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

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1 Abstract Quadrature Amplitude Modulation (QAM) is used for wireless and cable data transmissions. While QAM is notoriously straightforward mathematically, it is in permanent need of amendments and trade-offs to work properly in the physical world. We discuss the C++ part of a transmitter/receiver (TX/RX) pair that is fed with binary data from a pipe, encodes into a QAM signal, performs DA and AD conversions on microcontrollers, and recovers the original data. It can be used with different analog channels, including fiber optic and long wave radio. The implementation goes a long way to cope with distortions, discretization artifacts, out-of-band noise (literally), in-band noise, out-of sync clocks, varying signal strength, and blocking input. It is vital to control error propagation during signal processing and apply forward error correction. Electromagnetic compatibility requirements restrain signal generation in software and call for low pass filters in hardware. As timing and speed are critical, timed Direct Memory Access (DMA) is mandatory for data transport to and from the Digital to Analog Conversion (DAC) and ADC units. Indeed, timing is so critical that when you put a finger on one of the quartz crystals that drive the DMA channels, observable program behavior changes. Cope with it in software! Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

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

3 Band-limited Signals '/tmp/adcrecord.monitor' using 2:3 '' using 4:5 '/tmp/adcrecord.monitor2' using 2: The red dots ( symbols ) are complex amplitudes of a 137kHz sine wave They change at a fixed rate, the symbol rate. Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

µcs, Amplifiers & Antenna 2x (TX&RX) µc boards with 12 bit ADC and DAC units running at 1Mhz, slow serial connection to PC, 96KB of RAM, 84 MHz clock speed, 32 bit words resonating magnetic loop

4 µcs, Amplifiers & Antenna 2x (TX&RX) µc boards with 12 bit ADC and DAC units running at 1Mhz, slow serial connection to PC, 96KB of RAM, 84 MHz clock speed, 32 bit words resonating magnetic loop antenna, 3-15 turns of wire (TX) and ferrite antenna (RX) amplifiers moderately shielded against capacitive coupling, voltage regulators Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

5 Maximum Viable Byproduct '/tmp/artifacts' u 1: reqirement: -60 decibels dampening Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

6 Spectrum Plot 5 '/tmp/artifactsspectrum' u 1: handle this with a hardware filter Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

7 Analogue and Digital Signal Processing Filters pass on different frequency components of the signal with 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 response can be created as digital convolution filters Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

8 1st Order Low Pass Filters analogue: x y τ = RC f 0 = 1 2πτ digital: 1 y += ( x y)>>k ; one clock cycle/sample τ T s f 0 f s 2 k 2 k 2π Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

9 RC-Frequency Response Function RC-filter transfer function's modulus RC-filter transfer function's phase *log 10 ( A ) arg(a) *log 10 (f/f 0 ) *log 10 (f/f 0 ) A(f ) = 1 1+i f f 0 C A: complex amplitude Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

10 RC-Impulse Response Function RC-filter impulse response function K(f ) = A(f ) = K = A y(t) = (x K)(t) = y = x K K(t) e 2πitf dt x(p) K(t p) dp K(t) = { 1 τ e t τ t 0 0 t < 0 K: 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 Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

11 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 signal if we transmit only the real part of W Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

12 Modulation Code (µc) 1 v o i d 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 t d : : a r r a y <b a s i c i n t C o m p l e x <i n t 3 2 t >, 2> LP = s t a t i c L P ; 7 unsigned t = s t a t i c T i m e ; 8 f o r ( i n t 3 2 t o u t e r = 0 ; o u t e r < numcomplexperchunk ; ++o u t e r ) { 9 b a s i c i n t C o m p l e x <i n t 3 2 t > BASEB32( baseband. a t ( o u 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 d e r T X ; ++i n n e r, ++d e s t i n a t i o n ) { 11 LP [ 0 ]. r e ( ) += (BASEB32. r e ( ) LP [ 0 ]. r e ( ) ) >> LPshiftTX ; 12 LP [ 0 ]. im ( ) += (BASEB32. im ( ) LP [ 0 ]. im ( ) ) >> LPshiftTX ; 13 LP [ 1 ]. r e ( ) += ( LP [ 0 ]. r e ( ) LP [ 1 ]. r e ( ) ) >> LPshiftTX ; 14 LP [ 1 ]. im ( ) += ( LP [ 0 ]. im ( ) LP [ 1 ]. im ( ) ) >> LPshiftTX ; 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 = ( w i r e S i g n a l >> p o s t S h i f t ) ; 19 } 20 } 21 baseband. p o p f r o n t ( numcomplexperchunk ), s t a t i c L P = LP, s t a t i c T i m e = t ; 22 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

13 De-modulation Code (µc) 1 v o i d ADC Handler ( ) { 2 s t d : : a r r a y <b a s i c i n t C o m p l e x <i n t 3 2 t >, 2> LP = s t a t i c L P ; 3 u i n t 1 6 t s o u r c e = &dmarx [ i c h u n k S i z e ] ; 4 i = ( i+ 1) % numchunks ; 5 ADC >ADC RNPR = ( u i n t 3 2 t )(&dmarx [ ( ( i +1)%numChunks ) c h u n k S i z e ] ) ; 6 ADC >ADC RNCR = c h u n k S i z e ; 7 i n t 3 2 t LS = l a s t S a m p l e, t = s t a t i c T i m e ; 8 unsigned i n t e = BBend. l o a d ( ) ; 9 u i n t 3 2 t d s 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 o u t e r =0; o u t e r <s r. s r i o b u f f e r. numcomplexrx;++outer,++ d s 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 ; ++i n n e r, s o u r c e+=s s t e p ) { 12 i n t 3 2 t a d c v a l = s o u r c e ; 13 i n t 3 2 t d i f f v a l = ( a d c v a l LS ) << p r e S h i f t, LS = a d c v a l ; 14 LP [ 0 ]. r e ( ) += (EXPTABLE [ t ]. r e ( ) d i f f v a l LP [ 0 ]. r e ( ) ) >> LPshiftRX ; 15 LP [ 0 ]. im ( ) += (EXPTABLE [ t ]. im ( ) d i f f v a l LP [ 0 ]. im ( ) ) >> LPshiftRX ; 16 t = 1 + ( t? t : EXPTABLE t : : LENGTH ) ; 17 LP [ 1 ]. r e ( ) += ( LP [ 0 ]. r e () LP [ 1 ]. r e ( ) ) >> LPshiftRX ; 18 LP [ 1 ]. im ( ) += ( LP [ 0 ]. im() LP [ 1 ]. im ( ) ) >> LPshiftRX ; 19 } ; 20 d s 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 c T i m e=t, l a s t S a m p l e=ls, s t a t i c L P=LP ; 23 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

14 Data Forwarding (µc) 1 v o i d l o o p ( ) { 2 s t a t i c unsigned BBbegin { 0 } ; 3 s t a t i c u i n t 3 2 t sequencenr { 0 } ; 4 unsigned e = BBend. l o a d ( ) ; 5 unsigned f i l l = ( ( numbbbuffersrx+e) BBbegin)%numBBbuffersRX ; 6 w h 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. RXMCUbufferFill = 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 a m p l e D i s t r i b u t i o n ; 10 g l o b a l S a m p l e D i s t r i b u t i o n = 0 ; 11 s r. w r i t e F r a m e ( 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 o a d ( ) ; 17 f i l l = ( ( numbbbuffersrx + e ) BBbegin ) % numbbbuffersrx ; 18 } 19 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

15 Delta Encoding (µc) Serial Connection on µc is slow, so save bandwidth where possible: 1 2 v o i d d i f f e r e n t i a t e ( i ntcomplex s o u r c e I t e r a t o r, b u f f e r T & 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 d e s t = w. data ; 6 f o r ( i n t s c = 0 ; s c < w. numcomplexrx ; ++s c ) { 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 d e s t = d e l t a y. r e ( ) ; 12 ++d e s t ; 13 d e s t = d e l t a y. im ( ) ; 14 ++d e 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 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

16 This is not a pipe '/tmp/adcrecord.monitor' using 2:3 '' using 4:5 '/tmp/adcrecord.monitor2' using 2: yet. Target: QAM256 modulation scheme + pipe interface Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

17 QAM256 y=imag(symbol) Energy x 2 + y 2 = r 2 r = length of symbol vector x=real(symbol) Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

18 Live Demo: Anything that can go wrong '/tmp/adcrecord.monitor' using 2:3 '' using 4:5 '/tmp/adcrecord.monitor2' using 2: Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

19 Trick: Clock Synchronization Allow multiple symbols represent the same info ( tile pattern) See for that the average complex amplitude of the signal is real & positive in TX Adjust RX clock speed according to the number of rotations performed by the average amplitude of the received signal. additional average energy/symbol: moderate additional maximal energy/symbol: very moderate -16 imaginary real Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

20 TX Phase Adjustment (PC) 1 v o i d e n c o d e r : : a d j u s t P h a s e ( i n 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 maxradius 5 && 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 maxradius 9 && 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 / realtau, r e a l a v g += r e / r e a l T a u ; 13 symbol. r e ( ) = re, symbol. im ( ) = im ; 14 i f ( ( r e a l a v g < 3. 5 ) && ( maxradius < g l o b a l M a x R a d i u s ) ) { 15 ++maxradius ; 16 } e l s e i f ( ( ( abs ( imagsum ) ) >= 64) && ( maxradius < g l o b a l M a x R a d i u s ) ) { 17 ++maxradius ; 18 } e l s e i f ( maxradius > g l o b a l R a d i u s M i n ) { 19 maxradius ; 20 } 21 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

21 RX Phase Adjustment & Clock Sync (PC) 1 auto BBS = c o m p l e x t y p e ( r a w S i g n a l. a t ( 0 ). r e ( ), r a w S i g n a l. a t ( 0 ). im ( ) ) ; 2 BBS += c o m p l e x t y p e ( 0. 5 ( 0. 5 / 65536), 0. 5 ( 0. 5 / ) ) ; 3 p e n d i n g += r o t a t i o n s P e r B B S exptable. s i z e ( ) ; 4 i n t d e l t a = ( p e n d i n g >= 1) ( p e n d i n g <= 1); 5 p e n d i n g = d e l t a ; 6 p h a s e I n d e x = ( p h a s e I n d e x+d e l t a+exptable. s i z e ( ) ) % exptable. s i z e ( ) ; 7 BBS = exptable. a t ( p h a s e I n d e x ) ; 8 TxTime += t i m e S u b d i v i s i o n + d e l t a ; 9 auto n o r m a l i z e d D e 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 n o r m a l i z e d D e l t a += LP [ 1 ] ; 15 n o r m a l i z e d D e l t a. n o r m a l i z e (LP [ 1 ] ) ; 16 p h a s e I n t e g r a l += n o r m a l i z e d D e l t a. imag ( ) ; 17 r o t a t i o n s P e r B B S = p h a s e I n t e g r a l r o t a t i o n s P e r B B S a n d P h i ; Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

22 Things to determine: signal Amplitude Intermission 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) Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

23 RX Baseband Filter Frequency Response '/tmp/rhoresponse.h' u 1:2 '' u 1:3 '' u 1: x-unit = symbol rate Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

24 RX Baseband Filter Convolution Kernel '/tmp/rhokernel.h' u 0: x-unit = 1 Baseband sample Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

25 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 c o n s t char NIBBLES END = 1 9 ; 3 c o n s t unsigned char codeword [ NIBBLES END ] { 0 b , 0 b , 4 0 b , 0 b , 0 b , 0 b , 0 b , 5 0 b , 0 b , 0 b , 0 b , 0 b , 6 0 b , 0 b , 0 b , 0 b , 0 b , 7 0 b , 0 b } ; 8 p u b l i c : 9 unsigned char fwd ( c o n s t u nsigned char x ) c o n s t { 10 r e t u r n codeword [ x ] ; 11 } 12 unsigned char r e v ( unsigned char c, unsigned & e r r o r s, b o o l v e r b o s e ) { } 14 } ; Hamming distance >= 3, i.e., one bit error is always corrected, sometimes two Values 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. Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

26 Gray Code 1 c l a s s g r a y C o d e t { 2 p u b l i c : 3 char r e v ( c o n s t char x ) c o n s t { 4 r e t u r n g r a y [ x ] ; 5 // Frank Gray ( ) 6 } 7 char fwd ( c o n s t char x ) c o n s t { 8 r e t u r n p i e d C i p h e r [ x ] ; 9 } 10 g r a y C o d e t ( ) { 11 f o r ( i n t c = 0 ; c < 1 6 ; ++c ) 12 p i e d C i p h e r [ g r a y [ c ] ] = c ; 13 } 14 p r i v a t e : 15 c o n s t char g r a y [ 1 6 ] = { 0 b0111, 0 b0101, 0 b0100, 0 b1100, 0 b1101, 0 b b1110, 0 b1010, 0 b1011, 0 b1001, 0 b1000, 0 b0000, 0 b0001, 0 b0011, 17 0 b0010, 0 b0110 } ; 18 char p i e d C i p h e r [ 1 6 ] = { } ; 19 } ; limits number of bit flips due to analogue off-by-one errors Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

27 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 by Fourier Transforms of Amplitude distributions: imag real 1 : big red dot 16 P(imag) 1 : small red dot 32 0 : elsewhere in the correct time slice Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

28 Fourier Transform 20 re(x/170) Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

29 Fourier Transform from sampled data '/tmp/adcrecord.ampstat' u 0: Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

30 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 ). timestamp ; 2 i n t c y c l e I n d e x = ( t s / t i m e S l i c e T i m e ) % s l i c e s P e r S y m b o l ; 3 r e a l t y p e d e l t a A m p l i t u d e = p r e s c a l e 4 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 x 5 exptable2. s i z e ( ) ; 6 r e a l t y p e amptimesfreq = d e l t a A m p l i t u d e / 2 ; 7 f o r ( i n t f r e q u e n c y = 0 ; f r e q u e n c y < ; ++f r e q u e n c y ) { 8 t i m e S l i c e [ c y c l e I n d e x ] [ f r e q u e n c y ] = ; 9 t i m e S l i c e [ c y c l e I n d e x ] [ f r e q u e n c y ] += 10 exptable2 [ u i n t 6 4 t ( s t d : : l r o u n d ( amptimesfreq ) ) % exptable2. s i z e ( ) ] ; 11 amptimesfreq += d e l t a A m p l i t u d e ; 12 } One timeslice[cycleindex] array contains the fourier transform shown above The negative peaks of the timeslice variable yield both the time slice and the amplitude reference. Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

31 Live Demo: This is a pipe 1 $ c a t o u t f i l e > /tmp/ s d r t x 2 3 $ r e p o r t 4 R : 4. 4 S : 7 5 # count down 3 6 # count down 2 7 # count down 1 8 # s t a r t o f t r a n s m i s s i o n 9 R : 5. 1 S : R : 15 S : 7 12 # count up 0 13 # count up 1 14 # count up 2 15 # r e c e i v e d b y t e s, 800 oob b y t e s 16 # 2 b i t e r r o r s d i c o v e r e d i n raw b i t s, e r r o r r a t e : e # oob 18 +i 3 18 # oob 18 +i 4 19 # oob 18 +i 5 Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

32 Copyright (c): Heiko Frederik Bloch (The slides have been slightly edited since November 2017.Today s improved version of the hardware is more resistant to electrostatic noise) References: J. G. Proakis, Masoud Salehi: Digital Communications, 5th ed., McGraw-Hill L. Grafakos: Classical Fourier Analysis 2nd ed., Springer J. Franz: EMV. Störungssicherer Aufbau elektronischer Schaltungen. 4. Aufl., Teubner/Vieweg Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

33 1 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 Roadmap Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

34 Appendix 1 Fast Fourier Transform 2 Cache-friendly Sine and Cosine Lookup Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

35 Fast Fourier Transform 1 // computation time : 6 ns modulerank l o g 2 ( modulerank ) 2 s t a t i c v o i d f a s t ( moduletype & out, c o n s t moduletype & i n ) { 3 s t d : : v e c t o r <r i n g t y p e > b u f f e r (2 modulerank ) ; 4 r i n g t p t r o l d d a t a = &b u f f e r [ 0 ], newdata = &b u 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 s w a p ( pos ) ] ; 7 f o r ( u i n t 3 2 t l e v = 0 ; l e v < numdyadiclevels ; ++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 t d : : swap ( o l d d a t a, 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 p h i = pos << ( numdyadiclevels 1 l e v ) ; 12! r e v e r s e ( p h i = p h i ) ; 13 auto acc = o l d d a t a [ pos l e v e l B i t ] ; 14 acc = r o o t O f U n i t y : : t t ( p h i ) ; 15 acc += o l d d a 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 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

36 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 o m p l e x r o o t o f u n i t y { 3 s t a t i c c o n s t 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 c o n s t u i n t 3 2 t c h u n k 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 a y <s t d : : complex<r e a l t y p e >, c h u n k s 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 a y <s t d : : complex<r e a l t y p e >, numchunks> e x p t a b l e h i ; 7 p u b 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 & ( c h u n k s 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 s w i t c h (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 } Heiko Frederik Bloch Quadrature Amplitude Modulation (QAM) in C++ November 21, / 35

Mapper & De-Mapper System Document

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