lec 5 direct digital synthesis2

7/27/2019 Lec 5 Direct Digital Synthesis2

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ECE 5674 -- Direct Digital

Synthesis

Srikathyayani

Srikanteswara

J. H Reed


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Overview

Introduction to Direct Digital

Synthesis

Approaches to DDS

Pulse output DDS

ROM lookup table Impulse response of a filter


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Overview

Advanced techniques: Bandpasssignal generation

Sources of spurious signals and theireffects

Techniques used to minimize

spurious signals Generation of Random Sequences

Summary and Future Trends


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Introduction to DDS

Direct digital synthesis (DDS) is the

process of generating deterministic

communication carrier/referencesignals directly in discrete time with the

use of digital hardware

Discrete time signals are thenconverted into analog signals (for

transmission) using a D/A converter


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Need for DDS systems

Overcome the limitations ofanalog synthesis

Speed, precision, size, flexibility,stability, and ease ofimplementation

Compatible with and desirablefor todays high speed digitalcommunication technology


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Early DDS Systems

First DDS designs date back to the early

70s

Tierney et. al. developed a technique forgenerating 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


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Early DDS Systems

Roke Manor laboratories in 1981of the then Plessey companysprototype 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


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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


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Analog Generation

Techniques

Direct Analog Synthesis (DAS)

Generate frequencies by mixing

frequencies from different crystaland/or using their harmonics

Idealsituation with tuning

capabilities of LC oscillator andstability and purity of a crystal

oscillator


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Characteristics

Advantages High purity, low spurious content: better

than -80 dB

Fast switching: .1 - 20 s Disadvantages

Bulky, expensive, high powerconsumption

Not suitable for portable equipment

Used in medical and radar imaging,spectroscopy and frequency hoppingsystems


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Analog Generation

Techniques

)(txinPhase

DetectorLoop Filter

Output

Voltage Controlled

Oscillator

Reference

signal Amplifier

Gain =

Phase Locked Loop


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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


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Digital Signal

Generation

Output is smooth when a frequency

change is executed, no transients

Possible to achieve continuous phasefrequency switching

Crucial to frequency hopping spread

spectrum systems

Switching frequencies less than 1 spossible


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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


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Properties of DDS

PrecisionAccurately set the output frequency

significant for narrowbandmodulation formats

Analog systems have poorfrequency resolution

Stability DDS system parameters and output

frequency does not vary with time


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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


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DDS Features

Possible to predict the performance ofthe digital components

Size DDS for sub Hz resolution can beimplemented as a fraction of the size ofan analog synthesizer

Disadvantages Spurious frequency components in theoutput signal

Bandwidth of the output signal


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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


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Basic Approaches to DDS

Impulse response of a filter

Impulse response of an IIR filter

with poles on the unit circle forsinusoidal generation

Impulse response of a FIR filter

for pulse generation


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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 otherwaveform


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Pulse Output DDS

Frequency word radded toaccumulator once every clock periodTclk

Accumulator overflows and counterresets on the average once every 2N/rclock periods

Pulse: carry output of the accumulator

Rectangular waveform: MSB of theaccumulator

Sawtooth: output of the accumulator


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Pulse Output DDS

Accumulator

Output

2N-1 Carryoutput

nTMSB

O/P

Frequency Word Fr

Pulse output

N - Bit AdderOutput

Input

N - Bit

Storage

Register

Fclk

Clock

Square wave output

B A

A+B

MSB

Carry

S(n)

Sawtooth Waveform

nT nT


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Calculation of Output

Frequency

Accumulator overflows and

counter resets on the average

once every 2N/rclock periods. Repetition interval is 2N/r (1/Fclk)

Frequency is Fclk

r/ 2N


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Calculation of Output

Frequency

Frequency resolution is thesmallest possible change ofr,i.e., r=1

Frequency resolution

F= Fclk

/ 2N

Output frequency will always bemultiples ofFclk/ 2

N


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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 revery clock cycle

Output of the accumulator used toincrement the address lines of the

ROM


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ROM Lookup Table

Frequency of the outputwaveform can be varied bychanging

r

Output resolution can beincreased by increasing thenumber of bits in the accumulator

It is possible to generate arbitrarywaveforms


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Disadvantages of ROM

Lookup Table Approach

Highest output frequency

is a fraction of the clockfrequency

Spurious components inthe output in the absenceof a very large ROM


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ROM Lookup Table

Phase

Increment

Value

Clock Fclk

2

FF

N

clkrout

2N

clkFF

ROM

Lookup

TableDA

C

Filter/

Amplifier

W N

Phase

Incremen

t

Register

A

ccumulator

Nbits

B = N-W

rFout

na


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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 (numberadded to the accumulator every clock cycle)

na = width of the ROM (ROM has 2na

quantization levels)


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Need for Phase

Truncation

Design DDS for: Fout = 2.5MHz, F = 1Hz

Fclk should be 10MHz (Fout Fclk/4) N = log2(Fclk/F)=24

Size of ROM = 224 or 16 Mbytes (or

4Mbytes if only 1/4 cycle stored)! W bits, (W < N, MSBs) are used to

address the ROM

2

FF

N

clkrout

2N

clkFF Basic formulas:


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Effect of Phase Truncation

Accumulator

Output

Address

Lines of ROM

Angle

(Degrees)

000 00 45

001 00 45

010 01 135

011 01 135

100 10 225101 10 225

110 11 315

111 11 315Accumu

latorSizeN

=3,

ROMSizeW=2


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IIR filter that has poles placed on the

unit circle at ej0

Approach 3: Impulse

Response of a Filter

0

ej

e-j

12

11

110

ZbZb1

Zaa)z(H


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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


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Effect of Coefficient

Quantization

Implemented as recursive filter on a

DSP

Accuracy of output frequency0dependent on the accuracy of filter

coefficients

depends on accuracy of cos(0T) difficult to implement in finite

precision arithmetic


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Effect of Coefficient

Quantization

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 regionscorresponding to 0 and radians

Re-0.5-1.0

Z plane

Im

0 rad. rad. 0 0.5 1.0

Direct Form

Implementation (3 bits

+ sign bit)


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Summary of the

ApproachesApproach Advantages Disadvantages

Pulse output

DDS

Simple design

Can generate basic

waveforms like sawtooth,

square, and pulse

waveforms

Need additional

circuitry to generate

standard

communicationwaveforms

ROM lookup

table

Best for generating

arbitrary waveforms

Spurious components

due to phase truncation

IIR filter Large frequency rangeHigh spectral purity

Simple design

Coefficientquantization can

change the pole

locations and hence the

output frequency


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Bandpass Signal

Generation

Used to generate waveforms above

Nyquist frequency

Sampled signals replicate at multiplesof the sampling frequency (FoutnFs)

To obtain output frequencies beyond

the Nyquist frequency, the replicatedimages can be filtered to extract the

desired image


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Bandpass Signal

Generation

Digital bandpass signal can be

obtained by zero padding by N

and bandpass filtering

fs0-fs

Filter

Response


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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

Bandpass Signal

Generation

fs0-fs Nfs-Nfs


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Disadvantages of

Bandpass DDS

Spurious components inherent

in DDS signals do not decay

according to the sin(x)/x

function

Due to non-linear phasetruncation and timing jitter


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Disadvantages of

Bandpass DDS

Spurious signals make it harder to

separate the desired signal at

frequencies higher than theNyquist frequency

Higher output frequencies require

higher quality DACs


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Sources of Error in DDS

Signals

Errors are injected into the systemat various points Causes spurious components in the

output spectrum

Accumu-

lator

ROM

Lookup

Table

DAC

P1(n)

Timing Jitter P2(n)Phase

Truncation

P3(n)Amplitude

Truncation

DAC Non-

linearities

P4(n)


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Phase truncation causes phasemodulation with a periodic sawtoothwaveform

Most of the time, the DDS is puttingout a frequency that is biased

On particular clock pulses, the ROM

input does not advance ROM causes the D/A converter todeliver the same voltage as on theprevious clock cycle

Effects of Phase

Truncation


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Effects of Phase

Truncation

Thus the phase is held back by 2/2W radians before continuing to

creep forward as before

time

phase

Ideal change in

phase Actual change in

phase

0

2


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Effects of Phase Truncation

Extent of the spurs depend on the

values ofN, W, andr

The first harmonic is generally thestrongest

Spurs move closer to the fundamental

as W decreases or amount of phasetruncation increases

Harder to filter out the spurs close to

the fundamental


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Phase Truncation Spurs

Output can be expressed as a seriesof rectangular pulses

Compute the Fourier transform ofthese pulses

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

radians

Amplitude

Can get verytedious

We will look atsome basicanalysis


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0 000 00 0

1 001

2 010 01 /23 011

4 100 10

5 1016 110 11 3/2

7 111

r=1, N=3, Y = 23=8W = 2, B = N-W = 1

2( ) sin2 2

r

W Bmy m

Output of DDS

can be expressed

as


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2 2( ) sin sin

2 2 2 2

2 2sin ( )2 2

r r

W B N B B

B

r

N B

m my m

m s m

where ( ) 1

2 2

r r

B B

m ms m

2 2 2 2( ) sin ( ) cos

2 2 2

B

r r

N N N

m my m s m


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Largest spurious amplitude

Detailed calculation of spuriouscomponents requires further

analysis

2 2 2 2( ) sin ( ) cos

2 2 2

B

r r

N N N

m my m s m

Desired Output Spurious Component

2 2 2

2 2

B

sp N WA


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Timing Jitter

Even in the absence of phase

truncation (N = W), periodicities

appear in signal depending on thevalue ofr


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Timing Jitter

0, 2, 4, 6, 8, 10, 12, 14, 0, 2, 4, 6, 8, 10, 12, 14,

first period second period

N=4, r= 2

0, 6, 12, 2, 8, 14, 4, 10, 0, 6, 12, 2, 8, 14, 4, 10, 0,

first

period

second

period

third

period

fourth

period

fifth

period

N=4, r= 6

Accumu-lator

Values

Perfectlyequal

periods

Different

period

lengths


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Location of Spurs

Time period of spurious componentsdue to periodic jitter alone

Example: N=4, r= 6, W = 4, Tout=16/6Tclk

three periods of the fundamentaloutput needed to return to theoriginal state

)2,gcd(

2N

r

N

clkspur

T

kT kis any integer

gcd = greatest common divisor


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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= 3*Tout

Thus spur frequency at 1/3fundamental and their harmonics

exist


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Location of Spurs

Component at 1/3 fundamental at

0.125 visible

Desired

1/3 desired

frequency

Folded Spectrum


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Tertiary Periodicities

Presence of a combination of the

above three sources of errorscould cause additive periodicities

which could result in strong

spurs


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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 frequencycan be more pronounced than the

others


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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 k*0.0312

Figure(2), W = 4, spur enhanced by

phase truncation = 0.2812 = 9*0.0312


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Figure (1)

DesiredSpurs due to

timing jitter

Figure (2)

Phase truncation

spur superimposed

on spur due to

timing jitter

Desired


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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 digitalsections of the DDS

E F D/A


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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 additionalbit used

Errors From D/A

Converter


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W/na Ratio

00

01

10

4

3

1

2

11

Sampling and

quantizing a sine

wave for W = 3 Output of the ROM (na = 3)

corresponding to the 8

sampling points

0 0 06

0 0 05

0 0 17

0 1 00

0 1 11

0 1 12

0 1 030 0 14


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W/na Ratio

Choosing the right W/na ratio is

very important

ForW= 3, only four distinct levels arepresent

na = 2 bits will suffice

na = W-1 orW-2 is optimumdepending on whether the entire sine

wave or 1/4 of it is stored in the ROM


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Example

ForW= 11, 1024 distinct levels

are present

na has to be at least 10 bits toavoid repetition of values

If only 1/4 of the cycle is stored,na has to be at least 9 bits

T h i f


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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.

T h i f


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Techniques for

Suppressing Spurs

Randomization (all harmonics

reduced)

E.g, Wheatleys procedure

PN sequence

Generation of randomsequences


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Hybrid Systems

DDS systems make a trade off between

the bandwidth and spectral purity

If Fclk is reduced, Nyquist frequency isreduced, 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


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Hybrid Systems

ROM lookup table DDS

High switching frequencies

Low power consumption, small size Resolution can be increased byincreasing N

However, for same spectral purity,size of ROM needs to be increased


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DDS - PLL System

PLL

Relatively high switching time

between output frequenciesConsume more power

Larger in size

Very good spectral

characteristics at the output


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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, controlmechanism acts to reducephase error


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PLL Components

)(txin

)(txvco

Phase

Detector

Loop FilterOutput

Voltage Controlled

Oscillator

Reference

signal Amplifier

Gain =

)(ted

)(txin

)(tevco


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PLL Operation

)](2cos[)( ttfAtx ccin

)](2sin[)( ttfAtxcvvco

Phase Detector Model

)]()(sin[)( ttAAte vcd

Low Pass

Filter

(gain = 2)

)(txin

)(txvco

)(ted)(1 ted

)](2sin[)](2cos[)(1 ttfttfAAte ccvcd


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Phase Error

Phase error, (t) = (t) - (t)

To uniquely identify the phase,

output of phase detector has to bean odd function of the phase error

VCO output has to be in

quadrature to the PLL input


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Calculation of VCO Phase

( ) ( ) ( ) ( ) ( )vco d d

e t e t f t e f t d

Iff(t) is the impulse response of the loop

filter

Output frequency of VCOevco(t)

)(2 teKdt

dvcod

frequencyoutputVCO

Kd = VCO constant with units Hertz/volt


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)(2 teKdt

dvc od

frequencyoutputVCO

Substituting for evco(t), we get

ddtfAAK

ddtfeK

deKt

t

vcd

d

t

d

t

vc od

)()]()(sin[2

)()(2

)(2)(



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ddtfGtt

)()]()(sin[)(

vcd AAKG 2Define

Relationship between (t) - (t) does not

depend on the carrier frequency fc


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Analysis of Linear Model

If phase error is small, a linear

approximation can be made

)()()]()(sin[ tttt ddtfGt

t

)()()()(

Taking the Laplace transform

s

sFsss

)()()()(


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Analysis of Linear Model

)(

)()(

)(

1

)(

)(

)(

sGFs

sGFsH

s

sF

G

s

sFG

s

s

H(s)functiontransferPLL

Relating phase error to input phase

)()(1

)(

)()(

)(

)(

sGFs

ssH

s

ss

s

s


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Steady State Phase Error

Using final value theorem for

Laplace transform

Steady state phase error

)()( limlim0

ssAtast

)()(

)(1)()(

2

0

00

lim

limlim

sGFs

ss

sHsssA

s

ss

ss


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Assuming phase deviation of the form

0

2 2)( ftRtt

fRt

dt

d

2

1

Corresponding frequency deviation inHertz is

If R=0 and f 0, frequency step isapplied


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First order system (F(s)=1)

For R=0 and f 0

Perfect second order system

Imperfect second order system

G

fss

2

s

asF 1)( 0ss

as

assF

)(

G

fss

2


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Costas Loop

Demodulated

Output

Voltage Controlled

Oscillator

Amplifier

Gain =

Low Pass

Filter)(txin

)(, tx Ivco

)(te

)(tyI

Loop

Filter

Low Pass

Filter

900

)(, tx Qvco

)(tyQ


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Costas Loop Operation

)(2cos)()( ttftmAtx ccin

m(t)=message signal

)()(cos)()( tttmAty cI )()(sin)()( tttmAty cQ

)(2cos)(, ttftx cIvco

)(2sin)(, ttftx cQvco


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Costas Loop Operation

BPSKNRZfor)()(2sin2

1

)()(2sin)(21

)()()(

2

22

ttA

tttmA

tytyte

c

c

QI

e(t) = Loop control signal

Assuming phase error is small

)()()( 2 ttAte c Operation similar tobasic PLL


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DDS - PLL System

Complementary characteristics ofDDS and PLLs led to developmentof hybrid structures

Retain the good qualities ofDDS as well as PLLs

Filtered output of DDS is used togenerate the reference frequencyfor the PLL

DDS PLL S t


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DDS - PLL System

DDS

)(txin

)(txvco

Phase

DetectorLoop Filter

Output

Voltage Controlled

Oscillator

Referencesignal Amplifier

Gain =

)(ted

)(txin

)(tevco

Bandpass

Filter

Optional

Divider/

Interpolator


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DDS - PLL System

Optional divider may be used to dividethe DDS output to improve its noiseand spurious characteristics

Output of the PLL Fout is related to thereference frequency Frefas: Fout= NFref

Output frequency can be varied bychanging Frefof DDS

DDS PLL S


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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


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DDS - PLL System

Disadvantages

More complex and bulkier

than individual systems

PLL has some finite settling

time


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Randomization

Spurs occur because of

periodicities in the output signal

Adding minimal noise can destroythe periodicities

The spurs are minimized at the

cost of generating a much highernoise floor


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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) sequencesto remove periodicities


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Randomization

Randomization is done by

changing one or more bits of

Output of the accumulatorFrequency setting word ( r)

Output of the ROM


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Wheatleys Procedure

Accumulator

Random

Number

Generator

ROM

N W na

+

-

DACX

2N

NZ 2

Overflow

]1,0[ rX

Wh tl P d


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Wheatleys Procedure

Optimal Procedure

At each overflow of the accumulator,add a random number to accumulator

and subtract previous value ofAverage 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

Effect of Wheatleys


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Effect of Wheatley s

Procedure

Basic DDS Wheatleys Procedure

Effect of Wheatleys


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Effect of Wheatley s

Procedure

Fclk= 1, N= 9, W= 5, r= 7,Fout= 0.0137

Wheatleys procedure shows afew dB improvement

Noise floor is generated

Better improvements can be seen

on larger systems and longer runs

ROM Compression


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MPRGMPRG

96

p

Techniques

Main sources of spurs in output

signal - phase truncation

Values stored in ROM arerepeated at the input to the D/A

converter

Impractical to have a very large

sized ROM

ROM Compression


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ROM Compression

Techniques

Solution: Compress moreinformation in ROM and use thatinformation to generate a moreperfect sine wave

Most techniques based oninterpolation of the sine wave

Simple compression approach

Store only sine wave

Sampling the Sine


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p g

Wave

1/4 of the sine wave stored and

replicated with sign inversion Sine wave has to be sampled correctly

to exploit symmetry

1 2

3 4

4

31

2

N2

2

ROM Compression


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99

ROM Compression

Techniques

Taylor Series Expansion

Use of trigonometric identitiesHutchison Algorithm

Sunderland Algorithm



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100


If is any angle and is a smallincrement then

If is sufficiently small, the higherorder terms can be ignored

If sin(1) and sin(2) are stored inthe ROM, in-between values can be

generated using the Taylor series

...!3

))(cos(

!2

))(sin(

)cos()sin()sin(

32



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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

G

G

Effect of Increasing theNumber of Terms in the


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Number of Terms in the

Series Expansion

4terms

7terms

MPRG

MPRGAffect on the Frequency


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q y

Spectrum

Desired Output

Desired Output

MPRG

MPRGUse of Trigonometric


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g

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

MPRG

MPRG H t hi Al ith


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Hutchison Algorithm

Partition the values of the sine function (of

the first quadrant) into coarse ROMand

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 forangles in between those contained in the

coarse ROM

MPRG

MPRG Hutchison Algorithm


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Hutchison Algorithm

Any angle can be decomposed as a+b 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

MPRG

MPRG Sunderland Algorithm


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Partition the values of the sine function

(of the first quadrant) into 3 sub-ROMs

Any angle can be decomposed as a+b+c

sin() = sin(a+b)cos(c) + cos(a+b)sin(c )

= [sin(a) cos(b) + cos(a) sin(b) ]cos(c)+ [cos(a) cos(b) - sin(a) sin(b) ] sin(c )

MPRG

MPRG Sunderland Algorithm


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108


Modification to allow two ROMS

(or one ROM with phase shift)

If b and c are sufficiently smallsin() sin(a) + b cos(a) + c

cos(a) - b c sin(a)

MPRG

MPRGUse of DDS in Digital


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g

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 arbitrarywaveforms

Essential for software radios

MPRG

MPRGUse of DDS in Digital


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g

Communication

Digitally generated signals with

the help of multirate filters can be

used to perform digital modulation

and pulse shaping

MPRG

MPRG P l Sh i


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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 atmultiples of the sampling interval

can satisfy the Nyquist criteria

MPRG

MPRG Pulse Shaping


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Pulse Shaping

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

Sampling Points

time

C

Ts

4 positive pulses

MPRG

MPRG Raised Cosine Filter


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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

MPRG



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114


Results in side lobes in the

frequency spectrum

Interpolating at the final stageminimizes the computation up

stream in the processing.

MPRG



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115


Impulse response

2

0

00

)4(1

)2cos(

2

)2sin(2)(

tf

tf

tf

tffthe

MPRG

MPRG Frequency Spectrum


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116

Frequency Spectrum

1)( fHe

f

ff

2

)|(|cos12/1 1

= 0

|f| B

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 thetransition band in the frequency spectrum

r = 0, pulse becomes rectangular in the frequency

domain

MPRG

MPRGUse ofRandom


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Sequences

Dithering Minimizing spurious components in

DDS signals

Spread spectrum systems

Spread data in direct

sequence spread spectrumsystems

MPRG

MPRGUse ofRandom


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118

Sequences

Choose the carrier frequency for

frequency hopping spread

spectrum systems

Scramble data for security and bit

synchronizers

MPRG

MPRGGeneration of Random


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119

Sequences

PN sequence

Maximal length sequence

properties and generation

Gold codes

Generation and properties

MPRG

MPRG

Types of Random


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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

MPRG

MPRG

Applications of PN


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Sequences

Spread Spectrum Systems

Users share the same frequency

band Separated from each other by using

different spreading codes

properties of the codes determinehow well the user's are separated

MPRG

MPRG

Applications of PN


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Sequences

Data scramblers

At transmitter multiply the PN

sequence by the data to randomizedata and help maintain

synchronization

Also used for security purposes,

where the PN code is not universally

known

MPRG

MPRG Generation of PN Sequences


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123

Generation of PN Sequences

y(n)

z-1

X(n)

y(n-1) y(n-2) y(n-m)

h1 h2hm-1

hm

Binary Digital Linear Feedback Shift Register

m

k

kkZhDh

1

1)(

m

k

k knyhny1

)()(

z-1 z-1

MPRG

MPRG Generation of PN Sequences


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124

Generation of PN Sequences

Different sets ofh 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 shiftregister

s(n) = [y(n-1) y(n-2) ....... y(n-m)]

m

k

k knyhny1

)()(

m

k

kkZhDh

1

1)(

MPRG

MPRGMaximal Length


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Sequences

Sequences have the maximum possible

period ( N = 2m-1)

Shift register will generate a maximal length

sequence only if its connection polynomialh(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

MPRG

MPRGMaximal Length


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GG

126

Sequences

A polynomial is said to be irreducible if

it cannot be factored into the product of

polynomials with binary coefficientsand degrees of at least 1

h(D) = 1 + D + D4 is irreducible

h(D) = 1 - D4

is reducible

MPRG

MPRG

Properties of Maximal

L th S


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127

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 thesequence is 2m-1-1

MPRG

MPRG

Properties of Maximal

L th S


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128

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 k1s and 0s, for1 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

MPRG

MPRG Gold Codes


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129

Gold Codes

Constructed by forming the modulo-2sum of two preferred maximum

sequences of equal length

pn sequence

generator 1

pn sequencegenerator 2

Code 1

Code 2

Code 3Clock

Gold code generator

MPRG

MPRG Gold Codes


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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


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MPRG

MPRG Properties of Gold Codes


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132

p

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

MPRG

MPRG Properties of Gold Codes


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133

p

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 andbounded

m odd: maximum value of the cross

correlation function between any pair ofGold sequences is Rmax= (2 N)

m even: Rmax= (N)

MPRG

MPRG Summary


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134

y

DDS systems rapidly gaining importance

Digital communication and software

radios

Advantageous in terms of size, switchingfrequency, resolution, stability and

accuracy

Available as convenient ASICs

MPRG

MPRG Summary


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135

y

Various techniques used to generate

DDS signals

ROM lookup table most commonly used

Techniques used to minimize spurs Hybrid architectures, randomization,

ROM compression

MPRG

MPRG Summary


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136

y

Applications

digital communication systems, spread

spectrum systems, digital modulation,

and pulse shaping Future Trends

Higher clock speeds

Lower spur levels

MPRG

MPRG References


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137

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

MPRG

MPRG References


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References

Goldberg, Bar-Giora, Digital techniques in frequencysynthesis, McGraw-Hill, 1996

Henry T. Nicholas, III, Henry, Samueli, An analysis of theoutput spectrum of direct digital frequency synthesizers inthe presence of phase-accumulator truncation, 41st

Annual Frequency Control Symposium, 1987

Tierney, Joseph., Rader, Charles M., Gold, Bernard., Adigital frequency synthesizer, IEEE Transactions on Audioand 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

lec 5 direct digital synthesis2

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