6.776
High Speed Communication Circuits
Lecture 22
Noise in Integer-N and Fractional-N Frequency
Synthesizers
Michael Perrott
Massachusetts Institute of Technology
May 3, 2005
Copyright © 2005 by Michael H. Perrott
Outline of PLL Lectures
ƒ
Integer-N Synthesizers
ƒ
Noise in Integer-N and Fractional-N Synthesizers
ƒ
Design of Fractional-N Frequency Synthesizers and
Bandwidth Extension Techniques
- Basic blocks, modeling, and design
- Frequency detection, PLL Type
- Noise analysis of integer-N structure
- Sigma-Delta modulators applied to fractional-N
structures
- Noise analysis of fractional-N structure
- PLL Design Assistant Software
- Quantization noise reduction for improved bandwidth
and noise
M.H. Perrott
MIT OCW 2
Frequency Synthesizer Noise in Wireless Systems
Zin
From Antenna
and Bandpass
Filter
PC board
trace
Zo
Mixer
Package
Interface
RF in
IF out
To Filter
LNA
LO signal
VCO
Reference
Frequency
Frequency
Synthesizer
Phase
Noise
fo
ƒ
ƒ
f
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
M.H. Perrott
MIT OCW 3
Impact of Synthesizer Noise on Transmitters
Sx(f)
Sy(f)
reduction
of SNR
out-of-band
emission
f
fIF
x(t)
f
fRF
y(t)
out(t)
Sout(f)
close-in
phase noise
far-away
phase noise
Synthesizer
fLO
ƒ
f
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 channel
ƒ This is the critical issue for transmitters
M.H. Perrott
MIT OCW 4
Impact of Remaining Portion of Transmitter
Sx(f)
Sy(f)
reduction
of SNR
out-of-band
emission
f
fIF
x(t)
f
fRF
Band Select
Filter
y(t)
PA
out(t)
Sout(f)
close-in
phase noise
far-away
phase noise
Synthesizer
fLO
ƒ
ƒ
To
Antenna
f
Power amplifier
- Nonlinearity will increase out-of-band emission and create
harmonic content
Band select filter
- Removes harmonic content, but not out-of-band emission
M.H. Perrott
MIT OCW 5
Why is Out-of-Band Emission A Problem?
Transmitter
1
Interfering
Channel
( Desired
Channel)
Relative
Power
Difference
(dB)
Transmitter
2
( Interfering
Channel )
ƒ
Desired
Channel
Base
Station
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
M.H. Perrott
MIT OCW 6
Specification of Out-of-Band Emissions
Maximum
RF Output
Emission
(dBm)
M0 dBm
M1 dBm
M3 dBm
Integration
Bandwidth
= R Hz
M2 dBm
f
fRF
Channel Spacing
= W Hz
ƒ
Maximum radiated power is specified in desired and
adjacent channels
- Desired channel power: maximum is M dBm
- Out-of-band emission: maximum power defined as
0
integration of transmitted spectral density over
bandwidth R centered at midpoint of each channel offset
M.H. Perrott
MIT OCW 7
Example – DECT Cordless Telephone Standard
ƒ
ƒ
Standard for many cordless phones operating at 1.8 GHz
Transmitter Specifications
- Channel spacing: W = 1.728 MHz
- Maximum output power: M = 250 mW (24 dBm)
- Integration bandwidth: R = 1 MHz
- Emission mask requirements
o
M.H. Perrott
MIT OCW 8
Synthesizer Phase Noise Requirements for DECT
ƒ
Calculations (see Lecture 16 of MIT OCW 6.976 for details)
Channel
Offset
0
1.728 MHz
Mask
Power
24 dBm
-8 dBm
3.456 MHz -30 dBm
5.184 MHz -44 dBm
ƒ
Maximum Synth. Noise
Power in Integration BW
Maximum Synth. Phase Noise
at Channel Offset
set by required transmit SNR
X1 = -29.6 dBc
-92 dBc/Hz
X2 = -51.6 dBc
X3 = -65.6 dBc
-114 dBc/Hz
-128 dBc/Hz
Graphical display of phase noise mask
Synthesizer
Spectrum
(dBc)
-92 dBc/Hz
-114 dBc/Hz
-128 dBc/Hz
fLO
M.H. Perrott
Channel Spacing = 1.728 MHz
f
MIT OCW 9
Critical Specification for Phase Noise
ƒ
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/decade
ƒ Corresponds to VCO phase noise characteristic
ƒ
For DECT transmitter synthesizer
- Critical specification is -128 dBc/Hz at 5.184 MHz offset
Synthesizer
Spectrum
(dBc)
0 dBc
Phase Noise
Rolloff: -20 dB/dec
-92 dBc/Hz
-114 dBc/Hz
Critical
Spec.
-128 dBc/Hz
fLO
M.H. Perrott
Channel Spacing = 1.728 MHz
f
MIT OCW 10
Receiver Blocking Performance
RF Input
(dBm)
Band Select Filter Must
Pass All Channels
IF Output
(dBm)
-39
-58
-73
fRF
f
Channel Filter
Bandwidth
f
fIF
Band Select
Filter
Channel
Filter
To
IF Processing
Stage
LNA
Synthesizer
Spectrum
(dBc/Hz)
0 dBc
Synthesizer
fLO
ƒ
ƒ
f
Radio receivers must operate in the presence of large
interferers (called blockers)
Channel filter plays critical role in removing blockers
ƒ Passes desired signal channel, rejects interferers
M.H. Perrott
MIT OCW 11
Impact of Nonidealities on Blocking Performance
RF Input
(dBm)
Band Select Filter Must
Pass All Channels
IF Output
(dBm)
-39
-58
-73
Channel Filter
Bandwidth
Synthesizer Noise
and Mixer/LNA Distortion
Produce Inband Interference
fRF
f
f
fIF
Band Select
Filter
Channel
Filter
To
IF Processing
Stage
LNA
Synthesizer
Spectrum
(dBc/Hz)
0 dBc
Synthesizer
fLO
Phase Noise
(dBc/Hz)
Spurious Noise
(dBc)
f
ƒ
Blockers leak into desired band due to
ƒ
In-band interference cannot be removed by channel filter!
- Nonlinearity of LNA and mixer (IIP3)
- Synthesizer phase and spurious noise
M.H. Perrott
MIT OCW 12
Quantifying Tolerable In-Band Interference Levels
RF Input
(dBm)
Band Select Filter Must
Pass All Channels
IF Output
(dBm)
-39
-58
-73
Channel Filter
Bandwidth
Synthesizer Noise
and Mixer/LNA Distortion
Produce Inband Interference
Min SNR: 15-20 dB
fRF
f
f
fIF
Band Select
Filter
Channel
Filter
To
IF Processing
Stage
LNA
Synthesizer
Spectrum
(dBc/Hz)
0 dBc
Synthesizer
fLO
Phase Noise
(dBc/Hz)
Spurious Noise
(dBc)
f
ƒ
Digital radios quantify performance with bit error rate (BER)
ƒ
Goal: design so that SNR with interferers is above SNRmin
- Minimum BER often set at 1e-3 for many radio systems
- There is a corresponding minimum SNR that must be achieved
M.H. Perrott
MIT OCW 13
Example – DECT Cordless Telephone Standard
ƒ
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
-
ƒ Sets the value of SNRmin
ƒ Perform receiver simulations to determine SNRmin
ƒ Assume SNRmin = 15 dB for calculations to follow
Strength of interferers for blocking test
M.H. Perrott
MIT OCW 14
Synthesizer Phase Noise Requirements for DECT
RF Input
(dBm)
-33 dBm
-39 dBm
-58 dBm
-73 dBm
In-Channel
IF Output
(dBm)
Channel Filter
Bandwidth
W = 1.73 MHz
Inband Interference
Produced by
Synth. Phase Noise
SNRmin: 15 dB
fRF
Channel Spacing
= 1.73 MHz
Synthesizer
Spectrum
(dBc)
X3 dBc
f
fIF
RF
f
IF
0 dBc
X0 dBc
X1 dBc
X2 dBc
fLO
LO
f
Channel Spacing
= 1.73 MHz
Channel Relative Blocking Maximum Synth. Noise
Offset
Power
Power at Channel Offset
Maximum Synth. Phase
Noise at Channel Offset
0
1.728 MHz
0 dB
Y1 = 15 dB
X0 = -15 dBc
X1 = -30 dBc
-77 dBc/Hz
3.456 MHz
Y2 = 34 dB
Y3 = 40 dB
X2 = -49 dBc
X3 = -55 dBc
-111 dBc/Hz
-117 dBc/Hz
5.184 MHz
M.H. Perrott
-92 dBc/Hz
MIT OCW 15
Graphical Display of Required Phase Noise Performance
ƒ
Mark phase noise requirements at each offset frequency
Synthesizer
Spectrum
(dBc)
0 dBc
Phase Noise
Rolloff: -20 dB/dec
-92 dBc/Hz
Critical
Spec.
-111 dBc/Hz
-117 dBc/Hz
fLO
ƒ
f
Channel Spacing = 1.728 MHz
Calculate critical specification for receive synthesizer
- Critical specification is -117 dBc/Hz at 5.184 MHz offset
ƒ Lower performance demanded of receiver synthesizer than
transmitter synthesizer in DECT applications!
M.H. Perrott
MIT OCW 16
Noise Modeling for Frequency Synthesizers
Sout(f)
PLL dynamics
set VCO
carrier frequency
vc(t)
Phase
Noise
Extrinsic noise
(from PLL)
Intrinsic
noise
vn(t)
vin(t)
f
fo
out(t)
To PLL
ƒ
PLL has an impact on VCO noise in two ways
ƒ
Focus on modeling the above based on phase deviations
- Adds noise from various PLL circuits
- Suppresses low frequency VCO noise through PLL feedback
- Simpler than dealing directly with PLL sine wave output
M.H. Perrott
MIT OCW 17
Phase Deviation Model for Noise Analysis
Note: Kv units are Hz/V
PLL dynamics
set VCO
carrier frequency
vc(t)
Frequency-domain view
Sout(f)
Phase
Noise
Extrinsic noise
(from PLL)
Intrinsic
noise
vn(t)
vin(t)
2πKv
s
Φvn(t)
f
fo
Φout
2cos(2πfot+Φout(t))
out(t)
To PLL
ƒ
Model the impact of noise on instantaneous phase
- Relationship between PLL output and instantaneous phase
- Output spectrum
M.H. Perrott
MIT OCW 18
Phase Noise Versus Spurious Noise
ƒ
Phase noise is non-periodic
Sout(f)
1
dBc/Hz
fo
-fo
SΦout(f)
f
- Described as a spectral density relative to carrier power
ƒ
Spurious noise is periodic
dBc
-fo
fspur
Sout(f)
1 dspur 2
2 fspur
1
fo
f
- Described as tone power relative to carrier power
M.H. Perrott
MIT OCW 19
Sources of Noise in Frequency Synthesizers
Reference
Jitter
Reference
Feedthrough
Charge Pump
Noise
VCO Noise
-20 dB/dec
f
T
ref(t)
1/T
f
f
e(t) Charge
PFD
Pump
Loop
Filter
f
v(t)
VCO
Divider
div(t)
Divider
Jitter
N
f
ƒ
Extrinsic noise sources to VCO
- Reference/divider jitter and reference feedthrough
- Charge pump noise
M.H. Perrott
MIT OCW 20
Modeling the Impact of Noise on Output Phase of PLL
Divider/Reference
Jitter
Reference
Feedthrough
Charge Pump
Noise
S Espur(f)
S Φjit(f)
VCO Noise
S Φvn(f)
S Ιcpn(f)
-20 dB/dec
f
0
0
Φjit[k]
Φref [k]
f
π
PFD
f
0
e(t)
f
0
Ιcpn(t)
espur(t)
α
Φdiv[k]
1/T
Φvn(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
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!
M.H. Perrott
MIT OCW
21
Simplified Noise Model
PFD-referred
Noise
S En(f)
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
0
1/T
f
Φvn(t)
en(t)
Φref [k]
α
π
Φdiv[k]
PFD
f
0
e(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
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
M.H. Perrott
MIT OCW 22
Impact of PFD-referred Noise on Synthesizer Output
PFD-referred
Noise
S En(f)
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
0
1/T
f
Φvn(t)
en(t)
Φref [k]
α
π
Φdiv[k]
PFD
f
0
e(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
Transfer function derived using Black’s formula
M.H. Perrott
MIT OCW 23
Impact of VCO-referred Noise on Synthesizer Output
PFD-referred
Noise
S En(f)
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
0
1/T
f
Φvn(t)
en(t)
Φref [k]
α
π
Φdiv[k]
PFD
f
0
e(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
Transfer function again derived from Black’s formula
M.H. Perrott
MIT OCW 24
A Simpler Parameterization for PLL Transfer Functions
PFD-referred
Noise
S En(f)
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
0
1/T
f
Φvn(t)
en(t)
Φref [k]
α
π
Φdiv[k]
PFD
f
0
e(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
Define G(f) as
Always has a gain
of one at DC
- A(f) is the open loop transfer function of the PLL
M.H. Perrott
MIT OCW 25
Parameterize Noise Transfer Functions in Terms of G(f)
ƒ
PFD-referred noise
ƒ
VCO-referred noise
M.H. Perrott
MIT OCW 26
Parameterized PLL Noise Model
PFD-referred
Noise
S En(f)
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
1/T
0
f
0
en(t)
π
α N G(f)
fo
f
Φvn(t)
fo
1-G(f)
Φnvco(t)
Φnpfd(t)
Φn(t)
Divider Control
Φc(t)
of Frequency Setting
(assume noiseless for now)
ƒ
ƒ
ƒ
Φout(t)
PFD-referred noise is lowpass filtered
VCO-referred noise is highpass filtered
Both filters have the same transition frequency values
- Defined as f
M.H. Perrott
o
MIT OCW 27
Impact of PLL Parameters on Noise Scaling
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
1/T
0
f
0
en(t)
f
2
π
α N S e n (f)
Radians2/Hz
PFD-referred
Noise
S En(f)
S Φvn(f)
Φvn(t)
f
0
fo
π
α N G(f)
fo
1-G(f)
Φnvco(t)
Φnpfd(t)
Φn(t)
Divider Control
Φc(t)
of Frequency Setting
(assume noiseless for now)
ƒ
ƒ
Φout(t)
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)
M.H. Perrott
MIT OCW 28
Optimal Bandwidth Setting for Minimum Noise
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
1/T
0
f
0
en(t)
f
2
π
α N S e n (f)
Radians2/Hz
PFD-referred
Noise
S En(f)
S Φvn(f)
Φvn(t)
f
0
fo
π
α N G(f)
fo
(fo)opt
1-G(f)
Φnvco(t)
Φnpfd(t)
Φn(t)
Divider Control
Φc(t)
of Frequency Setting
(assume noiseless for now)
ƒ
Φout(t)
Optimal bandwidth is where scaled noise sources meet
- Higher bandwidth will pass more PFD-referred noise
- Lower bandwidth will pass more VCO-referred noise
M.H. Perrott
MIT OCW 29
Resulting Output Noise with Optimal Bandwidth
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
1/T
0
f
0
en(t)
f
2
π
α N S e n (f)
Radians2/Hz
PFD-referred
Noise
S En(f)
S Φvn(f)
Φvn(t)
f
0
fo
1-G(f)
Φnvco(t)
Φnpfd(t)
Φn(t)
Divider Control
Φc(t)
of Frequency Setting
(assume noiseless for now)
Φout(t)
S Φnpfd(f)
Radians2/Hz
fo
π
α N G(f)
S Φnvco(f)
f
0
ƒ
(fo)opt
(fo)opt
PFD-referred noise dominates at low frequencies
- Corresponds to close-in phase noise of synthesizer
ƒ VCO-referred noise dominates at high frequencies
Corresponds to far-away phase noise of synthesizer
M.H. Perrott
MIT OCW
30
Analysis of Charge Pump Noise Impact
Charge Pump
Noise
VCO Noise
S Φvn(f)
S Ιcpn(f)
-20 dB/dec
PFD-referred
Noise
α
π
Φdiv[k]
PFD
e(t)
f
0
Ιcpn(t)
en(t)
Φref [k]
f
0
Φvn(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
We can refer charge pump noise to PFD output by
simply scaling it by 1/Icp
M.H. Perrott
MIT OCW 31
Calculation of Charge Pump Noise Impact
Charge Pump
Noise
VCO Noise
S Φvn(f)
S Ιcpn(f)
-20 dB/dec
PFD-referred
Noise
α
π
Φdiv[k]
PFD
e(t)
f
0
Ιcpn(t)
en(t)
Φref [k]
f
0
Φvn(t)
v(t)
Icp
H(f)
Charge
Pump
Loop
Filter
KV
Φout(t)
jf
VCO
1
N
Divider
ƒ
Contribution of charge pump noise to overall output noise
- Need to determine impact of I
cp
M.H. Perrott
on SIcpn(f)
MIT OCW 32
Impact of Transistor Current Value on its Noise
Id
Ibias
id2bias
current
bias
M1
id2 W
L
M2
Cbig
current
source
ƒ
Charge pump noise will be related to the current it
creates as
ƒ
Recall that gdo is the channel resistance at zero Vds
- At a fixed current density, we have
M.H. Perrott
MIT OCW 33
Impact of Charge Pump Current Value on Output Noise
ƒ
Recall
ƒ
Given previous slide, we can say
- Assumes a fixed current density for the key transistors
ƒ
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
M.H. Perrott
MIT OCW 34
Fractional-N Frequency Synthesizers
M.H. Perrott
MIT OCW
Bandwidth Constraints for Integer-N Synthesizers
1/T
ref(t)
PFD
(1/T = 20 MHz)
Loop Filter
Bandwidth << 1/T
Loop
Filter
out(t)
Divider
N[k]
ƒ
PFD output has a periodicity of 1/T
ƒ
Loop filter must have a bandwidth << 1/T
- 1/T = reference frequency
- PFD output pulses must be filtered out and average value
extracted
Closed loop PLL bandwidth often chosen to be a
factor of ten lower than 1/T
M.H. Perrott
MIT OCW 36
Bandwidth Versus Frequency Resolution
1/T
ref(t)
(1/T = 20 MHz)
PFD
Loop Filter
Bandwidth << 1/T
Loop
Filter
out(t)
Divider
N[k]
Sout(f)
91
N[k]
90
1/T
out(t)
frequency resolution = 1/T
ƒ
1.80 1.82
GHz
Frequency resolution set by reference frequency (1/T)
- Higher resolution achieved by lowering 1/T
M.H. Perrott
MIT OCW 37
Increasing Resolution in Integer-N Synthesizers
1/T
20 MHz
ref(t)
100
(1/T = 200 kHz)
PFD
Loop Filter
Bandwidth << 1/T
Loop
Filter
out(t)
Divider
N[k]
Sout(f)
9001
N[k]
9000
1/T
out(t)
frequency resolution = 1/T
ƒ
1.80 1.8002
GHz
Use a reference divider to achieve lower 1/T
Leads to a low PLL bandwidth ( < 20 kHz here )
M.H. Perrott
MIT OCW 38
The Issue of Noise
1/T
20 MHz
ref(t)
100
(1/T = 200 kHz)
PFD
Loop Filter
Bandwidth << 1/T
Loop
Filter
out(t)
Divider
N[k]
Sout(f)
9001
N[k]
9000
1/T
out(t)
frequency resolution = 1/T
ƒ
1.80 1.8002
GHz
Lower 1/T leads to higher divide value
Increases PFD noise at synthesizer output
M.H. Perrott
MIT OCW 39
Modeling PFD Noise Multiplication
VCO-referred
Noise
S Φvn(f)
2
-20 dB/dec
1/T
0
f
0
en(t)
π
α N S e n (f)
Radians2/Hz
PFD-referred
Noise
S En(f)
f
S Φvn(f)
Φvn(t)
f
0
fo
1-G(f)
Φnvco(t)
Φnpfd(t)
Φn(t)
Divider Control
Φc(t)
of Frequency Setting
(assume noiseless for now)
S Φnpfd(f)
Radians2/Hz
fo
π
α N G(f)
Φout(t)
S Φnvco(f)
f
0
ƒ
(fo)opt
(fo)opt
Influence of PFD noise seen in above model
- PFD spectral density multiplied by N
output phase noise
High divide values
M.H. Perrott
2
before influencing PLL
high phase noise at low frequencies
MIT OCW 40
Fractional-N Frequency Synthesizers
ref(t)
(1/T = 20 MHz)
PFD
out(t)
Loop
Filter
91
Divider
90
Nsd[k]
Dithering
Modulator
N[k]
Sout(f)
Nsd[k] 91
1/T
90
out(t)
frequency resolution << 1/T
ƒ
Break constraint that divide value be integer
ƒ
Want high 1/T to allow a high PLL bandwidth
1.80 1.82
GHz
- Dither divide value dynamically to achieve fractional values
- Frequency resolution is now arbitrary regardless of 1/T
M.H. Perrott
MIT OCW 41
A Nice Dithering Method: Sigma-Delta Modulation
Time Domain
M-bit Input
Digital Σ−∆
Modulator
Analog Output
1-bit
D/A
Frequency Domain
Digital Input
Spectrum
Quantization
Noise
Analog Output
Spectrum
Input
ƒ
Σ−∆
Sigma-Delta dithers in a manner such that resulting
quantization noise is “shaped” to high frequencies
M.H. Perrott
MIT OCW 42
Linearized Model of Sigma-Delta Modulator
S r(ej2πfT)= 1
r[k]
NTF Hn(z)
1
Σ−∆
y[k]
x[k]
z=ej2πfT
q[k]
STF
x[k]
12
y[k]
Hs(z)
z=ej2πfT
S q(ej2πfT)= 1 |H n(ej2πfT)| 2
12
ƒ
Composed of two transfer functions relating input and
noise to output
- Signal transfer function (STF)
ƒ Filters input (generally undesirable)
- Noise transfer function (NTF)
ƒ Filters (i.e., shapes) noise that is assumed to be white
M.H. Perrott
MIT OCW 43
Example: Cutler Sigma-Delta Topology
x[k]
u[k]
H(z) - 1
ƒ
ƒ
ƒ
y[k]
e[k]
Output is quantized in a multi-level fashion
Error signal, e[k], represents the quantization error
Filtered version of quantization error is fed back to
input
- H(z) is typically a highpass filter whose first tap value is 1
ƒ i.e., H(z) = 1 + a z + a z "
- H(z) – 1 therefore has a first tap value of 0
1
-1
2
-2
ƒ Feedback needs to have delay to be realizable
M.H. Perrott
MIT OCW 44
Linearized Model of Cutler Topology
r[k]
x[k]
u[k]
H(z) - 1
ƒ
y[k]
e[k]
x[k]
u[k]
H(z) - 1
y[k]
e[k]
Represent quantizer block as a summing junction in
which r[k] represents quantization error
- Note:
ƒ
It is assumed that r[k] has statistics similar to white
noise
- This is a key assumption for modeling – often not true!
M.H. Perrott
MIT OCW 45
Calculation of Signal and Noise Transfer Functions
r[k]
x[k]
u[k]
H(z) - 1
ƒ
y[k]
e[k]
x[k]
u[k]
H(z) - 1
y[k]
e[k]
Calculate using Z-transform of signals in linearized
model
- NTF:
- STF:
M.H. Perrott
Hn(z) = H(z)
Hs(z) = 1
MIT OCW 46
A Common Choice for H(z)
8
m=3
7
6
Magnitude
5
m=2
4
3
m=1
2
1
0
0
M.H. Perrott
1/(2T)
Frequency (Hz)
MIT OCW 47
Example: First Order Sigma-Delta Modulator
ƒ
x[k]
Choose NTF to be
u[k]
H(z) - 1
ƒ
y[k]
e[k]
Plot of output in time and frequency domains with
input of
Amplitude
Magnitude (dB)
1
0
0
M.H. Perrott
Sample Number
200
0
Frequency (Hz)
1/(2T)
MIT OCW 48
Example: Second Order Sigma-Delta Modulator
ƒ
x[k]
Choose NTF to be
u[k]
H(z) - 1
ƒ
y[k]
e[k]
Plot of output in time and frequency domains with
input of
Magnitude (dB)
Amplitude
2
1
0
-1
0
M.H. Perrott
Sample Number
200
0
Frequency (Hz)
1/(2T)
MIT OCW 49
Example: Third Order Sigma-Delta Modulator
ƒ
x[k]
Choose NTF to be
u[k]
H(z) - 1
ƒ
y[k]
e[k]
Plot of output in time and frequency domains with
input of
4
Magnitude (dB)
Amplitude
3
2
1
0
-1
-2
-3
0
M.H. Perrott
Sample Number
200
0
Frequency (Hz)
1/(2T)
MIT OCW 50
Observations
ƒ
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
ƒ Quantization noise, r[k], fails to be white
ƒ
Higher 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
M.H. Perrott
MIT OCW 51
Warning: Higher Order Modulators May Still Have Tones
ƒ
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 noise
ƒ
Worst input for tone generation are DC signals that are
rational with a low valued denominator
- Examples (third order modulator):
Magnitude (dB)
x[k] = 0.1 + 1/1024
Magnitude (dB)
x[k] = 0.1
0
M.H. Perrott
Frequency (Hz)
1/(2T)
0
Frequency (Hz)
1/(2T)
MIT OCW 52
Cascaded Sigma-Delta Modulator Topologies
x[k]
Σ∆M1[k]
q1[k]
M
y1[k]
Σ∆M2[k]
q2[k]
1
Σ∆M3[k]
y2[k]
1
y3[k]
y[k]
Digital Cancellation Logic
Multibit
output
ƒ
ƒ
Achieve higher order shaping by cascading low order
sections and properly combining their outputs
Advantage over single loop approach
- Allows pipelining to be applied to implementation
ƒ High speed or low power applications benefit
ƒ
Disadvantages
- Relies on precise matching requirements when combining
outputs (not a problem for digital implementations)
- Requires multi-bit quantizer (single loop does not)
M.H. Perrott
MIT OCW 53
MASH topology
x[k]
Σ∆M1[k]
r1[k]
M
y1[k]
Σ∆M2[k]
r2[k]
1
Σ∆M3[k]
y2[k]
y3[k]
(1-z-1)2
1-z-1
u[k]
ƒ
ƒ
1
y[k]
Cascade first order sections
Combine their outputs after they have passed through
digital differentiators
M.H. Perrott
MIT OCW 54
Calculation of STF and NTF for MASH topology (Step 1)
x[k]
Σ∆M1[k]
r1[k]
M
Σ∆M2[k]
y1[k]
r2[k]
1
Σ∆M3[k]
y2[k]
1
y3[k]
(1-z-1)2
1-z-1
u[k]
y[k]
ƒ
Individual output signals of each first order modulator
ƒ
Addition of filtered outputs
M.H. Perrott
MIT OCW 55
Calculation of STF and NTF for MASH topology (Step 1)
x[k]
Σ∆M1[k]
r1[k]
Σ∆M2[k]
M
r2[k]
1
Σ∆M3[k]
y2[k]
y1[k]
y3[k]
(1-z-1)2
1-z-1
u[k]
ƒ
1
y[k]
Overall modulator behavior
- STF: H (z) = 1
- NTF: H (z) = (1 – z )
s
n
M.H. Perrott
-1 3
MIT OCW 56
Sigma-Delta Frequency Synthesizers
Fout = M.F Fref
Fref
ref(t)
e(t) Charge
PFD
Pump
Loop
Filter
v(t)
out(t)
VCO
Divider
div(t)
Nsd[m]
N[m]
Σ−∆
Modulator
M+1
M
Σ−∆
Quantization
Noise
ƒ
Riley et. al.,
JSSC, May 1993
f
Use Sigma-Delta modulator rather than accumulator to
perform dithering operation
- Achieves much better spurious performance than classical
fractional-N approach
M.H. Perrott
MIT OCW 57
Background: The Need for A Better PLL Model
PFD-referred
Noise
S En(f)
0
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
f
1/T
0
en(t)
Φref [k]
α
e(t)
π
Φdiv[k]
PFD
v(t)
Icp
H(f)
Loop
Charge
Pump Divider Filter
KV
f
Φvn(t)
Φout(t)
jf
VCO
1
N
N[k]
ƒ
Classical PLL model
- Predicts impact of PFD and VCO referred noise sources
- Does not allow straightforward modeling of impact due
to divide value variations
ƒ This is a problem when using fractional-N approach
M.H. Perrott
MIT OCW 58
A PLL Model Accommodating Divide Value Variations
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
PFD-referred
Noise
S En(f)
PFD
Φref [k]
Tristate: α=1
XOR: α =2
α
1/T
0
en(t)
0
C.P.
e(t)
T
f
2π
Icp
Loop
Filter
H(f)
VCO
v(t)
KV
f
Φvn(t)
Φout(t)
jf
Divider
Φdiv[k]
1
1
T
Nnom
Φd [k]
n[k]
ƒ
-1
z
2π
1 - z-1
j2πfT
z=e
See derivation in Perrott et. al., “A Modeling Approach for
Sigma-Delta Fractional-N Frequency Synthesizers …”,
JSSC, Aug 2002
M.H. Perrott
MIT OCW 59
Parameterized Version of New Model
Φjit[k]
Noise
VCO-referred
PFD-referred
Noise
Noise
S En(f)
S Φvn(f)
-20 dB/dec
αT
π
espur(t)
Ιcpn(t)
1/T
0
f
0
en(t)
1
π
α NnomG(f)
fo
Ι
n[k]
Divide
value
variation
2π
Φ (t)
d
z-1
1 - z-1
z=ej2πfT
fo
1-G(f)
Φn(t)
Φc(t)
Φout(t)
fo
Alternate Representation
G(f)
Fc(t)
n[k]
fo
D/A and Filter Freq.
M.H. Perrott
Φvn(t)
Φnvco(t)
Φnpfd(t)
T G(f)
f
1
jf
Φc(t)
Phase
MIT OCW 60
Spectral Density Calculations
case (a): CT
case (b): DT
case (c): DT
ƒ
Case (a):
ƒ
Case (b):
ƒ
Case (c):
M.H. Perrott
CT
DT
CT
y(t)
x(t)
H(f)
x[k]
j2πfT
H(e
y[k]
)
y(t)
x[k]
H(f)
MIT OCW 61
Example: Calculate Impact of Ref/Divider Jitter (Step 1)
S Φjit(e j2πfT)
Div(t)
T
t
(∆tjit)rms= β sec.
∆tjit[k]
π
π 2 2
β
T
0
f
Φjit[k]
T
ƒ
Assume jitter is white
- i.e., each jitter value independent of values at other time
instants
ƒ
Calculate spectra for discrete-time input and output
- Apply case (b) calculation
M.H. Perrott
MIT OCW 62
Example: Calculate Impact of Ref/Divider Jitter (Step 2)
S Φjit(e j2πfT)
Div(t)
T
t
(∆tjit)rms= β sec.
∆tjit[k]
π
SΦn(f)
2 π 2
1
β2
T Nnom
T
T
π 2 2
β
T
0
Φjit[k]
f
0 fo
T NnomG(f)
Φjit[k]
T
ƒ
f
Φn (t)
fo
Compute impact on output phase noise of synthesizer
- We now apply case (c) calculation
- Note that G(f) = 1 at DC
M.H. Perrott
MIT OCW 63
Now Consider Impact of Divide Value Variations
Φjit[k]
Noise
VCO-referred
PFD-referred
Noise
Noise
S En(f)
S Φvn(f)
-20 dB/dec
αT
π
espur(t)
Ιcpn(t)
1/T
0
f
0
en(t)
1
π
α NnomG(f)
fo
Ι
n[k]
Divide
value
variation
2π
Φ (t)
d
z-1
1 - z-1
z=ej2πfT
fo
1-G(f)
Φn(t)
Φc(t)
Φout(t)
fo
Alternate Representation
G(f)
Fc(t)
n[k]
fo
D/A and Filter Freq.
M.H. Perrott
Φvn(t)
Φnvco(t)
Φnpfd(t)
T G(f)
f
1
jf
Φc(t)
Phase
MIT OCW 64
Divider Impact For Classical Vs Fractional-N Approaches
Classical Synthesizer
1
1/T
G(f)
n(t)
1
T
Fout(t)
n[k]
fo
D/A and Filter
Fractional-N Synthesizer
1
1/T
G(f)
nsd(t)
ƒ
1
T
nsd[k]
Dithering
Modulator
Fout(t)
n[k]
fo
D/A and Filter
Note: 1/T block represents sampler (to go from CT to DT)
M.H. Perrott
MIT OCW 65
Focus on Sigma-Delta Frequency Synthesizer
n[k]
Fout(t)
nsd[k]
1
1/T
G(f)
nsd(t)
1
T
nsd[k]
Σ−∆
Fout(t)
n[k]
fo
D/A and Filter
freq=1/T
ƒ
Divide value can take on fractional values
ƒ
PLL dynamics act like lowpass filter to remove much
of the quantization noise
- Virtually arbitrary resolution is possible
M.H. Perrott
MIT OCW 66
Quantifying the Quantization Noise Impact
VCO-referred
Noise
S Φvn(f)
PFD-referred
Noise
S En(f)
S r(e
)= 1
12
j2πfT
r[k]
Σ−∆
Quantization
Noise
1/T
0
-20 dB/dec
f
Φvn(t)
En(t)
S q(ej2πfT)
fo
f
0
π
α NnomG(f)
1-G(f)
fo
NTF Hn(z)
q[k]
STF
nsd[k]
Hs(z)
n[k]
z=ej2πfT
ƒ
f
0
2π
Φ [k]
n
z-1
1 - z-1
z=ej2πfT
T G(f)
Φdiv(t)
Φtn,pll(t)
Φout(t)
fo
Σ−∆
Calculate by simply attaching Sigma-Delta model
- We see that quantization noise is integrated and then
lowpass filtered before impacting PLL output
M.H. Perrott
MIT OCW 67
A Well Designed Sigma-Delta Synthesizer
fo = 84 kHz
-60
PFD-referred
noise
Spectral Density (dBc/Hz)
-70
-80
SΦ
out,En
-90
(f)
-100
-110
-120
-130
VCO-referred
noise
Σ−∆
noise
SΦ
SΦ
out,vn
(f)
(f)
out,∆Σ
-140
-150
-160
10 kHz
100 kHz
f0
1 MHz
Frequency
10 MHz
1/T
ƒ
Order of G(f) is set to equal to the Sigma-Delta order
ƒ
Bandwidth of G(f) is set low enough such that synthesizer
noise is dominated by intrinsic PFD and VCO noise
- Sigma-Delta noise falls at -20 dB/dec above G(f) bandwidth
M.H. Perrott
MIT OCW 68
Impact of Increased PLL Bandwidth
fo = 84 kHz
fo = 160 kHz
-60
PFD-referred
noise
Spectral Density (dBc/Hz)
-70
-80
SΦ
-90
out,En
(f)
-100
-110
Σ−∆
noise
-120
-130
VCO-referred
noise
SΦ
SΦ
out,vn
(f)
(f)
out,∆Σ
-140
-70
-80
-90
PFD-referred
noise
SΦ
out,En
(f)
Σ−∆
noise
-100
SΦ
-110
-120
-130
(f)
out,∆Σ
VCO-referred
noise
SΦ
out,vn
(f)
-140
-150
-150
-160
10 kHz
100 kHz
f0
ƒ
ƒ
ƒ
Spectral Density (dBc/Hz)
-60
1 MHz
Frequency
-160
10 kHz
10 MHz
1/T
100 kHz
1 MHz
f0
Frequency
10 MHz
1/T
Allows more PFD noise to pass through
Allows more Sigma-Delta noise to pass through
Increases suppression of VCO noise
M.H. Perrott
MIT OCW 69
Impact of Increased Sigma-Delta Order
m=2
m=3
-60
-60
PFD-referred
noise
-80
SΦ
-90
out,En
-70
(f)
-100
-110
-120
-130
VCO-referred
noise
Σ−∆
noise
SΦ
SΦ
out,vn
(f)
(f)
out,∆Σ
-140
-150
-80
PFD-referred
noise
SΦ
out,En
-90
(f)
-100
VCO-referred
noise
-110
-130
SΦ
Σ−∆
noise
-120
SΦ
out,vn
(f)
(f)
out,∆Σ
-140
-150
-160
10 kHz
100 kHz
f0
ƒ
ƒ
Spectral Density (dBc/Hz)
Spectral Density (dBc/Hz)
-70
1 MHz
Frequency
-160
10 kHz
10 MHz
1/T
100 kHz
f0
1 MHz
Frequency
10 MHz
1/T
PFD and VCO noise unaffected
Sigma-Delta noise no longer attenuated by G(f) such
that a -20 dB/dec slope is achieved above its bandwidth
M.H. Perrott
MIT OCW 70