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Design of a Rom-Less Direct Digital Frequency Synthesizer in
65nm CMOS Technology
Master thesis performed in Electronic Devices
Author: Golnaz Ebrahimi Mehr
Report number: LiTH-ISY-EX--13/4657--SE
Linköping, April 2013
Design of a ROM-Less Direct Digital Frequency Synthesizer
in 65nm CMOS Technology
............................................................................
............................................................................
Master thesis Performed in Electronic Devices
at Linköping Institute of Technology
by Golnaz Ebrahimi Mehr
...........................................................
LiTH-ISY-EX--13/4657--SE
Supervisor: Dr. Behzad Mesgarzadeh
Examiner: Professor Atila Alvandpour
Linköping, April 2013
Presentation Date
Department and Division
16 April 2013
Publishing Date (Electronic version)
Department of Electrical Engineering
Electronic Devices
29 April 2013
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× Degree thesis
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ISRN: LiTH-ISY-EX--13/4657--SE
65
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URL, Electronic Version
http://www.ep.liu.se
Publication Title
Design of a Rom-Less Direct Digital Frequency Synthesizer in 65nm CMOS technology.
Author(s)
Golnaz Ebrahimi Mehr
Abstract
A 4 bit, Rom-Less Direct Digital Frequency Synthesizer (DDFS) is designed in 65nm CMOS technology.
Interleaving with Return-to-Zero (RTZ) technique is used to increase the output bandwidth and synthesized
frequencies. The performance of the designed synthesizer is evaluated using Cadence Virtuoso design tool. With
3.2 GHz sampling frequency, the DDFS achieves the spurious-free dynamic range (SFDR) of 60 dB to 58 dB for
synthesized frequencies between 200 MHz to 1.6 GHz. With 6.4 GHz sampling frequency, the synthesizer
achieves the SFDR of 46 dB to 40 dB for synthesized frequencies between 400 MHz to 3.2 GHz. The power
consumption is 80 mW for the designed mixed-signal blocks.
Keywords
Rom-Less DDFS, Current Steering Digital-to-Analog Converter, Interleaved DACs, Return-to-Zero, Sine weighted DAC
Abstract
A 4 bit, Rom-Less Direct Digital Frequency Synthesizer (DDFS) is designed in 65nm CMOS
technology. Interleaving with Return-to-Zero (RTZ) technique is used to increase the output
bandwidth and synthesized frequencies. The performance of the designed synthesizer is
evaluated using Cadence Virtuoso design tool. With 3.2 GHz sampling frequency, the DDFS
achieves the spurious-free dynamic range (SFDR) of 60 dB to 58 dB for synthesized frequencies
between 200 MHz to 1.6 GHz. With 6.4 GHz sampling frequency, the synthesizer achieves the
SFDR of 46 dB to 40 dB for synthesized frequencies between 400 MHz to 3.2 GHz. The power
consumption is 80 mW for the designed mixed-signal blocks.
Key words: Rom-Less DDFS, Current Steering Digital-to-Analog Converter, Interleaved DACs,
Return-to-Zero, Sine-weighted DAC
Acknowledgment
I would like to express my deepest appreciation to all the people who have helped me during the
conduction of this thesis work.
I would like to thank my supervisor Dr. Behzad Mesgarzadeh for his valuable ideas, guidance
and help during this project. Thank you so much for the experience.
I also would like to express my gratitude to Associate Professor Dr. J Jacob Wikner, for all his
insightful discussions and guidance which was given with the most passion and generosity in
time and knowledge.
I am also grateful for all the valuable help that I have received from Petter källström, Ph.D.
student of Electronic System division, Ameya Bhide, Ph.D. student of Electronic Devices
division, and Farrokh Ghani Zadegan, Ph.D. student of Embedded Systems Laboratory. I am also
thankful from all other researchers who have helped me during this project.
My special thanks to Amin Ojani, Ph.D. student of Electronic Devices division, for his
discussions and technical help throughout this work. His enthusiasm and curiosity is admirable.
A big thank you goes to my beloved family, my parents and my dear sister Golpooneh, for their
unconditional love and support. I am also thankful to all my friends, who have enriched my life
with love and joy.
My acknowledgments to Linköping University, for providing all the resources that I needed to
learn and grow.
2
Contents
Abstract ......................................................................................................................................................... 1
Acknowledgment .......................................................................................................................................... 2
Table of Figures ............................................................................................................................................ 5
Table of Tables ............................................................................................................................................. 7
Chapter1 Introduction ................................................................................................................................... 9
1.1
Motivation ..................................................................................................................................... 9
1.2
Thesis Organization .................................................................................................................... 10
1.3
List of Acronyms ........................................................................................................................ 11
Chapter 2 DDFS Principles and Architectures ........................................................................................... 13
2.1
Conventional DDFS .................................................................................................................... 13
2.1.1 The Phase Accumulator ............................................................................................................. 14
2.1.2 The phase to amplitude converter .............................................................................................. 17
2.1.3 The Digital to Analog Converter................................................................................................ 18
2.1.4 Anti-aliasing Filter ..................................................................................................................... 21
2.2
ROM-Less Direct Digital Synthesizers....................................................................................... 23
2.2.1 Direct digital synthesizer using a sine weighted DAC ............................................................... 23
2.2.2 Direct digital synthesizer using triangle to sine wave converter ................................................ 24
Chapter 3 Noise Analysis of DDFS output spectrum ................................................................................. 26
3.1
Spurious related to the phase truncation error............................................................................. 26
3.2
Spurious related to the DAC’s finite resolution .......................................................................... 28
3.3
Spurious related to the nonlinearities of the DAC ..................................................................... 28
3.3.1 Static performance ..................................................................................................................... 28
3.3.2 Dynamic performance ................................................................................................................ 32
3.3.3 Output spectrum of the digital to analog converter .................................................................... 35
3.4
The phase noise of the DDFS .................................................................................................... 37
Chapter 4 DAC Interleaving ....................................................................................................................... 38
4.1
DAC limitations for high frequency performance ..................................................................... 38
4.2
Different approaches for DACs interleaving ............................................................................. 39
4.2.1 Hold Interleaved DACs .............................................................................................................. 39
4.2.2 Data Interleaving DACs ............................................................................................................. 40
4.2.3 Data and Hold Interleaving DACs ............................................................................................. 41
3
4.3
The Interleaving and Return to Zero approach used in this project ........................................... 42
Chapter 5 Designed Direct digital frequency synthesizer ........................................................................... 46
5.1
System Overview ....................................................................................................................... 46
5.2
High Level Simulation Results .................................................................................................. 52
5.3
Transistor Level Simulation Results ........................................................................................... 55
Future Work ................................................................................................................................................ 64
References ................................................................................................................................................... 65
Appendix A ................................................................................................................................................. 68
4
Table of Figures
Figure2-3 Pipelined Phase Accumulator [18]........................................................................................... 16
Figure 2-4 Logic to exploit quarter wave symmetry [1]. ............................................................................ 18
Figure 2-5 N bit binary weighted current steering DAC. ......................................................................... 19
Figure 2-6 N bit thermometer coded current steering DAC. ...................................................................... 20
Figure 2-7 N bit segmented current steering DAC ...................................................................................... 20
Figure 2-8 Frequency response of DDFS (a) Ideal anti-aliasing filter (b), Realistic anti-aliasing filter (c) [5].
.................................................................................................................................................................... 22
Figure 2-9 DDFS Block Diagram using sine weighted DAC [2]..................................................................... 24
Figure 2-10 DDFS Block Diagram using triangle to sine wave converter [3]............................................... 25
Figure 2-11 TSC schematic (a), TSC transfer function (b) [18]. ................................................................... 25
Figure3-1 DDFS Spur Sources [1]. ............................................................................................................... 26
Figure 3-2 Transfer Characteristic of a DAC [20]. ....................................................................................... 29
Figure 3-3 Thermometer DAC with finite output current source impedance [7]. ...................................... 32
Figure 3-4 DAC’s full scale transition [20]. .................................................................................................. 32
Figure 3-5 The Current Cell of Current Steering DAC ................................................................................. 35
Figure 3-6 Image replicas and nonlinearities in a DAC [9] .......................................................................... 36
Figure 3-7. The ZOH and Sinc function of DAC............................................................................................ 36
Figure 4-1 Hold Interleaving DAC. ............................................................................................................... 40
Figure4-2 Data Interleaved DAC ................................................................................................................. 41
Figure4-3 Data and Hold Interleaving [9].................................................................................................... 42
Figure4-4 Interleaved DAC block diagram [9]. ............................................................................................ 43
Figure4-5 Image replicas of the first DAC (a ), Image replicas of the second DAC (b) and Image replicas
of the Interleaved DACs (c) [21]. ................................................................................................................. 44
Figure 4-6 Return-to-Zero Effect[21] .......................................................................................................... 45
Figure 5-1The block diagram of the designed DDFS. .................................................................................. 47
Figure 5-2 The Block Diagram of Flash ADC ................................................................................................ 47
Figure 5-3 The comparator of ADC ............................................................................................................. 47
Figure 5-4 The block diagram of sine weighted DAC. ................................................................................ 49
Figure 5-5 The Current Cell of Current Steering DAC ................................................................................. 50
Figure 5-6 The implemented System .......................................................................................................... 51
Figure 5-7 Return to Zero, using discharge transistors ............................................................................... 52
Figure 5-8 The generated triangle and sine waves with FCW=1. ............................................................... 53
Figure5-9 The output spectrum of 1.75 GHz output frequency. ................................................................ 54
Figure5-10 The SFDR versus FCW ............................................................................................................... 54
Figure5-11 The sine wave generated with FCW=2,
MHz and 3.2 GHz sampling frequency. .... 55
Figure 5-12 The sine wave generated with FCW=2,
MHz and 6.4 GHz sampling frequency. ... 56
Figure 5-13 The output spectrum of DDFS
MHz output frequency with 3.2 GHz sampling frequency.
.................................................................................................................................................................... 57
Figure 5-14 The spectrum of
MHz, with 6.4 GHz sampling frequency .................................... 58
5
Figure 5-15 The output spectrum of DDFS at Nyquist frequency (1.6 GHz) with 3.2 GHz sampling
frequency. ................................................................................................................................................... 59
Figure 5-16 The Nyquist frequency output spectrum with 6.4 GHz sampling frequency. ......................... 60
6
Table of Tables
Table 5-1 The simulation results of this work ............................................................................................. 61
Table 5-2 The simulation results of previous works ................................................................................... 62
Table 5-3 The measurement results of previous works.............................................................................. 63
7
8
Chapter1 Introduction
1.1
Motivation
A direct digital frequency synthesizer (DDFS) uses digital signal processing to generate
frequency and phase tunable output signals. The generated output frequency is a division of the
reference clock frequency. The division factor is set in a binary tuning word [5]. The DDFS has
the advantages of fast frequency switching, fine frequency resolution, direct digital phase and
frequency modulation in the digital domain and low phase noise. DDFS has a variety of
applications from instrumentations and measurements to modern digital communication systems.
For example, they can be utilized as a clock generator, which produces
output frequencies
with N the resolution of its phase accumulator. This characteristic is useful for the systems that
need multiple clock frequencies with no integer relationship between them and they need to be
changed rapidly and frequently [5]. In modern communication systems, DDFS seems to be an
alternative to phase-locked loops (PLL). Fast switching speed is becoming more and more
important in today’s wireless communication systems, such as in spread spectrum
communication systems. The limitation of the tuning speed of the PLL comes from the produced
delay due to its internal feedback [1].
Aside from these advantages, DDFS is only capable of producing the exact integer division of
the reference clock frequency when the FCW is 2 to the power of an integer. However, PLL has
the ability to lock its output to the input phase of a reference clock. Moreover, PLL is capable of
producing higher output frequencies. In order to take advantages of both PLL and DDFS, some
applications use a hybrid frequency synthesizer, combining PLL and DDFS [5]. Moreover,
conventional direct digital frequency synthesizers are considered power hungry systems due to
the use of ROM look up table in their architecture [2]. Consequently, ROM-Less architectures
has been introduced [2] [3]. The first approach in ROM-Less DDFS architecture was to use all
thermometer sine-weighted DAC [4]. However, this approach needed a huge number of current
cells. Therefore, to decrease the number of current cells segmentation algorithm for nonlinear
DAC was proposed [12]. The segmentation of nonlinear DAC is more complicated than the
linear ones and this architecture suffers from more complexity. The second approach in ROM9
Less DDFS design is to use the triangle to sine wave conversion. This method uses the parabolic
approximation, and utilizes the exponential current-voltage relationship of the transistors to
implement it electronically. This method shows a moderate precision in triangle to sine wave
conversion [3].
In the DDFS design, the most important performance parameters are sampling rate, power
consumption and spectral purity. However, a new figure of merit was introduced in [2] to also
take in to account the amplitude resolution information. In order to achieve high sampling rates
and high synthesized frequencies, the direct digital synthesizers are mostly designed in indium
phosphide (InP) HBT, silicon germanium (SiGe) HBT and SiGe BiCMOS technology [3]. The
effort in this project was to design a DDFS with multi-GHz sampling rate in 65nm CMOS
technology, with high spectral purity. Interleaving with return to zero (RTZ) technique has been
used to achieve a high bandwidth.
1.2
Thesis Organization
The organization of this thesis is as follow. In chapter two, the principles of the DDFS will be
discussed through explaining the conventional DDFS architecture and the functionality of each
block. Moreover, ROM-less architectures including the ones using nonlinear DAC and triangle
to sine wave converter will be introduced. In chapter three, the error sources of the DDFS
including the phase truncation error, the phase to amplitude conversion error, the introduced
errors due to the nonlinearities of the DAC and the DDFS phase noise will be discussed. Chapter
four will cover the limitations of DAC for having a wide bandwidth. In this chapter DAC
interleaving principle and its different approaches will also discussed. In chapter five, the
designed architecture in this project will be discussed and it will be followed by high level and
transistor level simulations. Moreover, the results of the previous works with different
architectures of DDFS will also be presented. The Verilog-A codes used for high level blocks
including the phase accumulator and complementor can be found in Appendix A.
10
1.3
List of Acronyms
ADC
Analog-to-Digital Converter
BiCMOS
Bipolar complementary metal-oxide-semiconductor
CLK
Clock
CMOS
Complementary metal-oxide-semiconductor
CORDIC
Co-ordinate digital computer
dB
Decibel
DAC
Digital-to-Analog Converter
DDFS
Direct Digital Frequency Synthesizer
DFF
Delay-flip-flop
DNL
Differential Nonlinearity
FCW
Frequency Control Word
FSK
Frequency-Shift Keying
GCD
Greatest Common Divisor
Third Order Distortion
INL
Integral Nonlinearity
LSB
Least Significant Bit
LUT
Look-up table
MSB
Most Significant Bit
MSK
Minimum-Shift Keying
PAC
Phase to Amplitude Converter
11
PLL
Phase Locked Loops
ROM
Read-only memory
RTZ
Return-to-Zero
SNR
Signal to Noise Ratio
SFDR
Spurious Free Dynamic Range
TSC
Triangle to Sine Converter
ZOH
Zero Order Hold
12
Chapter 2 DDFS Principles and Architectures
As it was stated earlier, Direct Digital Frequency Synthesizer, DDFS, uses digital signal
processing to generate frequency and phase tunable output signals. In order to change the
frequency of the output signal, frequency control word (FCW) or the frequency of the reference
clock can be changed. In this chapter the DDFS principles are described through explaining
conventional DDFS architecture. Also, the most common DDFS architectures will be presented.
2.1
Conventional DDFS
The block diagram of a conventional DDFS is shown in figure 2-1. The DDFS consists of a
phase accumulator, a phase to sinusoid amplitude converter (PAC) and a digital to analog
converter (DAC) followed by a filter. The phase accumulator consists of a counter and a register.
The register restores the frequency control word (FCW), which is the jump size of the counter.
With each clock cycle, the over flow of the counter is added to the FCW. The result of this
counting is the production of the phase information of the sine wave. The output of the phase
accumulator will be fed to PAC, which converts the phase information of the sine wave to
amplitude. The discrete-time, discrete-amplitude information of the sine will be converted to
analog by passing through a DAC. The final block of the system is an ant-aliasing filter. The
functionality of each block is described in more details in the following sections.
FCW
Phase
Accumulator
Phase to Amplitude
Converter
Digital to Analog
Converter
CLK
Figure 2-1 The Block Diagram of conventional DDS.
13
Filter
2.1.1 The Phase Accumulator
The phase accumulator is basically a counter which has the responsibility of generating the
phase information of the sine wave. In order to understand how the frequency is synthesized
using a phase accumulator, consider the phase wheel in figure 2-2.
M = Jump Size
N
Number of points:
8
256
12
4096
16
65535
20
1046576
24
16777216
28
268435456
32
4294967296
Figure 2-2 Digital phase wheel [5].
14
Each point on the phase wheel is correspondent to an equivalent phase of the sine wave. A
complete rotation of the phase wheel with constant speed will generate one complete period of a
sine wave. In every clock cycle, the over flow of the counter is added to the FCW which is stored
in the phase accumulator register. Consequently, FCW determines how fast the counter travels
around the phase wheel. As a result of a higher jump size, the counter completes one rotation
around the phase wheel faster, and consequently a higher output frequency will be synthesized.
The resolution of the phase accumulator (N) determines how many phase points the phase wheel
contains, and consequently it determines the resolution of the synthesized output frequency. For
example, if N is taken to be 32, then the FCW of 0000…0001 will result the counter to overflow
after
reference clock cycles (a complete rotation) and gives the lowest possible output
frequency. The FCW of 0111…1111 will result the counter to overflow after only two reference
clock cycles (a complete rotation). The relation between the reference clock frequency
output frequency
,
the FCW and resolution of the phase accumulator is given in equation 2-1.
Equation 2-1
=
According to Nyquist theorem, we need at least two samples per cycle in order to reconstruct the
sine wave; consequently, the highest output frequency that we can achieve is equal to
The frequency resolution of the synthesizer (
.
is found when the FCW is set to one:
Equation2-2
As the phase accumulator is not able to complete multi bit addition in a short clock period, in
order to run the DDFS in high speeds, pipelined phase accumulator is usually used. A pipelined
phase accumulator is shown in the figure 2-3 [1].
15
As it can be understood from the above, all the signal processing operations which are needed for
synthesizing and tuning the output frequency of the DDFS, is done in digital domain. This is why
the direct digital synthesizers are so attractive for digital modulation techniques, such as FSK
and MSK.
DFF
Adder
DFF
clk
DFF
clk
clk
DFF
clk
DFF
clk
clk
Adder
DFF
clk
clk
Adder
clk
Adder
clk
Figure2-3 Pipelined Phase Accumulator [18].
16
1’s
Com
plime
ntor
2.1.2 The phase to amplitude converter
After the phase information is generated by the phase accumulator, it will be fed to the phase to
amplitude converter, which is a ROM look up table in the conventional DDFS. The look up table
contains the amplitude information correspondent with each of the phase points of the phase
wheel. In order to avoid a very large look up table, it is common to use only a fraction of the
most significant bits of the phase accumulator information In order to produce a sine wave. In
this case we say that the DDFS is truncated from k bits to j bits, for example from 32 bits to 12
bits. The truncation results in spurs in the output spectrum of the DDFS, which will be discussed
in the next chapter. However, 12 bits still results in a large look up table. A large look up table
decreases the speed of the synthesizer and increase the power consumption and die area,
moreover a high resolution DAC will be needed to design. Therefore, a tremendous work has
been done to reduce the size of the look up table. A very basic one is to use the quarter wave
symmetry of the sine wave. The block diagram of this method is shown in the figure 2-4 [1]. In
this case only the amplitude information of the 0 to π/2 of the sine wave is stored in the ROM,
and the two most significant bits of the phase accumulator output are used to distinguish the
quarter of the sine wave. The most significant bit illustrates the sign of the sine wave amplitude
and the second most significant bit is used to determine weather the amplitude is increasing or
decreasing.
Other ROM compression techniques include the Sunderland architecture, Nicholas architecture,
polynomial approximation and CORDIC algorithm. In the Sunderland architecture the large look
up table is divided in to two smaller memories. The Nicholas architecture has improved the
Sunderland architecture and hence has achieved a higher ROM compression. In the Polynomial
approximations, the coefficient of the polynomial is stored in the ROM. In this method the
interval of [0,
is divided in smaller divisions and the sine/cosine is produced in for each of
them. The CORDIC algorithm has its advantage over ROM when the needed accuracy is more
than 9 bits. Using this algorithm the needed hardware is not growing exponentially when the
output word size is increasing. For more information about this methods please refer to [1].
17
MSB
nd
2 MSB
Phase
Accumulator
π/2 sine look up
table
Complementor
Complementor
2π
π/2
0
0
Figure 2-4 Logic to exploit quarter wave symmetry [1].
2.1.3 The Digital to Analog Converter
As it is shown in the figure 2-1, the discrete-time, discrete-amplitude information of the sine
wave is fed to a digital to analog converter to be converted to a continuous-amplitude,
continuous-time sine- wave. The current steering DACs are the best choice for high speed
applications because of their fast switching speed. They can be implemented in binary weighted,
thermometer coded and segmented architectures. The segmented architecture combines the
binary weighted and thermometer coded architectures to take advantage of the benefits of both
architectures. It uses thermometer coded for its most significant bits (MSB) and binary weighted
for its least significant (LSB) bits. The binary weighted, thermometer coded and segmented
architectures are shown in the figures 2-5, 2-6 and 2-7 respectively. As it can be seen in the
figure 2-5, in the binary weighted architecture, each current source is as twice as much of the
previous one. According to the digital input code a combination of the current sources will be
switched to the output. This architecture has the advantage of small area and low power
consumption. However, it suffers from differential nonlinearity (DNL) and the presence of
18
glitches, degrades its dynamic performance. On the other hand, thermometer coded architecture
has more complexity and higher power consumption, but it has improved DNL, low glitches and
small switching errors. In this architecture, all the current sources are equal. The digital input
code is first fed to a thermometer decoder, and the thermometer code turns on the switches
accordingly. Definitions and sources of the DAC nonlinearities will be presented in the next
chapter. The segmented architecture uses the thermometer coded for its most significant bits,
which are more responsible for the dynamic performance, and binary weighted for its least
significant bits. A dummy decoder should be used for the binary weighted part to compensate for
the delay of the thermometer decoder of the thermometer decoded part [9]. The considerations
on how the resolution of each of architectures should be chosen in the segmented current steering
DACs can be found in [19]. It has to be noted that in the DDFS the dynamic performance of the
DAC plays a significant role in the spectral purity of the output spectrum, which will be
discussed in more details in the next chapter.
I
4I
2I
Figure 2-5 N bit binary weighted current steering DAC.
19
I
I
I
Figure 2-6 N bit thermometer coded current steering DAC.
……………
..
Thermometer
Decoder
……………
.. Decoder
Dummy
Switch Driver
Switch Driver
Unary Switches
Binary Switches
Unary Current
Sources
Binary Current
Sources
Figure 2-7 N bit segmented current steering DAC
20
I
2.1.4 Anti-aliasing Filter
As it will be discussed in more details in the next chapter, the DDFS is a sampling system.
Therefore, there will be images at the frequencies of
the output frequency and
of the output spectrum, with
the sampling clock. As the result of the zero order hold
functionality of the DAC, the amplitude of the images are weighted by the
function.
For most applications, these images are undesirable. In order to remove these images, a filter by
an inverse
function called anti-aliasing filter is used at the end of the system. Ideally,
this filter should have unity response over the Nyquist bandwidth and zero beyond that.
However, designing such a filter is not practical; consequently, some percentage of available
bandwidth will be unusable. Therefore, the synthesized output frequency of DDFS is usually
limited to less than 3/8 of sampling frequency [2]. Figure 2-8(a) shows the spectrum of the
DDFS, taking in to account only the image replicas. Figure 2-8 (b) and (c) show the effect of the
ideal and non ideal filter on the output spectrum of the synthesizer. The design of this filter is
beyond the scope of this thesis.
21
Amplitude
Frequency
(a)
Amplitude
Ideal filter response
No aliases
Frequency
Amplitude
No aliases
Frequency
𝑜
2
/2
(b)
Amplitude
Realistic filter response
Supressed
aliases
Frequency
Amplitude
No aliases
Frequency
𝑜
/2
2
(c)
Figure 2-8 Frequency response of DDFS (a) Ideal anti-aliasing filter (b), Realistic anti-aliasing filter (c) [5].
22
2.2
ROM-Less Direct Digital Synthesizers
As it was stated earlier, the ROM look up table is the speed, power and area bottleneck of direct
digital synthesizers. Although a tremendous work has been done to compress the ROM look up
table, direct digital synthesizers using this method still have high power consumption and
limitations in higher frequency operations [3]. Consequently, ROM-Less architectures has been
introduced. The two most common ones are described briefly in the following section.
2.2.1 Direct digital synthesizer using a sine weighted DAC
In order to reduce the power consumption of direct digital synthesizers, ROM-Less architectures
based on sine weighted DACs has been proposed [4]. The block diagram of DDFS using a sine
weighted DAC is shown in the figure 2-9. In this architecture the sine/cosine mapping and the
digital to analog conversion are performed in a same block, called sine weighted DAC. The
design challenges of the sine weighted DAC is mostly the same with the linear DAC. The main
difference between the sine weighted DAC and linear DAC is that in the linear DAC the current
sources are identical with each other or they are a power of two weighted. However, in the sine
weighted DAC the current sources are weighted according the amplitude of the sine wave [2]. In
this architecture, for each phase of the sine wave the sine weighted DAC switches the
corresponding amount of current to the output. The most two significant bits are used to exploit
the quarter wave symmetry of the sine wave. Initially, these architectures used all thermometer
sine weighted DACs. In order to reduce the number of DAC cells, segmentation techniques were
proposed [12]. Segmentation techniques for nonlinear DACs are more complicated than for
linear ones, and these architectures suffer from complexity when the resolution is high [3].
23
MSB
nd
2
FCW
Phase
Accumulator
MSB
Complementor
Nonlinear DAC
CLK
2
0
Figure 2-9 DDFS Block Diagram using sine weighted DAC [2].
2.2.2 Direct digital synthesizer using triangle to sine wave converter
The block diagram of a DDFS using triangle to sine wave converter is shown in the figure 2-10.
This architecture uses the most significant bit to exploit the half wave symmetry of the sine
wave; consequently, it decreases the truncation error. The output of the complementor will then
fed to a linear DAC. The linear DAC produces a triangle wave which contains the analog phase
information of the sine wave. The triangle wave is then converted to a sine wave using an analog
sine-mapping methodology. This methodology uses the parabolic approximation, which is
implemented electronically by using the exponential current-voltage relationship of the
transistors [3] [18]. Figure 2-11 shows the schematic and transfer function of the triangle to sine
wave converter [18]. This architecture has a simple and low power structure and shows a
moderate precision in triangle to sine wave conversion [3].
24
MSB
FCW
Phase
Accumulator
Complementor
Triangle to Sine
Converter
Linear DAC
CLK
Figure 2-10 DDFS Block Diagram using triangle to sine wave converter [3].
Output signal
Transfer function
Analog triangle from DAC
(a)
(b)
Equation2-3:
.
Equation2-4:
Equation2-5:
-1
Figure 2-11 TSC schematic (a), TSC transfer function (b) [18].
25
Chapter 3 Noise Analysis of DDFS output spectrum
The direct digital frequency synthesizer has four sources of spurs, which is shown in the figure
3-1. These error sources include the truncation error of the phase accumulator, the phase to
amplitude conversion error, the errors due to the nonlinearity of the DAC and also the phase
noise. In this chapter these error sources and their effect on the output spectrum of the DDFS are
discussed.
FCW
Phase Accumulator
Phase to Amplitude
Converter
Digital to Analog
Converter
Reference Clock
Reference Clock Spurs Noise
Phase Truncation Spurs
Angle to Amplitude Error Spurs
Nonlinearity of the DAC’s Spurs
Figure3-1 DDFS Spur Sources [1].
3.1
Spurious related to the phase truncation error
As it was stated earlier, in order to have fine frequency resolution we would like to increase the
resolution of the phase accumulator. However, this would result in large circuits that are needed
to convert the phase data to amplitude data. Therefore, the output of the phase accumulator is
usually truncated from N bits in to P bits. This truncation will result in a phase error between the
generated phase by the accumulator, and the phase that is used by the PAC for amplitude
generation; consequently, there will be an error in the generated amplitude. This error is periodic
26
in the time domain and hence shows itself as spurs in the frequency domain [5]. The periodic
nature of the error is due to the fact that after sufficient rotation of the phase wheel the
accumulator phase and the truncated phase will coincide and there will be no phase error. The
pattern will continue as the phase accumulator continues to count. However, certain frequency
control words result in the maximum level of the phase truncation spurs while some result in no
error. The control words that yield the maximum spurs level should satisfy the following
equation [5]:
Equation3-1
(FCW,
)=
Where, GCD denotes the greatest common divisor between the two variables in the parentheses.
Hence, any control word with 1 in the bit position of
and 0 in all other least significant
bit positions will result in the maximum truncation spurs level. Moreover, the control word that
yield to no truncation error should satisfy the following equation [5]:
Equation3-2
(FCW,
)=
Hence, any control word with 1 in the bit position of
and 0 in all other least significant
bit positions will result in no phase truncation spurs. The generated spurs due to the phase
truncation are the most significant spurs, if we consider the DAC ideal. They will be mixed by
the DDFS output frequency, and will generate spurs at multiples of the output frequency, which
is calculated by the following equation [1] [2]:
Equation3-3
27
3.2
Spurious related to the DAC’s finite resolution
The finite resolution of the DAC and consequently the finite number of quantization levels of the
DAC will result in an error, called the quantization error. The quantization error is basically the
difference between the amplitude of the reconstructed sine wave and the ideal sine wave, which
is due to the limited resolution of the. This error will show itself as spurs in the output spectrum
of DDFS. The quantization error can be decreased by increasing the resolution of the DAC. The
relationship between the resolution of the DAC and the amount of quantization distortion can be
quantified with the following equation [5]:
Equation3-4
1.76 + 6.02P
Where, P is the number of bits of the DAC and SQR is the ratio of the signal power to
quantization noise power. It should be noted that this equation does not provide any information
about the total SFDR of the system, and only considers the spurs due to the quantization error.
3.3
Spurious related to the nonlinearities of the DAC
The most dominant spurs in the output spectrum of the DDFS is the spurs related to the
nonlinearities of the DAC. Both static and dynamic nonlinearities will be discussed in the
following section; however, in high sampling rates circuits the dynamic nonlinearities play the
significant role and being statically linear is the prerequisite for the DAC to have a good dynamic
linearity [6].
3.3.1 Static performance
The static specifications of a digital to analog converter include offset error, gain error, integral
nonlinearity (INL) and differential nonlinearity (DNL). These errors will result a nonlinear
28
relation between the actual output level produced by the DAC and the ideal output level that the
designer expects; consequently, there will be harmonic distortions at the output spectrum of the
digital to analog converter. Figure 3-2 shows the ideal and actual transfer functions of a three bit
DAC, together with the correspondent static nonlinearities, which is briefly discussed in the
following section.
Analog Output
Ideal Straight Line
Actual transfer function
1 LSB
DNL
INL
Ideal transfer characteristic
Offset
Digital Input
000 001
010
011
100
101
110
111
Figure 3-2 Transfer Characteristic of a DAC [20].
Offset error: offset error is the shift in the transfer function of the DAC on the vertical axis, and
it shows that for an input value of zero, the DAC will output an analog value, not equal to zero.
Gain error: In the transfer function of the DAC, the difference between the actual slope and the
ideal slop is defined as the gain error. The gain error is not of a big concern when a single
converter is being used, because rather than the absolute accuracy, the relative accuracy is of
concern [6].
29
Monotonicity: The monotonicity of a digital to analog converter is its ability to decrease or
increase in the same direction of its input signal.
Integral nonlinearity (INL) and differential nonlinearity (DNL): If we consider a line that
passes through the end points of the transfer function of the DAC, the integral nonlinearity (INL)
would be the maximum deviation between that line and the actual analog output of the DAC.
The differential nonlinearity (DNL) is the difference between the actual step size and the ideal
one least significant bit step size in the transfer function of the DAC. These errors are shown in
the figure 3-2. The differential nonlinearity can be given in terms of least significant bit step
sizes with the normalized form according to the following equation [20]:
Equation3-5
In the above formula,
and
are the analog outputs corresponding to two successive
codes of the converter. Moreover, the integral nonlinearity can be given as the accumulation of
previous differential nonlinearity errors according to the following equation [20]:
Equation3-6
In the above formula,
and
=
are the actual and ideal analog outputs of the converter.
Mostly, transistor mismatch in the current source of the DAC cells and the finite output
impedance of the current sources are responsible for INL and DNL [6]. Care has to be taken
when designing the current cells to have as much matching as possible, to meet the static
specifications. This can be done by using sufficient gate area for current source transistors, short
distance between the transistors and equal environments by using dummy transistors [7].
Moreover, there have been introduced some techniques, such as dynamic element matching,
calibration technique and trimming to overcome the matching problem of the DAC. However, as
these techniques introduce more complicated circuits and consequently more parasitic
capacitance, they do more harm than good in high frequencies [7]. The finite output impedance
of the DAC current sources will also result in distortion in the output spectrum of the DAC.
Figure 3-3 shows a thermometer coded DAC. The current sources are considered ideal with
current value of
and finite output impedance of
30
. N corresponds to total number of current
sources and n is equal to the digital input code. With different input codes, different number of
current sources will be connected in parallel with the output load
and consequently, the total
effective load impedance will be dependent on the input signal [7]; hence, the output voltage will
be signal dependant which will produce distortion in the DAC’s output spectrum. The produced
output voltage
correspondent to the digital input word (n) in the figure 3-3 is equal to [7]:
Equation3-7
}
For full swing condition (n=N) the expression for the third order distortion can be written as [7]:
Equation3-8
As it can be seen from the above formula, for having a low third order distortion, a high output
impedance from each current source is needed. In low frequencies this can be achieved by using
cascade transistors; however, for high frequencies this is not a sufficient solution, as will be
discussed in the next section.
31
Figure 3-3 Thermometer DAC with finite output current source impedance [7].
3.3.2 Dynamic performance
The dynamic errors of the digital to analog converter include glitches, settling time and feed
through effects. These errors are shown in the figure 3-4. Dynamic errors have a significant
impact on the performance of the DAC and they even become more critical for higher output
frequencies and sampling rates. These errors are presented in the following section.
Analog Output
Glitch
Clock Feedthrough
Ideal Transition
Settling Time
Time
Figure 3-4 DAC’s full scale transition [20].
32
Glitches: Glitches happen as a result of an unmatched switching time between different bits,
which can be due to skew between bits in the digital part or the timing mismatch in the switches
of the DAC. The result is a signal dependant error from the inputs to the output of the DAC
during the code transitions. For example, consider the case that the input code is changing from
0111 to 1000. If the switching time of all the current cells do not be synchronized, it is possible
that we get the analog converted of 111 for a very short period in the output; consequently, a
glitch will be occurred in the output. Figure2-3 shows this situation. This phenomenon is much
severe in high frequencies. Careful layout and using thermometer decoding can be used to
degrade this effect.
Settling time: is defined as the time which is needed for the analog output to settle between the
accepted error band of its final value and is due to the parasitic capacitances of the circuit. The
settling time should be kept as small as possible to have a low distortion on the analog output
signal.
Feed through effects: feed through effects have two sources in a DAC cells. The first one is
the feed through of the digital signal through
or
of the switch transistors, which actually
results in distortion in the Nyquist bandwidth of the output spectrum, since it’s a code dependant
error. This error can be minimized by a careful layout and switches sizing. The second one is the
feed through of the clock to the analog output, which also can be reduced by minimizing the size
of the switches and hence reducing the capacitive coupling of the switches to the output. All the
dynamic nonlinearities associated with the switches can be addressed by using return to zero
(RTZ) technique, which can be implemented both with analog or digital solutions. In analog
return to zero technique the output of the current cells is forced to zero when the clock is low and
their current is switched to the output only when the clock is high; consequently, the switching
transients do not appear in the DAC’s output.
As it was stated earlier, finite output impedance of the DAC will also result in dynamic
nonlinearities. A current cell of a current steering DAC is shown in Figure3-5. In the previous
part it was assumed that the output impedance of the current sources are purely resistive.
However, at higher frequencies this impedance is modeled as the resistor in parallel with the
effective output capacitance [7]:
33
Equation3-9
)
Accordingly, the third order distortion can be calculated as [7]:
Equation3-10
Where, N is the number of current sources. As it can be seen, higher frequencies will result in
higher third order harmonics. Moreover, it can be seen in the above equation that the frequency
of the DAC is limited due to the minimum achievable amount of
. This is one of the limiting
factors for designing high speed DACs, which will be discussed in more details in section the
next chapter.
As it can be understood from the above explanations, sizing the transistor’s properly plays an
important role in both static and dynamic performances of current steering DACs. For current
source (M2) the high output impedance and matching is of concern. Therefore, large sized
transistors together with cascade transistors (M1), with both M1 and M2 working in the
saturation region, is the desirable choice. However, in order to reduce the parasitic capacitances
at the sources of the switches, the cascade transistors (M1) should be just large enough to be able
to support the current. For switches (M) we would like to have small on resistance and minimum
feed through effect.
As the switches work with high gate-source voltage, minimum sized
transistors can be chosen for them [7].
34
Digital
Circuit
s
M
Digital
Circuit
s
M
4
M1
Biasing
Circuits
M2
Figure 3-5 The Current Cell of Current Steering DAC
3.3.3 Output spectrum of the digital to analog converter
The output spectrum of a DAC is shown in the figure 3-6. As it can be seen, the output spectrum
consists of harmonics and image replicas. Harmonics rise from the DAC’s nonlinearities, as it
was discussed previously. They will be mixed between the clock and the signal and will result in
spurious components at the locations of
, for integer values of m and n [9]. Image
replicas rise from the sampling characteristics of the DAC. Taking in to account the Poisson
summation, the frequency domain of an ideal sampled signal is written as:
Equation3-11
,with the
)
the sampling frequency. Consequently, the output spectrum will be the summation of
the fundamental signal and all the images locating at
. It also should be noted that the
amplitude of these images are decreasing in time. This is because of the zero order hold (ZOH)
behavior of the DAC’s response in the time domain. As the DAC samples the input in every
clock cycle, the analog output will have a zero order hold function, with the Sinc function as its
Fourier transform. The time domain ZOH function and frequency domain Sinc function is shown
in the figure 3-7. The amplitude of the Sinc function decreases in time, which leads to the
35
decrease of the amplitude of the images. It should be noted that the harmonics do not follow the
Sinc function and it is not possible to predict their magnitude [5]. According to the Nyquist
theorem at least two samples is required per cycle, in order to reconstruct the desired output
waveform.
Signal
Image Replica
Nonlinearity
Hold Distortion
Figure 3-6 Image replicas and nonlinearities in a DAC [9]
-2π/
2π/
Frequency Domain
Time Domain
Figure 3-7. The ZOH and Sinc function of DAC
36
SFDR and SNR are the most common terminologies that are used to describe the performance of
the output spectrum of the DAC. SFDR stands for spurious free dynamic range and is the ratio
between the signal power and the strongest spurious component in the output spectrum. SNR
stands for signal to noise ratio and is the ratio between the signal power and the total power of
the summation of all spectral components, excluding the harmonics. The SNR of an N bit DAC
is approximately given by [9]:
𝑜
Equation 3-12
Where, PN is the power of the total noise produced by the noise of the circuit (including the
thermal noise and flicker noise) and the quantization noise. the lower limit of the SNR is
contributed by the quantization noise[20].
The SFDR is calculated as:
Equation 3-13
SFDR= 10log (PS/PH)
Where, PH is the power of the strongest distortion component.
3.4
The phase noise of the DDFS
The dominant contributor to the DDFS phase noise is the phase noise of the reference clock. In
fact, because DDFS is a divider of the sampling clock, the purity of its output spectrum is
directly affected by the purity of its reference clock. However, DDFS has a great advantage over
PLL regarding to its phase noise. This is because PLL multiplies the phase noise of the reference
clock in its feedback loop, but DDFS is a feed forward system, which its output is a fractional
division of the reference clock; consequently, the phase noise which presents in the output
spectrum of DDFS decreases by 20 log (N), where N is the division ratio. Moreover, as DDFS is
a sampling system and the time interval between the samples are important, and the jitter of the
reference clock will have an important role on the output spectral purity.
37
Chapter 4 DAC Interleaving
Interleaving has shown promising impacts on expanding the usable bandwidth of analog to
digital converters. Consequently, interleaving the digital to analog converters also became an
interesting approach for increasing their high sampling rate performance. In interleaved ADCs
the same input signal is fed to all sub ADCs, which are sampled at multiple phases of the clock.
However, because of the zero order hold response of digital to analog converters, this approach
is not as straight forward as ADCs for them. DAC interleaving categorized in two groups, data
interleaving and hold interleaving [9]. In this chapter, after discussing the performance
limitations of the digital to analog converters for high sampling rates and wide bandwidth,
different methods of interleaving DACs will be briefly described and this will be followed by
discussing the interleaving and return to zero technique used in this project and its influence on
the output bandwidth.
4.1
DAC limitations for high frequency performance
High frequency performance of digital to analog converters is limited for the following three
main reasons [9]:
1) As it was stated in the previous chapter, DAC has a zero order hold behavior in the time
domain and consequently a Sinc frequency response with a null located at
This will
result in amplitude distortion for high frequency generated signals.
2) According to the Nyquist theorem the applicable bandwidth of the digital to analog
converters is limited to
/2.
3) As it was mentioned in the last chapter, the expression for the third order distortion is
equal to:
Equation4-1
38
With
the output impedance equal to:
Equation4-2
)
Consequently, it can be seen that increase of the frequency will result in
the other words, the requirement on the
consequently on
degredation. In
will give us a requirement on the
. When the frequency increases, the required value of
and
will decrease and it
will be harder to achieve [7].
4.2
Different approaches for DACs interleaving
Interleaving of digital to analog converters is categorized in two different groups of hold
interleaving and data interleaving, this gives us different approaches for interleaving the DACs.
In the following section, each approach is described briefly.
4.2.1 Hold Interleaved DACs
The block diagram of hold interleaving DAC is shown in the figure 4-1. In the hold interleaved
DACs, the same data is used for all sub DACs, each clocked at time shifted version of the
sampling clock, meaning that each digital data is sampled by each phase of the sampling clock. It
was shown in [13] that this technique will lead to cancelling and lowering the aliases of the DAC
by choosing the phase shifts of the sampling clock for each DAC correctly. However, this
approach does not give significant advantages for wide band applications.
39
3
1
DAC1
1
=
3
a
a
3
2
DAC2
2
1
2
3
4
2
+
DAC3
3
1
4
b
b
4
c
c
DAC4
4
d
d
Figure 4-1 Hold Interleaving DAC.
4.2.2 Data Interleaving DACs
Data interleaving approach is shown in the figure 4-2. This approach was developed to suppress
the Nyquist images so that the produced frequencies near Nyquist could be used without
stringent requirements on the filter [14]. This suppression is done by having the Nyquist images
with 180 phase shift, and yet the fundamentals with the same phase. However, this approach
requires the sampling rate of each DAC to be N
40
.
3
1 2
=
1
/4
DAC1
a
2
3
DAC2
2
1
3
b
4
1
2
3
4
4
+
2
DAC3
3
1 2
c
4
DAC4
d
/4
Figure4-2 Data Interleaved DAC
4.2.3 Data and Hold Interleaving DACs
The block diagram of data and hold interleaving DACs are shown in the figure 4-3. In this
approach each DAC is fed with the interleaved samples of the signal and each DAC samples at
interleaved time instants. Consequently, the first N-1 image replicas and nonlinearity sours are
cancelled, and the wide band operation will be achieved [9].
41
DAC1
1
1
=
a
2
DAC2
b
b
+
3
DAC3
3
c
c
DAC2
b
+
DAC3
c
4
DAC4
4
=
a
a
2
DAC1
DAc4
d
d
d
3
2
1
4
1
2
3
4
4
2
3
1 2
Figure4-3 Data and Hold Interleaving [9]
4.3
The Interleaving and Return to Zero approach used in this project
The block diagram of both data and hold interleaving DAC used in this project is shown in the
figure 4-4. As it can be seen from the figure, each DAC works at interleaved phases of the clock
and holds its output for the entire period of the clock. Consequently, if we consider that the
second DAC is working on the phase shifted clock of the first DAC, the frequency domain
components of the first and second DAC can be written as the following equations with taking in
to account the effect of Sinc function on the amplitudes [9]:
42
Equation 4-3
DAC(I):
Equation 4-4
DAC(II):
)
)
DAC (I)
2
Digital
inputs
+
DAC (II)
Figure4-4 Interleaved DAC block diagram [9].
As it can be seen from the above formulas, the images of the two DACs have the same sign for
even values of k, while having opposite signs for odd values. Consequently, the images with odd
values of k cancel each other when the outputs of the two DACs are added, (the analog addition
is done by return to zero technique). Therefore, the resulting spectrum will be like a single DAC
running with twice sampling rate and it will be possible to generate output frequencies beyond
the Nyquist frequency of the individual DACs. This is shown in figure 4-5. However, since the
zero order hold function of each DAC is not changing with this technique and they will have a
null at
which is the new Nyquist frequency, the usable bandwidth will not improve as much.
In order to push this null to 2
, the return to zero technique as an analog switch is used. With
return to zero technique, each DAC is only active for the half of the clock cycle, so the sinc
function will have its null at 2
which is shown in the figure 4-6. Consequently, it can be
concluded that two interleaved DACs with return to zero technique is equivalent to a single DAC
which is running with twice sampling rate [9] [21].
43
f
f
(b)
(a)
f
(c
Figure4-5 Image replicas of the first DAC (a), Image replicas of the second DAC (b) and Image replicas of the
Interleaved DACs (c) [21].
However, it should be noted that in time interleaved digital to analog converters time alignment
plays an important role. The cancellation of the images will not happen properly and unwanted
spikes within the Nyquist band will occur if there will be any deviation from the ideal half
sample time delay. Moreover, the two DACs must be balanced in terms of their amplitude.
Correction algorithm for amplitude balance and time alignment might be applied to the system
by the use of calibration filters if needed. The design of this filter is beyond the scope of this
thesis, and aligned input clocks have applied to each block [9].
44
2
Figure 4-6 Return-to-Zero Effect[21]
45
Chapter 5 Designed Direct digital frequency synthesizer
5.1
System Overview
A four bit direct digital frequency synthesizer is designed in 65nm CMOS technology. In order
to increase the bandwidth and sampling rate of the system interleaving with return to zero
technique described in the previous chapter is used. Moreover, instead of the conventional digital
thermometer decoder, a simple combination of a binary weighted DAC and a Flash ADC is used.
This combination cuts the number of sine weighted current cells in half, with taking advantage of
oversampling. Oversampling is known to put more requirements on the system and also using
more power consumption. However, since the synthesized output frequency of the DDFS is
restricted to be less than 3/8 of the sampling clock, due to limitations of the filter design in the
real implementation [2], this drawback does not show a strong impact in our application. It also
should be mentioned that the mismatch and dynamic nonlinearities of the binary weighted DAC
such as glitches, will not affect the output spectrum of the synthesizer, since they will be
tolerated by the comparators of the Flash ADC. However, the primarily purpose for choosing this
architecture was to reach to a different contribution. The Block diagram of the designed DDFS is
shown in the figure 5-1. The Phase Accumulator and the Complementor are behaviorally
modeled in Verolg-A. The Verilog-A codes can be found in Appendix A.
This system exploits the half wave symmetry of the sine wave by using the most significant bit
of the phase accumulator; consequently, the sine weighted DAC only needs to produce the
corresponding currents for the phases over the range of 0 to π. When the most significant bit
turns to 1, the complementor inverts its input digital bits, so that a decreasing ramp will be
achieved. The output of the complementor will be the input to a binary weighted DAC. The
binary weighted DAC produces the analog information of the phases of the sine wave. The
analog phase information of the sine wave will be the input to a flash ADC, so that the
thermometer representation of the phase information is achieved. The block diagram of a 2 bit
flash ADC is shown in the figure 5-2. The Flash ADC is known to be used in applications with
tens of GHz sampling rates since all the conversion is done in one clock cycle. The Flash ADC
converts the analog input to a thermometer code. Usually the thermometer code is fed to a
thermometer to binary decoder to be converted to a digital binary code, which is not the case in
46
our system. As for the N bit flash ADC
comparators are needed, usually their resolutions
are restricted to maximum 8 bits. The circuit shown in the figure 5-3 followed by buffer is used
as the comparator of the ADC. The produced thermometer codes will then fed to a sine weighted
DAC, and turns on the correspondent number of cells. The currents of these cells are added
together and a sine wave is produced at the output.
MS
B
FCW
Phase
Accumulator
Complementor
Binary Weighted
DAC
Flash ADC
Sine weighted
DAC
(thermometer)
CLK
Figure 5-1The block diagram of the designed DDFS.
Vdd
Vdd
R
R
R
R
Figure 5-2 The Block Diagram of Flash ADC
Figure 5-3 The comparator of ADC
47
Filter
The reference voltages of the flash ADC is calculated according to the following formula [11]:
Equation 5-1
=
-
Equation 5-2
=
+
Figure 5-4 shows the block diagram of sine-weighted DAC. The currents of the sine weighted
DAC is calculated as:
𝑜
Equation 5-3
]
𝑜
]
𝑜
𝑜
48
]
Figure 5-4 The block diagram of sine weighted DAC.
The circuit of one current cell of the sine weighted DAC is shown in the figure 5-5. In order to
achieve high spectral purity it is important that the output current of each of the current cells be
as precise as possible. In order to decrease the impact of voltage variation of the drain-source of
the current sources on the output current, long channel transistors have been used. Moreover, as
for linear DAC, flip flops are used before the switches of the current cells, so that all the currents
are switched to the output synchronized. The load resistor is selected 25Ω, to avoid the problem
of the headroom in the interleaved system.
49
Digital
Circuits
M
M
Digital
Circuits
4
M2
Biasing
Circuits
M1
Figure 5-5 The Current Cell of Current Steering DAC
Figure 5-6 shows the system level implementation using the interleaving approach. As it was
discussed in more details in the previous chapter, when the output spectrum of two interleaved
DACs are added with each other, the odd images and harmonics of the spectrums of each DAC
cancel each other, consequently the resultant spectrum will be like the spectrum of one DAC
with twice sampling rate and each DAC can produce output frequencies beyond their Nyquist
frequency. However, as the zero order hold function of each DAC will not change with
interleaving, to widening the bandwidth the return to zero should be used. With return to zero
each DAC is only active for its half clock cycle, consequently, the null of the Sinc function will
be pushed to the 2
(with
the sampling rate of each DAC) from
The return to zero can be
done by injecting zeros to the input of binary weighted DAC for each half clock cycle of the
sampling frequency or by discharging the switch transistor of the current cells, which is shown in
the figure 5-7.
50
1 3 5 7
1 3
FCW
Phase
Accumulator
5 7
RTZ
Data
φ DAC
RTZ
Data
φ DAC
Flash
ADC
Sine
DAC
Complementor
MUX
CLK
4
6
2
1
3
5
Flash
ADC
7
2
4 6
2 4
Figure 5-6 The implemented System
51
6
Sine
DAC
Biasing
Circuits
CLKB
Biasing
Circuits
M3 M4
Digital Circuits
M5
Digital Circuits
M
Biasing
Circuits
CLK
M6
M
M2
M1
Figure 5-7 Return to Zero, using discharge transistors
5.2
High Level Simulation Results
The architecture described in the previous section was first simulated behaviorally in Verilog-A,
with Cadence Virtuoso design tool. The phase Accumulator was designed for 4 bits. With 3 bit
flash ADC and sine weighted DAC and 4 GHz sampling clock frequency, the frequency
resolution of
in the Nyquist bandwidth is achieved.
Equation 5-4
= 250 MHz
52
Figure 5-8 shows the triangle and sine waves generated with FCW=1. Figure 5-9, shows the
output spectrum of the synthesized frequency for FCW= 7, resulting in 1.750 GHz output
frequency.
Equation 5-5
The image replica (
) of
=
4 GHz= 1.75 GHz
is shown in the spectrum.
Equation 5-6
=
= 4GHz- 1.75 GHz= 2.25 GHz
The SFDR in the Nyquist-Band is equal to:
Equation 5-7
75.3 (dB) - 9.25 (dB) = 66.05 (dB)
This design is followed by a simple low pass RC filter.
Figure 5-10 shows the SFDR versus different control words .As it can be seen from this figure,
the SFDR is highest for the low synthesized frequencies (73 dB) and decreases to (64 dB) for
near Nyquist synthesized frequency.
Figure 5-8 The generated triangle and sine waves with FCW=1.
53
SFDR= 66 dB
2.25 GHz, -11.38
dB)
(1.75 GHz, -25.39)
dB
Figure5-9 The output spectrum of 1.75 GHz output frequency.
Figure5-10 The SFDR versus FCW
54
5.3
Transistor Level Simulation Results
The described system was designed and implemented in 65 nm CMOS technology, with 4 bit
phase accumulator and 3 bit flash ADC and Sine Weighted DAC. The DDFS has been sampled
with 3.2 GHz and 6.4 GHz clock frequencies. With 3.2 GHz the synthesizer was able to
synthesize outputs in 8 levels with each level 200 MHz. Obviously, with 6.4 GHz sampling
frequency the frequency resolution was 400 MHz. Figures 5-11 and 5-12 illustrates the generated
sine-waveforms for FCW=2, with 3.2 GHz and 6.4 GHz sampling frequencies respectively.
Figure5-11 The sine wave generated with FCW=2,
55
MHz and 3.2 GHz sampling frequency.
Figure 5-12 The sine wave generated with FCW=2,
MHz and 6.4 GHz sampling frequency.
Figure 5-13 shows the output spectrum of DDFS with 400 MHz output frequency and 3.2 GHz
clock frequency. In the output spectrum the worst case spurs in the Nyquist Bandwidth is the
third harmonic. The DDFS showed the SFDR of 60 dB. The first image replica at 2.8 GHz is
also shown in the figure.
Equation5-8
85.62(dB)- 25.39(dB)= 60.23 (dB)
Equation 5-9
Equation 5-10
=
=
3.2 GHz= 400 MHz
= 3.2GHz- 400 MHz= 2.8 GHz
56
(Synthesized Frequency: 400 MHz, -25.39 dB)
SFDR=60dB
B)
(Image Replica: 2.8 GHz, -56.03dBdB)
(Third Harmonic: 1.2 GHz, -85.62dB)
Figure 5-13 The output spectrum of DDFS
MHz output frequency with 3.2 GHz sampling frequency.
57
Figure 5-14 shows the output spectrum of DDFS with 800 MHz output frequency and 6.4 GHz
clock frequency. In the output spectrum the worst case spurs in the Nyquist Bandwidth is the
second harmonic. The DDFS showed the SFDR of 45 dB.
Equation 5-11
75(dB)- 30(dB)= 45 (dB)
Equation 5-12
=
6.4 GHz= 800 MHz
(Synthesized Frequency: 800 MHz, -30.52dB)
SFDR= 45(dB)
(Second Harmonic: 1.6 GHz, -75 dB)
Figure 5-14 The spectrum of
MHz, with 6.4 GHz sampling frequency
58
Figure 5-15 shows the output spectrum of DDFS at Nyquist output frequency and 3.2 GHz clock
frequency. The DDFS showed the SFDR of 58 dB.
(Synthesized Frequency: 1.6 GHz, -34.6 dB)
SFDR= 58(dB)
Figure 5-15 The output spectrum of DDFS at Nyquist frequency (1.6 GHz) with 3.2 GHz sampling frequency.
59
Figures 5-16 shows the output spectrum of Nyquist output frequency of 3.2 GHz with 6.4 GHz
clock frequency. In this synthesized frequency, the DDFS shows the SFDR of 40 dB.
(Synthesized Frequency: 3.2 GHz, -39.36 dB)
SFDR= 40(dB)
Figure 5-16 The Nyquist frequency output spectrum with 6.4 GHz sampling frequency.
As the phase accumulator is behaviorally modeled with Verilog-A, the power consumption of the
system could not be measured precisely. However, the power consumption without the phase
accumulator was 80 mW.
60
The simulation results of this work can be found in table 5-1.
Table 5-1 The simulation results of this work
Specification
This Work
Technology
65 nm CMOS
Clock Frequency
3.2 GHz
6.4 GHz
DAC Amplitude Resolution
3 bits
SFDR (Nyquist Frequency)
=3.2 GHz
58 dB
=6.4 GHz
40 dB
SFDR (
=3.2 GHz,
= 400 MHz)
60 dB
SFDR (
=6.4 GHz,
= 800 MHz)
46 dB
Power Supply
1.2 V
61
Table 5-2 shows the published transistor level simulation results for DDFS in different
technologies. As it can be seen, the best achieved SFDR is 44 dB with 4 GHz sampling
frequency in 90nm CMOS technology.
Table 5-2 The simulation results of previous works
Specification
[15]
[16]
[17]
Technology
350 nm CMOS
180 nm CMOS
90 nm CMOS
Maximum Clock
2 GHz
1 GHz
4 GHz
7 bits
8 bits
7 bits
35 dBc
20dB
44 dB
Power Consumption
820 mW
271mW
462mW
Power Efficiency
0.41 W/GHz
0.271 W/GHz
0.116 W/GHz
Power Supply
1.8 V
3.3 V
1.2 / 2.8 V
Frequency
DAC Amplitude
Resolution
SFDR(Nyquist
Range)
62
Table 5-3 shows the published measurements results for DDFS in different technologies and
with different architectures. The difference in the power consumption of the nonlinear DAC
architecture and the architecture using TSC converter does worth the attention.
Table 5-3 The measurement results of previous works
Specification
[3]
[2]
[18]
Technology
350 SiGe BiCMOS
0.13 SiGe BiCMOS
0.18 CMOS
Maximum Clock
5 GHz
8.6GHz
1 GHz
8 bits
10 bits
8 its
45.7 dB
33 dB
44dB
Power Consumption
460 mW
4.8W
50 mW
Power Supply
3.3 V
3.3 V
1.8 V
DDFS Architecture
TSC Converter
Nonlinear DAC
TSC Converter
Frequency
DAC Amplitude
Resolution
SFDR(Nyquist
Range)
63
Future Work
In this project a 4 bit direct digital frequency synthesizer was designed in 65nm CMOS
technology. Interleaving with return to zero has been used in order to increase the bandwidth and
synthesized frequencies of the synthesizer. The designed synthesizer was simulated using
Cadence design tool. With 3.2 GHz sampling rate, simulation results showed the SFDR of 60 dB
for 400 MHz synthesized frequency. This SFDR decreased to 58 dB for Nyquist synthesized
frequency. With 6.4 GHz sampling frequency the synthesizer showed the SFDR of 40 dB for 800
MHz and 3.2 GHz synthesized frequencies respectively. In order to make it possible to have
better comparison with measurement results of the previous works provided in table 5-3, it is
likely to increase the amplitude resolution of the nonlinear DAC to at least 7 or 8 bits and run the
Monte Carlo simulations. Moreover, simulations should be run for taking in to account the
process variations.
As it was mentioned in the text, the phase accumulator and the complementor are behaviorally
modeled in Verilog- A in this project. However, CMOS technology digital blocks also show
limitations for working with high frequencies. In [17] CML logic blocks have been used to make
the digital blocks run with 4 GHz sampling frequencies. However, CML logics increase the
power consumption of the system. Therefore, for designing high sampling rate DDFS,
limitations of the digital blocks should also be considered.
Nonidealities of the time interleaved DACs, with the most important ones the gain mismatch and
time mismatch between the channels is another important aspect to be considered in the proposed
synthesizer. As it was stated in the text, any deviation from the ideal half sample time delay will
result in imperfect cancellation of the images and spikes will be occurred in the output spectrum
of DDFS. In our design ideal clocks have applied to the system with 30ps rise time and fall time.
Correction algorithm for amplitude balance and time alignment might be applied to the system
by the use of calibration filters if needed.
64
References
[1]. Direct Digital Synthesizer, Theory, Design and Applications. Jouko Vankka / Nov 2000
[2]. An 11-Bit 8.6 GHz Direct Digital Synthesizer MMIC With 10-Bit Segmented Sine-weighted
DAC/ IEEE Journal of Solid-States Circuits/February 2010
[3]. A 5-GHz Direct Digital Frequency Synthesizer Using an Analog-Sine-Mapping Technique
in 0.35-um / IEEE Journal of Solid-States Circuits/September 2011
[4]. Design of Low-Power Rom-Less Direct digital Frequency synthesizer using nonlinear
Digital-to-Analog Converter/ IEEE Journal of Solid-States Circuits/ October 1999
[5]. A technical tutorial on Digital Signal Synthesis/ Analog Devices/1999
[6]. Basic Principles of Digital-to-Analog Conversion / Behzad Razavi
[7]. A 12 Bit GS/s DAC With IM3 < -60 dBc Beyond 1 GHz in 65nm CMOS / IEEE Journal of
Solid-States Circuits/September 2009
[8]. A CMOS 8-Bit 1.6-GS/s DAC With Digital Random Return-to-Zero / IEEE Transactions on
Circuits and Systems-II/January 2011
[9]. Systematic Analysis of Interleaved Digital-to-analog Converters/ IEEE Transactions on
Circuits and Systems-II/December 2011
[10]. Analog Devices/ Analog Dialogue, Single Chip Direct Digital Synthesis vs. the Analog
PLL by Jim Surber and Leo McHugh
[11]. Direct Digital Frequency Synthesis by Analog Interpolation / IEEE Transactions on
Circuits and Systems-II/ November 2006
[12]. Segmented SineWave Digital-to-Analog Converters for Frequency Synthesizer / IEEE
ISCAS 2001
65
[13]. On the attenuation of DAC aliases through multiphase clocking / IEEE Transactions on
Circuits and Systems-II / March 2009
[14]. Parallel-path Digital-to-Analog Converters for Nyquist Signal Generation / IEEE Journal of
Solid-States Circuits/ July 2004
[15]. 2 GHz 8-Bit CMOS Rom-Less Direct Digital Frequency synthesizer / IEEE Int. Symp.
Circuits and Systems / 2005
[16]. A High-Speed Rom-Less Direct Digital Frequency Synthesizer realized by a Segmented
Nonlinear-DAC / IEEE Region 10 Conference / 2007
[17]. A 4 GHz Direct Digital Frequency Synthesizer utilizing a nonlinear sine-weighted DAC in
90 nm CMOS / IEEE Asia Pacific Conference on Circuits and Systems
[18]. A Low-Power Direct Digital Frequency Synthesizer Using an Analogue-Sine-Conversion
Technique/ IEEE ISLPED /2011
[19]. A 10-bit 500-Msamples/s CMOS DAC in 65 nm/ IEEE Journal of Solid-States
Circuits/September 1998
[20]. Current Steering Digital to Analog Converters: Functional Specifications, Digital Basics
and Behavioral Modeling.
[21].DAC Interleaving in Ultra-High-Speed Arbs
http://www.evaluationengineering.com/articles/200912/dac-interleaving-in-ultra-high-speedarbs.php
66
67
Appendix A
`include "constants.vams"
`include "disciplines.vams"
module digital_input_4(out0, out1, out2, out3,
, cout, codeout, clk, enable,reset);
output out0;
electrical out0;
output out1;
electrical out1;
output out2;
electrical out2;
output out3;
electrical out3;
output cout;
electrical cout;
output codeout;
electrical codeout;
input clk;
electrical clk;
input enable;
electrical enable;
input reset;
electrical reset;
// parameter description
parameter real millivolt = 0.001;
parameter real vlogic_high= 1.2;
parameter real vlogic_low= 0;
parameter real vtrans_clk= 0.6;
parameter real vtrans_reset= 0.6;
parameter real vtrans_enable= 0.6;
parameter real tdel= 0;
parameter real trise= 1f;
parameter real tfall= 1f;
//local integer variables
integer reset_flag;
integer count;
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integer code;
integer d[0:4];
integer i;
parameter integer step= 1;
integer decrease;
analog begin
@ (initial_step) begin
for (i=0; i<5; i=i+1)begin
d[i]=0;
end
count=0;
end
if (V(reset) < vtrans_reset) begin
reset_flag =1;
count=0;
end
@ (cross( V(clk) - vtrans_clk, +1)) begin
if (V(enable) > vtrans_enable) begin
reset_flag=0;
begin
count=count+step;
end
code=count%16;
if(code > (8 + decrease)) begin
code = code - decrease;
end
for (i=4; i>=0;i=i-1) begin
if (code>15) begin
d[i]=1;
code=code-16;
end
else begin
d[i]=0;
end
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code= code*2;
end
end
end
V(out0) <+transition (vlogic_high*d[0]*!reset_flag, tdel, trise,tfall);
V(out1) <+transition (vlogic_high*d[1]*!reset_flag, tdel, trise, tfall);
V(out2) <+transition (vlogic_high*d[2]*!reset_flag, tdel, trise, tfall);
V(out3) <+transition (vlogic_high*d[3]*!reset_flag, tdel, trise, tfall);
V(cout) <+ transition (vlogic_high*d[0]*d[1]*d[2]*d[3], tdel, trise,tfall);
V(codeout) <+ transition (millivolt*code);
end
endmodule
Based on Verilog-A written for comparator in the vhdllib of cadence.
include "constants.h"
// sigin: (val,flow)
// sigref: reference to which 'sigin' is compared (val,flow)
// sigout:
comparator output (val,flow)
//
// INSTANCE parameters
// sigout_high = maximum output of the comparator (val)
// sigout_low = minimum output of the comparator (val)
// sigin_offset = subtracted from 'sigin' before comparason to sigref (val)
// comp_slope = determines the sensitivity of the comparator []
module comparator(sigin, sigref, sigout);
input sigin, sigref;
output sigout;
electrical sigin, sigref, sigout;
parameter real sigout_high = 1.2;
parameter real sigout_low = 0;
70
parameter real sigin_offset = 0;
analog begin
@ ( initial_step ) begin
if (sigout_high <= sigout_low) begin
$display("Range specification error. sigout_high = (%E) less than sigout_low = (%E).\n",
sigout_high, sigout_low );
$finish;
end
end
V(sigout) <+1.2 * (sigout_high - sigout_low)
* tanh(comp_slope*(V(sigin, sigref)- sigin_offset))
+ (sigout_high + sigout_low)/2;
end
endmodule
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