VDEC デザイナーズフォーラム 2011・デザインアワード

1

Self-Dithered Digital Delta-Sigma Modulators for Fractional-N PLL
Zule XU, Jun Gyu LEE, and Shoichi MASUI†
Research Institute of Electrical Communication, Tohoku University†
E-mail: shomasui@riec.tohoku.ac.jp, Tel: 022-217-5472

SUMMARY Digital delta-sigma modulators (DDSMs) applied in
fractional-N frequency synthesizers suffer from spurious tones which
undermine the synthesizer’s spectral purity. We propose a solution
featuring no hardware overhead while achieving equivalent spur
elimination effect as using LFSR-dithering. This method can be
implemented on MASH and single-loop DDSMs of 3rd- and 2nd-order.
key words: Digital delta-sigma modulator (DDSM), fractional-N PLL,
dither.

1. Introduction
Digital Delta Sigma Modulators (DDSMs) are widelyused in fractional-N phase locked loops (PLLs) which
are indispensable in modern communication systems [1].
In a fractional-N PLL, shown in Fig. 1, the DDSM
quantizes n-bit DC input X into an m-bit time-variant
integer Y to instantly modulate the programmabledivider’s modulo (n > m). On a long run, the average
modulo becomes (N + X/2n), and the average of PLL’s
output frequency fvco becomes the product of thereference frequency fref and average modulo; thus theoutput frequency with high resolution is available. The
quantization noise is shaped into high frequencies andsuppressed by the loop filter.
The basic building block of a DDSM is an accumulator,
or 1st-order DDSM, unused in PLL applications due toits non-white quantization noises. Higher-order DDSMs
are preferred for better noise-shaping and whither
quantization noise. Higher-order DDSMs are generallyclassified into two types – MASH and single-loop. The
former possesses advantages of stability andconsiderable noise-shaping; the latter is conditionallystable but enables a flexibly optimized noise transfer
function (NTF).
The DDSM suffers from spurious tones which throughthe loop appear at the PLL’s output spectrum, known as“fractional spurs”, higher than the noise floor and hard tobe suppressed, degrading the PLL’s spectral purity [2].
To solve this problem, several solutions have been
proposed which will be discussed in the following
sections. In this paper, we propose an effective
alternative, called self-dithering, featuring extremely
simple implementation and no hardware overhead.
2. Spurious Tones from DDSMs
DDSMs are implemented with accumulators with a DC
input so that periodic patterns are prone to appear at their

fref
fdiv

Phase/Frequency
Detector

Loop
FilterCharge
Pump

VCO

fvco

Programmable Divider÷ (N + Y )

X

n-bit

Digital DeltaSigma
Modulator

m-bit

Y

Fig. 1 Block diagram of a fractional-N PLL.

output sequences. In other words, in the DDSM’s output
spectrum more power converges onto some certain
frequency tones. Even for higher-order DDSMs of bothtypes, spurious tones are observed for some certain DC
inputs. Since the PLL acts as a low-pass filter to thequantization noise, shrinking the loop bandwidth lowers
the out-of-band spur level. By doing this, however, spurs
are not eliminated and the settling of the loop may beintolerably slowed, failing to achieve the spec. Therefore,techniques for eliminating the DDSM’s spurious tones
are highly demanded.
3. Previous Solutions
Previous solutions are classified into two types –deterministic [3][4] and stochastic [2][5]. Deterministic
approaches modify the DDSM’s structure or set up an
irrational condition to disrupt the modulator’s internal
states. The advantages of this type of solutions are thecontrollable sequence length, or period, and no lowfrequency noise floor incurred, yet so far they are mainly
used for MASH. On the other hand, the stochastic
solution, also called dithering, is used for both types ofmodulators. It exerts a 1-bit pseudorandom (PN)
sequence to scramble the DDSM’s input. Since dithering
introduces an elevated noise floor at low frequencies,
shaped dithering has been proposed to transfer the
additive noise into a high-pass characteristic [5].
An unwanted task to implement the dithering is that thePN generator has to be specifically designed with extra
circuits. It is commonly realized with linear feedback
shift registers (LFSR) [4], while a recently publishedmethod employs another accumulator together with an
11-stage LFSR to dither the DDSM’s coefficients [2].
Indeed, such extra endeavor can be avoided by applying
the method proposed in the following section.

VDEC デザイナーズフォーラム 2011・デザインアワード
2
Dn = E[n-1]4. Proposed Self-Dithering Method

unshaped

The proposed method circumvents the design of the PN
generator; instead, utilizes the DDSM’s own quantizationnoise which is generally approximated to be white, thus
suitable as a dithering source. For a MASH DDSM, thequantization noise of interest is generated at its last-stage
accumulator’s sum, while for a single-loop modulator, itlies at the n LSBs of the data before the quantizer, where
n is the DDSM’s input bit width. Fig. 2 and Fig. 3
illustrate the implementations of 3rd-order self-dithered
DDSMs of MASH and single-loop, respectively, wherein the adder, A, B, S, Ci and Co are n-bit inputs, sum of A
and B, carry input, and carry output, respectively. The ztransfer functions of Fig. 2 and Fig. 3 of unshaped andshaped ditherings are shown in (1) ~ (4).

B

(2)

YMASH , shaped ( z ) = X ( z ) + (1 − z −1 ) Dn ( z ) + (1 − z −1 ) 3 E ( z )

(3)
(4)

A

Ci S
Co

B

Ci S
Co

z −1
Y

A
B

z −1

m bitsz −1

5. Simulations
MASH and single-loop modulators with proposed selfdithering are simulated with MATALB and Simulink.
The DDSM’s input bit width is 16, i.e. input values range
from 0 to 65535. The number of stages of LFSR is 16.
The power spectral density (PSD) of DDSM’s output iscalculated using FFT of Welch method with 65536-point
hanning window and 50% overlapping [4], for which fclk
is 20 MHz. Thus, SLSB equals -170 dB/Hz calculated
from (5).
Fig. 4 shows the PSDs of a 3rd-order single-loop DDSMs
with the input equal to 1024. Without dithering, tones arehigher than the spectrums with LFSR-dithering and selfdithering. Unshaped ditherings elevate low-frequency
noise levels both around -170 dB/Hz as same as SLSB. As
to shaped ditherings, entire spectrums of LFSR-dithering

E[n-1:0]n bitsz −1

Dn = E[n-1]XAn bits1 bit

shaped

z

Ci S
B

A

−1

z

Ci S
B

E[n-1:0]n bits

A

−1

z

S

Y

−1
m bitsB

332nFig. 3 Self-dithered 3rd-order single-loop DDSM.

-50

undithered-100

shaped-150

unshaped-200

PSD [dB/Hz]

-50

7

6

5

10

10

(a)

10

10

undithered-100

shaped-150

unshaped

-200
7

6

5

4

10

10

Frequency [Hz]
(b)

10

10

PSD [dB/Hz]

Fig. 4 PSDs of 3rd-order single-loop with undithering, (a)unshaped/shaped LFSR-dithering, and (b) unshaped/shaped self-dithering.
-50
-100
-150

undithered
HK-MASH
4

10

PSD [dB/Hz]

It can be seen from Fig. 2 and Fig. 3 that not any extra
circuits but only 1-bit wire is required in our proposal.

S

z −1

unshaped

-50
-100
-150

unshaped
410

PSD [dB](5)

Co

Fig. 2 Self-dithered 3rd-order MASH DDSM.

4

As shown in Fig. 2 and Fig. 3, the quantization noise’s
most significant bit Dn is extracted and fed back to the Ci
of 1st-stage for unshaped dithering. This 1-bit sequence
introduces a noise level elevation whose power spectral
density (PSD) is calculated with (5), where fclk is clock
frequency of the DDSM. To lower this noise, shaped
dithering can be applied by feeding Dn into Ci of 2ndstage.
 2 −2 n 
S LSB = 10 × log  f clk 

1bitshapedA(1)

YSL ,unshaped ( z ) = z −3 ( X ( z ) + Dn ( z )) + (1 − z −1 ) 3 E ( z )YSL , shaped ( z ) = z −3 X ( z ) + z −2 (1 − z −1 ) Dn ( z ) + (1 − z −1 ) 3 E ( z )n bitsPSD [dB/Hz]

YMASH ,unshaped ( z ) = X ( z ) + Dn ( z ) + (1 − z −1 ) 3 E ( z )

X

-50
-100

unshaped

5

10

6

(a)

10

7

10

undithered

5

10

6

(b)

10

7

10

undithered

-150
4

10

5

10
Frequency [Hz]
(c)

6

10

7

10

Fig. 5 PSDs of 2nd-order MASH with undithering, (a) HK_MASH, (b)unshaped LFSR-dithering, and (c) unshaped self-dithering.

and self-dithering are equivalent.
Fig. 5 illustrates the PSDs of a 2nd-order MASH DDSM
with the input equal to 64. Since there is no guaranteedshaped dithering scheme for 2nd-order DDSMs, unshaped
dithering is applied. The results are compared with using
LFSR and HK-MASH, a MASH DDSM withdeterministic spur reduction [3]. We do not consider

VDEC デザイナーズフォーラム 2011・デザインアワード

3

0.5

Table 1

0.5

Amplitude

0.4
0.3

0

0.2

Amplitude

-0.5
0.1
0

-2

-1

0
Index
(b)

1

2
5

x 10

0.5

Amplitude

-0.1
-0.2

0

-0.3
-0.4

-0.5
-2

-1

0
Index
(a)

1

2
5

x 10

-2

-1

0
Index
(c)

1

2
5

x 10

Fig. 6 Auto-correlations of 2nd-order (a) HK-MASH, LFSRdithered MASH, and (c) self-dithered MASH DDSMs.

PSD [dBc/Hz]

-60
-80
-100

undithered
shapedtheoretical

-120
-140
5
10

PSD [dBc/Hz]

-60
-80
-100

6

10
(a)

7

10

LFSR

HK-MASH0~655350~65535

1~65521

Mean Value
Deviation (max)

0.5/216

0.5/216

15/216

HardwareOverheadNone

16 flip-flops +some logic gates

1 multiplexer +some flip-flops

set the loop bandwidth to 303 kHz. In Fig. 7, undithered
DDSM contributes fractional spurs with maximum levelof -68 dBc/Hz violating the spec. For each frequency bin,
LFSR-dithering and self-dithering remove spurs, and the
maximum noise powers are around -100 dBc/Hz. Also,
two spectrums are both similar to the theoretical one
calculated with (6). If we try to decrease the spur levels
to below -100 dBc/Hz by decreasing the loop bandwidth
to 72 kHz while maintaining the loop stability, thesettling time drastically increases to around 20 µs, 4times of the spec requirement. Therefore, shrinking theloop bandwidth is ineffective; instead, dithered DDSMs
are necessary.
(2π ) 2
1 − z −1

⋅
2

1
G(s)
⋅ N⋅
N2
1 + G(s)theoretical2

S qn = E ( z ) ⋅ NTF ( z ) =

-120
-140
5
10

Proposed
Input RangeΓ∆Σ , div = S qn ⋅undithered
shapedSummary of 16-bit 2nd-order MASH DDSMs

6

10
Frequency [Hz]
(b)

7

10

Fig. 7 Fractional-N PLL phase noises due to the quantization noises of
3rd-order single-loop DDSMs in theory, with undithering, (a) shaped
LFSR-dithering, and (b) shaped self-dithering.

initial-condition method since it is ineffective for 2ndorder MASH according to [4]. Seen from Fig. 4, threemethods all effectively suppress the spurs but HKMASH’s spectrum is “thicker” meaning that the noise
floor is less whitened than the other two.
Fig. 6 shows the autocorrelations of 2nd-order MASH
DDSM output under the same condition as in Fig. 5.
Except for the spike at the center, the 1st spike’s x-index
represents the sequence length. In Fig. 6, no spikes areobserved for LFSR-dithering and self-dithering so that
their sequence lengths are longer than the number ofsimulation samples, 218, while for HK-MASH we obtainthe sequence length as 65521, less than one fourth of 218.
Table 1 summarizes characteristics of the 16-bit 2ndorder MASH DDSMs for three solutions.
Fig. 7 shows the PLL’s phase noises attributed to the 3rdorder DDSM quantization. According to [6], PLL’s phasenoise due to the quantization noise is calculated from (6),where N, is divider’s division ratio, G(s) is the open-looptransfer function, and Sqn is the PSD of the DDSM,
theoretically defined in (7). DDSM spurious tones can bemapped to the PLL fractional spurs using (6). Our PLL
loop is designed for a fast settling application used in awireless sensor network with the spec of 5 µs settling
time and -80 dBc/Hz spur level. To achieve the spec, we

2

12⋅ NTF ( z )12 f ref(6)(7)

6. Implementations and Experimental Results
The test element group (TEG) of 3rd-order 16-bit MASH
and single-loop DDSMs has been implemented with
0.18um CMOS process. The TEG also includes a 16stage LFSR, a 41-bit shift register, and some control
logic.
Fig. 8 shows the TEG’s block diagram and its data flow.
The DDSM’s input and the control bits are seriallyshifted into the TEG. Control bits and control logic
enable one DDSM, set its input and dithering scheme
which is either LFSR-dithering or self-dithering. The 4bit output of the enabled DDSM is output in parallel andrecorded by an external controller for spectrum plotting.
This design follows a typical digital IC design flow from
RTL to chip integration. The power consumption
estimated by Synopsis IC Compiler is 630 µW at 1.8V
power supply and 26MHz operating frequency.
Fig. 9 shows the die photo of the TEG. Its active area is
0.06 mm2. Since the area of flip-flops is dominant, eachblock’s area can be estimated with number of flip-flops.
There are 48 in one DDSM, 16 in the LFSR, 41 in theshift register, and more than 32 in the control logic forthe input and output data latching. Therefore, one DDSM
occupies less than 0.02 mm2 area. On the other hand,compared with LFSR-dithered DDSM, self-dithered
DDSM reduces the area by 1/4 which is the portion of
the 16-stage LFSR.
The TEG is mounted on VDEC-MOT48-160FP socket

VDEC デザイナーズフォーラム 2011・デザインアワード
4board [7], and measured at the power supply of 3.3 V for
IOs and 1.8 V for the core. An FPGA evaluation board
XtremeDSP™ Starter Platform powered by Spartan®-3A
DSP 1800A is employed as the external controller which
provides the TEG with 41-bit serial data, reset signal,
clocks, etc.. It transfers the DDSM’s parallel output to
PC via RS232 cable. These output data is processed by
MATLAB programs on PC.
Fig. 10 and Fig. 11 show the PSDs of fabricated singleloop and MASH DDSMs. Equivalent performances canbe observed by using LFSR-dithering and self-dithering.

DDSM’s output
Control Logic
Single-Loop
DDSM

MASH DDSM

16-Stage LFSR
Control Logic
Shift Register

7. Conclusion
Serial input(DDSM’s input and control bits)

Spur suppression techniques are important to fractionalN PLLs. Conventional LFSR-dithering and HK-MASH
require extra hardware. We have proposed self-dithering
method holding the equivalent performance of LFSRdithering featuring no hardware overhead. Therefore, fora 16-bit DDSM with dithering, 1/4 area can be reduced
by removing 16-stage LFSR circuits.

Fig. 8 DDSM TEG and data flow

Acknowledgement

Fig. 9 Die photo of DDSM TEG
PSD of fabricated single-loop with 16-stage LFSR-dithering-50

PSD [dB/Hz]

This work is supported by VLSI Design and Education
Center (VDEC), the University of Tokyo in collaboration
with Synopsys, Inc, Cadence Design Systems, Inc,
Mentor Graphics, Inc. The VLSI chip in this study has
been fabricated in the chip fabrication program of VLSI
Design and Education Center (VDEC), the University ofTokyo in collaboration with Rohm Corporation andToppan Printing Corporation.

Undithered
Dithered
Shaped-dithered

-100
-150
-200

References

5

6

10

10

7

10

PSD of fabricated single-loop with self-dithering

PSD [dB/Hz]-50

Undithered
Dithered
Shaped-dithered

-100
-150
-200
4

10

5

6

10

Frequency [Hz]

10

7

10

Fig. 10 PSDs of fabricated single-loop DDSM with undithering,
shaped/unshaped LFSR-dithering, and shaped/unshaped self-dithering.
PSD of fabricated MASH1-1-1 with 16-stage LFSR-dithering

PSD [dB/Hz]-50

Undithered
Dithered
Shaped-dithered

-100
-150
-200
4

10

5

6

10

10

7

10

PSD of fabricated MASH1-1-1 with self-dithering-50
PSD [dB/Hz][1] T. Riley, M. Copeland, and T. Kwasniewsky, “Delta-Sigma
Modulation in Fractional-N Frequency Synthesis,” IEEE J. SolidState Circuits, vol. 28, 1993, pp. 553-559.
[2] F. Maloberti, E. Bonizzoni, and A. Surano, “Time Variant Digital
Sigma-Delta
Modulator
for
Fractional-N
Frequency
Synthesizers,” IEEE International Symposium on RadioFrequency Integration Technology, 2009, pp. 111–114.
[3] K. Hosseini and M. P. Kennedy, “Maximum Sequence Length
MASH Digital Delta-Sigma Modulator,” IEEE Tran. on Circuits
and Systems – I, vol. 54, 2007, pp. 2628-2638.
[4] M. Kozak and I. Kale, Oversampled Delta-Sigma Modulators,
Springer, New York, 2003.
[5] S. Pamarti and I. Galton, “LSB Dithering in MASH Delta-Sigma
D/A Converters,” IEEE Trans. on Circuits and Systems–I, vol. 54,
2007, pp. 779-790.
[6] B. D. Muer and M. Steyaert, CMOS Fractional-N Synthesizers,
Springer, New York, 2003.
[7] VLSI Design and Education Center (VDEC), http://www.vdec.utokyo.ac.jp/

410Undithered
Dithered
Shaped-dithered

-100
-150
-200
4

10

5

10

6

Frequency [Hz]

10

7

10

Fig. 11 PSDs of fabricated MASH DDSM with undithering,
shaped/unshaped LFSR-dithering, and shaped/unshaped self-dithering.