6.976
High Speed Communication Circuits and Systems
Lecture 17
Advanced Frequency Synthesizers
Michael Perrott
Massachusetts Institute of Technology
Copyright © 2003 by Michael H. Perrott
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
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]
N[k]
Sout(f)
91
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
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]
N[k]
Sout(f)
9001
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
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]
N[k]
Sout(f)
9001
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
Modeling PFD Noise Multiplication
VCO-referred
Noise
S Φvn(f)
2
-20 dB/dec
f
1/T
0
0
en(t)
π
α N S e n (f)
Radians2/Hz
PFD-referred
Noise
S En(f)
f
S Φvn(f)
Φvn(t)
0
fo
1-G(f)
Φnvco(t)
Φnpfd(t)
Φn(t)
Divider Control
Φc(t)
of Frequency Setting
(assume noiseless for now)
ƒ
S Φnpfd(f)
Radians2/Hz
fo
π
α N G(f)
(fo)opt
f
Φout(t)
S Φnvco(f)
0
(fo)opt
f
Influence of PFD noise seen in model from Lecture 16
- PFD spectral density multiplied by N
2
before influencing PLL
output phase noise
High divide values
M.H. Perrott
high phase noise at low frequencies
MIT OCW
Dual-Loop Frequency Synthesizer
ref1
PFD
cos(w1t)
Loop
Filter
sin(w1t)
VCO
out
Divider
N
ref2
PFD
cos(w2t)
Loop
Filter
sin(w2t)
VCO
Divider
Single-sideband
Mixer
M
ƒ
Overall synthesizer output
ƒ
From trigonometry: cos(A-B) = cosAcosB+sinAsinB
M.H. Perrott
MIT OCW
Advantage #1: Avoids Large Divide Values
ref1
PFD
cos(w1t)
Loop
Filter
sin(w1t)
VCO
out
Divider
N
ref2
PFD
cos(w2t)
Loop
Filter
sin(w2t)
VCO
Divider
ƒ
Single-sideband
Mixer
M
Choose top synthesizer to provide coarse tuning and
bottom synthesizer to provide fine tuning
- Choose w to be high in frequency
ƒ Set ref to be high to avoid large N
- Choose w to be low in frequency
1
1
low resolution
2
M.H. Perrott
ƒ Allows ref2 to be low without large M
high resolution
MIT OCW
Advantage #2: Provides Suppression of VCO Noise
ref1
PFD
cos(w1t)
Loop
Filter
sin(w1t)
VCO
out
Divider
N
ref2
PFD
cos(w2t)
Loop
Filter
sin(w2t)
VCO
Divider
Single-sideband
Mixer
M
ƒ
Top VCO has much more phase noise than bottom VCO
due to its much higher operating frequency
- Suppress top VCO noise by choosing a high PLL
bandwidth for top synthesizer
ƒ High PLL bandwidth possible since ref1 is high
M.H. Perrott
MIT OCW
Alternate Dual-Loop Architecture
ref1
PFD
out
cos(w1t)
Loop
Filter
sin(w1t)
VCO
Divider
y
N
ref2
PFD
cos(w2t)
Loop
Filter
sin(w2t)
VCO
Divider
Single-sideband
Mixer
M
ƒ
Calculation of output frequency
M.H. Perrott
MIT OCW
Advantage of Alternate Dual-Loop Architecture
ref1
PFD
out
cos(w1t)
Loop
Filter
sin(w1t)
VCO
Divider
y
N
ref2
PFD
cos(w2t)
Loop
Filter
sin(w2t)
VCO
Divider
Single-sideband
Mixer
M
ƒ
ƒ
Issue: a practical single-sideband mixer implementation
will produce a spur at frequency w1 + w2
PLL bandwidth of top synthesizer can be chosen low
enough to suppress the single-sideband spur
- Negative: lower suppression of top VCO noise
M.H. Perrott
MIT OCW
Direct Digital Synthesis (DDS)
clk
Counter
ƒ
ƒ
ROM
DAC
LPF
out
Encode sine-wave values in a ROM
Create sine-wave output by indexing through ROM
and feeding its output to a DAC and lowpass filter
- Speed at which you index through ROM sets frequency
of output sine-wave
ƒ Speed of indexing is set by increment value on counter
(which is easily adjustable in a digital manner)
M.H. Perrott
MIT OCW
Pros and Cons of Direct Digital Synthesis
clk
Counter
ƒ
ƒ
ROM
DAC
LPF
out
Advantages
- Very fast adjustment of frequency
- Very high resolution can be achieved
- Highly digital approach
Disadvantages
- Difficult to achieve high frequencies
- Difficult to achieve low noise
- Power hungry and complex
M.H. Perrott
MIT OCW
Hybrid Approach
clk
Counter
ROM
DAC
LPF
ref
PFD
out
Loop
Filter
VCO
Divider
N
ƒ
ƒ
ƒ
Use DDS to create a finely adjustable reference
frequency
Use integer-N synthesizer to multiply the DDS output
frequency to much higher values
Issues
- Noise of DDS is multiplied by N
- Complex and power hungry
2
M.H. Perrott
MIT OCW
Fractional-N Frequency Synthesizers
ref(t)
(1/T = 20 MHz)
PFD
out(t)
Loop
Filter
91
Nsd[k]
Divider
90
Dithering
Modulator
N[k]
Sout(f)
Nsd[k] 91
1/T
90
out(t)
frequency resolution << 1/T
1.80 1.82
ƒ
Break constraint that divide value be integer
ƒ
Want high 1/T to allow a high PLL bandwidth
GHz
- Dither divide value dynamically to achieve fractional values
- Frequency resolution is now arbitrary regardless of 1/T
M.H. Perrott
MIT OCW
Classical Fractional-N Synthesizer Architecture
ref(t)
PFD
div(t)
frac[k]
Accumulator
e(t)
Loop
Filter
out(t)
N/N+1
1-bit
carry_out[k]
Nsd[k] = N + frac[k]
ƒ
Use an accumulator to perform dithering operation
- Fractional input value fed into accumulator
- Carry out bit of accumulator fed into divider
M.H. Perrott
MIT OCW
Accumulator Operation
clk(t)
frac[k]
M-bit
Accumulator
M-bit
1-bit
residue[k]
carry_out[k]
residue[k]
frac[k] =.25
carry_out[k]
ƒ
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
M.H. Perrott
MIT OCW
Fractional-N Synthesizer Signals with N = 4.25
carry_out(t)
out(t)
div(t)
ref(t)
e(t)
phase error(t)
ƒ
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
ƒ Achieved by dividing by 5 every 4 reference cycles
M.H. Perrott
MIT OCW
The Issue of Spurious Tones
ref(t)
PFD
div(t)
frac[k]
Accumulator
e(t)
Loop
Filter
out(t)
N/N+1
1-bit
carry_out[k]
Nsd[k] = N + frac[k]
ƒ
PFD error is periodic
- Note that actual PFD waveform is series of pulses – the
sawtooth waveform represents pulse width values over time
ƒ
Periodic error signal creates spurious tones in synthesizer
output
Ruins noise performance of synthesizer
M.H. Perrott
MIT OCW
The Phase Interpolation Technique
e(t)
ref(t)
Loop
Filter
PFD
div(t)
out(t)
N/N+1
α
D/A
frac[k]
M-bit
Accumulator residue[k]
M-bit
ƒ
1-bit
carry_out[k]
Phase error due to fractional technique is predicted
by the instantaneous residue of the accumulator
- Cancel out phase error based on accumulator residue
M.H. Perrott
MIT OCW
The Problem With Phase Interpolation
e(t)
ref(t)
Loop
Filter
PFD
div(t)
out(t)
N/N+1
α
D/A
frac[k]
M-bit
Accumulator residue[k]
M-bit
ƒ
1-bit
carry_out[k]
Gain matching between PFD error and scaled D/A
output must be extremely precise
- Any mismatch will lead to spurious tones at PLL output
M.H. Perrott
MIT OCW
Is There a Better Way?
A Better 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
Σ−∆
ƒ
Sigma-Delta dithers in a manner such that resulting
quantization noise is “shaped” to high frequencies
M.H. Perrott
MIT OCW
Linearized Model of Sigma-Delta Modulator
S r(ej2πfT)= 1
r[k]
NTF Hn(z)
1
x[k]
Σ−∆
y[k]
x[k]
12
z=ej2πfT
q[k]
STF
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
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 L
- 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
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
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
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
Frequency (Hz)
1/(2T)
MIT OCW
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
M.H. Perrott
0
Sample Number
200
0
Frequency (Hz)
1/(2T)
MIT OCW
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
1
0
-1
M.H. Perrott
Magnitude (dB)
Amplitude
2
0
Sample Number
200
0
Frequency (Hz)
1/(2T)
MIT OCW
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
M.H. Perrott
0
Sample Number
200
0
Frequency (Hz)
1/(2T)
MIT OCW
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
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
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]
Digital Cancellation Logic
ƒ
ƒ
y[k]
Multibit
output
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
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
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
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
Sigma-Delta Frequency Synthesizers
Fout = M.F Fref
Fref
ref(t)
e(t) Charge
PFD
Pump
Loop
Filter
v(t)
out(t)
VCO
div(t)
Nsd[m]
Divider
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
Background: The Need for A Better PLL Model
PFD-referred
Noise
S En(f)
0
Φref [k]
α
f
1/T
0
en(t)
e(t)
π
Φdiv[k]
VCO-referred
Noise
S Φvn(f)
-20 dB/dec
PFD
Icp
H(f)
Loop
Charge
Pump Divider Filter
v(t)
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
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
0
en(t)
1/T
e(t)
T
2π
f
0
C.P.
Icp
Loop
Filter
H(f)
v(t)
VCO
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
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
0
espur(t)
Ιcpn(t)
1
1/T
0
en(t)
π
α NnomG(f)
fo
Ι
f
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
Spectral Density Calculations
case (a): CT
CT
case (b): DT
DT
case (c): DT
ƒ
Case (a):
ƒ
Case (b):
ƒ
Case (c):
M.H. Perrott
CT
x(t)
x[k]
x[k]
y(t)
H(f)
j2πfT
H(e
H(f)
)
y[k]
y(t)
MIT OCW
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
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
Φjit[k]
T NnomG(f)
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
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
0
espur(t)
Ιcpn(t)
1
1/T
0
en(t)
π
α NnomG(f)
fo
Ι
f
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
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
nsd(t)
1
T
nsd[k]
G(f)
Dithering
Modulator
n[k]
Fout(t)
fo
D/A and Filter
ƒ
Note: 1/T block represents sampler (to go from CT to DT)
M.H. Perrott
MIT OCW
Focus on Sigma-Delta Frequency Synthesizer
n[k]
Fout(t)
nsd[k]
1
1/T
G(f)
nsd(t)
1
T
nsd[k]
Σ−∆
freq=1/T
Fout(t)
n[k]
fo
D/A and Filter
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
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
S q(ej2πfT)
fo
nsd[k]
q[k]
Hs(z)
0
n[k]
z=ej2πfT
f
0
En(t)
NTF Hn(z)
STF
1/T
0
-20 dB/dec
π
α NnomG(f)
f
Φvn(t)
1-G(f)
fo
f
Φ [k]
-1
n
2π z -1
1-z
z=ej2πfT
T G(f)
Φdiv(t)
Φtn,pll(t)
Φout(t)
fo
Σ−∆
ƒ
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
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
Σ−∆
noise
-120
-130
SΦ
VCO-referred
noise
SΦ
(f)
out,vn
(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
Impact of Increased PLL Bandwidth
fo = 84 kHz
fo = 160 kHz
-60
PFD-referred
noise
Spectral Density (dBc/Hz)
-70
-80
SΦ
out,En
-90
(f)
-100
-110
Σ−∆
noise
-120
-130
SΦ
VCO-referred
noise
SΦ
(f)
out,vn
(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
f0
1 MHz
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
Impact of Increased Sigma-Delta Order
m=2
m=3
-60
-60
PFD-referred
noise
-80
SΦ
out,En
-90
-70
(f)
-100
-110
Σ−∆
noise
-120
-130
SΦ
out,∆Σ
VCO-referred
noise
SΦ
(f)
out,vn
(f)
-140
-150
-80
PFD-referred
noise
SΦ
out,En
-90
(f)
-100
VCO-referred
noise
-110
-120
-130
Σ−∆
noise
SΦ
-140
out,∆Σ
SΦ
out,vn
(f)
(f)
-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