ECE 5674 Direct Digital Synthesis

1
ECE 5674 -- Direct Digital Synthesis

• Srikathyayani Srikanteswara
• J. H Reed

2
Overview

• Introduction to Direct Digital Synthesis
• Approaches to DDS
• Pulse output DDS
• ROM lookup table
• Impulse response of a filter

3
Overview

• Advanced techniques Bandpass signal generation
• Sources of spurious signals and their effects
• Techniques used to minimize spurious signals
• Generation of Random Sequences
• Summary and Future Trends

4
Introduction to DDS

• Direct digital synthesis (DDS) is the process of
  generating deterministic communication
  carrier/reference signals directly in discrete
  time with the use of digital hardware
• Discrete time signals are then converted into
  analog signals (for transmission) using a D/A
  converter

5
Need for DDS systems

• Overcome the limitations of analog synthesis
• Speed, precision, size, flexibility, stability,
  and ease of implementation
• Compatible with and desirable for todays high
  speed digital communication technology

6
Early DDS Systems

• First DDS designs date back to the early 70s
• Tierney et. al. developed a technique for
  generating audio signals
• Used a Read Only Memory (ROM) to store sine waves
• Stored values were used to drive a D/A followed
  by analog interpolation filter

7
Early DDS Systems

• Roke Manor laboratories in 1981 of the then
  Plessey companys prototype DDS
• Occupied several complete boards of logic laid
  out on the bench
• Clocked at 10MHz
• Output frequency of up to 3MHz
• Spurious responses about 40 dB below the desired
  output

8
Modern DDS Systems

• Gained importance in the early 80s with the
  widespread use of digital communication systems
• Have incorporated a lot of changes and
  improvements making them a practical alternative
  to analog signal sources
• GHz frequencies possible, spurs of -60 to -80 dB
  or lower

9
Analog Generation Techniques

• Direct Analog Synthesis (DAS)
• Generate frequencies by mixing frequencies from
  different crystal and/or using their harmonics
• Ideal situation with tuning capabilities of LC
  oscillator and stability and purity of a crystal
  oscillator

10
Characteristics

• Advantages
• High purity, low spurious content better than
  -80 dB
• Fast switching .1 - 20 ?s
• Disadvantages
• Bulky, expensive, high power consumption
• Not suitable for portable equipment
• Used in medical and radar imaging, spectroscopy
  and frequency hopping systems

11
Analog Generation Techniques
Phase Locked Loop
12
Analog Generation Techniques

• Advantages of PLL
• fine frequency resolution
• low levels of spurious outputs, though not as low
  as DAS
• comparatively low cost
• Disadvantages
• slow switching times due to loop filter settling
  time

13
Digital Signal Generation

• Output is smooth when a frequency change is
  executed, no transients
• Possible to achieve continuous phase frequency
  switching
• Crucial to frequency hopping spread spectrum
  systems
• Switching frequencies less than 1 ?s possible

14
Comparison of DDS with Analog Generation

• DDS overcomes most problems of DAS and PLLs
• Superior in terms of precision, stability, ease
  of implementation, flexibility, and size

15
Properties of DDS

• Precision
• Accurately set the output frequency
• significant for narrowband modulation formats
• Analog systems have poor frequency resolution
• Stability
• DDS system parameters and output frequency does
  not vary with time

16
DDS Features

• Ease of implementation
• Basic structure easy to realize with ROM, clock,
  and DAC
• Implemented in hardware, software, or combination
  of both
• Easier to interface with computers for control

17
DDS Features

• Possible to predict the performance of the
  digital components
• Size
• DDS for sub Hz resolution can be implemented as a
  fraction of the size of an analog synthesizer
• Disadvantages
• Spurious frequency components in the output
  signal
• Bandwidth of the output signal

18
Basic Approaches to DDS

• Pulse output DDS
• Generates square, sawtooth, and pulse waveforms
• ROM lookup table
• Standard method
• Can generate sinusoidal as well as arbitrary
  waveforms

19
Basic Approaches to DDS

• Impulse response of a filter
• Impulse response of an IIR filter with poles on
  the unit circle for sinusoidal generation
• Impulse response of a FIR filter for pulse
  generation

20
Approach 1 Pulse Output DDS

• One of the simplest forms of DDS
• Used to generate pulse, sawtooth, or rectangular
  waveforms
• Use these basic waveforms to generate sinusoidal
  or other waveform

21
Pulse Output DDS

• Frequency word ?r added to accumulator once every
  clock period Tclk
• Accumulator overflows and counter resets on the
  average once every 2N/?r clock periods
• Pulse carry output of the accumulator
• Rectangular waveform MSB of the accumulator
• Sawtooth output of the accumulator

22
Pulse Output DDS
Carry output
2N-1
Accumulator Output
nT
MSB O/P
nT
nT
Square wave output
Frequency Word Fr
N - Bit Storage Register
MSB
B
Output
A
N - Bit Adder
S(n)
Input
AB
Carry
Clock
Sawtooth Waveform
Pulse output
Fclk
23
Calculation of Output Frequency

• Accumulator overflows and counter resets on the
  average once every 2N/?r clock periods.
• Repetition interval is 2N/?r (1/Fclk)
• Frequency is Fclk ?r / 2N

24
Calculation of Output Frequency

• Frequency resolution is the smallest possible
  change of ?r, i.e., ?r 1
• Frequency resolution
• DF Fclk / 2N
• Output frequency will always be multiples of Fclk
  / 2N

25
Approach 2 ROM Lookup Table

• Sine values are stored in a ROM and periodically
  output through a D/A converter
• Contents of N bit accumulator is incremented by
  ?r every clock cycle
• Output of the accumulator used to increment the
  address lines of the ROM

26
ROM Lookup Table

• Frequency of the output waveform can be varied by
  changing ?r
• Output resolution can be increased by increasing
  the number of bits in the accumulator
• It is possible to generate arbitrary waveforms

27
Disadvantages of ROM Lookup Table Approach

• Highest output frequency is a fraction of the
  clock frequency
• Spurious components in the output in the absence
  of a very large ROM

28
ROM Lookup Table
Clock Fclk
Fout
? r
W ? N
ROM Lookup Table
DAC
Phase Increment Register
Accumulator N bits
B N-W
Filter/ Amplifier
na
Phase Increment Value
29
Definitions of Variables

• Fclk Clock frequency
• Fout Output frequency
• ?F Frequency resolution
• N Number of bits in the accumulator
• W Number of bits used to address the ROM (W ?
  N)
• ?r Phase increment step size (number added to
  the accumulator every clock cycle)
• na width of the ROM (ROM has 2na quantization
  levels)

30
Need for Phase Truncation
Basic formulas

• Design DDS for Fout 2.5MHz, ?F 1Hz
• Fclk should be 10MHz (Fout ? Fclk/4) ?
• N log2(Fclk/DF)24
• Size of ROM 224 or 16 Mbytes (or 4Mbytes if
  only 1/4 cycle stored)!
• W bits, (W lt N, MSBs) are used to address the ROM

31
Effect of Phase Truncation
Accumulator Size N3, ROM Size W2
32
Approach 3 Impulse Response of a Filter

• IIR filter that has poles placed on the unit
  circle at e?j?0

33
Filter Coefficients

• Output frequency ?0
• Cosine wave h(n) cos(?0T) u(n)
• a0 1, a1 cos(?0T)
• b1 2cos(?0T), b2 -1
• Sine wave h(n) sin(?0T) u(n)
• a0 0, a1 sin(?0T)
• b_1 2cos( ?0T), b2 -1

34
Effect of Coefficient Quantization

• Implemented as recursive filter on a DSP
• Accuracy of output frequency ?0 dependent on the
  accuracy of filter coefficients
• depends on accuracy of cos(?0T)
• difficult to implement in finite precision
  arithmetic

35
Effect of Coefficient Quantization
Im
Direct Form Implementation (3 bits sign bit)
Z plane
0 rad.
? rad.
Re
-0.5
-1.0
0
0.5
1.0

• Uniform quantization of filter coefficients
• Possible to obtain only certain output
  frequencies (pole locations)
• Pole locations more closely spaced around ?/2
  radians than in the regions corresponding to 0
  and ? radians

36
Summary of the Approaches
37
Bandpass Signal Generation

• Used to generate waveforms above Nyquist
  frequency
• Sampled signals replicate at multiples of the
  sampling frequency (Fout ? nFs)
• To obtain output frequencies beyond the Nyquist
  frequency, the replicated images can be filtered
  to extract the desired image

38
Bandpass Signal Generation

• Digital bandpass signal can be obtained by zero
  padding by N and bandpass filtering

Filter Response
fs
0
-fs
39
Bandpass Signal Generation

• Roll off in the amplitude of replicated images
  follows the sin(x)/x function due to finite width
  pulses
• Spurious harmonics generated by DAC are generally
  much lower in amplitude

40
Disadvantages of Bandpass DDS

• Spurious components inherent in DDS signals do
  not decay according to the sin(x)/x function
• Due to non-linear phase truncation and timing
  jitter

41
Disadvantages of Bandpass DDS

• Spurious signals make it harder to separate the
  desired signal at frequencies higher than the
  Nyquist frequency
• Higher output frequencies require higher quality
  DACs

42
Sources of Error in DDS Signals

• Errors are injected into the system at various
  points
• Causes spurious components in the output spectrum

43
Effects of Phase Truncation

• Phase truncation causes phase modulation with a
  periodic sawtooth waveform
• Most of the time, the DDS is putting out a
  frequency that is biased
• On particular clock pulses, the ROM input does
  not advance
• ROM causes the D/A converter to deliver the same
  voltage as on the previous clock cycle

44
Effects of Phase Truncation

• Thus the phase is held back by 2p /2W radians
  before continuing to creep forward as before

45
Effects of Phase Truncation

• Extent of the spurs depend on the values of N, W,
  and Dr
• The first harmonic is generally the strongest
• Spurs move closer to the fundamental as W
  decreases or amount of phase truncation increases
• Harder to filter out the spurs close to the
  fundamental

46
Phase Truncation Spurs

• Output can be expressed as a series of
  rectangular pulses
• Compute the Fourier transform of these pulses

• Can get very tedious
• We will look at some basic analysis

47
Phase Truncation Spurs
?r1, N3, Y 238 W 2, B N-W 1
Output of DDS can be expressed as
48
Phase Truncation Spurs
49
Phase Truncation Spurs
Spurious Component
Desired Output

• Largest spurious amplitude
• Detailed calculation of spurious components
  requires further analysis

50
Timing Jitter

• Even in the absence of phase truncation (N W),
  periodicities appear in signal depending on the
  value of ?r

51
Timing Jitter
N4, ?r 2
0, 2, 4, 6, 8, 10, 12, 14, 0, 2, 4, 6, 8, 10, 12,
14,
Perfectly equal periods
first period
second period
Accumu-lator Values
N4, ?r 6
0, 6, 12, 2, 8, 14, 4, 10, 0, 6, 12, 2, 8, 14, 4,
10, 0,
Different period lengths
first period
second period
third period
fourth period
fifth period
52
Location of Spurs

• Time period of spurious components due to
  periodic jitter alone
• Example N4, ?r 6, W 4, Tout 16/6Tclk
• three periods of the fundamental output needed to
  return to the original state

k is any integer
gcd greatest common divisor
53
Example contd.

• Will create a harmonic at 1/3 of the fundamental
• Verify from formula
• Period of spurs 24/gcd(6,16)Tclk 16/2 8Tclk
  3Tout
• Thus spur frequency at ?1/3 fundamental and their
  harmonics exist

54
Location of Spurs

• Component at 1/3 fundamental at 0.125 visible

Folded Spectrum
Desired
1/3 desired frequency
55
Tertiary Periodicities

• Presence of a combination of the above three
  sources of errors could cause additive
  periodicities which could result in strong spurs

56
Tertiary Periodicities

• In the presence of more than one independent set
  of periodicities, the least common multiple (lcm)
  of the independent periodicities is another spur
  frequency
• Spurs at a particular frequency can be more
  pronounced than the others

57
Tertiary Periodicities

• Spurs due to phase truncation and timing jitters
  can superimpose and cause stronger spurs
• Example Fclk 1, N 5, ?r 7, na 32, Fout
  0.2188
• Figure(1) W 5, spur due to timing jitter alone
  at k0.0312
• Figure(2), W 4, spur enhanced by phase
  truncation 0.2812 90.0312

58
Tertiary Periodicities
Phase truncation spur superimposed on spur due to
timing jitter
Desired
Desired
Spurs due to timing jitter
Figure (2)
Figure (1)
59
Errors From D/A Converter

• Inherent non-linearities
• Difficult to manufacture high speed D/A
  converters that are accurate
• Difficult to predict and quantify the errors
  accurately unlike the digital sections of the DDS

60
Errors From D/A Converter

• Experimental findings
• as a rule of thumb, when number of D/A converter
  bits (Da) is greater than seven, spurious outputs
  decrease by 6dB per each additional bit used

61
W/na Ratio
Sampling and quantizing a sine wave for W 3
Output of the ROM (na 3) corresponding to the 8
sampling points
62
W/na Ratio

• Choosing the right W/na ratio is very important
• For W 3, only four distinct levels are present
• na 2 bits will suffice
• na W-1 or W-2 is optimum depending on whether
  the entire sine wave or 1/4 of it is stored in
  the ROM

63
Example

• For W 11, 1024 distinct levels are present
• na has to be at least 10 bits to avoid repetition
  of values
• If only 1/4 of the cycle is stored, na has to be
  at least 9 bits

64
Techniques for Suppressing Spurs

• Use of hybrid systems (PLL filtering of
  harmonics)
• DDS-PLL systems
• ROM compression techniques
• Taylor series expansions
• Trigonometric expansions
• Sunderland, Hutchison etc.

65
Techniques for Suppressing Spurs

• Randomization (all harmonics reduced)
• E.g, Wheatleys procedure
• PN sequence
• Generation of random sequences

66
Hybrid Systems

• DDS systems make a trade off between the
  bandwidth and spectral purity
• If Fclk is reduced, Nyquist frequency is
  reduced, hence reducing the bandwidth
• Lower clock frequencies allow higher resolution
  and better spectral purity for a given number of
  bits in the accumulator (N) and a given ROM size

67
Hybrid Systems

• ROM lookup table DDS
• High switching frequencies
• Low power consumption, small size
• Resolution can be increased by increasing N
• However, for same spectral purity, size of ROM
  needs to be increased

68
DDS - PLL System

• PLL
• Relatively high switching time between output
  frequencies
• Consume more power
• Larger in size
• Very good spectral characteristics at the output

69
Phase Locked Loop

• Synchronizing circuit
• Synchronize output of a system with reference
  frequency
• Phase error at a minimum when system is in lock
• If phase error builds up, control mechanism acts
  to reduce phase error

70
PLL Components
71
PLL Operation
Phase Detector Model
72
Phase Error

• Phase error, ?(t) ?(t) - ?(t)
• To uniquely identify the phase, output of phase
  detector has to be an odd function of the phase
  error
• VCO output has to be in quadrature to the PLL
  input

73
Calculation of VCO Phase
If f(t) is the impulse response of the loop filter
Output frequency of VCO ? evco(t)
Kd VCO constant with units Hertz/volt
74
Calculation of VCO Phase
Substituting for evco(t), we get
75
Calculation of VCO Phase
Relationship between ?(t) - ?(t) does not depend
on the carrier frequency fc
76
Analysis of Linear Model

• If phase error is small, a linear approximation
  can be made

Taking the Laplace transform
77
Analysis of Linear Model
Relating phase error to input phase
78
Steady State Phase Error

• Using final value theorem for Laplace transform
• Steady state phase error

79
Steady State Phase Error
Assuming phase deviation of the form
Corresponding frequency deviation in Hertz is
If R0 and ?f ? 0, frequency step is applied
80
Steady State Phase Error

• First order system (F(s)1)
• For R0 and ?f ? 0
• Perfect second order system
• Imperfect second order system

81
Costas Loop
Low Pass Filter
Demodulated Output
Low Pass Filter
900
Voltage Controlled Oscillator
Amplifier Gain µ
Loop Filter
82
Costas Loop Operation
m(t)message signal
83
Costas Loop Operation
e(t) Loop control signal
Assuming phase error is small
? Operation similar to basic PLL
84
DDS - PLL System

• Complementary characteristics of DDS and PLLs led
  to development of hybrid structures
• Retain the good qualities of DDS as well as PLLs
• Filtered output of DDS is used to generate the
  reference frequency for the PLL

85
DDS - PLL System
Optional Divider/ Interpolator
Bandpass Filter
DDS
Reference signal
Phase Detector
Loop Filter
Output
Amplifier Gain µ
Voltage Controlled Oscillator
86
DDS - PLL System

• Optional divider may be used to divide the DDS
  output to improve its noise and spurious
  characteristics
• Output of the PLL Fout is related to the
  reference frequency Fref as Fout N Fref
• Output frequency can be varied by changing Fref
  of DDS

87
DDS - PLL System

• Advantages
• Has very high resolution and high switching
  speeds
• Spectral purity of the output is largely defined
  by the spectral purity of the PLL subsystem
• Higher than that of the DDS sub-system

88
DDS - PLL System

• Disadvantages
• More complex and bulkier than individual systems
• PLL has some finite settling time

89
Randomization

• Spurs occur because of periodicities in the
  output signal
• Adding minimal noise can destroy the
  periodicities
• The spurs are minimized at the cost of generating
  a much higher noise floor

90
Randomization

• Optimal procedures do not increase the total
  energy contained in the spurs
• Wheatleys procedure
• Sub-optimal procedures can increase the total
  noise energy
• Using Pseudo Noise (PN) sequences to remove
  periodicities

91
Randomization

• Randomization is done by changing one or more
  bits of
• Output of the accumulator
• Frequency setting word ( ?r)
• Output of the ROM

92
Wheatleys Procedure
-
ROM
Accumulator
DAC
X

2N
Overflow
Random Number Generator
93
Wheatleys Procedure

• Optimal Procedure
• At each overflow of the accumulator, add a random
  number to accumulator and subtract previous value
  of
• Average of X(i) X(i 1) 0
• No net noise added
• Average output frequency does not change
• Not easy to implement in high speed logic

94
Effect of Wheatleys Procedure
Basic DDS
Wheatleys Procedure
95
Effect of Wheatleys Procedure

• Fclk 1, N 9, W 5, ?r 7, Fout 0.0137
• Wheatleys procedure shows a few dB improvement
• Noise floor is generated
• Better improvements can be seen on larger systems
  and longer runs

96
ROM Compression Techniques

• Main sources of spurs in output signal - phase
  truncation
• Values stored in ROM are repeated at the input to
  the D/A converter
• Impractical to have a very large sized ROM

97
ROM Compression Techniques

• Solution Compress more information in ROM and
  use that information to generate a more perfect
  sine wave
• Most techniques based on interpolation of the
  sine wave
• Simple compression approach
• Store only ¼ sine wave

98
Sampling the Sine Wave

• 1/4 of the sine wave stored and replicated with
  sign inversion
• Sine wave has to be sampled correctly to exploit
  symmetry

99
ROM Compression Techniques

• Taylor Series Expansion
• Use of trigonometric identities
• Hutchison Algorithm
• Sunderland Algorithm

100
Taylor Series Expansion

• If ? is any angle and ?? is a small increment
  then
• If ?? is sufficiently small, the higher order
  terms can be ignored
• If sin(?1) and sin(?2) are stored in the ROM,
  in-between values can be generated using the
  Taylor series

101
Taylor Series Expansion

• Series expansion can be implemented in a
  dedicated DSP, FPGA, or combinatorial logic up to
  desired number of terms

Using W 2 bits, N 12, and two terms of the
series expansion, results in remarkable
improvements
102
Effect of Increasing the Number of Terms in the
Series Expansion
4 terms
7 terms
103
Affect on the Frequency Spectrum
Desired Output
Desired Output
104
Use of Trigonometric Identities

• Use trigonometric identities to interpolate
  between two values of the sine function
• Most of these methods work well only if the
  increment from the known angle is very small
• Need additional circuitry to perform interpolation

105
Hutchison Algorithm

• Partition the values of the sine function (of the
  first quadrant) into coarse ROM and fine ROM
• Coarse ROM contains values of sine function for a
  certain number of angles at a fixed step size
• Fine ROM has values of sine function for angles
  in between those contained in the coarse ROM

106
Hutchison Algorithm

• Any angle ? can be decomposed as ? ? ab
• sin(a) is contained in the coarse ROM and sin(b)
  is contained in fine ROM
• sin(? ) sin(a) cos(b) cos(a) sin(b)
• Example
• Coarse ROM has sine values from 00 - 900 in steps
  of 100
• Fine ROM has values from 10 - 90
• To evaluate sin(55), a 50, b 5

107
Sunderland Algorithm

• Partition the values of the sine function (of the
  first quadrant) into 3 sub-ROMs
• Any angle ? can be decomposed as ? ? abc
• sin(? ) sin(ab)cos(c) cos(ab)sin(c )
• sin(a) cos(b) cos(a) sin(b) cos(c)
• cos(a) cos(b) - sin(a) sin(b) sin(c )

108
Sunderland Algorithm

• Modification to allow two ROMS (or one ROM with
  phase shift)
• If b and c are sufficiently small
• sin(? ) ? sin(a) b cos(a) c cos(a) - b c
  sin(a)

109
Use of DDS in Digital Communication

• Used to generate signals for paging radios,
  mobile telephones, and multi-mode radios
• Spread spectrum frequency hopping systems
  require fast switching with good spectral purity
• Used for creating custom and arbitrary waveforms
• Essential for software radios

110
Use of DDS in Digital Communication

• Digitally generated signals with the help of
  multirate filters can be used to perform digital
  modulation and pulse shaping

111
Pulse Shaping

• Used to minimize intersymbol interference (ISI)
  and bandwidth
• Nyquist Criteria
• Intersymbol interference can be eliminated by
  using special pulse shapes
• Magnitude of the impulse response of the pulse
  shaping filter should be zero at multiples of the
  sampling interval
• can satisfy the Nyquist criteria

112
Pulse Shaping
Sampling Points
C
time
Ts

• hp(kTs) C, k 0
• 0, k ? 0
• k is an integer, Ts is the sampling interval
• hp can have any non-zero value between the
  sampling intervals
• Infinitely long pulse shapes can satisfy the
  Nyquist criteria

4 positive pulses
113
Raised Cosine Filter

• Satisfies the Nyquist criteria for eliminating
  ISI commonly used in pulse shaping
• Ideal raised cosine pulse
• Infinite duration in time domain
• Practical applications

114
Raised Cosine Filter

• Results in side lobes in the frequency spectrum
• Interpolating at the final stage minimizes the
  computation up stream in the processing.

115
Raised Cosine Filter
Impulse response

116
Frequency Spectrum
f? B - f0, f1 f0 - f?, r f?/f0 fo
6dB Bandwidth of the raised cosine filter B
absolute bandwidth of the filter r roll off
factor determines the width of the transition
band in the frequency spectrum r 0, pulse
becomes rectangular in the frequency domain

117
Use of Random Sequences

• Dithering
• Minimizing spurious components in DDS signals
• Spread spectrum systems
• Spread data in direct sequence spread spectrum
  systems

118
Use of Random Sequences

• Choose the carrier frequency for frequency
  hopping spread spectrum systems
• Scramble data for security and bit synchronizers

119
Generation of Random Sequences

• PN sequence
• Maximal length sequence
• properties and generation
• Gold codes
• Generation and properties

120
Types of Random Sequences

• An ideal binary random sequence
• Infinite sequence of independent, identically
  distributed, random variables each taking on
  values 0 or 1 with probability 0.5
• Pseudo-noise (PN) sequences
• Finite length sequences, which closely
  approximate an ideal random sequence

121
Applications of PN Sequences

• Spread Spectrum Systems
• Users share the same frequency band
• Separated from each other by using different
  spreading codes
• properties of the codes determine how well the
  user's are separated

122
Applications of PN Sequences

• Data scramblers
• At transmitter multiply the PN sequence by the
  data to randomize data and help maintain
  synchronization
• Also used for security purposes, where the PN
  code is not universally known

123
Generation of PN Sequences
hm
hm-1
h1
h2
X(n)
y(n-2)
y(n-m)
y(n-1)
y(n)
Binary Digital Linear Feedback Shift Register
124
Generation of PN Sequences

• Different sets of h gives rise to different
  connection polynomial h(D)
• m degree of the polynomial
• State of the PN sequence generator is defined as
  the contents of the shift register
• s(n) y(n-1) y(n-2) ....... y(n-m)

125
Maximal Length Sequences

• Sequences have the maximum possible period ( N
  2m-1)
• Shift register will generate a maximal length
  sequence only if its connection polynomial h(D)
  is primitive
• A necessary but not sufficient condition for a
  connection polynomial h(D), of degree m, to be
  primitive is that it be irreducible

126
Maximal Length Sequences

• A polynomial is said to be irreducible if it
  cannot be factored into the product of
  polynomials with binary coefficients and degrees
  of at least 1
• h(D) 1 D D4 is irreducible
• h(D) 1 - D4 is reducible

127
Properties of Maximal Length Sequences

• Different settings of h gives rise to different
  kinds of sequences
• Maximal length sequences are common
• Number of 1s in a period of the sequence is 2m-1
• Number of 0s in a period of the sequence is 2m-1-1

128
Properties of Maximal Length Sequences

• In a period of the sequence, there should be
• Sequence of consecutive m 1s, and (m-1) 0s
• 2m-k-2 sequences of consecutive k 1s and 0s, for
  1 ? k ? m-2
• No sequences of consecutive (m-1) 1s or
  consecutive m 0s
• Periodic autocorrelation function
• R(n) 1, for n 0
• R(n) 0, otherwise

129
Gold Codes
pn sequence generator 1
Code 1
Code 3
Clock
pn sequence generator 2
Code 2
Gold code generator

• Constructed by forming the modulo-2 sum of two
  preferred maximum sequences of equal length

130
Gold Codes

• Preferred m-sequences are maximum length
  sequences that have certain specific desirable
  correlation properties
• Though constructed from a maximal sequence code,
  it is not a maximal sequence code

131
Properties of Gold Codes

• A different Gold code is generated by shifting
  the one of the sequence relative to the other
• Gold codes allow construction of families of 2m
  -1 codes from pairs of m stage shift registers

132
Properties of Gold Codes

• Gold code are useful because of the large number
  of codes they supply although they require only
  one pair of feedback tap sets
• Multiple register gold code generator can
  generate
• (2m - 1)r non- maximum length sequences
• r maximum length sequences
• r number of registers, m register length

133
Properties of Gold Codes

• Gold codes can be chosen so that over a set of
  codes available from a given generator, the
  cross-correlation between the codes is uniform
  and bounded
• m odd maximum value of the cross correlation
  function between any pair of Gold sequences is
  Rmax ?(2 N)
• m even Rmax ?(N)

134
Summary

• DDS systems rapidly gaining importance
• Digital communication and software radios
• Advantageous in terms of size, switching
  frequency, resolution, stability and accuracy
• Available as convenient ASICs

135
Summary

• Various techniques used to generate DDS signals
• ROM lookup table most commonly used
• Techniques used to minimize spurs
• Hybrid architectures, randomization, ROM
  compression

136
Summary

• Applications
• digital communication systems, spread spectrum
  systems, digital modulation, and pulse shaping
• Future Trends
• Higher clock speeds
• Lower spur levels

137
References

• Dixon, Robert C, Spread spectrum systems with
  commercial applications,Third edition,Wiley
  Interscience, 1994
• Gilmore Robert, Kornfeld, Hybrid PLL/DDS
  frequency synthesis, Proceedings RF Technology
  Expo. 90, pp. 419 - 436, January 1990
• Goldberg, Bar-Giora DDS part 1, Reviewing
  various techniques for synthesiszing signals,
  Microwaves and RF, pp. 181 - 185, May 1996
• Goldberg, Bar-Giora, DDS part 2, Enhancing the
  performance of DDS signal sources, Microwaves
  and RF, pp. 110-116, June 1996

138
References

• Goldberg, Bar-Giora, Digital techniques in
  frequency synthesis, McGraw-Hill, 1996
• Henry T. Nicholas, III, Henry, Samueli, An
  analysis of the output spectrum of direct digital
  frequency synthesizers in the presence of
  phase-accumulator truncation, 41st Annual
  Frequency Control Symposium, 1987
• Tierney, Joseph., Rader, Charles M., Gold,
  Bernard., A digital frequency synthesizer,
  IEEE Transactions on Audio and Electroacoustics,
  vol. AU-19, no. 1, pp. 48 - 57, March, 1971
• Viterbi, Andrew, J., CDMA, Principles of spread
  spectrum communication, Addison Wesley Longman,
  Inc, Reading, MA, 1995
• Wheatley, lll, C E., Spurious suppression in
  direct digital synthesizers, Proceedings 35th
  Annual Frequency Control Symposium, pp. 428 -
  435, May 1981