quadrature amplitude modulation (qam) in c++ · we discuss the c++ part of a transmitter/receiver...

Abstract

Quadrature Amplitude Modulation (QAM) is used for wireless and cable datatransmissions. While QAM is notoriously straightforward mathematically, it is inpermanent need of amendments and trade-offs to work properly in the physicalworld.We discuss the C++ part of a transmitter/receiver (TX/RX) pair that is fed withbinary data from a pipe, encodes into a QAM signal, performs DA and ADconversions on microcontrollers, and recovers the original data. It can be used withdifferent analog channels, including fiber optic and long wave radio.The implementation goes a long way to cope with distortions, discretizationartifacts, out-of-band noise (literally), in-band noise, out-of sync clocks, varyingsignal strength, and blocking input. It is vital to control error propagation duringsignal processing and apply forward error correction.Electromagnetic compatibility requirements restrain signal generation in softwareand call for low pass filters in hardware. As timing and speed are critical, timedDirect Memory Access (DMA) is mandatory for data transport to and from theDigital to Analog Conversion (DAC) and ADC units.Indeed, timing is so critical that when you put a finger on one of the quartz crystalsthat drive the DMA channels, observable program behavior changes. Cope with itin software!

Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, 2017 1 / 35

Quadrature Amplitude Modulation (QAM) in C++

Heiko Frederik Bloch

November 21, 2017


Band-limited Signals

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'/tmp/adcrecord.monitor'using2:3''using4:5

'/tmp/adcrecord.monitor2'using2:3

The red dots (”symbols”) are complex amplitudes of a 137kHz sine wave

They change at a fixed rate, the symbol rate.


µCs, Amplifiers & Antenna

2x (TX&RX) µC boards with 12 bitADC and DAC units running at 1Mhz,slow serial connection to PC, 96KB ofRAM, 84 MHz clock speed, 32 bitwords

resonating magnetic loop antenna,3-15 turns of wire (TX) and ferriteantenna (RX)

amplifiers moderately shielded againstcapacitive coupling,

voltage regulators


Maximum Viable Byproduct

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reqirement: -60 decibels dampening


Spectrum Plot

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handle this with a hardware filter


Analogue and Digital Signal Processing

Filters pass on different frequency components of the signalwith different amplification A(f)They

help create low-noise signals in the first place.

recover the original signal from a distorted and noisy channel

Filters are needed for both

baseband signals, here: −300Hz < f < 300Hz , before modulation, digital

bandpass signals, here: : 136, 4kHz < f < 137kHz , after modulation, analogue

Filters with arbitrary frequency responsecan be created as digital convolution filters


1st Order Low Pass Filters

analogue:x y

τ = RC

f0 =1

2πτ

digital:

1 y += ( x−y)>>k ;

I one clock cycle/sample

Iτ

Ts≈ 2k

I

f0fs

≈ 2−k

2π


RC-Frequency Response Function

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20*log 10(|A|)

20*log10(f/f0)

RC-filtertransferfunction'smodulus

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arg(A)

20*log10(f/f0)

RC-filtertransferfunction'sphase

A(f ) = 11+i· ff0

∈ C A: complex amplitude


RC-Impulse Response Function

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RC-filterimpulseresponsefunction

K = A

K (f ) = A(f ) =

∫ ∞−∞

K (t) · e−2πitf · dt

y = x ∗ K

y(t) = (x∗K )(t) =

∫ ∞−∞

x(p)·K (t−p)·dp

K (t) =

{1τ e− t

τ t ≥ 0

0 t < 0K : convolution kernel

This generalizes to arbitrary frequency response filters

digital kernels may live on both sides of the y-axis

digital kernels are actually discrete


Modulation

Modulation is multiplication in the complex plane ,

W (t) = e iωt ·B(t), t = time, B(t) ∈ C

with a complex oscillator.

e iωt = cos(ωt) + i · sin(ωt)

de-modulation is modulation with the oscillator in reverse gear

B(t) = e−iωt ·W (t)

but we need low pass filters after demodulation to come back to the original signalif we transmit only the real part of W


Modulation Code (µC)

1 vo id DACC Handler ( ) {2 v o l a t i l e u i n t 1 6 t ∗ d e s t i n a t i o n = &dmaTX[ i ∗ chunkSizeTX ] ;3 i = (1 + i ) % numChunks ;4 DACC−>DACC TNPR = ( u i n t 3 2 t )&dmaTX [ ( ( 1 + i )%numchunks )∗ chunkSizeTX ] ;5 DACC−>DACC TNCR = chunkSizeTX ;6 s td : : a r r ay<ba s i c i n tComp l e x<i n t 3 2 t >, 2> LP = s t a t i c LP ;7 uns igned t = s t a t i cT ime ;8 f o r ( i n t 3 2 t ou t e r = 0 ; ou t e r < numComplexPerChunk ; ++ou t e r ) {9 ba s i c i n tComp l e x<i n t 3 2 t> BASEB32( baseBand . at ( ou t e r ) ) ;

10 f o r ( i n t 3 2 t i n n e r = 0 ; i n n e r < d i v i de rTX ; ++inne r , ++d e s t i n a t i o n ) {11 LP [ 0 ] . r e ( ) += (BASEB32 . r e ( ) − LP [ 0 ] . r e ( ) ) >> LPshi f tTX ;12 LP [ 0 ] . im ( ) += (BASEB32 . im ( ) − LP [ 0 ] . im ( ) ) >> LPshi f tTX ;13 LP [ 1 ] . r e ( ) += (−LP [ 0 ] . r e ( ) − LP [ 1 ] . r e ( ) ) >> LPshi f tTX ;14 LP [ 1 ] . im ( ) += (−LP [ 0 ] . im ( ) − LP [ 1 ] . im ( ) ) >> LPshi f tTX ;15 i n t 3 2 t w i r e S i g n a l = EXPTABLE[ t ] . r e ( ) ∗ (LP [ 1 ] . r e ( ) >> p r e S h i f t )16 − EXPTABLE[ t ] . im ( ) ∗ (LP [ 1 ] . im ( ) >> p r e S h i f t ) ;17 t = −1 + ( t ? t : EXPTABLE t : : LENGTH) ;18 ∗ d e s t i n a t i o n = 2048 + ( w i r e S i g n a l >> p o s t S h i f t ) ;19 }20 }21 baseBand . p op f r o n t ( numComplexPerChunk ) , s t a t i c LP = LP , s t a t i cT ime = t ;22 }


De-modulation Code (µC)

1 vo id ADC Handler ( ) {2 s td : : a r r ay<ba s i c i n tComp l e x<i n t 3 2 t >, 2> LP = s t a t i c LP ;3 u i n t 1 6 t ∗ s ou r c e = &dmaRX[ i ∗ chunkS i ze ] ;4 i = ( i+ 1) % numChunks ;5 ADC−>ADC RNPR = ( u i n t 3 2 t )(&dmaRX [ ( ( i+1)%numChunks )∗ chunkS i ze ] ) ;6 ADC−>ADC RNCR = chunkS i ze ;7 i n t 3 2 t LS = las tSamp le , t = s t a t i cT ime ;8 uns igned i n t e = BBend . l o ad ( ) ;9 u i n t 3 2 t ∗ ds t = &baseBand [ s r . s r i o b u f f e r . numComplexRX ∗ e ] ;

10 f o r ( i n t 3 2 t ou t e r =0; oute r<s r . s r i o b u f f e r . numComplexRX;++oute r ,++ds t ){11 f o r ( i n t 3 2 t i n n e r = 0 ; i n n e r < d i v i d e r ; ++inne r , s ou r c e+=Sstep ) {12 i n t 3 2 t ad c va l = ∗ s ou r c e ;13 i n t 3 2 t d i f f v a l = ( adc va l − LS) << p r e Sh i f t , LS = adcva l ;14 LP [ 0 ] . r e ( ) += (EXPTABLE[ t ] . r e ( ) ∗ d i f f v a l−LP [ 0 ] . r e ( ) ) >> LPshi f tRX ;15 LP [ 0 ] . im ( ) += (EXPTABLE[ t ] . im ( ) ∗ d i f f v a l−LP [ 0 ] . im ( ) ) >> LPshi f tRX ;16 t = −1 + ( t ? t : EXPTABLE t : : LENGTH) ;17 LP [ 1 ] . r e ( ) += (−LP [ 0 ] . r e ()−LP [ 1 ] . r e ( ) ) >> LPshi f tRX ;18 LP [ 1 ] . im ( ) += (−LP [ 0 ] . im()−LP [ 1 ] . im ( ) ) >> LPshi f tRX ;19 } ;20 ∗ ds t =(( u i n t 3 2 t (LP [ 1 ] . r e ())>>16)) |((( u i n t 3 2 t (LP [ 1 ] . im())>>16))<<16);21 }22 BBend=(( e+1)%numBBbuffersRX ) , s t a t i cT ime=t , l a s tSamp l e=LS , s t a t i c LP=LP ;23 }


Data Forwarding (µC)

1 vo id l oop ( ) {2 s t a t i c uns igned BBbegin {0} ;3 s t a t i c u i n t 3 2 t sequenceNr {0} ;4 uns igned e = BBend . l oad ( ) ;5 uns igned f i l l = ( ( numBBbuffersRX+e)−BBbegin)%numBBbuffersRX ;6 wh i l e ( f i l l ) {7 s r . s r i o b u f f e r . SEQ NR = sequenceNr ;8 s r . s r i o b u f f e r . RXMCUbuf ferF i l l = f i l l ;9 s r . s r i o b u f f e r . DIAGNOSTIC = g l o b a l S amp l eD i s t r i b u t i o n ;

10 g l o b a l S amp l eD i s t r i b u t i o n = 0 ;11 s r . wr i teFrame (12 ( intComplex ∗) &baseBand [ s r . s r i o b u f f e r . numComplexRX∗BBbegin ]13 ) ;14 BBbegin = ( BBbegin + 1) % numBBbuffersRX ;15 ++sequenceNr ;16 e = basebandEnd . l oad ( ) ;17 f i l l = ( ( numBBbuffersRX + e ) − BBbegin ) % numBBbuffersRX ;18 }19 }


Delta Encoding (µC)

Serial Connection on µC is slow, so save bandwidth where possible:

12 vo id d i f f e r e n t i a t e ( intComplex ∗ s o u r c e I t e r a t o r , bu f f e rT & w) {3 intComplex l a s t y = l a s t y v a l u e ;4 w. complex ( ) = l a s t y ;5 i n t 8 t ∗ de s t = w. data ;6 f o r ( i n t s c = 0 ; sc < w. numComplexRX ; ++sc ) {7 intComplex y (∗ s o u r c e I t e r a t o r ) ;8 auto d e l t a y = y ;9 ++s o u r c e I t e r a t o r ;

10 d e l t a y −= l a s t y ;11 ∗ de s t = d e l t a y . r e ( ) ;12 ++de s t ;13 ∗ de s t = d e l t a y . im ( ) ;14 ++de s t ;15 l a s t y = y ;16 }17 l a s t y v a l u e = l a s t y ;18 }


This is not a pipe

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... yet. Target: QAM256 modulation scheme + pipe interface


QAM256

y=imag(symbol)

x=real(symbol)-10 10

Energy ∼ x2 + y2 = r2

r = length of symbol vector


Live Demo: Anything that can go wrong ...

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Clock SynchronizationTrick:

Allow multiple symbols represent the same info( tile pattern)

See for that the average complex amplitude of thesignal is real & positive in TX

Adjust RX clock speed according to the number ofrotations performed by the average amplitude ofthe received signal.

additional average energy/symbol: moderate

additional maximal energy/symbol: very moderate

imaginary

real-16


TX Phase Adjustment (PC)

1 vo id encode r : : ad j u s tPha s e ( in tComplex & symbol ) {2 i n t im = symbol . im ( ) , r e = symbol . r e ( ) ;3 i n t newReal = r e + t i l e ;4 i f ( newReal ∗ newReal + im ∗ im <= maxRadius ∗ maxRadius5 && newReal < t i l e )6 r e = newReal ;7 i n t newImag = im + t i l e ∗ ( i n t (0 > imagsum ) − i n t (0 < imagsum ) ) ;8 i f ( r e ∗ r e + newImag ∗ newImag <= maxRadius ∗ maxRadius9 && abs ( newImag ) < t i l e )

10 im = newImag ;11 imagsum += im ;12 r e a l a v g −= r e a l a v g / rea lTau , r e a l a v g += re / rea lTau ;13 symbol . r e ( ) = re , symbol . im ( ) = im ;14 i f ( ( r e a l a v g < 3 . 5 ) && (maxRadius < g loba lMaxRad ius ) ) {15 ++maxRadius ;16 } e l s e i f ( ( ( abs ( imagsum ) ) >= 64) && (maxRadius < g loba lMaxRad ius ) ) {17 ++maxRadius ;18 } e l s e i f ( maxRadius > g l oba lRad iu sMin ) {19 −−maxRadius ;20 }21 }


RX Phase Adjustment & Clock Sync (PC)

1 auto BBS = complextype ( r awS i gna l . a t ( 0 ) . r e ( ) , r awS i gna l . a t ( 0 ) . im ( ) ) ;2 BBS += complextype ( 0 . 5 − ( 0 . 5 / 65536) , 0 . 5 − ( 0 . 5 / 65536 ) ) ;3 pend ing += rota t i onsPe rBBS ∗ expTable . s i z e ( ) ;4 i n t d e l t a = ( pend ing >= 1) − ( pend ing <= −1);5 pend ing −= de l t a ;6 pha se Index = ( phase I ndex+d e l t a+expTable . s i z e ( ) ) % expTable . s i z e ( ) ;7 BBS ∗= expTable . a t ( pha se Index ) ;8 TxTime += t imeSubd i v i s i o n + d e l t a ;9 auto no rma l i z e dDe l t a = −LP [ 1 ] ;

10 f o r ( i n t c = 0 ; c != LP . s i z e ( ) ; ++c ) {11 LP [ c ] −= LP [ c ] / LPtau [ c ] ;12 LP [ c ] += ( c ? LP [ c − 1 ] : BBS) / LPtau [ c ] ;13 }14 no rma l i z e dDe l t a += LP [ 1 ] ;15 no rma l i z e dDe l t a . n o rma l i z e (LP [ 1 ] ) ;16 p h a s e I n t e g r a l += no rma l i z e dDe l t a . imag ( ) ;17 ro ta t i onsPe rBBS = ph a s e I n t e g r a l ∗ ro ta t ionsPerBBSandPh i ;


Intermission

Things to determine:

signal Amplitude

When to sample and round the baseband signal? (Every symbol duration time,but in which time slice?)

clock speed difference !

signal Phase !

Things to deal with:

distortions, e. g., by resonators

arbitrary noise outside the transmission band (large amplitude)

random noise inside the transmission band (small amplitude)


RX Baseband Filter Frequency Response

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RX Baseband Filter Convolution Kernel

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Error Correction Code

1 c l a s s e r r o r C o r r e c t i o n t {2 s t a t i c const char NIBBLES END = 19 ;3 const uns igned char codeword [ NIBBLES END ] { 0b01110000 , 0b11010001 ,4 0b10100010 , 0b01100011 , 0b10110100 , 0b00010101 , 0b01010110 ,5 0b10000111 , 0b11101000 , 0b10111001 , 0b11011010 , 0b00011011 ,6 0b10001100 , 0b00101101 , 0b00111110 , 0b01001111 , 0b00000000 ,7 0b11111111 , 0b11100101 } ;8 pub l i c :9 uns igned char fwd ( const uns igned char x ) const {

10 r e t u r n codeword [ x ] ;11 }12 uns igned char r e v ( uns igned char c , uns igned & e r r o r s , boo l v e r bo s e ) {13 . . . }14 } ;

Hamming distance >= 3, i.e., one bit error is always corrected, sometimes two

Values 0..15 are used for data transmission, 16,17,18 for control purposes

Error rate 10−5 in the uncorrected stream (near field longwave version)

Noise from switched-mode power supplies introduces many errors.


Gray Code

1 c l a s s g rayCode t {2 pub l i c :3 char r e v ( const char x ) const {4 r e t u r n gray [ x ] ;5 // Frank Gray (1887−1969)6 }7 char fwd ( const char x ) const {8 r e t u r n p i e dC i ph e r [ x ] ;9 }

10 g rayCode t ( ) {11 f o r ( i n t c = 0 ; c < 16 ; ++c )12 p i e dC i ph e r [ g ray [ c ] ] = c ;13 }14 p r i v a t e :15 const char gray [ 1 6 ] = { 0b0111 , 0b0101 , 0b0100 , 0b1100 , 0b1101 , 0b1111 ,16 0b1110 , 0b1010 , 0b1011 , 0b1001 , 0b1000 , 0b0000 , 0b0001 , 0b0011 ,17 0b0010 , 0b0110 } ;18 char p i e dC i ph e r [ 1 6 ] = { } ;19 } ;

limits number of bit flips due to analogue off-by-one errors


Amplitude & Time Slice Gauge

Can be done only after clock sync.

1. Amplitude Approximation by signal root mean square (RMS)

1% accuracy required.

2. Sampling Time Slice & Amplitude Approximation byFourier Transforms of Amplitude distributions:

imag

real

0.5 7.5-7.5

P(imag) ≈

1

16: big red dot

1

32: small red dot

0 : elsewhere

in the correct time slice


Fourier Transform

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re(x/170)


Fourier Transform from sampled data

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'/tmp/adcrecord.ampStat'u0:1


Amplitude & Time Slice Gauge

1 i n t 6 4 t t s = f i l t e r e d S i g n a l . a t ( c ) . t imestamp ;2 i n t c y c l e I n d e x = ( t s / t imeS l i c eT ime ) % s l i c e sP e r S ymbo l ;3 r e a l t y p e de l t aAmp l i t ude = p r e s c a l e4 ∗ f i l t e r e d S i g n a l . a t ( c ) . s i g . imag ( ) / f r e q 1 i n d e x5 ∗ expTable2 . s i z e ( ) ;6 r e a l t y p e ampTimesFreq = de l t aAmp l i t ud e / 2 ;7 f o r ( i n t f r e qu en c y = 0 ; f r e qu en c y < 512 ; ++f r equ en c y ) {8 t im e S l i c e [ c y c l e I n d e x ] [ f r e qu en c y ] ∗= 0 . 9 9 7 ;9 t im e S l i c e [ c y c l e I n d e x ] [ f r e qu en c y ] +=

10 expTable2 [ u i n t 6 4 t ( s t d : : l r o und ( ampTimesFreq ) ) % expTable2 . s i z e ( ) ] ;11 ampTimesFreq += de l t aAmp l i t ude ;12 }

One timeSlice[cycleIndex] array contains the fourier transform shown above

The negative peaks of the timeSlice variable yieldboth the time slice and the amplitude reference.


Live Demo: This is a pipe

1 $ ca t o u t f i l e > /tmp/ s d r t x23 $ r e p o r t4 R : 4 . 4 S : 75 # count−down 36 # count−down 27 # count−down 18 # s t a r t o f t r a n sm i s s i o n9 R : 5 . 1 S : 7

10 . . .11 R : 15 S : 712 # count−up 013 # count−up 114 # count−up 215 # r e c e i v e d 100800 bytes , 800 oob by t e s16 # 2 b i t e r r o r s d i c o v e r e d i n 1612832 raw b i t s , e r r o r r a t e : 1 .24005 e−0617 # oob 18 +i ∗ 318 # oob 18 +i ∗ 419 # oob 18 +i ∗ 5


Copyright (c): Heiko Frederik Bloch

(The slides have been slightly edited since November 2017.Today’s improved version ofthe hardware is more resistant to electrostatic noise)

References:

www.longhaulmail.de/misc/sdr2.pdf

J. G. Proakis, Masoud Salehi: Digital Communications, 5th ed., McGraw-Hill

L. Grafakos: Classical Fourier Analysis 2nd ed., Springer

J. Franz: EMV. Storungssicherer Aufbau elektronischer Schaltungen. 4. Aufl.,Teubner/Vieweg


http://www.longhaulmail.de/misc/sdr2.pdf

Roadmap1 Band-limited Signals

2 Microcontroller Hardware

3 Maximum Viable Byproduct

4 Signal Processing

5 Modulation

6 Serial Interface

7 This is not a pipe

8 QAM256

9 Clocks

10 Intermission

11 RX Baseband Filters

12 Errors

13 Gauge

14 This is a pipe

15 Copyright & References

16 Appendix


Appendix

1 Fast Fourier Transform

2 Cache-friendly Sine and Cosine Lookup


Fast Fourier Transform

1 // computat ion t ime : 6 ns ∗ moduleRank ∗ l o g 2 ( moduleRank )2 s t a t i c vo id f a s t ( moduletype & out , const moduletype & i n ) {3 s td : : v e c to r<r i n g t yp e> b u f f e r (2 ∗ moduleRank ) ;4 r i n g t p t r o l dda t a = &bu f f e r [ 0 ] , newdata = &bu f f e r [ moduleRank ] ;5 f o r ( u i n t 3 2 t pos = 0 ; pos < moduleRank ; ++pos )6 newdata [ pos ] = i n [ b i t swap ( pos ) ] ;7 f o r ( u i n t 3 2 t l e v = 0 ; l e v < numDyadicLeve l s ; ++l e v ) {8 auto l e v e l B i t = u i n t 3 2 t (1 ) << l e v ;9 s td : : swap ( o lddata , newdata ) ;

10 f o r ( u i n t 3 2 t pos = 0 ; pos < moduleRank ; ++pos ) {11 u i n t 3 2 t ph i = pos << ( numDyadicLeve l s − 1 − l e v ) ;12 ! r e v e r s e | | ( ph i = −ph i ) ;13 auto acc = o l dda t a [ pos | l e v e l B i t ] ;14 acc ∗= roo tOfUn i t y : : t t ( ph i ) ;15 acc += o ldda t a [ pos & ˜ l e v e l B i t ] ;16 newdata [ pos ] = acc ;17 }18 }19 out . a s s i g n ( newdata , newdata + moduleRank ) ;20 }


Cache-friendly Sine and Cosine Lookup

1 template<c l a s s r e a l t y p e , u i n t 3 2 t N>2 c l a s s c omp l e x r o o t o f u n i t y {3 s t a t i c const u i n t 3 2 t numChunks = ( ( u i n t 3 2 t ) 1) << (N / 2 − 1 ) ;4 s t a t i c const u i n t 3 2 t chunk s i z e = ( ( u i n t 3 2 t ) 1) << (N − N / 2 − 1 ) ;5 s t a t i c s t d : : a r r ay<s t d : : complex<r e a l t y p e >, chunks i z e> e x p t a b l e l o ;6 s t a t i c s t d : : a r r ay<s t d : : complex<r e a l t y p e >, numChunks> e x p t a b l e h i ;7 pub l i c :8 s t a t i c s t d : : complex<r e a l t y p e> t t ( u i n t 3 2 t k ) {9 auto c = e x p t a b l e l o [ k & ( chunks i z e −1)]

10 k >>= (N − N / 2 − 1 ) ;11 c ∗= e x p t a b l e h i [ k & ( numChunks−1) ] ;12 k >>= (N / 2 − 1 ) ;13 sw i tch (3 & k ) {14 case 1 : r e t u r n {−c . imag ( ) , c . r e a l ( ) } ;15 case 2 : r e t u r n −c ;16 case 3 : r e t u r n {c . imag () ,− c . r e a l ( ) } ;17 }18 r e t u r n c ;19 }20 }


quadrature amplitude modulation (qam) in c++ · we discuss the c++ part of a transmitter/receiver...

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