiseless for now)
PFD-referred noise is lowpass filteredVCO-referred noise is highpass filteredBoth filters have the same transition frequency values- Defined as fo
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28M.H. Perrott MIT OCW
Impact of PLL Parameters on Noise Scaling
Φvn(t)en(t)
Φout(t)Φc(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
foG(f)�πNα
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
Divider Controlof Frequency Setting
(assume noiseless for now)
Sen(f)�πNα
2
Rad
ians
2 /Hz
SΦvn(f)
f0
PFD-referred noise is scaled by square of divide value and inverse of PFD gain- High divide values lead to large multiplication of this noise
VCO-referred noise is not scaled (only filtered)
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29M.H. Perrott MIT OCW
Optimal Bandwidth Setting for Minimum Noise
Φvn(t)en(t)
Φout(t)Φc(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
foG(f)�πNα
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
Divider Controlof Frequency Setting
(assume noiseless for now)
Sen(f)�πNα
2
Rad
ians
2 /Hz
SΦvn(f)
f(fo)opt0
Optimal bandwidth is where scaled noise sources meet- Higher bandwidth will pass more PFD-referred noise- Lower bandwidth will pass more VCO-referred noise
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30M.H. Perrott MIT OCW
Resulting Output Noise with Optimal Bandwidth
Φvn(t)en(t)
Φout(t)Φc(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
foG(f)�πNα
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
Divider Controlof Frequency Setting
(assume noiseless for now)
Sen(f)�πNα
2
Rad
ians
2 /Hz
SΦvn(f)
f(fo)opt0
Rad
ians
2 /Hz SΦnpfd(f)
SΦnvco(f)
f(fo)opt0
PFD-referred noise dominates at low frequencies- Corresponds to close-in phase noise of synthesizer
VCO-referred noise dominates at high frequencies- Corresponds to far-away phase noise of synthesizer
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31M.H. Perrott MIT OCW
Analysis of Charge Pump Noise Impact
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
en(t) Φvn(t)Ιcpn(t)
Icp
VCO Noise
f0
SΙcpn(f)
Charge PumpNoise
f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
PFD-referredNoise
We can refer charge pump noise to PFD output by simply scaling it by 1/Icp
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32M.H. Perrott MIT OCW
Calculation of Charge Pump Noise Impact
Contribution of charge pump noise to overall output noise
- Need to determine impact of Icp on SIcpn(f)
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
en(t) Φvn(t)Ιcpn(t)
Icp
VCO Noise
f0
SΙcpn(f)
Charge PumpNoise
f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilter
Divider
VCO
PFD-referredNoise
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33 M.H. Perrott MIT OCW
Impact of Transistor Current Value on its Noise
M2M1
Ibias
currentsource
current bias
Id
idbias2 id
2
Cbig
WL
Charge pump noise will be related to the current it creates as
Recall that gdo is the channel resistance at zero Vds- At a fixed current density, we have
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34M.H. Perrott MIT OCW
Impact of Charge Pump Current Value on Output Noise
Recall
Given previous slide, we can say
- Assumes a fixed current density for the key transistors in the charge pump as Icp is varied
Therefore
- Want high charge pump current to achieve low noise- Limitation set by power and area considerations
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Fractional-N Frequency Synthesizers
M.H. Perrott MIT OCW
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36M.H. Perrott MIT OCW
Bandwidth Constraints for Integer-N Synthesizers
PFD LoopFilter(1/T = 20 MHz)
ref(t) out(t)
N[k]
Divider
1/T Loop FilterBandwidth << 1/T
PFD output has a periodicity of 1/T- 1/T = reference frequency
Loop filter must have a bandwidth << 1/T- PFD output pulses must be filtered out and average value
extracted
Closed loop PLL bandwidth often chosen to be afactor of ten lower than 1/T
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37M.H. Perrott MIT OCW
Bandwidth Versus Frequency Resolution
PFD LoopFilter
1.80 1.82 GHz
(1/T = 20 MHz)ref(t) out(t)
out(t)
Sout(f)
N[k]
N[k]9091
Divider
frequency resolution = 1/T
1/T
1/T Loop FilterBandwidth << 1/T
Frequency resolution set by reference frequency (1/T)- Higher resolution achieved by lowering 1/T
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38M.H. Perrott MIT OCW
Increasing Resolution in Integer-N Synthesizers
PFD LoopFilter
1.80 1.8002 GHz
(1/T = 200 kHz)ref(t) out(t)
out(t)
Sout(f)
N[k]
N[k]90009001
Divider
frequency resolution = 1/T
1/T
1/T Loop FilterBandwidth << 1/T
10020 MHz
Use a reference divider to achieve lower 1/T- Leads to a low PLL bandwidth ( < 20 kHz here )
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39M.H. Perrott MIT OCW
The Issue of Noise
PFD LoopFilter
1.80 1.8002 GHz
(1/T = 200 kHz)ref(t) out(t)
out(t)
Sout(f)
N[k]
N[k]90009001
Divider
frequency resolution = 1/T
1/T
1/T Loop FilterBandwidth << 1/T
10020 MHz
Lower 1/T leads to higher divide value- Increases PFD noise at synthesizer output
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40M.H. Perrott MIT OCW
Modeling PFD Noise Multiplication
Influence of PFD noise seen in above model- PFD spectral density multiplied by N2 before influencing PLL
output phase noise
Φvn(t)en(t)
Φout(t)Φc(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
foG(f)�πNα
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
Divider Controlof Frequency Setting
(assume noiseless for now)
Sen(f)�πNα
2
Rad
ians
2 /Hz
SΦvn(f)
f(fo)opt0
Rad
ians
2 /Hz SΦnpfd(f)
SΦnvco(f)
f(fo)opt0
High divide values high phase noise at low frequencies
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41M.H. Perrott MIT OCW
Fractional-N Frequency Synthesizers
Break constraint that divide value be integer- Dither divide value dynamically to achieve fractional values- Frequency resolution is now arbitrary regardless of 1/TWant high 1/T to allow a high PLL bandwidth
DitheringModulator
PFD LoopFilter
1.80 1.82 GHz
(1/T = 20 MHz)ref(t) out(t)
out(t)
Sout(f)
N[k]Nsd[k]
Nsd[k]90
90
91
91Divider
frequency resolution << 1/T
1/T
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42M.H. Perrott MIT OCW
A Nice Dithering Method: Sigma-Delta Modulation
Sigma-Delta dithers in a manner such that resulting quantization noise is “shaped” to high frequencies
M-bit Input 1-bitD/A
Analog Output
Input
QuantizationNoise
Digital InputSpectrum
Analog OutputSpectrum
Time Domain
Frequency Domain
Σ−∆
Digital Σ−∆Modulator
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43M.H. Perrott MIT OCW
Linearized Model of Sigma-Delta Modulator
x[k] y[k] y[k]x[k]q[k]
r[k]
z=ej2πfT
z=ej2πfT
NTF
STF
Σ−∆
Hn(z)
Hs(z)
1
Sr(ej2πfT)= 112
Sq(ej2πfT)= |Hn(ej2πfT)|2112
Composed of two transfer functions relating input and noise to output- Signal transfer function (STF)
Filters input (generally undesirable)- Noise transfer function (NTF)
Filters (i.e., shapes) noise that is assumed to be white
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44M.H. Perrott MIT OCW
Example: Cutler Sigma-Delta Topology
x[k] u[k]
e[k]
y[k]
H(z) - 1
Output is quantized in a multi-level fashionError signal, e[k], represents the quantization errorFiltered version of quantization error is fed back to input- H(z) is typically a highpass filter whose first tap value is 1
i.e., H(z) = 1 + a1z-1 + a2 z-2
- H(z) – 1 therefore has a first tap value of 0Feedback needs to have delay to be realizable
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45M.H. Perrott MIT OCW
Linearized Model of Cutler Topology
x[k] u[k]
e[k]
y[k]
H(z) - 1
x[k] u[k]r[k]
e[k]
y[k]
H(z) - 1
Represent quantizer block as a summing junction in which r[k] represents quantization error- Note:
It is assumed that r[k] has statistics similar to white noise- This is a key assumption for modeling – often not true!
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46M.H. Perrott MIT OCW
Calculation of Signal and Noise Transfer Functions
x[k] u[k]
e[k]
y[k]
H(z) - 1
x[k] u[k]r[k]
e[k]
y[k]
H(z) - 1
Calculate using Z-transform of signals in linearizedmodel
- NTF: Hn(z) = H(z)- STF: Hs(z) = 1
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47M.H. Perrott MIT OCW
A Common Choice for H(z)
m = 1
m = 2
m = 3
Mag
nitu
de
0Frequency (Hz)
1/(2T)
8
7
6
5
4
3
2
1
0
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48M.H. Perrott MIT OCW
Example: First Order Sigma-Delta Modulator
Choose NTF to be
Plot of output in time and frequency domains with input of
00
Am
plitu
de
Mag
nitu
de (d
B)
Sample Number 200 0
1
Frequency (Hz) 1/(2T)
x[k] u[k]
e[k]
y[k]
H(z) - 1
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49M.H. Perrott MIT OCW
Example: Second Order Sigma-Delta Modulator
Choose NTF to be
Plot of output in time and frequency domains with input of
x[k] u[k]
e[k]
y[k]
H(z) - 1
Mag
nitu
de (d
B)
0 Sample Number 200 0-1
Am
plitu
de
2
1
0
Frequency (Hz) 1/(2T)
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50M.H. Perrott MIT OCW
Example: Third Order Sigma-Delta Modulator
Choose NTF to be
Plot of output in time and frequency domains with input of
x[k] u[k]
e[k]
y[k]
H(z) - 1
Mag
nitu
de (d
B)
0 0Sample Number 200
Am
plitu
de
4
3
2
1
0
-1
-2
-3Frequency (Hz) 1/(2T)
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51M.H. Perrott MIT OCW
Observations
Low order Sigma-Delta modulators do not appear to produce “shaped” noise very well- Reason: low order feedback does not properly
“scramble” relationship between input and quantization noise
Quantization noise, r[k], fails to be whiteHigher order Sigma-Delta modulators provide much better noise shaping with fewer spurs- Reason: higher order feedback filter provides a much
more complex interaction between input and quantization noise
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52M.H. Perrott MIT OCW
Warning: Higher Order Modulators May Still Have Tones
Quantization noise, r[k], is best whitened when a “sufficiently exciting” input is applied to the modulator- Varying input and high order helps to “scramble”
interaction between input and quantization noiseWorst input for tone generation are DC signals that are rational with a low valued denominator- Examples (third order modulator):
Mag
nitu
de (d
B)
0 Frequency (Hz) 1/(2T)
Mag
nitu
de (d
B)
0 Frequency (Hz) 1/(2T)
x[k] = 0.1 x[k] = 0.1 + 1/1024
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53M.H. Perrott MIT OCW
Cascaded Sigma-Delta Modulator Topologies
x[k]
y[k]
q1[k]Σ∆M1[k]
y1[k]
q2[k]Σ∆M2[k]
y2[k]
Σ∆M3[k]
y3[k]
M 1 1
Digital Cancellation LogicMultibitoutput
Achieve higher order shaping by cascading low order sections and properly combining their outputsAdvantage over single loop approach- Allows pipelining to be applied to implementation
High speed or low power applications benefitDisadvantages- Relies on precise matching requirements when combining
outputs (not a problem for digital implementations)- Requires multi-bit quantizer (single loop does not)
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54M.H. Perrott MIT OCW
MASH topology
x[k]
y[k]
r1[k]Σ∆M1[k]
y1[k]
r2[k]Σ∆M2[k]
y2[k]
u[k]
Σ∆M3[k]
y3[k]
M 1 1
1-z-1 (1-z-1)2
Cascade first order sectionsCombine their outputs after they have passed through digital differentiators
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55M.H. Perrott MIT OCW
Calculation of STF and NTF for MASH topology (Step 1)
Individual output signals of each first order modulator
Addition of filtered outputs
x[k]
y[k]
r1[k]Σ∆M1[k]
y1[k]
r2[k]Σ∆M2[k]
y2[k]
u[k]
Σ∆M3[k]
y3[k]
M 1 1
1-z-1 (1-z-1)2
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56M.H. Perrott MIT OCW
Calculation of STF and NTF for MASH topology (Step 1)
x[k]
y[k]
r1[k]Σ∆M1[k]
y1[k]
r2[k]Σ∆M2[k]
y2[k]
u[k]
Σ∆M3[k]
y3[k]
M 1 1
1-z-1 (1-z-1)2
Overall modulator behavior
- STF: Hs(z) = 1- NTF: Hn(z) = (1 – z-1)3
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57M.H. Perrott MIT OCW
Sigma-Delta Frequency Synthesizers
PFD ChargePump
Nsd[m]
out(t)e(t)
Σ−∆Modulator
v(t)
N[m]
LoopFilter
DividerVCO
ref(t)
div(t)
MM+1
Fout = M.F Fref
f
Σ−∆Quantization
Noise
Fref
Riley et. al.,JSSC, May 1993
Use Sigma-Delta modulator rather than accumulator to perform dithering operation- Achieves much better spurious performance than classical
fractional-N approach
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58M.H. Perrott MIT OCW
Background: The Need for A Better PLL Model
Φdiv[k]
Φref [k] KV
jf
v(t) Φout(t)H(f)
1N
�πα e(t)
Φvn(t)en(t)
Icp
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn(f)
-20 dB/dec
PFDChargePump
LoopFilterDivider
VCO
N[k]
Classical PLL model- Predicts impact of PFD and VCO referred noise sources- Does not allow straightforward modeling of impact due
to divide value variationsThis is a problem when using fractional-N approach
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59M.H. Perrott MIT OCW
A PLL Model Accommodating Divide Value Variations
Φdiv[k]
Φref[k] KV
jf
v(t) Φout(t)H(f)
1Nnom
2π z-1
z=ej2πfT1 - z-1n[k]
T1
Φd[k]
T2π
e(t)
PFD
Divider
LoopFilterC.P. VCO
α
Tristate: α=1XOR: α=2
Icp
en(t)f
0
SEn(f)
PFD-referredNoise
1/TΦvn(t)
VCO-referredNoise
f0
SΦvn(f)
-20 dB/dec
See derivation in Perrott et. al., “A Modeling Approach for Sigma-Delta Fractional-N Frequency Synthesizers …”, JSSC, Aug 2002
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60M.H. Perrott MIT OCW
Parameterized Version of New Model
Φvn(t)
T G(f)
2π z-1
1 - z-1
n[k]
en(t)
Φout(t)Φc(t)Φd(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
fo
fo
G(f)�πNnom
z=ej2πfT
α
T�πα
1Ι
espur(t)
Φjit[k]
Ιcpn(t)
G(f)n[k] Φc(t)
jf1Fc(t)
Freq. PhaseD/A and Filter
Alternate Representation
fo
Noise
Dividevalue
variation
f0
SEn(f)
PFD-referredNoise
1/T
VCO-referredNoise
f0
SΦvn(f)
-20 dB/dec
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61M.H. Perrott MIT OCW
Spectral Density Calculations
y[k]x[k]H(ej2πfT)
y(t)x[k]H(f)
y(t)x(t)H(f)case (a): CT CT
case (b): DT DT
case (c): DT CT
Case (a):
Case (b):
Case (c):
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62M.H. Perrott MIT OCW
Example: Calculate Impact of Ref/Divider Jitter (Step 1)
∆tjit[k]
(∆tjit)rms= β sec.
�π Φjit[k]T
T�πT f
0
SΦjit(ej2πfT)
t
Div(t)
β22
Assume jitter is white- i.e., each jitter value independent of values at other time
instantsCalculate spectra for discrete-time input and output- Apply case (b) calculation
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63M.H. Perrott MIT OCW
Example: Calculate Impact of Ref/Divider Jitter (Step 2)
Φjit[k]fo
fo
G(f)Nnom
Φn (t)
SΦn(f)
Nnom
f0
2
T
TT1 �π
Tβ2
2
∆tjit[k]
(∆tjit)rms= β sec.
�π Φjit[k]T
T�πT f
0
SΦjit(ej2πfT)
t
Div(t)
β22
Compute impact on output phase noise of synthesizer- We now apply case (c) calculation
- Note that G(f) = 1 at DC
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64M.H. Perrott MIT OCW
Now Consider Impact of Divide Value Variations
Φvn(t)
T G(f)
2π z-1
1 - z-1
n[k]
en(t)
Φout(t)Φc(t)Φd(t)Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
fo
fo
G(f)�πNnom
z=ej2πfT
α
T�πα
1Ι
espur(t)
Φjit[k]
Ιcpn(t)
G(f)n[k] Φc(t)
jf1Fc(t)
Freq. PhaseD/A and Filter
Alternate Representation
fo
Noise
Dividevalue
variation
f0
SEn(f)
PFD-referredNoise
1/T
VCO-referredNoise
f0
SΦvn(f)
-20 dB/dec
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65M.H. Perrott MIT OCW
Divider Impact For Classical Vs Fractional-N Approaches
G(f)n[k] Fout(t)
D/A and Filter
T1n(t)
1/T1
fo
G(f)n[k] Fout(t)
D/A and Filter
T1
1/T1
foDitheringModulator
nsd(t) nsd[k]
Classical Synthesizer
Fractional-N Synthesizer
Note: 1/T block represents sampler (to go from CT to DT)
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66M.H. Perrott MIT OCW
Focus on Sigma-Delta Frequency Synthesizer
G(f)n[k]
n[k]
nsd[k]
nsd[k]
nsd(t) Fout(t)
Fout(t)
D/A and Filter
T1
1/T1
foΣ−∆
freq=1/T
Divide value can take on fractional values- Virtually arbitrary resolution is possible
PLL dynamics act like lowpass filter to remove much of the quantization noise
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67M.H. Perrott MIT OCW
Quantifying the Quantization Noise Impact
Calculate by simply attaching Sigma-Delta model- We see that quantization noise is integrated and then
lowpass filtered before impacting PLL output
Φvn(t)
T G(f)
2π z-1
1 - z-1
n[k]nsd[k] Φn[k]
En(t)
Φout(t)Φdiv(t)Φtn,pll(t)
fo1-G(f)
fo
fo
G(f)�πNnom
z=ej2πfT
α
f0 f0
-20 dB/decSEn
(f)
Sq(ej2πfT)
SΦvn(f)
1/T
f0
z=ej2πfT
Σ−∆
q[k]
r[k]
NTF
STF
Hn(z)
Hs(z)
Sr(ej2πfT)= 112
PFD-referredNoise
VCO-referredNoise
Σ−∆Quantization
Noise
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68M.H. Perrott MIT OCW
A Well Designed Sigma-Delta Synthesizer
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
Frequency
Spe
ctra
l Den
sity
(dB
c/H
z)
f0 1/T
10 kHz 100 kHz 1 MHz 10 MHz
PFD-referrednoise
SΦout,En(f)
VCO-referred noise
SΦout,vn(f)
Σ−∆noise
SΦout,∆Σ(f)
fo = 84 kHz
Order of G(f) is set to equal to the Sigma-Delta order- Sigma-Delta noise falls at -20 dB/dec above G(f) bandwidth
Bandwidth of G(f) is set low enough such that synthesizer noise is dominated by intrinsic PFD and VCO noise
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69M.H. Perrott MIT OCW
Impact of Increased PLL Bandwidth
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
Frequency
Spe
ctra
l Den
sity
(dB
c/H
z)
f0 1/T
10 kHz 100 kHz 1 MHz 10 MHz
PFD-referrednoise
SΦout,En(f)
VCO-referred noise
SΦout,vn(f)
Σ−∆noise
SΦout,∆Σ(f)
f0 1/T
10 kHz 100 kHz 1 MHz 10 MHz
Frequency
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
Spe
ctra
l Den
sity
(dB
c/H
z)
PFD-referrednoise
SΦout,En(f)
VCO-referred noise
SΦout,vn(f)
Σ−∆noise
SΦout,∆Σ(f)
fo = 84 kHz fo = 160 kHz
Allows more PFD noise to pass throughAllows more Sigma-Delta noise to pass throughIncreases suppression of VCO noise
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70M.H. Perrott MIT OCW
Impact of Increased Sigma-Delta Order
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
Frequency
Spe
ctra
l Den
sity
(dB
c/H
z)
f0 1/T
10 kHz 100 kHz 1 MHz 10 MHz
PFD-referrednoise
SΦout,En(f)
VCO-referred noise
SΦout,vn(f)
Σ−∆noise
SΦout,∆Σ(f)
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
-60
Spe
ctra
l Den
sity
(dB
c/H
z)f0 1/T
10 kHz 100 kHz 1 MHz 10 MHz
PFD-referrednoise
SΦout,En(f)
VCO-referred noise
SΦout,vn(f)
Σ−∆noise
SΦout,∆Σ(f)
Frequency
m = 2 m = 3
PFD and VCO noise unaffectedSigma-Delta noise no longer attenuated by G(f) such that a -20 dB/dec slope is achieved above its bandwidth