Enthalpy Posted May 5, 2019 Author Posted May 5, 2019 I forgot to explain in the last message that, when a Dirac is negative in the convolving sequence, and has the position p after multiplication by the harmonic rank, I replace it with a positive one at position T/2-p or T/2+p. The effect on the harmonic's amplitude is the same. Diracs all positive can then be grouped just by their spacings. ========== Here's a algebraic proof that H5=H7=0 in the (210, 5, 14, 16) sequence I proposed and depicted onMar 04, 2018 9:05 pm and Mar 17, 2018 8:33 pm I multiply the positions by 5 respectively 7, replace the negative Diracs positions p by positive ones at T/2-p, compute modulo T=210, and sort by increasing positions: ================================ + - + - + - + H1 -16 -14 -5 0 5 14 16 ================================ H5 130 175 185 105 25 35 80 25 35 80 105 130 175 185 ================================ H7 98 203 175 105 35 7 112 7 35 98 105 112 175 203 ================================ Then I group the Diracs: H5 (25 130) (80 185) (35 105 175) H7 (7 112) (98 203) (35 105 175) The pairs are T/2=105 ticks apart, the triplets T/3=70, so the sums are zero. This very short sequence was manageable by looking at the numbers. The previous (180, 4, 10, 16, 30, 36) needed drawings. Sequences with 27 transitions need some better help. ========== The proofs explain why highly composite sequence lengths like 180 or 210 favour the suppression harmonics. They don't explain why the rank of a suppressed harmonic divides the length of the found sequences. Was it a coincidence? Marc Schaefer, aka Enthalpy
Enthalpy Posted May 5, 2019 Author Posted May 5, 2019 Detail improvement let my dumb programSearch2931357.zip find T=210 waveforms with 29 and 31 transitions per half-period. They increase the H1 amplitude or squeeze the H3 further, as compared with the waveforms ofMar 03, 2018 that make the top of the table here. H1 H3 H5 H7 H9 | T a b c d e f g h i j k l m n o ============================================================================ 0.73 -104 nil nil -23 | 210 2 7 14 16 19 20 26 28 42 43 0.34 -114 nil nil 6 | 210 3 6 7 14 22 35 36 38 43 45 46 0.59 -111 nil nil -19 | 210 5 6 10 14 16 17 19 20 29 32 44 46 0.38 -114 nil nil -8 | 210 1 2 4 6 10 19 25 34 35 39 41 43 46 ============================================================================ 0.61 -111 nil nil -19 | 210 3 4 5 6 11 14 15 17 26 27 29 31 43 45 0.42 -115 nil nil -9 | 210 1 2 4 9 16 19 24 26 29 31 35 36 41 43 51 0.30 -116 nil nil -6 | 210 2 9 18 21 22 24 28 30 32 33 38 40 43 45 51 ============================================================================ T=180 is bad but here T=210 improves a bit. Marc Schaefer, aka Enthalpy
Enthalpy Posted May 7, 2019 Author Posted May 7, 2019 T was twice longer than needed onJan 21, 2018 9:58 pmT=90 and T=210 allow H3=H5=0, so here's an aggiornamento: H1 H3 H5 H7 H9 H11 | T a b c d e ==================================================== 0.42 nil nil -2 2 -3 | 90 2 5 8 15 18 0.71 nil nil -9 -11 -3 | 90 4 9 10 15 16 0.79 nil nil -20 nil 1 | 210 3 10 15 30 32 ==================================================== H3=0 for the first time with T=210. More harmonics show that the second waveform outperforms the first one. The third suggests two matched resistors to squeeze H7. (210, 5, 14, 16) suggested already two matched resistors to squeeze H3 and H9 at once.Mar 04, 2018 9:05 pm
Enthalpy Posted May 7, 2019 Author Posted May 7, 2019 Here's an algebraic proof that H3=H5=0 for the waveform (210, 3, 10, 15, 30, 32) ofJan 21, 2018 9:58 pm updated May 07, 2019 ------------------------------------------------------------------------- Dirac sign | + - + - + - + - + - + Positions | [ -32 -30 -15 -10 -3 0 ] 3 10 15 30 32 ------------------------------------------------------------------------- *3, +T/2 if - | 114 15 165 75 201 105 9 135 45 195 96 Increasing | 9 15 45 75 96 105 114 135 165 195 201 Cycles | (15 45 75 105 135 165 195) (96 201) (9 114) ------------------------------------------------------------------------- *5, +T/2 if - | 50 165 135 55 195 105 15 155 75 45 160 Increasing | 15 45 50 55 75 105 135 155 160 165 195 Cycles | (15 45 75 105 135 165 195) (50 155) (55 160) ------------------------------------------------------------------------- First cycles with 7 members. Marc Schaefer, aka Enthalpy
Enthalpy Posted May 11, 2019 Author Posted May 11, 2019 Algebraic proof that H5=H7=0 for some long waveforms listed onMay 05, 2019 06:35 pm Since T=210 divides by 5 and 7, I don't multiply the Diracs' positions by 5 or 7 any more; instead, I keep them but compute modulo 42 or 30 respectively. ---------- Waveform (210, 2, 7, 14, 16, 19, 20, 26, 28, 42, 43) Waveform ------------------------------------------------------------- Dirac sign | + - + - + - + - + - + Positions | 0 2 7 14 16 19 20 26 28 42 43 | -2 -7 -14 -16 -19 -20 -26 -28 -42 -43 ------------------------------------------------------------- Harmonic 5 ------------------------------------------------------------- Modulo T/5=42 | 0 2 7 14 16 19 20 26 28 0 1 | 40 35 28 26 23 22 16 14 0 41 ------------------------------------------------------------- +21 if -Dirac | 0 23 7 35 16 40 20 5 28 21 1 | 19 35 7 26 2 22 37 14 21 41 ------------------------------------------------------------- Ordered | 0 1 2 5 7 7 14 16 19 20 21 | 21 22 23 26 28 35 35 37 40 41 ------------------------------------------------------------- Cycles | (0 7 14 21 28 35) (7 21 35) | (1 22) (2 23) (5 26) | (16 37) (19 40) (20 41) ------------------------------------------------------------- Harmonic 7 ------------------------------------------------------------- Modulo T/7=30 | 0 2 7 14 16 19 20 26 28 12 13 | 28 23 16 14 11 10 4 2 18 17 ------------------------------------------------------------- +15 if -Dirac | 0 17 7 29 16 4 20 11 28 27 13 | 13 23 1 14 26 10 19 2 3 17 ------------------------------------------------------------- Ordered | 0 1 2 3 4 7 10 11 13 13 14 | 16 17 17 19 20 23 26 27 28 29 ------------------------------------------------------------- Cycles | (0 10 20) (3 13 23) (7 17 27) | (1 16) (2 17) (4 19) | (11 26) (13 28) (14 29) ------------------------------------------------------------- ---------- Waveform (210, 1, 2, 4, 6, 10, 19, 25, 34, 35, 39, 41, 43, 46) Waveform ------------------------------------------------------------------------- Dirac sign | - + - + - + - + - + - + - + Positions | 0 1 2 4 6 10 19 25 34 35 39 41 43 46 | -1 -2 -4 -6 -10 -19 -25 -34 -35 -39 -41 -43 -46 ------------------------------------------------------------------------- Harmonic 5 ------------------------------------------------------------------------- Modulo T/5=42 | 0 1 2 4 6 10 19 25 34 35 39 41 1 4 | 41 40 38 36 32 23 17 8 7 3 1 41 38 ------------------------------------------------------------------------- +21 if -Dirac | 21 1 23 4 27 10 40 25 13 35 18 41 22 4 | 41 19 38 15 32 2 17 29 7 24 1 20 38 ------------------------------------------------------------------------- Ordered | 1 1 2 4 4 7 10 13 15 17 18 19 20 21 | 22 23 24 25 27 29 32 35 38 38 40 41 41 ------------------------------------------------------------------------- Cycles | (1 15 29) (4 18 32) (7 21 35) | (10 24 38) (13 27 41) | (1 22) (2 23) (4 25) | (17 38) (19 40) (20 41) ------------------------------------------------------------------------- Harmonic 7 ------------------------------------------------------------------------- Modulo T/7=30 | 0 1 2 4 6 10 19 25 4 5 9 11 13 16 | 29 28 26 24 20 11 5 26 25 21 19 17 14 ------------------------------------------------------------------------- +15 if -Dirac | 15 1 17 4 21 10 4 25 19 5 24 11 28 16 | 29 13 26 9 20 26 5 11 25 6 19 2 14 ------------------------------------------------------------------------- Ordered | 1 2 4 4 5 5 6 9 10 11 11 13 14 15 | 16 17 19 19 20 21 24 25 25 26 26 28 29 ------------------------------------------------------------------------- Cycles | (1 6 11 16 21 26) (4 9 14 19 24 29) | (5 15 25) | (2 17) (4 19) (5 20) | (10 25) (11 26) (13 28) ------------------------------------------------------------------------- ---------- Waveform (210, 3, 4, 5, 6, 11, 14, 15, 17, 26, 27, 29, 31, 43, 45) Waveform ----------------------------------------------------------------------------- Dirac sign | + - + - + - + - + - + - + - + Positions | 0 3 4 5 6 11 14 15 17 26 27 29 31 43 45 | -3 -4 -5 -6 -11 -14 -15 -17 -26 -27 -29 -31 -43 -45 ----------------------------------------------------------------------------- Harmonic 5 ----------------------------------------------------------------------------- Modulo T/5=42 | 0 3 4 5 6 11 14 15 17 26 27 29 31 1 3 | 39 38 37 36 31 28 27 25 16 15 13 11 41 39 ----------------------------------------------------------------------------- +21 if -Dirac | 0 24 4 26 6 32 14 36 17 5 27 8 31 22 3 | 18 38 16 36 10 28 6 25 37 15 34 11 20 39 ----------------------------------------------------------------------------- Ordered | 0 3 4 5 6 6 8 10 11 14 15 16 17 18 20 | 22 24 25 26 27 28 31 32 34 36 36 37 38 39 ----------------------------------------------------------------------------- Cycles | (3 10 17 24 31 38) (4 11 18 25 32 39) | (0 14 28) (6 20 34) (8 22 36) | (5 26) (6 27) (15 36) (16 37) ----------------------------------------------------------------------------- Harmonic 7 ----------------------------------------------------------------------------- Modulo T/7=30 | 0 3 4 5 6 11 14 15 17 26 27 29 1 13 15 | 27 26 25 24 19 16 15 13 4 3 1 29 17 15 ----------------------------------------------------------------------------- +15 if -Dirac | 0 18 4 20 6 26 14 0 17 11 27 14 1 28 15 | 12 26 10 24 4 16 0 13 19 3 16 29 2 15 ----------------------------------------------------------------------------- Ordered | 0 0 0 1 2 3 4 4 6 10 11 12 13 14 14 | 15 15 16 16 17 18 19 20 24 26 26 27 28 29 ----------------------------------------------------------------------------- Cycles | (0 10 20) (4 14 24) (6 16 26) | (0 15) (0 15) (1 16) (2 17) (3 18) | (4 19) (11 26) (12 27) (13 28) (14 29) ----------------------------------------------------------------------------- ---------- Waveform (210, 2, 9, 18, 21, 22, 24, 28, 30, 32, 33, 38, 40, 43, 45, 51) Waveform --------------------------------------------------------------------------------- Dirac sign | - + - + - + - + - + - + - + - + Positions | 0 2 9 18 21 22 24 28 30 32 33 38 40 43 45 51 | -2 -9 -18 -21 -22 -24 -28 -30 -32 -33 -38 -40 -43 -45 -51 --------------------------------------------------------------------------------- Harmonic 5 --------------------------------------------------------------------------------- Modulo T/5=42 | 0 2 9 18 21 22 24 28 30 32 33 38 40 1 3 9 | 40 33 24 21 20 18 14 12 10 9 4 2 41 39 33 --------------------------------------------------------------------------------- +21 if -Dirac | 21 2 30 18 0 22 3 28 9 32 12 38 19 1 24 9 | 40 12 24 0 20 39 14 33 10 30 4 23 41 18 33 --------------------------------------------------------------------------------- Ordered | 0 0 1 2 3 4 9 9 10 12 12 14 18 18 19 20 | 21 22 23 24 24 28 30 30 32 33 33 38 39 40 41 --------------------------------------------------------------------------------- Cycles | (0 14 28) (4 18 32) (10 24 38) | (0 21) (1 22) (2 23) (3 24) (9 30) (9 30) | (12 33) (12 33) (18 39) (19 40) (20 41) --------------------------------------------------------------------------------- Harmonic 7 --------------------------------------------------------------------------------- Modulo T/7=30 | 0 2 9 18 21 22 24 28 0 2 3 8 10 13 15 21 | 28 21 12 9 8 6 2 0 28 27 22 20 17 15 9 --------------------------------------------------------------------------------- +15 if -Dirac | 15 2 24 18 6 22 9 28 15 2 18 8 25 13 0 21 | 28 6 12 24 8 21 2 15 28 12 22 5 17 0 9 --------------------------------------------------------------------------------- Ordered | 0 0 2 2 2 5 6 6 8 8 9 9 12 12 13 15 | 15 15 17 18 18 21 21 22 22 24 24 25 28 28 28 --------------------------------------------------------------------------------- Cycles | (2 12 22) (2 12 22) (5 15 25) (8 18 28) (8 18 28) | (0 15) (0 15) (2 17) (6 21) | (6 21) (9 24) (9 24) (13 28) --------------------------------------------------------------------------------- Marc Schaefer, aka Enthalpy
Enthalpy Posted May 13, 2019 Author Posted May 13, 2019 33 transitions improve the waveforms with T=210. T=180 stays bad. The reduction in H3 to -124dBc is compatible with luck and a uniform distribution of the harmonic voltage amplitude. The many trials result from minor optimizations, overclocking, and patience. H1 H3 H5 H7 H9 H11 | T a b c d e f g h i j k l m n o p ==================================================================================== 0.48 -115 nil nil -10 -23 | 210 1 2 7 10 11 12 14 15 20 24 35 38 45 46 47 48 0.35 -124 nil nil +4 -12 | 210 3 4 9 10 12 14 16 20 21 23 36 41 42 43 50 51 ==================================================================================== An different attempt up to T=554 and at T=630 with 15 transitions was sterile. Marc Schaefer, aka Enthalpy
Enthalpy Posted May 13, 2019 Author Posted May 13, 2019 Algebraic proof that H5=H7=0 for the waveform with H3=-124dBc, having 33 transitions and T=210, fromMay 13, 2019 here Waveform ------------------------------------------------------------------------------------- Dirac sign | + - + - + - + - + - + - + - + - + Positions | 0 3 4 9 10 12 14 16 20 21 23 36 41 42 43 50 51 | -3 -4 -9 -10 -12 -14 -16 -20 -21 -23 -36 -41 -42 -43 -50 -51 ------------------------------------------------------------------------------------- Harmonic 5 ------------------------------------------------------------------------------------- Modulo T/5=42 | 0 3 4 9 10 12 14 16 20 21 23 36 41 0 1 8 9 | 39 38 33 32 30 28 26 22 21 19 6 1 0 41 34 33 ------------------------------------------------------------------------------------- +21 if -Dirac | 0 24 4 30 10 33 14 37 20 0 23 15 41 21 1 29 9 | 18 38 12 32 9 28 5 22 0 19 27 1 21 41 13 33 ------------------------------------------------------------------------------------- Ordered | 0 0 0 1 1 4 5 9 9 10 12 13 14 15 18 19 20 | 21 21 22 23 24 27 28 29 30 32 33 33 37 38 41 41 ------------------------------------------------------------------------------------- Cycles | (0 14 28) (1 15 29) (4 18 32) (5 19 33) | (9 23 37) (10 24 38) (13 27 41) | (0 21) (0 21) (1 22) (9 30) (12 33) (20 41) ------------------------------------------------------------------------------------- Harmonic 7 ------------------------------------------------------------------------------------- Modulo T/7=30 | 0 3 4 9 10 12 14 16 20 21 23 6 11 12 13 20 21 | 27 26 21 20 18 16 14 10 9 7 24 19 18 17 10 9 ------------------------------------------------------------------------------------- +15 if -Dirac | 0 18 4 24 10 27 14 1 20 6 23 21 11 27 13 5 21 | 12 26 6 20 3 16 29 10 24 7 9 19 3 17 25 9 ------------------------------------------------------------------------------------- Ordered | 0 1 3 3 4 5 6 6 7 9 9 10 10 11 12 13 14 | 16 17 18 19 20 20 21 21 23 24 24 25 26 27 27 29 ------------------------------------------------------------------------------------- Cycles | (0 10 20) (3 13 23) (7 17 27) | (1 16) (3 18) (4 19) (5 20) (6 21) (6 21) | (9 24) (9 24) (10 25) (11 26) (12 27) (14 29) ------------------------------------------------------------------------------------- Marc Schaefer, aka Enthalpy
Enthalpy Posted May 18, 2019 Author Posted May 18, 2019 I suggested to add virtual points to a constellation to prove its sum is zeroMay 05, 2019 12:50 am It hasn't been necessary up to now, so here's an artificial case to illustrate it.
Enthalpy Posted June 1, 2019 Author Posted June 1, 2019 35 transitions improve further the waveforms with T=210: -135dBc. H1 H3 H5 H7 H9 H11 H13 | T a b c d e f g h i j k l m n o p q ============================================================================================= 0.52 -126 nil nil -11 -6 -8 | 210 2 7 8 10 12 14 15 16 19 27 28 31 33 40 42 50 51 0.47 -135 nil nil -28 -4 -8 | 210 5 6 9 12 14 17 25 27 31 33 36 37 42 43 50 51 52 ============================================================================================= T=140 and 180 with 35 transitions aren't quite as good as T=210. Neither did 15 transitions provide good waveforms with T<=702, T=840 nor T=1050. Marc Schaefer, aka Enthalpy
Enthalpy Posted June 2, 2019 Author Posted June 2, 2019 Algebraic proof that H5=H7=0 for the waveform with H3=-135dBc, having 35 transitions and T=210, here onJune 01, 2019 Waveform ----------------------------------------------------------------------------------------- Dirac sign | - + - + - + - + - + - + - + - + - + Positions | 0 5 6 9 12 14 17 25 27 31 33 36 37 42 43 50 51 52 | -5 -6 -9 -12 -14 -17 -25 -27 -31 -33 -36 -37 -42 -43 -50 -51 -52 ----------------------------------------------------------------------------------------- Harmonic 5 ----------------------------------------------------------------------------------------- Modulo T/5=42 | 0 5 6 9 12 14 17 25 27 31 33 36 37 0 1 8 9 10 | 37 36 33 30 28 25 17 15 11 9 6 5 0 41 34 33 32 ----------------------------------------------------------------------------------------- +21 if -Dirac | 21 5 27 9 33 14 38 25 6 31 12 36 16 0 22 8 30 10 | 37 15 33 9 28 4 17 36 11 30 6 26 0 20 34 12 32 ----------------------------------------------------------------------------------------- Ordered | 0 0 4 5 6 6 8 9 9 10 11 12 12 14 15 16 17 20 | 21 22 25 26 27 28 30 30 31 32 33 33 34 36 36 37 38 ----------------------------------------------------------------------------------------- Cycles | (0 14 28) (6 20 34) (8 22 36) | (0 21) (4 25) (5 26) (6 27) (9 30) | (9 30) (10 31) (11 32) (12 33) (12 33) | (15 36) (16 37) (17 38) ----------------------------------------------------------------------------------------- Harmonic 7 ----------------------------------------------------------------------------------------- Modulo T/7=30 | 0 5 6 9 12 14 17 25 27 1 3 6 7 12 13 20 21 22 | 25 24 21 18 16 13 5 3 29 27 24 23 18 17 10 9 8 ----------------------------------------------------------------------------------------- +15 if -Dirac | 15 5 21 9 27 14 2 25 12 1 18 6 22 12 28 20 6 22 | 25 9 21 3 16 28 5 18 29 12 24 8 18 2 10 24 8 ----------------------------------------------------------------------------------------- Ordered | 1 2 2 3 5 5 6 6 8 8 9 9 10 12 12 12 14 15 | 16 18 18 18 20 21 21 22 22 24 24 25 25 27 28 28 29 ----------------------------------------------------------------------------------------- Cycles | (2 12 22) (2 12 22) (5 15 25) | (8 18 28) (8 18 28) | (1 16) (3 18) (5 20) (6 21) (6 21) | (9 24) (9 24) (10 25) (12 27) (14 29) ----------------------------------------------------------------------------------------- Marc Schaefer, aka Enthalpy
johnywhy Posted March 22, 2023 Posted March 22, 2023 Fascinating work. I don't understand most of it, but i know a simple sine generator would be very useful. Your output is a essentially a stepped, multi-level approximate sine aka "modified" sine, correct? My application is audio, and i need all sub-20 kHz harmonics at 96 db quieter than a fundamental at 20 Hz. Is there a very-low-parts-count hardware implementation?
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