High precision WSPR

This directory contains software to implement a WSPR beacon on a Raspberry Pico.

The unusual/novel aspects of this software include:

a) disciplining the transmit frequency to GPS with very high precision (< 10 ppb)

b) implementing very fine frequency adjustments so that the transition from one FSK frequency to another can be made smoothly, even at VHF (or even UHF) frequencies. With this code, millihertz resolution at >100MHz is possible using the Si5351. At higher frequencies, this is 3-5 orders of magnitude better than with simpler approaches.

The first point is supported by a novel architecture using the PWM, PIO and DMA capabilities to achieve true 32 bit frequency counting referenced to the PPS output of a GPS module. The jitter on the period for this count is < 50ns and there is no reset offset.

Both points are supported by the use of high precision rational arithmetic for computing the settings on the Si5351 used to generate the transmitted frequency.

This repository includes some WIP related to support for the RP2350. That code is in the src/machine_x directory and can safely be ignored if you are here for the high resolution frequency synthesis.

High resolution counting

In order to discipline a synthesized frequency generator, it is necessary to first measure the current output frequency and then to adjust it. The RP2040 (and RP2350) has built in PWM hardware that can act as a frequency counter that can be used to measure frequencies but there are serious limitations. The most serious issues are that the PWM counters are only 16 bits long, there isn't an easy way to sample the PWM counters from an external time reference and most software for sampling the PWM counters is subject to several microseconds of jitter in the response time even if there are no surprises such as garbarge collection.

You can find other efforts to use the Pico as a frequency counter in Richard Kendall's blog, or in Jeremy Bentham's work. These other efforts use a variety of methods to solve the problem of resolution, but generally still suffer from the problem of jitter.

The novelty in my efforts is in chaining PWMs together in hardware instead of software to get high resolution. At the same time, I use the PIO and DMA systems of the Pico to get very stable, low-jitter values of microsecond time and counter state into a buffer without worrying about counter wraparound between samples.

How the PWM hardware works

The RP2040 and RP2350 have a number of PWM cores that include a counter that counts up to set value (known as TOP) and then either reverse or are reset directly to zero. The clock input for these counters is typically the system clock. That allows these counters can be used to generate pulse trains with variable duty cycle by setting a threshold value (known as THRESH). While the count is below the threshold, the output is low and when the count passes the threshold the output is high. Each PWM core has two outputs, A and B, which share the same counter, but which can have independent thresholds.

The clock for the counter normally comes from the internal processor clock which runs at 133MHz in the RP2040 and at 150MHz for the RP2350. This internal clock can be replaced by an external clock which is taken from the B input for the PWM core. Either way, the clock can be scaled down if slower PWM outputs are desired.

Interestingly, the A output of a PWM core can still be used as an output when the B is used as an input for the clock. This means that PWM cores can be daisy-chained by connecting the A output from one PWM to the B input on the next. This can be problematic in that there is a delay of several internal clock periods from when the fast PWM is reset to when the slow PWM is incremented, but there are ways to compensate for this effect. One compensation will be discussed later. They key here is that the resolution of the counters can be extended 16 bits at a time to truly absurd levels. Even with only 32 bits, a 100MHz input (which is possible if you clock the CPU at over 200MHz) will take 42 seconds to roll over. If we go for the absurd and use 48 bits, the same 100MHz counter won't roll over for over a month.

So then, the question is how to read these composited PWM counters at a precise moment in time and still retain as much accuracy while preserving all this lovely precision.

Reading the counters

Reading the counters is harder than it looks. First, there is fact that we can only read different counters one after another which means we physically can't read them at exactly at the same time. That can lead to confusion during the moments when the lower order counter is rolling over and incrementing the higher order counter because we can't tell whether we are getting the value of the slow counter before or after it is incremented.

We can, however, detect that a rollover has occurred, even in the presence of a short delay before the higher counter increments by reading the counters more than once. If B is the slow (higher order) counter and A is the fast (lower order) counter, we can read a sequence of values values B1 A1 B2 A2. Based on observation with live signals, the time between successive samples is roughly 75ns or about 10 clock cycles at 133MHz. In contrast, the delay from the wrap-around of A and the incrementing of B is no more than half that, or about 5 cycles. That is important because we know for certain that the slow counter can increment at most once between measurements of the fast counter.

Because of the timings involved, we can determine a 32-bit count by breaking things down into the following two cases:

• B1 = B2. No increment of B happened between B1 and A1 so (B1 A1) is a good measurement of the state at the time of the A1. Note that we don't know that B didn't increment between B1 and B2, but we do know that B could only have incremented after B2-5 clocks and thus could only have incremented after we measured A1 (if it incremented at all) since A1 is at about B2-10 clocks.

• B1 = B2-1. The logic here is tricky because B may have incremented before or after A1. If B incremented between B1 and A1, then (B2, A1) is the correct thing to return. If B incremented between A1 and B2, however, then (B1, A1) is the best answer. We can distinguish these two sub-cases by examining A1 and A2. If A1 is near 0xffff and A2 is near zero, then A rolled over after A1 and B could only have incremented after that so the answer is (B1, A1). On the other hand, if A1 <= A2, then both A1 and A2 were read after the increment of B so the answer is (B2, A1).

The upshot is that we can read the counters and get the value of both B and A as of the time of the A1 sample. The remaining problem is how can place that A1 sample in time as accurately as possible? The basic answer is to this same trick to read the high and low portions of the fast system clock on the Pico immediately before or after reading the state of the counter. If the time between these two measurements can have very low jitter, then we know the value at a precise moment in time.

Thus, the remaining problem is to control this measurement jitter and the jitter between an external event and the measurement.

Controlling sample time jitter

The answer to controlling jitter in the sample time of A1 is to use the specialized hardware of the RP2040 to trigger and conduct the entire sampling process without any software intervention. One way to get all of the required samples in a clean sequence is to use DMA chaining. In this process, one DMA channel reads a memory buffer containing control blocks that, in turn, set the control registers in a second DMA channel. Those control blocks each contain a source and destination address which, when written to second DMA, will cause a 32 bit value to be transfered. After that transfer, the second DMA triggers the first DMA to write another control block. That cause anothers transfer and another trigger until a null control block is encountered. In the system built here, the control blocks read from the following locations:

TIMERAWH
TIMERAWL   <-- t1
TIMERAWH
TIMERAWL
PWM1.CTR
PWM0.CTR   <-- t2
PWM1.CTR
PWM0.CTR

This lets us reconstruct a microsecond timestamp at time t1 and to get a sample of the composite counter at time t2 roughly 40 clocks later (actually 300±15ns). The DMA system transfers data at a 48MHz tempo with three transfers required for each value above.

The remaining problem then boils down to triggering this DMA transfer from an external signal with a constant delay with very low jitter. This can't be done in software because that introduces a massive amount of jitter. This also can't be done using the DMA hardware alone because a GPIO can't be used as a data read trigger on a DMA channel. The solution is to use a short PIO program and a third DMA channel. The PIO program monitors the input pin that the GPS is driving once per second. Within one system clock after a rising transition is detected, the PIO writes a value to its FIFO and a third DMA channel transfers that value from the PIO to the transfer count of the first DMA, triggering the transfer process.

In testing a live system on a 30MHz input, the resulting counts appear to indicate that the sample initiation process is stable to within 30ns. Over a 10 second period, this means that we should be able to get 3 ppb jitter for this 30 MHz input which means counting errors will be limited only by the size of the count. For 30MHz-50MHz, this gives us about 100 mHz error. This method is limited on the Raspberry Pico to measuring frequencies less than roughly 50MHz, but if we use Si5351 clock generator, we can generate multiple signals from the same internal or external oscillator. That let's us generate a 40MHz signal for calibrating the oscillator in the SI5351 and transfer that calibration to a separate signal at a higher frequency. We can also use harmonics of the SI5351 output to generate disciplined signals above the 200MHz frequency limit of the SI5351. All together, this system allows signal generation in the 2m (144-148MHz), 1.25m (220-222MHz) and 70cm (420-450MHz) bands with 200-600mHz resolution with high accuracy and stability.

Turning frequency measurements into frequency control

Once we get an accurate and high-resolution frequency measurement, the question becomes one of whether we can control a frequency generator precisely enough to get precisely the output we want.

The system described here uses an Si5351 clock generator to generate the transmitted signal. This chip generates a (nearly) square-wave signal on each of three outputs. These outputs can be driven from either of two internal phase-locked-loops (PLL) via a integer + fractional divider that has 20 bit resolution for the numerator and denominator in the divider. The PLLs can be configured to run in the range from 600-900MHz referenced to an integrated crystal reference that runs in this design at 25MHz via a similar integer + fractional divider. The output frequency can be set by changing the frequency of the PLL or by changing the divider after the PLL, but the question of resolution applies equally. At frequencies above 50MHz, phase noise is lower if you control the PLL frequency and set the divider to an even integer value.

It is common in HF band WSPR beacons based on the Si5351 to set the denominator of the frequency divider to a fixed value chosen to allow stepping the generated frequency by exactly the required WSPR tone separation. For example, to transmit on the 30m band, the PLL can be set to 900MHz and the denominator can be set to the somewhat magical number 778730 to get frequency spacing that is very close to 1/10 of the nominal 1.4648Hz separation in the WSPR specification. This works, and it even allows for the generated tone to be moved smoothly from one tone to the next rather than jumping it directly to the next value. In the 10m band, however, this method can produce steps no finer than the full channel spacing making the microtone adjustments from the WSPR spec impossible. At higher frequencies, this method doesn't even allow steps as small as the tone spacing. In any case, compensation for small errors in oscillator frequency is also not possible.

This problem is illustrated in the following table which shows the integer and fractional settings on the divider (a, b and c, respectively) and the resulting output frequencies with a PLL at 900MHz.

a	b	c	Frequency (Hz)	Δf (Hz)
88	2	778730	10,227,272.429	0.0
88	1	778730	10,227,272.578	0.149
88	0	778730	10,227,272.727	0.298
32	2	600000	28,124,997.07	0.0
32	1	600000	28,124,998.54	1.47
32	0	600000	28,125,000.00	2.93

None of the WSPR bands above 10m can be addressed by this mechanism and anywhere above about 10MHz, the solution is marginal.

Number theory to the rescue

There is a way to fix this problem of resolution at higher frequencies. The trick is to vary both numerator and denominator to control the PLL frequency. The specific values for the numerator and denominator can be chosen using continued fractions to find an optimal rational approximation of our desired value. This is the same method that is used by Glenn Elmore to produce high quality disciplined oscillators.

It would appear at first that simply setting the denominator (c) to as large a value as possible would give the finest resolution, but this is not true. The following table shows how this works.

a	b	c	Frequency (Hz)	Δf (Hz)
31	15611	31250	28124600.00000	0.00000
31	311219	622996	28124600.14648	0.14648
31	111031	222261	28124600.29296	0.29296
31	302517	605576	28124600.43944	0.43944

The values of b and c are now seemingly chosen at random, but the frequencies generated are spot on the desired values.

This same approach can work in the 2m band

a	b	c	Frequency (Hz)	Δf (Hz)
34	16943	25000	144490500.00000	0.00000
34	97938	144511	144490500.14647	0.14647
34	425657	628072	144490500.29296	0.29296
34	89701	132357	144490500.43947	0.43947

The precision here is kind of amazing. Using this method, we can control the output with a resolution of 0.1mHz. This is 12 significant digits which seems like the result of dark arts.

So how can these seemingly magical values be determined? The algorithm for finding these values of a, b and c is actually fairly simple.

The fundamental idea is based on continued fractions as a way to find the best fractional representation of a number with a limited size for the denominator. To generate a frequency $f = 144,490,500.14647Hz$, we figure out the required up conversion for the PLL given a fixed down conversion of 6x. This gives a PLL frequency of about 867MHz and which is about 34.68 times the 25MHz clock on the generator.

Skipping to the final answer, this reduces to a call to NearestFraction with the following parameters

NearestFraction(34_677_720_035_152, 1_000_000_000_000, 1<<20)

This is asking for the closest approximation of the slightly outrageous fractional representation of our intended factor 34_677_720_035_152/1_000_000_000_000 with a denominator in the result no larger than 2^20.

Inside the NearestFraction function, we rewrite the desired fraction $a/b$ using $$ a/b = \lfloor a / b \rfloor + {a \bmod b} / b $$ We abbreviate this a little bit by setting $k_1 = \lfloor a / b \rfloor$ and $a_1 = {a \bmod b}$ $$ a/b = k_1 + a_1 / b $$ At this point, we know that $a_1 \lt b$ so we can repeat this trick by expanding $b/a_1$ using $k_2 = \lfloor b/a_1 \rfloor$ and $b_1 = b \bmod a_1$ $$ a/b = k_1 + \frac 1 {k_2 + b_1 / a_1} $$ But $b_1 \lt a_1$ do this again (and again and again!) with $k_3 = \lfloor a_1 / b_1 \rfloor$ and $a_2 = a_1 \bmod b_1$ $$ a/b = k_1 + \frac 1 {k_2 + \frac 1 {k_3 + a_2/b_1}} $$ This seems like busywork, but the cool thing is that the continued fractions formed from more and more elements of the sequence of $k_1, k_2, k_3 \ldots$ are the best possible fractional approximations of $a/b$ with larger and larger denominations (strictly speaking we may need to increment the last $k_n$ to get the best value). All we have to do is to know when to stop. The function NearestFraction continues this process until going one more step would result in too large a denominator.

For the ratio that we started with at the beginning of this section, if we apply this process we get a sequence $k_1 \ldots k_9 = (34, 1, 2, 9, 1, 2, 1, 1, 4)$ that represents the fraction $33464/965$. This is only $1.72 \times 10^-7$ away from the correct value which is impressive since the denominator is less than a thousand.