frequency modulation gmsk modulation

Chapter 21Chapter 21

Frequency Modulation Frequency Modulation

GMSK ModulationGMSK Modulation

DSP C5000DSP C5000

Copyright © 2003 Texas Instruments. All rights reserved.Copyright © 2003 Texas Instruments. All rights reserved.

Copyright © 2003 Texas Instruments. All rights reserved.

ESIEE, Slide 2

Learning ObjectivesLearning Objectives

Overview of Digital ModulationOverview of Digital Modulation Understanding GMSK Modulation Understanding GMSK Modulation Learning how to Implement a GMSK MLearning how to Implement a GMSK M

odulator on a C54odulator on a C54


ESIEE, Slide 3

Digital ModulationsDigital Modulations

Baseband and bandpass signalling are used to Baseband and bandpass signalling are used to transmit data on physical channels such as transmit data on physical channels such as telephone cables or radiofrequency channels.telephone cables or radiofrequency channels. The source of data (bits or symbol) may be a The source of data (bits or symbol) may be a

computer file or a digitized waveform (speech, computer file or a digitized waveform (speech, video…)video…)

The transmitted signal carries information about The transmitted signal carries information about the data and its characteristic changes at the same the data and its characteristic changes at the same rate as the data.rate as the data.

When the signal carrying the data information When the signal carrying the data information Extends from 0 Hz upwards, the term baseband Extends from 0 Hz upwards, the term baseband

signalling is used.signalling is used. Has its power centered on a central frequency fc, Has its power centered on a central frequency fc,

the term bandpass signalling or the term bandpass signalling or modulationmodulation is is used.used.


ESIEE, Slide 4

Digital ModulationDigital Modulation

Modulation is used because:Modulation is used because: The channel does not include the 0 Hz The channel does not include the 0 Hz

frequency and baseband signalling is frequency and baseband signalling is impossibleimpossible

The bandwidth of the channel is split The bandwidth of the channel is split between several channels for frequency between several channels for frequency multiplexingmultiplexing

For wireless radioFor wireless radio--communications, the communications, the size of the antenna decreases when the size of the antenna decreases when the transmitted frequency increases.transmitted frequency increases.


ESIEE, Slide 5

Carrier FrequencyCarrier Frequency

In simple modulation schemes, a single In simple modulation schemes, a single frequency signal, the carrier, is frequency signal, the carrier, is modified at the rate of the data.modified at the rate of the data.

The carrier is commonly written as:The carrier is commonly written as:

cos 2 cA f t ffcc= Carrier frequency= Carrier frequency

A = Carrier AmplitudeA = Carrier Amplitude

= Carrier phase= Carrier phase

The 3 main parameters of the carrier: The 3 main parameters of the carrier: amplitude, phase and frequency can be amplitude, phase and frequency can be modified to carry the information modified to carry the information leading to: amplitude, phase and leading to: amplitude, phase and frequency modulation.frequency modulation.


ESIEE, Slide 6

Digital ModulationDigital Modulation CPM: Continuous Phase ModulationCPM: Continuous Phase Modulation

Frequency or phase modulationFrequency or phase modulation Example: MSK, GMSKExample: MSK, GMSK Characteristic: constant envelope modulationCharacteristic: constant envelope modulation

QAM: Quadrature Amplitude ModulationQAM: Quadrature Amplitude Modulation Example: QPSK, OQPSK, 16QAMExample: QPSK, OQPSK, 16QAM Characteristic: High spectral efficiencyCharacteristic: High spectral efficiency

Multicarrier ModulationMulticarrier Modulation Example: OFDM, DMTExample: OFDM, DMT Characteristic: Muti-path delay spread tolerance, Characteristic: Muti-path delay spread tolerance,

effectivness against channel distortioneffectivness against channel distortion


ESIEE, Slide 7

What is the Complex Envelope What is the Complex Envelope z(t) z(t) of a of a Modulated Signal Modulated Signal x(t)x(t) ? ?

.2sin)(2)cos()()( 2 tftztftzetztx cQcItfj c

ffcc= Carrier frequency= Carrier frequency

2 ( )( ) ( ) ( ) ( ) ( ) ( )cj f t j tH I Qz t x t jx t e z t jz t A t e z(t)z(t) = Complex envelope of = Complex envelope of x(t)x(t)

zzII(t), z(t), zQQ(t)(t) are the baseband components are the baseband components

xxHH(t) (t) = Hilbert transform of = Hilbert transform of x(t) = x(t) = x(t)x(t) with a phase shift ofwith a phase shift of /2/2

.)()(21)( czczx ffSffSfS

SSxx(f)(f) = Power spectral density of = Power spectral density of x(t)x(t)


ESIEE, Slide 8

Complex Envelope z of a Modulated Signal xComplex Envelope z of a Modulated Signal xFrequency DomainFrequency Domain

0 1 2

0 1 2

0 1 2

f

f

f

X(f)

Xa(f)

Z(f) 2

1

2


ESIEE, Slide 9

GMSK ModulationGMSK Modulation

GGaussian aussian MMinimum inimum SShift hift KKeying.eying.

Used in GSM and DECT standards.Used in GSM and DECT standards. Relevant to mobile communications Relevant to mobile communications

because of constant envelope because of constant envelope modulation:modulation: Quite insensitive to non-linearities of power Quite insensitive to non-linearities of power

amplifieramplifier Robust to fading effectsRobust to fading effects

But moderate spectral efficiency.But moderate spectral efficiency.


ESIEE, Slide 10

What is GMSK Modulation?What is GMSK Modulation?

Continuous phase digital frequency Continuous phase digital frequency modulationmodulation

Modulation index h=1/2Modulation index h=1/2 Gaussian Frequency Shaping FilterGaussian Frequency Shaping Filter GMSK = MSK + Gaussian filterGMSK = MSK + Gaussian filter Characterized by the value of BTCharacterized by the value of BT

T = bit durationT = bit duration B = 3dB Bandwidth of the shaping filterB = 3dB Bandwidth of the shaping filter BT = 0.3 for GSMBT = 0.3 for GSM BT = 0.5 for DECTBT = 0.5 for DECT


ESIEE, Slide 11

GMSK Modulation, Expression for the GMSK Modulation, Expression for the Modulated Signal x(t)Modulated Signal x(t)

( ) cos 2 ( ) with:

( ) 2 ( )

c

t

kk

x t f t t

t h a s kT d

2

1)(

ds

NormalizationNormalization

aakk = Binary data = +/- 1 = Binary data = +/- 1

hh = Modulation index = 0.5 = Modulation index = 0.5

s(t)s(t) = Gaussian frequency shaping filter = Gaussian frequency shaping filter

s(t)s(t)= Elementary frequency pulse= Elementary frequency pulse


ESIEE, Slide 12

GMSK Elementary Phase PulseGMSK Elementary Phase Pulse

Elementary phase pulse = ( )

( ) 2 ( ) 2 ( ) .t

t

t hq t h s d

( ) ( ) .t

q t s d

For , ( 1) ( ) 2 ( ) ( )

( ) cos 2 ( ) cos 2 ( ) .

n n

k kk k

n

c c kk

t nT n T t h a q t kT a t kT

x t f t t f t a t kT


ESIEE, Slide 13

Architecture of a GMSK ModulatorArchitecture of a GMSK Modulator

C o d e r B i t s a k

r t ( ) V C O

h

x t ( ) h t ( )

G a u s s i a n f i l t e r

G M S K m o d u l a t o r u s i n g a V C O

kk

a s t k T kk

a t k T

( ) ( ) * ( )s t r t h t

R e c t a n g u l a r f i l t e r

x t ( ) Coder Bits ak

s t ( )

2 h

t ( ) t

cos()

sin()

+ -

s t r t h t ( ) ( ) * ( )

GMSK modulator without VCO

kk

a t kT

cos 2 cf t

sin 2 cf t


ESIEE, Slide 14

Equation for the Gaussian Filter h(t)Equation for the Gaussian Filter h(t)

2 22

22

2 2( ) exp

ln(2) ln(2)

ln(2)( ) exp

2

Bh t B t

H f fB

The duration The duration MTMTbb of the gaussian pulse of the gaussian pulse

is truncated to a value inversely is truncated to a value inversely proportional to B.proportional to B.

BT = 0.5, BT = 0.5, MTMTbb = 2 = 2TTbb

BT = 0.3, BT = 0.3, MTMTbb = 3 or 4 = 3 or 4TTbb


ESIEE, Slide 15

Frequency and Phase Elementary PulsesFrequency and Phase Elementary Pulses

-2 -1 0 1 2 0

0.1

0.2

0.3

0.4

0.5 BT b

BT b 0 5 ,

BT b 0 3 ,

t in number of bit periods Tb

T g t b ( ) Elementary frequency pulse

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t in number of bit period Tb

BTb

BTb

0 3.

BTb

0 5.

Elementary phase pulse ( )t

/2

The elementary frequency pulse is The elementary frequency pulse is the convolution of a square pulse the convolution of a square pulse

r(t) with a gaussian pulse h(t).r(t) with a gaussian pulse h(t).

Its duration is Its duration is (M+1)T(M+1)Tbb..


ESIEE, Slide 16

GMSK SignalsGMSK Signals

Binary sequence

GMSK modulated Signal

( )t

z t tI ( ) cos ( )

z t tQ ( ) sin ( )

5

1

0 5 10 15 20-1

0

1

0 5 10 15 20-1

0

1

t

0 5 10 15 20

1

-1

0 t

0 5 10 15 20-5

0 t

0 5 10 15 20-1

0 t

t

in rd


ESIEE, Slide 17

Power Spectral Density of GMSK SignalsPower Spectral Density of GMSK Signals

0 0.5 1 1.5 2 2.5 3 3.5 4 -140

-120

-100

-80

-60

-40

-20

0

20

Power spectral density of the complex envelope of GMSK, BT=0.3, fe=8, T=1. Logarithmic scale

Frequency normalized by 1/T


ESIEE, Slide 18

Implementing a GMSK Modulator on a DSPImplementing a GMSK Modulator on a DSP

Quadrature modulation can be used:Quadrature modulation can be used: The DSP calculates the phase The DSP calculates the phase and the 2 and the 2

baseband components baseband components zzII and and zzQQ and sends and sends them to the DAC.them to the DAC.

Or a modulated loop can be used. Or a modulated loop can be used. In this case, the DSP generates the In this case, the DSP generates the

instantaneous frequency instantaneous frequency ffinstinst signal that is signal that is sent to the DAC.sent to the DAC.

inst ( ).kk

f h a s kT


ESIEE, Slide 19

Calculation of the Instantaneous FrequencyCalculation of the Instantaneous Frequency

ffinstinst is obtained by a simple filtering of is obtained by a simple filtering of the bit sequence athe bit sequence akk by a FIR filter of by a FIR filter of impulse response s(n). impulse response s(n).

Open Matlab routine pul_phas.m to calculate s(n).

Explanation of parameters are given in following slides.


ESIEE, Slide 20

Expression for the Baseband ComponentsExpression for the Baseband Components

The baseband components The baseband components zzII and and zzQQ are are modulated in amplitude by the 2 modulated in amplitude by the 2 quadrature carriers.quadrature carriers.

( ) cos 2 ( ) cos ( ) cos 2 sin ( ) sin 2c c cx t f t t t f t t f t

( ) ( )cos 2 ( )sin 2I c Q cx t z t f t z t f t


ESIEE, Slide 21

Baseband components Baseband components

The carriers are generally RF analog The carriers are generally RF analog signals generated by analog oscillatorssignals generated by analog oscillators

However, we will show how they could However, we will show how they could be generated digitally if the value of be generated digitally if the value of ffcc were not too high.were not too high.

Baseband Components and CarriersBaseband Components and Carriers

( ) cos ( )

( ) sin ( )

I

Q

z t t

z t t

I

Q

Carrier cos 2

Carrier sin 2

c

c

f t

f t


ESIEE, Slide 22

Calculation of the Baseband Calculation of the Baseband Components on a DSPComponents on a DSP

Calculate the phase Calculate the phase at time at time mTmTSS

TTss the sampling frequency. the sampling frequency.

Read the value of cos(Read the value of cos() and sin() and sin() ) from a table.from a table.

For , ( 1) ( ) ( ).

( ) ( ).

n

kk

n

S k Sk

t nT n T t a t kT

mT a mT kT


ESIEE, Slide 23

Calculation of the Elementary Pulse Phase Calculation of the Elementary Pulse Phase with Matlabwith Matlab

The elementary pulse phase The elementary pulse phase (mT(mTSS)) is is calculated with Matlab and stored in calculated with Matlab and stored in memory.memory. The duration The duration LTLT of the evolutive part of of the evolutive part of

(mT(mTSS)) depends on the value of BT. depends on the value of BT. is called phi in the matlab routine.is called phi in the matlab routine.

( ) 2 ( )

( ) 0 0

( )

t

t h s d

t t

t h t LT


ESIEE, Slide 24

Calculation of the Elementary Pulse Phase Calculation of the Elementary Pulse Phase with Matlab 1/2with Matlab 1/2

function [phi,s]=pul_phas(T_over_Ts,L,BT,T) Ts=T/ T_over_Ts ; % calculates the number of samples in phi in the interval 0 - LT Nphi=ceil(T_over_Ts*L); % calculates the number of samples in T Nts=ceil(T_over_Ts); phi=zeros(1,Nphi); s=zeros(1,Nphi); sigma=sqrt(log(2))/2/pi/(BT/T)

Open Matlab routine pul_phas.m

Beginning of the matlab routine


ESIEE, Slide 25

Calculation of the Elementary Pulse Phase Calculation of the Elementary Pulse Phase with Matlab 2/2with Matlab 2/2

t=[-L*T/2:Ts:L*T/2]; t=t(1:Nphi); ta=t+T/2; tb=t-T/2; Qta=0.5*(ones(1,Nphi)+erf(ta/sigma/sqrt(2))); Qtb=0.5*(ones(1,Nphi)+erf(tb/sigma/sqrt(2))); expta=exp(-0.5*((ta/sigma).^2))/sqrt(2*pi)*sigma; exptb=exp(-0.5*((tb/sigma).^2))/sqrt(2*pi)*sigma; phi=pi/T/2*(ta.*Qta+expta-tb.*Qtb-exptb); s=1/2/T*(Qta-Qtb);

End of the matlab routine


ESIEE, Slide 26

Using the Matlab RoutineUsing the Matlab Routine

Start MatlabStart Matlab UseUse::

T_over_Ts=8T_over_Ts=8 L=4L=4 T=1T=1 BT=0.3BT=0.3

ccall the routine usingall the routine using to calculate the to calculate the phase pulse phi (phase pulse phi ( )) and the pulse sand the pulse s:: [phi,s]=pul_phas(T_over_Ts,L,BT,T)[phi,s]=pul_phas(T_over_Ts,L,BT,T)

Plot the phase phi and the shaping pulse sPlot the phase phi and the shaping pulse s plot(phi)plot(phi) plot(s)plot(s)


ESIEE, Slide 27

Results of Matlab RoutineResults of Matlab Routine


ESIEE, Slide 28

Results for phiResults for phi

For L=4 and T_over_Ts=8, we obtain 32 For L=4 and T_over_Ts=8, we obtain 32 samples for phi. Matlab gives the samples for phi. Matlab gives the following values for phi:following values for phi:

Phi= [0.0001, 0.0002, 0.0005, 0.0012, 0.0028, Phi= [0.0001, 0.0002, 0.0005, 0.0012, 0.0028, 0.0062, 0.0127, 0.0246, 0.0446, 0.0763, 0.1231, 0.0062, 0.0127, 0.0246, 0.0446, 0.0763, 0.1231, 0.1884, 0.2740, 0.3798, 0.5036, 0.6409, 0.7854, 0.1884, 0.2740, 0.3798, 0.5036, 0.6409, 0.7854, 0.9299, 1.0672, 1.1910, 1.2968, 1.3824, 1.4476, 0.9299, 1.0672, 1.1910, 1.2968, 1.3824, 1.4476, 1.4945, 1.5262, 1.5462, 1.5581, 1.5646, 1.5680, 1.4945, 1.5262, 1.5462, 1.5581, 1.5646, 1.5680, 1.5696, 1.5703, 1.5706]1.5696, 1.5703, 1.5706]

After this evolutive part of phi, phi stays After this evolutive part of phi, phi stays equal to equal to /2 (1.57)./2 (1.57).

To calculate To calculate , the evolutive part and , the evolutive part and the constant part of the elementary the constant part of the elementary phase pulse phi are treated separately.phase pulse phi are treated separately.


ESIEE, Slide 29

1

1 1

For ,( 1)

( ) ( ) ( ) ( ) with:

( ) 0 0

( )2

( ) ( ) phimem( ) ( )2

n n L n

k k kk k k n L

n L n n

k k kk k n L k n L

t nT n T

t a t kT a t kT a t kT

t t

ht h t LT

t a a t kT n a t kT

Calculation of Calculation of

Separation of evolutive and constant Separation of evolutive and constant parts of phi.parts of phi.


ESIEE, Slide 30

Memory Part and Evolutive Part of Memory Part and Evolutive Part of

For ,( 1)

phimem( ) phimem( 1) ( ) .2

t nT n T

n n a n L

1

1

( ) phimem( ) ( )

( ) phimem( ) ( )

n

kk n L

S

n

S k Sk n L

t n a t kT

t mT

mT n a mT kT


ESIEE, Slide 31

Recursive Calculation of Recursive Calculation of

Names of variablesNames of variables Phase = Phase = phi = array of evolutive part of phi = array of evolutive part of ,,

Nphi samples = LT/TsNphi samples = LT/Ts

Ns = number of samples per bit = T/TsNs = number of samples per bit = T/Ts an = binary sequence (+/- 1)an = binary sequence (+/- 1)

2 calculation steps:2 calculation steps: Calculate Calculate Calculate zCalculate zII=cos(=cos() and z) and zQQ=sin(=sin() by table ) by table

readingreading


ESIEE, Slide 32

Calculation of Calculation of : Initialization: Initialization

Initialization step:Initialization step: L first bit periods, phimem = 0L first bit periods, phimem = 0 At bit L+1, phimem is set to aAt bit L+1, phimem is set to a11 /2./2.

Principle of the initialization processing:Principle of the initialization processing: FOR i=1 to i=LFOR i=1 to i=L

for j=(i-1)Ns+1 to j=(i-1)Ns+Nphifor j=(i-1)Ns+1 to j=(i-1)Ns+Nphi phase((i-1)Ns+j)= phase((i-1)Ns+j)+phi(j)an(i)phase((i-1)Ns+j)= phase((i-1)Ns+j)+phi(j)an(i)

EndforEndfor

endFORendFOR phimem=an(1) phimem=an(1) /2/2


ESIEE, Slide 33

Calculation of Calculation of : After Initialization : After Initialization

FORFOR i=L+1 to last_bit i=L+1 to last_bit ForFor j=(i-1)Ns+1 to j=(i-1)Ns+Nphi j=(i-1)Ns+1 to j=(i-1)Ns+Nphi

phase((i-1)Ns+j)= phase((i-1)Ns+j)+phi(j)an(i)phase((i-1)Ns+j)= phase((i-1)Ns+j)+phi(j)an(i)

endForendFor ForFor j=(i-1)Ns+1 to j=i Ns j=(i-1)Ns+1 to j=i Ns

phase((i-1)Ns+j)= phase((i-1)Ns+j)+phimemphase((i-1)Ns+j)= phase((i-1)Ns+j)+phimem xi=cos(phase(i-1)Ns+j)xi=cos(phase(i-1)Ns+j) xq=sin(phase((i-1)Ns+j)xq=sin(phase((i-1)Ns+j)

endForendFor phimem=phimem+pi/2 an(i+1-L)phimem=phimem+pi/2 an(i+1-L)

endFORendFOR


ESIEE, Slide 34

Coding and Wrapping the Phase TableCoding and Wrapping the Phase Table

The phase value in [-The phase value in [-,,[ is represented [ is represented by the 16-bit number Iphaseby the 16-bit number Iphase Minimum phase = -Minimum phase = -, Iphase = -2, Iphase = -21515.. Maximum phase = Maximum phase = (1-2(1-21515), Iphase = 2), Iphase = 21515-1.-1.

152 phaseIphase


ESIEE, Slide 35

Index for the table in Phase CalculationIndex for the table in Phase Calculation

Phase increment between 2 table values:Phase increment between 2 table values: 22 / Ncos and on Iphase 2 / Ncos and on Iphase 21616/Ncos/Ncos For Iphase, the index of the table is:For Iphase, the index of the table is:

i= Ncos Iphase 2i= Ncos Iphase 2-16-16

Here Ncos 2Here Ncos 2-16-16 = 2 = 2-9-9,, So the index in the table is given by shifting So the index in the table is given by shifting

Iphase 9 bits to the right.Iphase 9 bits to the right.


ESIEE, Slide 36

Coding of phiCoding of phi

The quantized values of phi can be The quantized values of phi can be obtained with Matlab:obtained with Matlab: phi= round(phi*2^15/pi);phi= round(phi*2^15/pi);

phi is defined and initialized in the file phi is defined and initialized in the file phi03.asm as:phi03.asm as:

.ref phi .sect "phi"

phi .word 1,2,5,13,29,65,133,256 .word 465,795,1284,1965,2858,3961,5252,6685

.word 8192,9699,11132,12423,13526,14419,15100,15589 .word 15919,16128,16251,16319,16355,16371,16379,16382


ESIEE, Slide 37

Calculation of sin() and cos() by Calculation of sin() and cos() by Table LookupTable Lookup

We use a table of cosine valuesWe use a table of cosine values Length of the table Ncos=128, circular Length of the table Ncos=128, circular

bufferbuffer Contents of the table: cosine of angles Contents of the table: cosine of angles ii

uniformly distributed between -uniformly distributed between - and and . .

1

2,

2coscos NN

iN

ii

deb_cos = first address of the tabledeb_cos = first address of the table mid_cos = address of the table middlemid_cos = address of the table middle cos(cos(ii ) is at address mid_cos + i ) is at address mid_cos + i


ESIEE, Slide 38

Calculation of sin() and cos() by Calculation of sin() and cos() by Table LookupTable Lookup

Conversion of the phase modulo 2Conversion of the phase modulo 2:: Wrapping phase in [-Wrapping phase in [-,,[ [ Done by a macro: testa02piDone by a macro: testa02pi

* Macro to wrap the phase between - et , phase is in ACCU A testa02pi .macro SUB #8000h,A,B ; sub -2^(15) BC suite1?, BGEQ ; test if phase is in [0,2pi[ SUB #8000h,A ; sub -2^(15) SUB #8000h,A ; SUB -2^(15) B suite? suite1? ADD #8000h,A,B ; add -2^(15) BC suite?,BLT ADD #8000h,A ; add -2^(15) ADD #8000h,A ; add -2^(15) suite? ; end of test .endm


ESIEE, Slide 39

Generating the Sine Values from a Table of Generating the Sine Values from a Table of Cosine ValuesCosine Values

To generate sin(To generate sin() from the table of ) from the table of cosine values, we use:cosine values, we use: cos(cos(/2 - /2 - )) or or cos(-cos(-/2 + /2 + ) or ) or cos(3cos(3/2+ /2+

)) If i is the index of the cosine, for the sine we If i is the index of the cosine, for the sine we

use:use: i-Ncos/4i-Ncos/4 if i>= -Ncos/4 if i>= -Ncos/4 i+3Ncos /4 i+3Ncos /4 if i < Ncos/4.if i < Ncos/4.


ESIEE, Slide 40

Implementation on a C54x, Test SequenceImplementation on a C54x, Test Sequence

We can test the GMSK modulation on a We can test the GMSK modulation on a given periodical binary sequence an:given periodical binary sequence an: an=[ 0 1 1 0 0 1 0 1 0 0]an=[ 0 1 1 0 0 1 0 1 0 0] These bits are stored in a circular buffer: These bits are stored in a circular buffer:

Size NB = 10Size NB = 10 Declaration of a section bits aligned on an Declaration of a section bits aligned on an

address which is a multiple of 16 bits.address which is a multiple of 16 bits.

AR5 deb_bit

NB

an


ESIEE, Slide 41

Testing the Generation of Testing the Generation of on the C54x, on the C54x,Results Buffer Results Buffer

We calculate We calculate and store it in a buffer and store it in a buffer pointed by AR1. The size of the result pointed by AR1. The size of the result buffer is 1000 words.buffer is 1000 words.

AR1 resu

1000

Buffer for the last 1000 result values of


ESIEE, Slide 42

Testing the Generation of Testing the Generation of on the C54x on the C54xBuffersBuffers

AR4 phi

Nphi

phi

Circular Buffer for phi (evolutive part) Allocated at an address multiple of 64

(Nphi = 32)

AR3 deb_phase

Nphi

phase

Circular Buffer for the variable phase Allocated at an address multiple of 64

(Nphi = 32)


ESIEE, Slide 43

Listing for the Calculation of Listing for the Calculation of DefinitionsDefinitions

.mmregs .global debut,boucle .global deb_cos, phi .global deb_phase .global resu

Nphi .set 32 Nphim .set -32 L .set 4 Ncos .set 128 Nsur2 .set 64 Nsur4 .set 32 NB .set 10 NS .set 8

.bss resu,1000


ESIEE, Slide 44

Listing for the Calculation of Listing for the Calculation of Macro for phase wrappingMacro for phase wrapping

* Definition of macro of phase wrapping testa02pi testa02pi .macro SUB #8000h,A,B ;sub -2^(15)

BC suite1?, BGEQ ;test if phase in [0,2pi[ SUB #8000h,A ;sub -2^(15) SUB #8000h,A ;sub -2^(15) B suite?

suite1? ADD #8000h,A,B ;add -2^(15) BC suite?,BLT ADD #8000h,A ;add -2^(15) ADD #8000h,A ;add -2^(15)

suite? ;end test wrapping [0,2pi[ .endm


ESIEE, Slide 45

Listing for the Calculation of Listing for the Calculation of Initialization of Registers and Buffers 1/2Initialization of Registers and Buffers 1/2

.text * Initialization of BK, DP, ACCUs, and ARi debut: LD #0, DP

LD #0, A LD #0, B STM #deb_bit,AR5 STM #deb_cos,AR2 STM #deb_phase ,AR3 STM #phi,AR4 STM #resu,AR1 STM +1,AR0 STM #Nphi, BK STM #(NS-1),AR7 RSBX OVM SSBX SXM


ESIEE, Slide 46

Listing for the Calculation of Listing for the Calculation of Initialization of Registers and Buffers 2/2Initialization of Registers and Buffers 2/2

* Initialization of the phase buffer RPT #(Nphi-1) STL A,*AR3+%

*initialization of phimem LD *AR5,T MPY #04000h,A ; -2^(14) an(1)(pi/2 an(1)) STL A,*(phimem)


ESIEE, Slide 47

Listing for the Calculation of Listing for the Calculation of Calculation for the first L bitsCalculation for the first L bits

* processing for the first L bits STM #(L-1),AR6 Ldeb STM #Nphi-1,BRC

RPTB fin-1 ; from k=0 to k=Nphi LD *AR3,A ; accu=phase(k) MAC *AR4+0%,*AR5,A ; A=phase(k)+an(i) phi(k) testa02pi STL A,*AR3+% ; A->phase(k) NOP

fin NOP STM #(NS-1), AR7

Nsbouc LD *AR3,A ; output of NS values STL A,*AR1+ ; and reset at 0 of NS words ST #0,*AR3+% BANZ nsbouc,*AR7- STM #NB, BK MAR *AR5+% STM #Nphi,BK BANZ Ldeb, *AR6-


ESIEE, Slide 48

Listing for the Calculation of Listing for the Calculation of Calculation of the Following Bits 1/2Calculation of the Following Bits 1/2

* begin of the infinite loop boucle STM #Nphi-1,BRC

RPTB fin2-1 ; for k=0 to Nphi LD *AR3,A ; accu=phase(k) MAC *AR4+0%,*AR5,A ; A=phase(k)+an(i) phi(k) testa02pi STL A,*AR3+% ; A->phase(k)

fin2 * adding phimem to the NS first points of the array * then taking them out and reset to 0 in phase

STM #NS-1,AR7 nsbouc2 LD *(phimem),B

ADD *AR3,B,A ; B=phimem+phase(k) testa02pi STL A,*AR1+ ST #0,*AR3+% ; 0->phase(k) BANZ nsbouc2,*AR7- ; ....


ESIEE, Slide 49

Listing for the Calculation of Listing for the Calculation of Calculation of the Following Bits 2/2Calculation of the Following Bits 2/2

fin3 * actualization of phimem

LD *(phimem),A STM #NB, BK MAR *+AR5(#unmL)% LD *AR5,T MAC #04000h,A ; -2^(14) an(1)(pi/2 an(1)) testa02pi STL A,*(phimem) MAR *+AR5(#L)% STM #Nphi,BK B boucle .end


ESIEE, Slide 50

Illustration of the Phase Result Wrapped Illustration of the Phase Result Wrapped Between -Between - and and

Plot under CCS the buffer of phase resultsPlot under CCS the buffer of phase results


ESIEE, Slide 51

Generation of the Baseband ComponentsGeneration of the Baseband ComponentszI=cos(zI=cos() and zQ=sin() and zQ=sin())

We calculate zI and zQ and we do not We calculate zI and zQ and we do not need to save the phase any more.need to save the phase any more.

We output ZI and Zq, in this example We output ZI and Zq, in this example on DXR0 and DXR1.on DXR0 and DXR1.

We need the table of constants for the We need the table of constants for the values of cosine.values of cosine. This table is defined in the file tabcos.asmThis table is defined in the file tabcos.asm The cosine values are given in format Q14.The cosine values are given in format Q14.


ESIEE, Slide 52

File tabcos.asmFile tabcos.asm

.def deb_cos,mid_cos

.sect "tab_cos" deb_cos .word -16384,-16364,-16305,-16207,-16069,-15893,-15679,-15426 .word -15137,-14811,-14449,-14053,-13623,-13160,-12665,-12140

.word -11585,-11003,-10394,-9760,-9102,-8423,-7723,-7005 .word -6270,-5520,-4756,-3981,-3196,-2404,-1606,-804 .word 0,804,1606,2404,3196,3981,4756,5520 .word 6270,7005,7723,8423,9102,9760,10394,11003 .word 11585,12140,12665,13160,13623,14053,14449,14811 .word 15137,15426,15679,15893,16069,16207,16305,16364 .word 16384,16364,16305,16207,16069,15893,15679,15426 .word 15137,14811,14449,14053,13623,13160,12665,12140 .word 11585,11003,10394,9760,9102,8423,7723,7005 .word 6270,5520,4756,3981,3196,2404,1606,804 .word 0,-804,-1606,-2404,-3196,-3981,-4756,-5520 .word -6270,-7005,-7723,-8423,-9102,-9760,-10394,-11003 .word -11585,-12140,-12665,-13160,-13623,-14053,-14449,-14811 .word -15137,-15426,-15679,-15893,-16069,-16207,-16305,-16364

mid_cos .set deb_cos+64


ESIEE, Slide 53

Listing for the Calculation of zI and zQListing for the Calculation of zI and zQ

The listing for definitions and The listing for definitions and initializations is the same as before.initializations is the same as before.

Processing of the L first bits,Processing of the L first bits, Processing of the other bits Processing of the other bits

(infinite loop)(infinite loop)


ESIEE, Slide 54

Processing of the First L Bits 1/2Processing of the First L Bits 1/2 * Processing of the L first bits

STM #(L-1),AR6 Ldeb STM #Nphi-1,BRC

RPTB fin-1 ; from k=0 to Nphi LD *AR3,A ; accu=phase(k) MAC *AR4+0%,*AR5,A ; A=phase(k)+an(i) phi(k) testa02pi STL A,*AR3+% ; A->phase(k)

fin NOP STM #(NS-1), AR7

Nsbouc LD *AR3,A ; output of NS values SFTA A,-9,A ADD #mid_cos,A,B STLM B,AR1 NOP NOP LD *AR1,B STL B,DXR0


ESIEE, Slide 55

Processing of the First L Bits 2/2Processing of the First L Bits 2/2 * Calculation of the sine

ADD #Nsur4,A,B BC sui,BGT ADD #(Nsur2),B B sui1

sui SUB #Nsur2,B sui1 ADD #mid_cos,B

STLM B,AR1 NOP NOP LD *AR1,A STL A,DXR1 ST #0,*AR3+% BANZ nsbouc,*AR7- STM #NB, BK MAR *AR5+% STM #Nphi,BK


ESIEE, Slide 56

Processing of the Following bits 1 of 2Processing of the Following bits 1 of 2

* begin the infinite loop boucle STM #Nphi-1,BRC

RPTB fin2-1 ; for k=0 to Nphi LD *AR3,A ; accu=phase(k) MAC *AR4+0%,*AR5,A ; A=phase(k)+an(i) phi(k) testa02pi STL A,*AR3+% ; A->phase(k)

fin2 *phimem is added to the NS first points of the array * then they are output and words are reset to 0 in buffer phase

STM #NS-1,AR7 nsbouc2 LD *(phimem),B

ADD *AR3,B,A ; A=phimem+phase(k) testa02pi SFTA A,-9,A ADD #mid_cos,A,B STLM B,AR1 NOP NOP LD *AR1,B

STL B,DXR0


ESIEE, Slide 57

Processing of the Following Bits 2 of 2Processing of the Following Bits 2 of 2 * Calculation of the sine

NOP ADD #Nsur4,A,B NOP NOP BC suii,BGT ADD #(Nsur2),B B suii1

suii SUB #(Nsur2) ,B suii1 ADD #mid_cos,B

STLM B,AR1 NOP NOP LD *AR1,A STL A,DXR1 ST #0,*AR3+% ; 0->phase(k) BANZ nsbouc2,*AR7-

fin3 * actualization of phimem

LD *(phimem),A STM #NB, BK MAR *+AR5(#unmL)% LD *AR5,T MAC #04000h,A ; -2^(14) an(1)(pi/2 an(1)) testa02pi STL A,*(phimem) MAR *+AR5(#L)% STM #Nphi,BK B boucle .end


ESIEE, Slide 58

Results Observed in CCSResults Observed in CCS


ESIEE, Slide 59

Generation of 2 Quadrature CarriersGeneration of 2 Quadrature Carriers

Generation of:Generation of: cos(2cos(2ffcct) and sin (2t) and sin (2ffcct)t) ffcc is the frequency of the carrier is the frequency of the carrier In this example we choose fIn this example we choose fcc=1/T=1/T Sampling frequency = 1/Ts = fSampling frequency = 1/Ts = fSS

In RF applications, the carriers are not In RF applications, the carriers are not generated by the DSP. It is only possible generated by the DSP. It is only possible to use the DSP for low values of fto use the DSP for low values of fcc..

The angle The angle (t)(t)= 2= 2ffcct is calculated then t is calculated then the value of cos() is read from a table.the value of cos() is read from a table.


ESIEE, Slide 60

Calculation of the Angle of the CarriersCalculation of the Angle of the Carriers

( ) 2 ( 1) 2 ( 1)S c S S c S snT f nT n T f T n T

Recursive calculation of the angle Recursive calculation of the angle ::

The precision of the generated The precision of the generated frequency depends on the precision of frequency depends on the precision of ..

The phase increment The phase increment corresponds to corresponds to an increment an increment II to the integer that to the integer that represents represents ..II=2=21616ffccTTSS rounded to the closest rounded to the closest

integer.integer.


ESIEE, Slide 61

Calculation of the Angle of the CarriersCalculation of the Angle of the Carriers

If the number of samples per period of If the number of samples per period of the carrier is a power of 2, 2the carrier is a power of 2, 2kk:: II=2=21616ffccTTSS=2=2(16-k)(16-k) is exact is exact Then the precision of the generated Then the precision of the generated

frequency depends only on the frequency depends only on the precision of the sampling frequency.precision of the sampling frequency.

Otherwise, there is an error dfOtherwise, there is an error dfcc in f in fcc:: |df|dfcc|<2|<2-17-17ffSS..

Error in the amplitudes of the carriers Error in the amplitudes of the carriers due to finite precision of the table due to finite precision of the table reading. (possible interpolation).reading. (possible interpolation).


ESIEE, Slide 62

Implementation on a C54xImplementation on a C54x

We use: fs/fc=8=Ns.We use: fs/fc=8=Ns. The cosine table has 128 Values in Q14.The cosine table has 128 Values in Q14.

AR1 deb_cos

128

Cosine values

Circular buffer,Table of cosine


ESIEE, Slide 63

Implementation on a C54xImplementation on a C54x

Here k=3 (8 samples per period) and the Here k=3 (8 samples per period) and the increment increment II=2=21616ffccTTSS=2=2(16-k)(16-k) =2 =21313

Simple case of fSimple case of fSS/f/fcc as an integer value, the as an integer value, the table is read with an offset from the pointer = table is read with an offset from the pointer = 16 = 216 = 277/2/23 3 to generate a cosine with 8 samples to generate a cosine with 8 samples per period.per period.

We can work directly on the index i and not We can work directly on the index i and not on I.on I.


ESIEE, Slide 64

Generation of the Cosine and Sine CarriersGeneration of the Cosine and Sine Carriers

For the cosine:For the cosine: The circular buffer containing the cosine The circular buffer containing the cosine

values (length N) is accessed with AR1.values (length N) is accessed with AR1. Incremented by the content of AR0=16.Incremented by the content of AR0=16. AR1 initialized with deb_cos.AR1 initialized with deb_cos.

For the sine:For the sine: Same circular buffer accessed by AR2.Same circular buffer accessed by AR2. AR2 initialized with deb_cos + Ncos/4.AR2 initialized with deb_cos + Ncos/4. Decremented by AR0=16.Decremented by AR0=16.

Outputs (cos and sin) are sent to DXR0 Outputs (cos and sin) are sent to DXR0 and DXR1.and DXR1.


ESIEE, Slide 65

Generation of Generation of quadrature quadrature 2 Carriers2 Carriers

File File porteuse.asmporteuse.asm A file associated with DXR0 and DXR1 A file associated with DXR0 and DXR1

is used to save visual results obtained is used to save visual results obtained with the CCS simulator.with the CCS simulator.

Here cosine table goes from:Here cosine table goes from: 0 to 20 to 2


ESIEE, Slide 66

Listing for the Generation of 2 Carriers 1 of 2Listing for the Generation of 2 Carriers 1 of 2

.mmregs .global debut,boucle

Ncos .set 128 Nsur2 .set 64 Nsur4 .set 32 Inc .set 16 * Definition and initialization of Table of cosine

.sect "tab_cos" deb_cos .word 16384,16364,16305,16207,16069,15893,15679,15426

.word 15137,14811,14449,14053,13623,13160,12665,12140 .word 11585,11003,10394,9760,9102,8423,7723,7005 .word 6270,5520,4756,3981,3196,2404,1606,804 .word 0,-804,-1606,-2404,-3196,-3981,-4756,-5520 .word -6270,-7005,-7723,-8423,-9102,-9760,-10394,-11003 .word -11585,-12140,-12665,-13160,-13623,-14053,-14449,-14811 .word -15137,-15426,-15679,-15893,-16069,-16207,-16305,-16364 .word -16384,-16364,-16305,-16207,-16069,-15893,-15679,-15426 .word -15137,-14811,-14449,-14053,-13623,-13160,-12665,-12140 .word -11585,-11003,-10394,-9760,-9102,-8423,-7723,-7005 .word -6270,-5520,-4756,-3981,-3196,-2404,-1606,-804 .word 0,804,1606,2404,3196,3981,4756,5520 .word 6270,7005,7723,8423,9102,9760,10394,11003 .word 11585,12140,12665,13160,13623,14053,14449,14811 .word 15137,15426,15679,15893,16069,16207,16305,16364


ESIEE, Slide 67

Listing for the Generation of 2 Carriers 2 of 2Listing for the Generation of 2 Carriers 2 of 2 .text * Initializations of DP, and of the phase Ialpha at 0 * The phase Ialpha is in ACCU A, and the index of table in B * attention we work in the part of ACCUs * Initialize AR1 at mid_cos and AR0 at Nsur4 debut:

LD #0, DP LD #0, A STM #Ncos,BK STM #deb_cos,AR1 STM #(deb_cos+Nsur4),AR2 STM #Inc, AR0

* endless loop boucle:

LD *AR1+0%,A STL A, DXR0 LD *AR2-0%,B STL B,DXR0

* Return to the beginning of the endless loop B boucle


ESIEE, Slide 68

Results Obtained with CCSResults Obtained with CCS


ESIEE, Slide 69

TutorialTutorial

The listing files for the precedent examples The listing files for the precedent examples can be found in the directory “tutorial”:can be found in the directory “tutorial”: Tutorial > Dsk5416 > Chapter 21 > Labs_modulationTutorial > Dsk5416 > Chapter 21 > Labs_modulation


ESIEE, Slide 70

Further ActivitiesFurther Activities

Application 5 for the TMS320C5416 Application 5 for the TMS320C5416 DSK and for the TMS320C5510. DSK and for the TMS320C5510.

Alien Voices. A very simple application showing the Alien Voices. A very simple application showing the effect of modulation on audio frequencies. It shows effect of modulation on audio frequencies. It shows how the carrier causes sum and difference how the carrier causes sum and difference frequencies to be generated. Here it is used to frequencies to be generated. Here it is used to generate the strange voices used for aliens in science generate the strange voices used for aliens in science fiction films and television.fiction films and television.

frequency modulation gmsk modulation

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