www.tyndall.ie
Michael Peter Kennedy FIEEE
Circuits and Systems Group
IEEE Circuits and Systems Society
Santa Clara Valley Chapter
28 July 2014
School of Engineering―Electrical & Electronic Engineering,
University College Cork, and Tyndall National Institute, Cork, Ireland
An Introduction to
Digital Delta-Sigma Modulators
www.tyndall.ie
Dedication
• William F. Egan
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Digital Delta Sigma Modulators
“Although the number of commercially
deployed DDSMs far exceeds that of analog
Δ∑ modulators, most of the published Δ∑
modulator analyses apply only to analog Δ∑
modulators.
Interestingly, most of these analyses do not
apply or even readily extend to the case
of DDSMs”
[PWG07]
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Outline
• The ideal DDSM The Big Idea
Signal processing assumptions
Applications of DDSMs
• The real DDSM DDSM architectures
Origins of spurious tones
Minimization of spurious tones
• Current research
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The ideal DDSM
• The Big Idea
• Signal processing assumptions
• Applications of DDSMs
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The ideal DDSM
• A bandlimited digital signal x (n bits) is
requantized to a shorter word y (m bits)
• The additive quantization noise eq is
highpass filtered for later removal by a
lowpass filter
n bits m bits
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The ideal DDSM 7/115
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The ideal DDSM
• x is bandlimited
• y includes highpass filtered quantization noise
• quantization noise can be removed by lowpass filtering
X
Eq X
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The ideal DDSM
• x is bandlimited
• y includes highpass filtered quantization noise
• quantization noise can be removed by lowpass filtering
X
X Eq
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The ideal DDSM
• The oversampled output has almost the
same S/N ratio as the input but much
fewer bits (e.g. 16 reduced to 1)
• Fewer bits means it’s easier to implement
the digital to analog conversion further
along the signal processing path
• Oversampling makes it easier to do
continuous-time reconstruction filtering
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The ideal DDSM
• One typically assumes that the Classical
Model of Quantization (CMQ) applies…
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Classical model of quantization (CMQ)
• eq is statistically independent of x
• eq is uniformly distributed in [-Δ/2, + Δ/2]
• eq is stationary with a flat power spectrum
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Classical model of quantization (CMQ)
“CMQ can be applied when the quantizer
input traverses several quantization levels
between two successive samples”
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The ideal DDSM
• The output depends on the signal x and the quantization noise eq
Y(z) = STF(z) X(z) + NTF(z) Eq(z)
• The signal is typically scaled by the DDSM
STF(z) = (1/M)
• The quantization noise is highpass filtered
NTF(z) = (1-z-1)l
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The ideal DDSM
• The output comprises the signal plus the
filtered quantization noise
Y(z) = (1/M)X(z) + Eq(z) (1-z-1)l
x[n] spectrum
y[n] spectrum
Shaped white quantization
noise
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The ideal DDSM
• The noise scales as f2l at low frequencies
|Eq(z) (1-z-1)l|2 = (Δ2/12) 22l sin2l(πf/fs)
≈ (Δ2/12) (2πf/fs)2l when f « fs
+20l dB/decade
CMQ approximation
Simulation
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The ideal DDSM
• x is bandlimited
• y includes highpass filtered quantization noise
• quantization noise can be removed by lowpass filtering
X
X Eq
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Applications of DDSMs
• Example 1: Class D audio amplifier
• Example 2: Oversampled DAC
• Example 3: Fractional-N frequency
synthesizer
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Applications of DDSMs
• Example 1: Class D audio amplifier
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Class D digital audio power amplifier
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Class D digital audio power amplifier
• Example: uniformly-sampled PWM; N=16
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Applications of DDSMs
• Example 2: Oversampled DAC
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Oversampled DAC
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Oversampled DAC
• By using oversampling and DDSM, the filter
rolloff and DAC linearity specifications are
relaxed
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Applications of DDSMs
• Example 3: Fractional-N frequency
synthesizer
[R90]
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Fractional-N Frequency Synthesis
Phase Detector
(PFD) LPF VCO
fout
fref
÷ (Nint+y)
fd
1800 MHz 13 MHz
DDSM
X
y
refnout fX
Nf )2
( int
n bits
m bits
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Fractional-N Frequency Synthesis
• DDSM sets the average division ratio
• The frequency resolution is determined by
the wordlength of the DDSM
refnout fX
Nf )2
( int
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Fractional-N Frequency Synthesis
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Fractional-N Frequency Synthesis
• GSM example:
fref = 13 MHz; fout = 1.8 GHz
N (INT )= 138
X (FRAC) = 30
M (MOD) = 65
fout = fref (INT + FRAC/MOD)
= 13 (138 + 30/65) MHz
= 1794 + 6 MHz
= 1800 MHz
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Fractional-N Frequency Synthesis
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Fractional-N Frequency Synthesis
• Output phase noise [PTS02]
Nint+y
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Fractional-N Frequency Synthesis
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Fractional-N Frequency Synthesis
• DDSM contributes little in-band noise; loop
filter attenuates out-of-band components
MASH 1-1 with 2nd order loop
+20(l-1) dB/decade
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Fractional-N Frequency Synthesis
• Quantization noise is first highpass filtered
by the DDSM’s NTF
• Shaped quantization noise is then
attenuated by the (lowpass CT) loop filter
• We get high frequency resolution for free!
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The ideal DDSM: Summary
• The Big Idea Linear; allpass STF, NTF attenuates quantization noise in the signal
band
• Signal processing assumptions CMQ applies, quantization noise is uncorrelated with the signal
• Applications of DDSMs Digital Power Amplifiers, DACs, Fractional-N Frequency
Synthesizers
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The real DDSM
• DDSM architectures
• Origins of spurious tones (spurs)
• Minimization of spurious tones
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The real DDSM
• The output depends on the signal x and
the quantization noise eq
Y(z) = STF(z) X(z) + NTF(z) Eq(z)
where STF(z) = 1/M and NTF(z) = (1-z-1)l
• Consider the simplest case l=1
Y(z) = (1/M)X(z) + (1-z-1) Eq(z)
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The real DDSM
• A 1st order DDSM can be implemented
using a digital accumulator
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The real DDSM
• 1st order Error Feedback Modulator (EFM1)
• x is the input; y (the carry out) is the
output; s (the value stored in the register)
is the state; M is the modulus
y
vM
1
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The real DDSM
• 1st order Error Feedback Modulator (EFM1)
s[n+1] = (x[n] + s[n]) mod M
y[n] = Q(x[n] + s[n])
y
vM
1
s[n]
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The real DDSM
• 1st order Error Feedback Modulator (EFM1)
Q(v[n]) = (1/M)v[n] + eq[n]
e[n] = v[n]- My[n] = -M eq[n]
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The real DDSM
• 1st order Error Feedback Modulator (EFM1)
y[n] = Q(x[n] + s[n])
= Q(x[n]+e[n-1])
= (1/M)(x[n] + e[n-1]) + eq[n]
= (1/M)(x[n] - Meq[n-1]) + eq[n]
= (1/M)(x[n]) + (eq[n] - eq[n-1])
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The real DDSM
• 1st order Error Feedback Modulator (EFM1)
Y(z) = (1/M)X(z) + Eq1(z) (1-z-1)1
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The real DDSM
Spurs
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The real DDSM
• Higher-order noise-shaping architectures Single Quantizer (SQ-DDSM)
Multi-Stage Noise Shaping(MASH)
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The real DDSM
• MASH: Higher order filtering with exact
cancellation of intermediate errors
Y(z) = (1/M)X(z) + Eql(z) (1-z-1)l
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The real DDSM
• MASH: Idealized power spectra (white
noise with 2nd and 3rd order filters)
+60 dB/decade
+40 dB/decade
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The real DDSM 48/115
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The real DDSM 49/115
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The real DDSM
• A MASH 1-1-1 with a constant input and an
even initial condition can produce spurious
tones (spurs)
MASH 1-1-1
wordlength: 18 bits
Spurs
+60 dB/decade
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The real DDSM
• In audio DACs, these are called idle tones
MASH 1-1-1
wordlength: 18 bits
Spurs
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The real DDSM
• Phase noise in fractional-N frequency
synthesizer with odd initial condition in
MASH 1-1-1
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The real DDSM
• Phase noise in fractional-N frequency
synthesizer with constant input and even
initial condition in MASH 1-1-1
Spurs
Spurs
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What can go wrong and why?
• Different wordlengths can produce
different spectra
• Different inputs can produce different
spectra
• Different initial conditions can produce
different spectra
• All of the above can cause spurs!
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What can go wrong and why?
• Different wordlengths produce different
spectra
MASH 1-1-1
green plot: 9 bits
magenta plot: 18 bits
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What can go wrong and why?
• Different inputs produce different spectra
MASH 1-1-1
wordlength: 17 bits
(i) X=1
(ii) X=216
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What can go wrong and why?
• Different initial conditions produce
different spectra MASH 1-1-1
Input = 8 (decimal)
accumulator word length: 14 bits
cycle length for green plot: 4096
cycle length for blue plot: 32768
red plot: CMQ approximation
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What can go wrong and why?
• Fundamental problem: Short cycles
produce strong tones
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What can go wrong and why?
“CMQ can be applied when the quantizer
input traverses several quantization levels
between two successive samples”
• When the input is constant or slowly
moving, CMQ does not apply…
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What can go wrong and why?
• The DDSM is a Finite State Machine (FSM)
• The FSM has a finite state space S
(containing Ns states) and a deterministic
rule GD (called the dynamic) that governs
the evolution of states
• The next state is determined completely
by the current state and the input:
s[n+1] = GD(s[n], x[n])
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What can go wrong and why?
• If the input is fixed, the most complex
trajectory visits each state in the state
space once before repeating; the longest
cycle has period N = Ns-1
• In the worst case, the trajectory in a MASH
1-1-1 repeats with period N = 4
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What can go wrong and why?
• Example worst case period: N = 4
MASH 1-1-1
wordlength: 17 bits
(i) X=1
(ii) X=216
fs/4
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What can go wrong and why?
• Parseval’s theorem
(1/N) ∑n=0,1,…,N-1 |x[n]|2 = ∑k=0,1,…,N-1 |X[k]|2
power in time domain = power in frequency domain
• X[k] are the Discrete Time Fourier Series
(DTFS) coefficients
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What can go wrong and why?
• Quantization noise power is distributed
over N tones
• Fewer tones results in greater power per
tone
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How to fix the problem?
• Problem: When the cycle length is short,
the quantization noise is distributed over a
small number of tones, resulting in high
average quantization noise power per tone
• Solution: Maximize the cycle length to
distribute the quantization noise power
over more tones, thereby reducing the
average noise per tone
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How to fix the problem?
• Stochastic approach Add dither
• Deterministic approaches Set initial states
Restrict inputs
Change the architecture
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Stochastic approach
• Make the dynamic stochastic
• The next state depends on the current
state s, the input x, and a random dither
signal d
s[n+1] = GS(s[n], x[n], d[n])
• Periodicity is destroyed
• Trajectories can be much longer than Ns
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Stochastic approach: LSB dither
• Additive “LSB” dither: V(z) = (1-z-1)R
DDSM
n0-bit
][nx
][ny
][nd
1-bit
V(z)
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Stochastic approach: LSB dither
• Dither spreads the power over more tones
9 bit without dither 9 bit with LSB dither
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Fractional-N frequency synthesizer
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Fractional-N frequency synthesizer
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Fractional-N frequency synthesizer
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Stochastic approach: LSB dither
• LSB dither is indistinguishable from x…
Y(z) = X(z) + Eql (z)(1-z-1)l + D(z) (1-z-1)R
DDSM
n0-bit
][nx
][ny
][nd
1-bit
V(z)
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Stochastic approach: LSB dither
• MASH 1-1-1 with LSB dither: V(z) = (1-z-1)R
Green: V(z)= 0
Red: V(z)=1
Blue: V(z)=1-z-1
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Stochastic approach: LSB dither
• Maximum filter order for LSB dither [PG07]
Type of
MASH Filter
V(z) = (1-z-1)R
allowed R
1-1 lowpass z-2(1-z-1)2 0
1-1-1, 1-2,
2-1 lowpass z-3(1-z-1)3 1
m1-m2-…-mk,
∑mj = l z-l(1-z-1)l l-2
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Deterministic approaches: ICs
• MASH 1-1-1: Choose s1[0] odd (“seeding”)
MASH 1-1-1
Input = 8 (decimal),
accumulator word length: 14 bits
cycle length for green plot: 4096
cycle length for blue plot: 32768
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Deterministic approaches: ICs
• MASH cycle lengths with s1[0] odd (M=2n)
Modulator
order l
Guaranteed
minimum N Maximum N
2 M/2 2M
3 2M 2M
4 2M 4M
5 4M 4M
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Deterministic approaches: Architecture
• HK-MASH: for maximum length cycles
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Deterministic approaches: Architecture
• Guaranteed minimum cycle lengths
Order (l) Conventional
MASH with odd
initial condition
HK-MASH
2 M/2 ≈ M2
3 (M=512)
2M (1024)
≈ M3
(5093 ≈ 132 x 106)
4 2M ≈ M4
5 4M ≈ M5
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Deterministic approaches: Architecture
• HK-MASH vs seeded and dithered MASH
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The real DDSM: Summary
• DDSM architectures Nonlinear, CMQ does not always apply
• Origins of spurious tones Constant and/or period inputs produce periodic quantization noise
Shorter cycles yield higher power per tone
Cycle lengths depend on the input, initial conditions, and word
length
• Minimization of spurious tones Stochastic: add filtered dither
Deterministic: choose odd input or initial condition, or modify
architecture
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Current research 1
• Error masking Deliberately introduce errors and then mask their effects spectrally
• Bus-splitting Save area and power by nesting DDSMs
• Mixed-radix divider controller Set frequency precisely by using mixed-radix fractions
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Current research 2
• Dither Determine the effects of periodic dither
• Nonlinearity Spurs regrow when the “noiselike” output of the DDSM encounters a
nonlinearity
• Successive requantizers Sequences which are less likely to produce tones when they
encounter nonlinearities
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Error Masking
• The output of a DDSM contains two terms:
Y(z) = STF(z)X(z) + NTF(z)Eq(z)
• The second (shaped noise) term is
removed by filtering further along the
signal processing chain
• Any additional quantization noise that lies
below the mask of the shaped noise will
also be removed by that filter!
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Error Masking
• The signal itself is also quantized,
providing a noise floor below which other
errors can be masked
Quantization noise (of DDSM)
Quantization noise (of input)
“Corner” frequency
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Error Masking
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Error Masking
• Exploit error masking to reduce the bus
width, thereby reducing area and power
and increasing speed
• We consider two ideas: MASH DDSMs with interstage quantization
Bus-splitting
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All modulators have the same wordlength N.
Exact 3rd order MASH architecture 88/115
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Exact 3rd order MASH architecture
• All stages have the same wordlength N
• The hardware requirement scales as 3N
• This is overkill…
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Later stages have smaller wordlengths: L<M<N.
Reduced complexity MASH DDSM 90/115
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Reduced complexity 3rd order MASH
• Successive stages have decreasing
wordlengths
• The hardware requirement scales less than
3N
• This is optimal…
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Reduced complexity 3rd order MASH
• 3rd order MASH with interstage errors
Y1(z) = (1/2N)X(z) + Eq1(z) (1-z-1)
Y2(z) = (1/2M)(-2MEq1(z) + Eq12(z)) + Eq2(z) (1-z-1)
Y3(z) = (1/2L)(-2LEq2(z) + Eq23(z)) + Eq3(z) (1-z-1)
Y(z) = Y1(z) + Y2(z) (1-z-1) + Y3(z) (1-z-1)2
= (1/2N)X(z) + Eq3(z) (1-z-1)3
+ (1/2M)Eq12(z) (1-z-1) + (1/2L)Eq23(z) (1-z-1)2
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Error Masking
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Reduced complexity 3rd order MASH
• Relative Hardware Consumption (RHC):
• For large N, RHC approaches 67%, representing a 33% saving in area relative to a MASH DDSM with identical EFM1 stages
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Bus-splitting
• We can split the bus multiple times…
• Example: Nested bus-splitting 1-2-3 DDSM3
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Bus-splitting accelerator
[PC12] Park & Cho, IEEE J. Solid-State Circuits, 2012.
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Bus-splitting
Y(z) = (1/2N)X(z) + (1/2(NMSB+NISB))Eq1(z) (1-z-1)
+ (1/2NMSB)Eq2(z) (1-z-1)2 + Eq3(z) (1-z-1)3
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Comparison
DDSM3 1-2-3 DDSM3
ENOB = 19.54 ENOB = 19.22
area, power = 1, 1 area, power = 0.62, 0.51
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Frequency synthesis: State of the art
• Conventional frac-N frequency synthesizer
Phase
Detector (PFD)
LPF VCO
fVCO
fPD
÷ (INT+FRAC/MOD)
fd
3999.999… MHz 122.88 MHz
2n DDSM
FRAC
y
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Frequency synthesis
• Mixed-radix frac-N frequency synthesizer
Phase Detector
(PFD) LPF VCO
fVCO
fPD
÷ (INT+FRAC/MOD+Δ)
fd
4000.0000 MHz 122.88 MHz
Mixed-Radix DDSM
FRAC
y
Δ
[KMFHSK14] Kennedy et al., IEEE J. Solid-State Circuits, 2014.
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Frequency synthesis
• Conventional
• Mixed-radix
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Frequency synthesis
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Frequency synthesis
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Frequency synthesis
[KFHSK14] Kennedy et al., J. Solid-State Circuits, 2014.
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Stochastic approach: LSB dither
• Dither second and third stages…
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Stochastic approach: LSB dither
• “Efficient dithering” [GDGAFVM10]
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Fractional-N frequency synthesizer
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Fractional-N frequency synthesizer
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Simplified model
Requantizer
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Simplified model
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Simplified model
[SNK13] Sadeghi et al., IEEE Trans. Circuits Syst. II, 2013.
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Simplified model
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Simplified model
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Spurs due to the PFD-CP nonlinearity
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Acknowledgements
• FIRB Italian National Research Program
RBAP06L4S5
• Irish Research Council for Science,
Engineering & Technology (IRCSET)
• Science Foundation Ireland Grants
02/IN1/I45 and 08/IN1/I1854
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• [FKM11] B. Fitzgibbon, M.P. Kennedy and F. Maloberti.
Hardware Reduction in Digital Delta-Sigma Modulators via
Bus-Splitting and Error Masking―Part I: Constant Input.
IEEE Trans. Circuits and Systems-Part I, 58(9):2137-2148,
Sep. 2011.
• [FKM12] B. Fitzgibbon, M.P. Kennedy and F. Maloberti.
Hardware Reduction in Digital Delta-Sigma Modulators via
Bus-Splitting and Error Masking―Part II: Non-constant
Input. IEEE Trans. Circuits and Systems-Part I,
59(9):1980-1991, Sep. 2012.
• [FPK11] B. Fitzgibbon, S. Pamarti and M.P. Kennedy. A
Spur-Free MASH DDSM with High-Order Filtered Dither.
IEEE Trans. Circuits and Systems-Part II, 58(9):585-588,
Sep. 2011.
References 116/115
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• [GDGAFVM10] V.R. Gonzalez-Diaz, M.A. Garcia-Andrade,
G.E. Flores-Verdad and F. Maloberti. Efficient Dithering
in MASH Sigma-Delta Modulators for Fractional-N
Frequency Synthesizers. IEEE Trans. Circuits and
Systems-Part I, 57(9):2394-2403, Sep. 2010.
• [GDPM11] V.R. Gonzalez-Diaz, A. Pena Perez and F.
Maloberti. Use of Time Variant Digital Sigma-Delta for
Fractional Frequency Synthesizers. In Proc. IEEE Int.
Symp. Circuits and Systems, pages 169-172, May 2011.
• [HK07a] K. Hosseini and M.P. Kennedy. Mathematical
Analysis of a Prime Modulus Quantizer MASH Digital
Delta-Sigma Modulator. IEEE Trans. Circuits and Systems-
Part II, 54(12):1105-1109, Dec. 2007.
References 117/115
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References
• [HK07b] K. Hosseini and M.P. Kennedy. Maximum
Sequence Length MASH Digital Delta-Sigma Modulators.
IEEE Trans. Circuits and Systems-Part I, 54(12):2628-
2638, Dec. 2007.
• [HK08] K. Hosseini and M.P. Kennedy. Architectures for
Maximum-Sequence-Length Digital Delta-Sigma
Modulators. IEEE Trans. Circuits and Systems-Part II,
55(11):1104-1108, Nov. 2008.
• [HK11] K. Hosseini and M.P. Kennedy. Minimizing Tones
in Digital Delta-Sigma Modulators. Springer, New York,
2011.
118/115
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References
• [K12] M.P. Kennedy. Recent Advances in the Design,
Analysis and Optimization of Digital Delta-Sigma
Modulators. NOLTA, 3(3):258-286, Jul. 2012.
• [KFHSK13] M.P. Kennedy, B. Fitzgibbon, A. Harney, H.
Shanan and M. Keaveney. High Speed, High Accuracy
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