Arduino vs. Significant Figures: Useful 64-bit Fixed Point

Devoting eight bytes to every fixed point number may be excessive, but having nine significant figures apiece for the integer and fraction parts pushes the frequency calculations well beyond the limits of the DDS hardware, without involving any floating point library routines. This chunk of code performs a few more calculations using the format laid out earlier and explores a few idioms that may come in handy later.

Rounding the numbers to a specific number of decimal places gets rid of the repeating-digit problem that turns 0.10 into 0.099999:

uint64_t RoundFixedPt(union ll_u TheNumber,unsigned Decimals) {
union ll_u Rnd;

  Rnd.fx_64 = (One.fx_64 / 2) / (pow(10LL,Decimals));
  TheNumber.fx_64 = TheNumber.fx_64 + Rnd.fx_64;
  return TheNumber.fx_64;
}

That pretty well trashes the digits beyond the rounded place, so you shouldn’t display any more of them:

void PrintFixedPtRounded(char *pBuffer,union ll_u FixedPt,unsigned Decimals) {
char *pDecPt;

  FixedPt.fx_64 = RoundFixedPt(FixedPt,Decimals);

  PrintIntegerLL(pBuffer,FixedPt);  // do the integer part

  pBuffer += strlen(pBuffer);       // aim pointer beyond integer

  pDecPt = pBuffer;                 // save the point location
  *pBuffer++ = '.';                 // drop in the decimal point, tick pointer

  PrintFractionLL(pBuffer,FixedPt);

  if (Decimals == 0)
    *pDecPt = 0;                    // 0 places means discard the decimal point
  else
    *(pDecPt + Decimals + 1) = 0;   // truncate string to leave . and Decimals chars
}

Which definitely makes the numbers look prettier:

  Tenth.fx_64 = One.fx_64 / 10;             // Likewise, 0.1
  PrintFixedPt(Buffer,Tenth);
  printf("\n0.1: %s\n",Buffer);
  PrintFixedPtRounded(Buffer,Tenth,9);                    // show rounded value
  printf("0.1 to 9 dec: %s\n",Buffer);

  TestFreq.fx_64 = RoundFixedPt(Tenth,3);                 // show full string after rounding
  PrintFixedPt(Buffer,TestFreq);
  printf("0.1 to 3 dec: %s (full string)\n",Buffer);

  PrintFixedPtRounded(Buffer,Tenth,3);                    // show truncated string with rounded value
  printf("0.1 to 3 dec: %s (truncated string)\n",Buffer);

0.1: 0.099999999
0.1 to 9 dec: 0.100000000
0.1 to 3 dec: 0.100499999 (full string)
0.1 to 3 dec: 0.100 (truncated string)

  CtPerHz.fx_64 = -1;                       // Set up 2^32 - 1, which is close enough
  CtPerHz.fx_64 /= 125 * MEGA;              // divide by nominal oscillator
  PrintFixedPt(Buffer,CtPerHz);
  printf("\nCt/Hz = %s\n",Buffer);

  printf("Rounding: \n");
  for (int d = 9; d >= 0; d--) {
    PrintFixedPtRounded(Buffer,CtPerHz,d);
    printf("     %d: %s\n",d,Buffer);
  }

Ct/Hz = 34.359738367
Rounding:
     9: 34.359738368
     8: 34.35973837
     7: 34.3597384
     6: 34.359738
     5: 34.35974
     4: 34.3597
     3: 34.360
     2: 34.36
     1: 34.4
     0: 34

Multiplying two scaled 64-bit fixed-point numbers should produce a 128-bit result. For all the values we (well, I) care about, the product will fit into a 64-bit result, because the integer parts will always multiply out to less than 232 and we don’t care about more than 32 bits of fraction. This function multiplies two fixed point numbers of the form a.b × c.d by adding up the partial products thusly: ac + bd + ad + bc. The product of the integers ac won’t overflow 32 bits, the cross products ad and bc will always be slightly less than their integer factors, and the fractional product bd will always be less than 1.0.

Soooo, just multiply ’em out as 64-bit integers, shift the products around to align the appropriate parts, and add up the pieces:


uint64_t MultiplyFixedPt(union ll_u Mcand, union ll_u Mplier) {
union ll_u Result;

  Result.fx_64  = ((uint64_t)Mcand.fx_32.high * (uint64_t)Mplier.fx_32.high) << 32; // integer parts (clear fract) 
  Result.fx_64 += ((uint64_t)Mcand.fx_32.low * (uint64_t)Mplier.fx_32.low) >> 32;   // fraction parts (always < 1)
  Result.fx_64 += (uint64_t)Mcand.fx_32.high * (uint64_t)Mplier.fx_32.low;          // cross products
  Result.fx_64 += (uint64_t)Mcand.fx_32.low * (uint64_t)Mplier.fx_32.high;

  return Result.fx_64;
}

This may be a useful way to set magic numbers with a few decimal places, although it does require keeping the decimal point in mind:

  TestFreq.fx_64 = (599999LL * One.fx_64) / 10;           // set 59999.9 kHz differently
  PrintFixedPt(Buffer,TestFreq);
  printf("\nTest frequency: %s\n",Buffer);
  PrintFixedPtRounded(Buffer,TestFreq,1);
  printf("         round: %s\n",Buffer);

Test frequency: 59999.899999999
         round: 59999.9

Contrary to what I thought, computing the CtPerHz coefficient doesn’t require pre-dividing both 232 and the oscillator by 2, thus preventing the former from overflowing a 32 bit integer. All you do is knock the numerator down by one little itty bitty count you’ll never notice:

  CtPerHz.fx_64 = -1;                       // Set up 2^32 - 1, which is close enough
  CtPerHz.fx_64 /= 125 * MEGA;              // divide by nominal oscillator
  PrintFixedPt(Buffer,CtPerHz);
  printf("\nCt/Hz = %s\n",Buffer);

Ct/Hz = 34.359738367

That’s also the largest possible fixed-point number, because unsigned:

  TempFX.fx_64 = -1;
  PrintFixedPt(Buffer,TempFX);
  printf("Max fixed point: %s\n",Buffer);

Max fixed point: 4294967295.999999999

With nine.nine significant figures in the mix, tweaking the 125 MHz oscillator to within 2 Hz will work:

Oscillator tune: CtPerHz
 Oscillator: 125000000.00
 -10 -> 34.359741116
  -9 -> 34.359741116
  -8 -> 34.359740566
  -7 -> 34.359740566
  -6 -> 34.359740017
  -5 -> 34.359740017
  -4 -> 34.359739467
  -3 -> 34.359739467
  -2 -> 34.359738917
  -1 -> 34.359738917
  +0 -> 34.359738367
  +1 -> 34.359738367
  +2 -> 34.359737818
  +3 -> 34.359737818
  +4 -> 34.359737268
  +5 -> 34.359737268
  +6 -> 34.359736718
  +7 -> 34.359736718
  +8 -> 34.359736168
  +9 -> 34.359736168
 +10 -> 34.359735619

So, all in all, this looks good. The vast number of strings in the test program bulk it up beyond reason, but in actual practice I think the code will be smaller than the equivalent floating point version, with more significant figures. Speed isn’t an issue either way, because the delays waiting for the crystal tester to settle down at each frequency step should be larger than any possible computation.

The results were all verified with my trusty HP 50g and HP-15C calculators, both of which wipe the floor with any other way of handling mixed binary / hex / decimal arithmetic. If you do bit-wise calculations, even on an irregular basis, get yourself a SwissMicro DM16L; you can thank me later.

The Arduino source code as a GitHub Gist:

// Fixed point exercise for 60 kHz crystal tester
#include <avr/pgmspace.h>
char Buffer[10+1+10+1]; // string buffer for long long conversions
#define GIGA 1000000000LL
#define MEGA 1000000LL
#define KILO 1000LL
struct ll_fx {
uint32_t low;
uint32_t high;
};
union ll_u {
uint64_t fx_64;
struct ll_fx fx_32;
};
union ll_u CtPerHz; // will be 2^32 / 125 MHz
union ll_u HzPerCt; // will be 125 MHz / 2^32
union ll_u One; // 1.0 as fixed point
union ll_u Tenth; // 0.1 as fixed point
union ll_u TenthHzCt; // 0.1 Hz in counts
union ll_u Oscillator; // nominal oscillator frequency
union ll_u OscOffset; // oscillator calibration offset
union ll_u TestFreq,TestCount; // useful variables
union ll_u TempFX;
//-----------
// Round scaled fixed point to specific number of decimal places: 0 through 8
// You should display the value with only Decimals characters beyond the point
// Must calculate rounding value as separate variable to avoid mystery error
uint64_t RoundFixedPt(union ll_u TheNumber,unsigned Decimals) {
union ll_u Rnd;
// printf(" round before: %08lx %08lx\n",TheNumber.fx_32.high,TheNumber.fx_32.low);
Rnd.fx_64 = (One.fx_64 / 2) / (pow(10LL,Decimals));
// printf(" incr: %08lx %08lx\n",Rnd.fx_32.high,Rnd.fx_32.low);
TheNumber.fx_64 = TheNumber.fx_64 + Rnd.fx_64;
// printf(" after: %08lx %08lx\n",TheNumber.fx_32.high,TheNumber.fx_32.low);
return TheNumber.fx_64;
}
//-----------
// Multiply two unsigned scaled fixed point numbers without overflowing a 64 bit value
// The product of the two integer parts mut be < 2^32
uint64_t MultiplyFixedPt(union ll_u Mcand, union ll_u Mplier) {
union ll_u Result;
Result.fx_64 = ((uint64_t)Mcand.fx_32.high * (uint64_t)Mplier.fx_32.high) << 32; // integer parts (clear fract)
Result.fx_64 += ((uint64_t)Mcand.fx_32.low * (uint64_t)Mplier.fx_32.low) >> 32; // fraction parts (always < 1)
Result.fx_64 += (uint64_t)Mcand.fx_32.high * (uint64_t)Mplier.fx_32.low; // cross products
Result.fx_64 += (uint64_t)Mcand.fx_32.low * (uint64_t)Mplier.fx_32.high;
return Result.fx_64;
}
//-----------
// Long long print-to-buffer helpers
// Assumes little-Endian layout
void PrintHexLL(char *pBuffer,union ll_u FixedPt) {
sprintf(pBuffer,"%08lx %08lx",FixedPt.fx_32.high,FixedPt.fx_32.low);
}
// converts all 9 decimal digits of fraction, which should suffice
void PrintFractionLL(char *pBuffer,union ll_u FixedPt) {
union ll_u Fraction;
Fraction.fx_64 = FixedPt.fx_32.low; // copy 32 fraction bits, high order = 0
Fraction.fx_64 *= GIGA; // times 10^9 for conversion
Fraction.fx_64 >>= 32; // align integer part in low long
sprintf(pBuffer,"%09lu",Fraction.fx_32.low); // convert low long to decimal
}
void PrintIntegerLL(char *pBuffer,union ll_u FixedPt) {
sprintf(pBuffer,"%lu",FixedPt.fx_32.high);
}
void PrintFixedPt(char *pBuffer,union ll_u FixedPt) {
PrintIntegerLL(pBuffer,FixedPt); // do the integer part
pBuffer += strlen(pBuffer); // aim pointer beyond integer
*pBuffer++ = '.'; // drop in the decimal point, tick pointer
PrintFractionLL(pBuffer,FixedPt);
}
void PrintFixedPtRounded(char *pBuffer,union ll_u FixedPt,unsigned Decimals) {
char *pDecPt;
//char *pBase;
// pBase = pBuffer;
FixedPt.fx_64 = RoundFixedPt(FixedPt,Decimals);
PrintIntegerLL(pBuffer,FixedPt); // do the integer part
// printf(" Buffer int: [%s]\n",pBase);
pBuffer += strlen(pBuffer); // aim pointer beyond integer
pDecPt = pBuffer; // save the point location
*pBuffer++ = '.'; // drop in the decimal point, tick pointer
PrintFractionLL(pBuffer,FixedPt);
// printf(" Buffer all: [%s]\n",pBase);
if (Decimals == 0)
*pDecPt = 0; // 0 places means discard the decimal point
else
*(pDecPt + Decimals + 1) = 0; // truncate string to leave . and Decimals chars
// printf(" Buffer end: [%s]\n",pBase);
}
//-- Helper routine for printf()
int s_putc(char c, FILE *t) {
Serial.write(c);
}
//-----------
void setup ()
{
Serial.begin (115200);
fdevopen(&s_putc,0); // set up serial output for printf()
Serial.println (F("DDS calculation exercise"));
Serial.println (F("Ed Nisley - KE4ZNU - May 2017\n"));
// set up useful constants
TempFX.fx_64 = -1;
PrintFixedPt(Buffer,TempFX);
printf("Max fixed point: %s\n",Buffer);
One.fx_32.high = 1; // Set up 1.0, a very useful constant
PrintFixedPt(Buffer,One);
printf("\n1.0: %s\n",Buffer);
Tenth.fx_64 = One.fx_64 / 10; // Likewise, 0.1
PrintFixedPt(Buffer,Tenth);
printf("\n0.1: %s\n",Buffer);
PrintFixedPtRounded(Buffer,Tenth,9); // show rounded value
printf("0.1 to 9 dec: %s\n",Buffer);
TestFreq.fx_64 = RoundFixedPt(Tenth,3); // show full string after rounding
PrintFixedPt(Buffer,TestFreq);
printf("0.1 to 3 dec: %s (full string)\n",Buffer);
PrintFixedPtRounded(Buffer,Tenth,3); // show truncated string with rounded value
printf("0.1 to 3 dec: %s (truncated string)\n",Buffer);
CtPerHz.fx_64 = -1; // Set up 2^32 - 1, which is close enough
CtPerHz.fx_64 /= 125 * MEGA; // divide by nominal oscillator
PrintFixedPt(Buffer,CtPerHz);
printf("\nCt/Hz = %s\n",Buffer);
printf("Rounding: \n");
for (int d = 9; d >= 0; d--) {
PrintFixedPtRounded(Buffer,CtPerHz,d);
printf(" %d: %s\n",d,Buffer);
}
HzPerCt.fx_64 = 125 * MEGA; // 125 MHz / 2^32, without actually shifting!
PrintFixedPt(Buffer,HzPerCt);
printf("\nHz/Ct: %s\n",Buffer);
TenthHzCt.fx_64 = MultiplyFixedPt(Tenth,CtPerHz); // 0.1 Hz as delta-phase count
PrintFixedPt(Buffer,TenthHzCt);
printf("\n0.1 Hz as ct: %s\n",Buffer);
printf("Rounding: \n");
for (int d = 9; d >= 0; d--) {
PrintFixedPtRounded(Buffer,TenthHzCt,d);
printf(" %d: %s\n",d,Buffer);
}
// Try out various DDS computations
TestFreq.fx_64 = One.fx_64 * (60 * KILO); // set 60 kHz
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TestFreq.fx_64 += Tenth.fx_64; // set 60000.1 kHz
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TestFreq.fx_64 -= Tenth.fx_64 * 2; // set 59999.9 kHz
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TestFreq.fx_64 = (599999LL * One.fx_64) / 10; // set 59999.9 kHz differently
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TempFX.fx_64 = RoundFixedPt(TestCount,0); // compute frequency from integer count
TestFreq.fx_64 = MultiplyFixedPt(TempFX,HzPerCt);
PrintFixedPt(Buffer,TestFreq);
printf("Int ct -> freq: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestFreq.fx_64 = One.fx_64 * (10 * MEGA); // set 10 MHz
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TempFX.fx_64 = RoundFixedPt(TestCount,0); // compute frequency from integer count
TestFreq.fx_64 = MultiplyFixedPt(TempFX,HzPerCt);
PrintFixedPt(Buffer,TestFreq);
printf("Int ct -> freq: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestFreq.fx_64 = One.fx_64 * (10 * MEGA); // set 10 MHz + 0.1 Hz
TestFreq.fx_64 += Tenth.fx_64;
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TempFX.fx_64 = RoundFixedPt(TestCount,0); // compute frequency from integer count
TestFreq.fx_64 = MultiplyFixedPt(TempFX,HzPerCt);
PrintFixedPt(Buffer,TestFreq);
printf("Int ct -> freq: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestFreq.fx_64 = One.fx_64 * (10 * MEGA); // set 10 MHz - 0.1 Hz
TestFreq.fx_64 -= Tenth.fx_64;
PrintFixedPt(Buffer,TestFreq);
printf("\nTest frequency: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
TestCount.fx_64 = MultiplyFixedPt(TestFreq,CtPerHz); // convert to counts
PrintFixedPt(Buffer,TestCount);
printf("Delta phase ct: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestCount,0);
printf(" round to int: %s\n",Buffer);
TempFX.fx_64 = RoundFixedPt(TestCount,0); // compute frequency from integer count
TestFreq.fx_64 = MultiplyFixedPt(TempFX,HzPerCt);
PrintFixedPt(Buffer,TestFreq);
printf("Int ct -> freq: %s\n",Buffer);
PrintFixedPtRounded(Buffer,TestFreq,1);
printf(" round: %s\n",Buffer);
Oscillator.fx_64 = One.fx_64 * (125 * MEGA);
Serial.println("Oscillator tune: CtPerHz");
PrintFixedPtRounded(Buffer,Oscillator,2);
printf(" Oscillator: %s\n",Buffer);
for (int i=-10; i<=10; i++) {
OscOffset.fx_64 = i * One.fx_64;
CtPerHz.fx_64 = 1LL << 63;
CtPerHz.fx_64 /= (Oscillator.fx_64 + OscOffset.fx_64) >> 33;
PrintFixedPt(Buffer,CtPerHz);
printf(" %+3d -> %s\n",i,Buffer);
}
}
//-----------
void loop () {
}
view raw DDSCalcTest.ino hosted with ❤ by GitHub
DDS calculation exercise
Ed Nisley - KE4ZNU - May 2017
Max fixed point: 4294967295.999999999
1.0: 1.000000000
0.1: 0.099999999
0.1 to 9 dec: 0.100000000
0.1 to 3 dec: 0.100499999 (full string)
0.1 to 3 dec: 0.100 (truncated string)
Ct/Hz = 34.359738367
Rounding:
9: 34.359738368
8: 34.35973837
7: 34.3597384
6: 34.359738
5: 34.35974
4: 34.3597
3: 34.360
2: 34.36
1: 34.4
0: 34
Hz/Ct: 0.029103830
0.1 Hz as ct: 3.435973831
Rounding:
9: 3.435973832
8: 3.43597383
7: 3.4359738
6: 3.435974
5: 3.43597
4: 3.4360
3: 3.436
2: 3.44
1: 3.4
0: 3
Test frequency: 60000.000000000
round: 60000.0
Delta phase ct: 2061584.302070550
round to int: 2061584
Test frequency: 60000.099999999
round: 60000.1
Delta phase ct: 2061587.738044382
round to int: 2061588
Test frequency: 59999.900000000
round: 59999.9
Delta phase ct: 2061580.866096718
round to int: 2061581
Test frequency: 59999.899999999
round: 59999.9
Delta phase ct: 2061580.866096710
round to int: 2061581
Int ct -> freq: 59999.914551639
round: 59999.9
Test frequency: 10000000.000000000
round: 10000000.0
Delta phase ct: 343597383.678425103
round to int: 343597384
Int ct -> freq: 10000000.014506079
round: 10000000.0
Test frequency: 10000000.099999999
round: 10000000.1
Delta phase ct: 343597387.114398935
round to int: 343597387
Int ct -> freq: 10000000.114506079
round: 10000000.1
Test frequency: 9999999.900000000
round: 9999999.9
Delta phase ct: 343597380.242451271
round to int: 343597380
Int ct -> freq: 9999999.914506079
round: 9999999.9
Oscillator tune: CtPerHz
Oscillator: 125000000.00
-10 -> 34.359741116
-9 -> 34.359741116
-8 -> 34.359740566
-7 -> 34.359740566
-6 -> 34.359740017
-5 -> 34.359740017
-4 -> 34.359739467
-3 -> 34.359739467
-2 -> 34.359738917
-1 -> 34.359738917
+0 -> 34.359738367
+1 -> 34.359738367
+2 -> 34.359737818
+3 -> 34.359737818
+4 -> 34.359737268
+5 -> 34.359737268
+6 -> 34.359736718
+7 -> 34.359736718
+8 -> 34.359736168
+9 -> 34.359736168
+10 -> 34.359735619
view raw DDSCalcTest.txt hosted with ❤ by GitHub