Sigma-Delta Fractional-N Frequency Synthesis
Scott MeningerMichael Perrott
Massachusetts Institute of TechnologyJune 7, 2004
Copyright © 2004 by Michael H. PerrottAll rights reserved.
Note: Much of this material is taken from MITOpenCourseWare
http://ocw.mit.eduCourse: 6.976
Outline
Integer-N synthesis- Bandwidth constraints
Fractional-N synthesis- Issue of fractional spursΣ∆ Fractional-N Synthesis- Quantization noise impact on the PLL
Recent developments for lowering the impact of quantization noiseConclusionsQ&A
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
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
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 )
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
Background: Classical Linearized 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 dynamic divide value variationsMore on this shortly …
Background: Classical Linearized 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]
Parameterizing in terms of G(f) helps visualize the nature (high-pass or low-pass) and gain of the noise transfer functions
Parameterized Version of Classical Model
Φvn(t)en(t)
Φout(t)Φc(t)
Φn(t)
Φnvco(t)Φnpfd(t)
fo1-G(f)
foG(f)
2πNα
VCO-referredNoise
f0
SEn(f)
PFD-referredNoise
1/T f0
SΦvn
(f)
-20 dB/dec
Divider Control
of Frequency Setting
(assume noiseless for now)
G(f) represents the PLL closed loop dynamicsG(f) is low-passNature of noise transfer very easily seen from the parameterized model
Modeling PFD Noise Multiplication
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
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/T
Want 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
Classical Fractional-N Synthesizer Architecture
1-bit
PFD LoopFilter
ref(t)
div(t)
out(t)
frac[k]Accumulator
N/N+1
carry_out[k]
e(t)
Nsd[k] = N + frac[k]
Use an accumulator to perform dithering operation- Fractional input value fed into accumulator- Carry out bit of accumulator fed into divider
Accumulator Operation
residue[k]
carry_out[k]
frac[k] =.25
1-bitM-bit
M-bitfrac[k]
Accumulatorcarry_out[k]
residue[k]
clk(t)
Carry out bit is asserted when accumulator residue reaches or surpasses its full scale value- Accumulator residue increments by input fractional
value each clock cycle
Fractional-N Synthesizer Signals with N = 4.25
phase error(t)
carry_out(t)
out(t)
div(t)
ref(t)
e(t)
Divide value set at N = 4 most of the time - Resulting frequency offset causes phase error to
accumulate- Reset phase error by “swallowing” a VCO cycle
Achieved by dividing by 5 every 4 reference cycles
The Issue of Spurious Tones
1-bit
PFD LoopFilter
ref(t)
div(t)
out(t)
frac[k]Accumulator
N/N+1
carry_out[k]
e(t)
Nsd[k] = N + frac[k]
PFD error is periodic- Note that actual PFD waveform is series of pulses – the
sawtooth waveform represents pulse width values over timePeriodic error signal creates spurious tones in synthesizer output- Ruins noise performance of synthesizer
The Phase Interpolation Technique
1-bitM-bit
M-bit
PFD LoopFilter
ref(t)
div(t)
out(t)
frac[k]Accumulator
N/N+1
carry_out[k]
e(t)
D/A
residue[k]
α
Phase error due to fractional technique is predicted by the instantaneous residue of the accumulator- Cancel out phase error based on accumulator residue
The Problem With Phase Interpolation
1-bitM-bit
M-bit
PFD LoopFilter
ref(t)
div(t)
out(t)
frac[k]Accumulator
N/N+1
carry_out[k]
e(t)
D/A
residue[k]
α
Gain matching between PFD error and scaled D/A output must be extremely precise- Any mismatch will lead to spurious tones at PLL output
Is There a Better Way?
A Better Dithering Method: Sigma-Delta Modulation
M-bit Input 1-bitD/A
Analog Output
Input
QuantizationNoise
Digital InputSpectrum
Analog OutputSpectrum
Time Domain
Frequency Domain
Σ−∆
Digital Σ−∆Modulator
Sigma-Delta dithers in a manner such that resulting quantization noise is “shaped” to high frequencies
Dither
The sigma-delta noise shaping analysis assumes a white quantization noise spectrumIn order to make the input look “sufficiently exciting” a dither signal can be added to itDithering methods are directly taken from sigma-delta ADC and DAC design- This makes sense since the synthesizer is really a DAC
(digital to phase)- Most common method is to add a random sequence to
the LSB’s of the input
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
Summary: Sources of Phase Noise in Σ∆ Synthesis
Charge-pump / Phase Detector / Reference- Low-pass filtered by PLL, dominant at low offset frequencies
VCO- High-pass filtered by PLL, dominant at high offset frequenciesΣ∆ dithered quantization noise- Low-pass filtered by PLL, noise/bandwidth tradeoff exists
-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
Digital Σ∆ N[k]Σ∆ Noise
InSynth Output
fc
Parasitic
Pole
A quick note on the linearized model
Non-linearities break the assumptions of the linear model - The shaped noise can be “folded down” to lower
frequencies due to non-linearities in the synthesizer PFD/Charge-pump design
This process is best seen through behavioral simulation
Digital Σ∆ N[k]Σ∆ Noise
InSynth Output
fc
Parasitic
Pole
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
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
3 GHz/1.2GHz