The waveforms of the SID in the c64 and c128 can be examined, because the SID provides a 8-bit output register of the waveform of voice 3 in register $1b. The exact waveform can also be examined from the "start" of the waveform, because the test-bit (bit 3 in register $12 for voice 3) can be used to reset the random-waveform.
To examine the data, we want to be able to sample the output in a consistent way. First step is to establish the sampling-rate to be used. Of course the sampling-rate must depend on the frequency used in registers $0e and $0f, so our job is to determ ine the dependency between the frequency and the waveform "wavelength" in cycles, i.e. the number of cycles between the waveform-values change.
To this end, we want to read the waveform output of register $1b as fast
as possible. With a REU it is possible to sample the value every cycle,
and with the program "Cyclewise", which samples $10000 bytes of
the waveform output into bank 0 of a REU, I have collected the data in table
1 which are valid for the noise waveform. The number in the first column
is the frequency put into registers $0e and $0f. The second number is the
number of cycles each value of the waveform at least lasted, while the third
number is the number of cycles the first value ($fe) lasted when the sampling
is done using the construct:
sta $d412 ;Start waveform
stx $df01 ;Start sampling
The STA and STX instructions take 4 cycles, but the data in the table suggests
that the sampling is delayed only 3 cycles. Furthermore the first value
count is probably 1.5 times longer than the waveform.
Table 1: Frequency, wavelength and initial delay
Frequency Wavelength 1st value cnt ---------------------------------------------- $ffff $10.001000 $16 $C000 $15.555555 $1D $AAAA $18.001800 $22 $8000 $20 $2D $6000 $2A.AAAAAB $3D $4000 $40 $6D $3222 $51.B3F644 $78 $3000 $55.555555 $7D $1000 $100 $17D $0100 $1000 $17FD
This leads to the conclusion that the frequencies can generated with a loop like this for the noise waveform:
void Frequency-generator(long freq) {
long delay=0x180000; /* C-notation for $180000 */
long cycle=0;
for (;;;) { /* Repeat forever */
delay= delay-freq;
if (delay0) {
delay= delay+0x100000; /* C-notation for $100000 */
waveform_output= calculate new value for waveform;
}
waveform[cycle]= waveform_output;
cycle= cycle+1;
}
}
Furthermore, we notice that the highest frequency that is 100% cycle-aligned is $8000, which gives us a wavelength of 32 cycles. In the following discussion this frequency is assumed to be used.
Notice that the above program is not garanteed to be 100% correct, since I haven't tested it. Specifically is the value of $180000 a quick hack and it might be $17ffff just as well.
The next step is to determine whether the noise waveform loops, since this knowledge is useful in respect to an algorithm based on manipulating an internal register. If the algorithm is based upon manipulation an internal register (by doing for instance shifting and exclusive-or), the values _have_ to restart sometime, because an internal register on n bits can only hold 2^n different values. In other words, we will try to establish whether the output stream from $d41b will repeat itself and if it doesn't in a set period of time, say n clockcycles, we will know that the internal register will be of at least log2(n/32) bit s, since the value changes each 32 clockcycles (note: log2(x) is the same as log(x)/log(2)).
The program "Loopchecker" checks for loops with an algorithm like this:
void Loopchecker() {
start_noise-waveform();
/* We want to sample *inside* the potential loop */
wait_a_while();
/* Sample 256 values */
for (i=0;i256;i=i+1)
data[i]= peek($d41b);
/* Sample until those 256 values arrive again */
do {
i=0;
while (peek($d41b) = data[i])
i=i+1;
}
while (i256);
}
This program will terminate if a sequence of 256 recorded bytes will appear again later. If the 256 bytes reappear, we can be quite certain that the waveform loops, since the chance of this happening with a totally random source is infinitely small (we ll below 1e-500). Marko Makela and I (Asger Alstrup) hacked a loopchecker together over the IRC, and our results with a 16 cycle sampling-rate was that the computer terminated after approximately 2 minutes and 15 seconds. However, our results weren't consistent, partly because of the 16 cycle sampling frequency, which isn't completely cycle-aligned and partly because the reseting of a waveform with the TEST-bit does not reset the waveform immediately, but rather $2000-$8000 cycles later (this figure varies greatly, d oes anybody know of a way to reset the waveform fast?).
The "Loopchecker" program given below fixes these errors, and
if you run this program, the computer will terminate after approximately
4 minutes and 32 seconds or 272 seconds. Firstly this means that the waveform
repeats.
Secondly, when the waveform changes every 32 cycles, this means that the
length of the loop is approximately
(272 sec * 980000 cyc/sec)/32 cyc/bytes ~= 8.0 Megabytes.
Thirdly, this implies an internal register length of log2(8.0 MB)= 23 bits.
And finally, we must admit that it would be a difficult task to sample the
entire 8 MB on a c64 with only 64KB memory. It is possible to sample the
lot in chunks and saving it on multiple disks, but this is not a trivial
task. Notice that the data probably can't be packed much since they are
random. It should be mentioned that by changing the delayloop after the
noisewaveform is selected, it can be demonstrated that the waveform indeed
loops after 8 megabytes of data nomatter where in the waveform cycle you
are.
With the implication of an internal register of 23 bits, the next step is to find a pattern in the data which might hint the algorithm used to produce the data. Since we know a shifting and eor scheme can be used, it might be useful to take a look at the data in binary:
11111110 11111100 11111100 11111100 11111000 11111000 11111000 11111000 11110000 11110000 11100000 11100000 11100000 11000000 11000000 11000000 11000000 10000001 New value for bit 0! 10000001 00000011 00000011 00000011 00000110 00000110 00000100 00000100 00001100 00001000 00011000 00011000 00011000 00110000 00110000 ...
It should be quite clear that some kind of shifting scheme is used. Further examination of the data suggests that the internal register indeed is on 23 bits, where the mapping between the 8 bits in the output and the internal 23 bit register is like this:
Internal register bit number 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Output bit from register $1b 7 6 5 4 3 2 1 0
The first data from the output can be reproduced with this layout if the internal register is leftshifted, and the initial value of the internal register is:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0
Now we need to explain the new bits appearing in bit 0 of the data. Even further examination of the data implies that an eor-gate is used to feed bit 0 of the internal register. I have found that the bits 22 and 17 of the internal register provides the feed for bit 0 through an eor-gate, which gives us the entire mechanism:
22-21-20-19-18-17-16-15-14-13-12-...-8-7-6-5-4-3-2-1-0 | | | +----->-----> eor ----->----->----->----->----->-----+
This can also be expressed as a C-program like this:
/* Test a bit. Returns 1 if bit is set. */
long bit(long val, byte bitnr) {
return (val & (1<<bitnr))? 1:0;
}
/* Generate output from noise-waveform */
void Noisewaveform {
long bit22; /* Temp. to keep bit 22 */
long bit17; /* Temp. to keep bit 17 */
long reg= 0x7ffff8; /* Initial value of internal register*/
/* Repeat forever */
for (;;;) {
/* Pick out bits to make output value */
output = (bit(reg,22) << 7) |
(bit(reg,20) << 6) |
(bit(reg,16) << 5) |
(bit(reg,13) << 4) |
(bit(reg,11) << 3) |
(bit(reg, 7) << 2) |
(bit(reg, 4) << 1) |
(bit(reg, 2) << 0);
/* Save bits used to feed bit 0 */
bit22= bit(reg,22);
bit17= bit(reg,17);
/* Shift 1 bit left */
reg= reg << 1;
/* Feed bit 0 */
reg= reg | (bit22 ^ bit17);
}
}
Every loop in the above program provides a new value for the noise waveform in the variable "output". Notice that the value will repeat for several cycles, just like the real SID. All in all, I find that the above program can sufficiently reproduce t he output from the noise-waveform in the SID.
When it comes to the quality of the pseudo-random generated numbers, one thing is clear: Since all bit-patterns of the 23 bits are generated, all values in the 8 bit output will appear equally many times over time. Furthermore, I believe this particul ar scheme is recognized as one of the better ones for doing pseudo-random numbers, and it is based on polynomiums, much like the CRCs used for checksumming. But since the values are reproducible, you'll have to be careful if you plan to use the noise waveform as a source for random numbers for games and such. Firstly, only reset the waveform when the games is loaded. And wait with the sampling until the user presses a key or fire, which will provide an offset into the stream, so t he values will be different from each run. Furthermore, be sure to read the values at an appropriate sampling-rate, for instance every 32 cycle with a frequency of $8000. One prefered method is to do a table of say $2000 values before the game starts , so that voice 3 can be used in the game-music or for sound-effects in the game.
;"Cyclewise" - records noise-waveform each cycle using REU. ;---------------------------------------------------------- * = $1000 freq = $8000 ;Define frequency here cycle jsr init ;Init screen lda #$08 ;Set testbit to reset the waveform. sta $d412 jsr pause ;Give it time to settle - nescessary on my machine. lda #<freq ;Set frequency ldx #>freq sta $d40e stx $d40f ;Set up REU ldx #$d41b ;Define REU read address to be $d41b. ldy #>$d41b stx $df02 sty $df03 lda #$00 ;Record into $0000 sta $df04 sta $df05 sta $df06 ;in bank 0. sta $df07 ;Transfer $10000 bytes. sta $df08 lda #$00 ;No interrupts from the REU. sta $df09 lda #%10000000 ;Fix c64 adress at $d41b. sta $df0a lda #$80 ldx #%10010000 ;REU command: Do transfer from c64->REU. ;start waveform and do the sampling sta $d412 ;Enable noise-waveform stx $df01 ;and start recording. jmp done ;After sampling, wrap it up ;Loopchecker - checks for a loop in the noise-waveform. ;------------------------------------------------------ * = $1100 block = $2000 loop jsr init ;Init screen lda #$08 ;Set testbit to reset the waveform. sta $d412 jsr pause ;Give it time to settle - nescessary on my machine. lda #$00 ;Set frequency to $8000 sta $d40e lda #$80 sta $d40f lda #$80 ;Start noise-waveform sta $d412 jsr pause ;Wait a while, so we get well into the waveform inc $d020 ;Signal that the recording starts now ldx #0 ;Sample 256 bytes at 32 cycles l1 lda $d41b sta block,x bit $ffff bit $ffff bit $ffff bit $ffff nop inx bne l1 nop ;Wait exactly 1 value bit $ea bit $ffff bit $ffff l2 ldy #$1d ;See if the 256 bytes repeat ldx #$00 l3 lda $d3ff,y ;(Trick to get 5 cycles execution time) bit $ffff bit $ffff bit $ffff bit $ffff cmp block,x bne l2 inx bne l3 jmp done ;And exit if they do ;Misc subroutines ;---------------- * = $1800 ;Init screen init sei ;Secure accurate timing by lda #$00 ;disabling sprites sta $d015 in1 lda $d011 ;and screen. bpl in1 and #$ef sta $d011 rts ;Wait 50 frames so waveform can be reset pause ldx #50 p1 bit $d011 bpl p1 p2 bit $d011 bmi p2 dex bne p1 rts ;Wraps it up done lda $d011 ;Reenable screen ora #$10 sta $d011 cli ;and exit. rts
Output from "cyclewise" with a frequency of $ffff (The data are correct: the wavelength is effectively 16 cycles for the first $10000 or so values):
Value (number of times) $fe ($16), $fc ($30), $f8 ($40), $f0 ($20), $e0 ($30), $c0 ($40), $81 ($20), $03 ($30), $06 ($20), $04 ($20), $0c ($10), $08 ($10), $18 ($30), $30 ($20) ...