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Most audio and radio signals are passed through some unknown linear
time-invariant system, such as linear circuits or multipath fading,
before we get them.  So it’s often to our advantage to try to detect
features of these signals that survive such mangling.  Those features
are sinusoids, or, more generally, complex exponentials.

If you want to detect sinusoidal components in a signal, the standard
approach is to use the Fourier transform.  This has a couple of
disadvantages:

1. It requires a substantial amount of computation per sample,
   specifically (for the radix-2 FFT) 2 lg N real multiplications
   where N is the window size and lg is the base-2 logarithm;
2. When a signal is only present during part of the window, its energy
   is smeared across the whole window, which may remove aspects of
   interest (you may want to know when it happened)
   and may push it below the noise floor;
3. It more or less requires floating-point computation, which
   increases the number of bit operations substantially.

There are a number of other approaches which can be used in cases
where the Fourier transform’s disadvantages are fatal, but I haven’t
found a good overview of them.  Here I’m going to talk about counting
zero crossings, SPLLs, CIC decimation, the
Minsky circle algorithm, the Goertzel algorithm,
Karplus-Strong delay line filtering,
linear predictive coding,
and uh
this thing I just came up with in the shower,
and finally an extended zero-crossing approach.

Counting zero crossings
-----------------------

A very simple and commonly used nonlinear filtering or demodulation
approach is to count the number of times the signal crosses the
X-axis.

If the majority of your signal power is in a single frequency (perhaps
because you’ve already filtered it) the count of zero
crossings is a reasonable approximation of the frequency.  A step
up from this is to measure the time interval of the last N zero
crossings, maybe the last 16 crossings or the last 64 crossings,
in order to get an estimate whose
precision is limited by your sampling rate rather than by both your
sampling rate and your time window.

The simplest approach is just this, assuming 8-bit samples being
returned from `getchar()` (unsigned, as is `getchar()`’s wont):

    volatile int crossings = 0;
    ...
    for (;;) {
        while (getchar() < 128);
        crossings++;
        while (getchar() >= 128);
        crossings++;
    }

However, this allows any arbitrarily small amount of noise near a slow
zero crossing to add extra zero crossings.  Normally in circuits we
deal with this by adding a Schmitt trigger, as described in Otto
Schmitt’s 1937 paper, “A Thermionic Trigger”; as Schmitt says, “in
another application, [the thermionic trigger circuit] acts as a
frequency meter more linear than one of the thyratron type, and one
immune to locking,” whatever that means.
The Schmitt trigger adds a little bit
of hysteresis through positive feedback, and indeed roughly any kind
of positive feedback on digital inputs nowadays is called a “Schmitt
trigger”.

In code, adding this hysteresis is even simpler:

    volatile int crossings = 0;
    ...
    for (;;) {
        while (getchar() < 128+16);
        crossings++;
        while (getchar() >= 128-16);
        crossings++;
    }

This requires the noise to move the needle by at least 32 counts
before it qualifies as a “crossing”, which you would think is 18 dB
below full-scale, but of course it’s not an average over the whole
signal, but an increment at that point.  Impulsive noise can overcome
this while totaling well below -18 dB by being at -∞ dB in between
impulses.

The other problem is that unless the actual signal is substantially
higher than those -18 dB, this will miss crossings, maybe all of them.
You can adjust the hysteresis to compensate with, for example, a peak
detector (note I haven’t tried this):

    volatile int crossings = 0;
    ...
    int threshold = 0, x;
    for (;;) {
        int peak = 0;
        while ((x = getchar()) < 128 + threshold) {
            if (128 - x > peak) peak = 128 - x;
        }
        crossings++;
        threshold = peak >> 3;

        peak = 0;
        while ((x = getchar()) > 128 - threshold) {
            if (x - 128 > peak) peak = x - 128;
        }
        crossings++;
        threshold = peak >> 3;
    }

This may count a few extra crossings at the beginning of the process
until it warms up; from then on, it notes the peak of each half-cycle
to use to adjust the hysteresis threshold for the next crossing.  If
the signal you’re detecting drops suddenly (by a factor of 8, as
specified by `>> 3`) within a single cycle, it could suddenly stop
counting cycles; similarly, if there is a noise spike that exceeds the
signal by a factor of 8, it could suddenly stop being able to detect
the signal.

Per sample, this requires two subtractions, two comparisons, a
conditional assignment, and a conditional jump, plus a couple more
operations after each zero crossing.

A more robust version of this algorithm would use perhaps the median
peak, the median, or some percentile over the last several cycles,
rather than simply the peak, and would decay the threshold toward 0
over time in order to recover from the kinds of situations described
above.  This requires a few more operations.

For frequency detection of a signal, the zero-crossing approach may be
superior to the linear FFT-based approaches discussed below.  For
example, if you want to control a synthesizer using your voice, you
need to be able to discriminate at least between musical notes.  If
you want to distinguish between 110 Hz (A) and 106 Hz (closer to G#)
with an FFT you need your frequency bins to be 4 Hz apart, which means
you need to be running the FFT over a 250 ms window.  But at 48 ksps,
the zero crossings of a 110-Hz square wave are 436 samples apart,
while the zero crossings on a 106-Hz square wave are more like
453 samples apart, and it seems like you could probably use the median
of the last 4–8 intervals between zero crossings.  8 half-waves at
106 Hz would be 38 ms.

However, I suspect that all of these variants are strictly inferior to
using a phase-locked loop, which is almost exactly the same amount of
computation but has truly impressive noise immunity, and can
additionally tell you the power of the frequency it’s detecting (at a
little extra computational expense.)

Software phase-locked loops
---------------------------

A phase-locked loop is a nonlinear filter that measures the frequency,
phase, and possibly power of a signal with constant or slowly changing
frequency.  You use a phase detector to set the frequency of a local
oscillator to match the frequency of the signal, and the phase
detector uses that local oscillator as its reference.  The simplest
phase detector is just a chopper and a low-pass filter; here’s a
software implementation in one line of C, suitable for tone frequency
tracking:

    /* A PLL in one line of C. arecord | ./tinypll | aplay */
    main(a,b){for(;;)putchar(b+=16+(a+=(b&256?1:-1)*getchar()-a/512)/1024);}

This is potentially quite efficient,
taking almost exactly the same amount of computation as counting zero crossings;
at its base, it involves four
additions or subtractions per sample, plus a bit test, a conditional,
and a couple of bitshifts.

This takes a signal (by default at 8 ksps) from the ALSA audio driver
with one sample on `getchar()`; `b` is the current phase of a local
oscillator with period of 512 counts, whose free-running frequency is
one cycle per 512/16 = 32 samples, so 250 Hz.  The phase detector
accumulates its error signal in `a`.  The input sample is chopped by
the ≈250Hz square wave from `b`, then either added to or subtracted
from `a`, which has an exponential low-pass filter applied to it by
way of subtracting a 512th of itself, yielding a time constant (I
think?) of 512 samples or 64 ms.  The range of `a` extends up to where
a == a+255-a/512, which is to say when a == 512*255 == 130560; the
lower limit is analogously -130560.  So `a` is scaled down by dividing
by 1024 to get `b`’s free-running increment, which means that in
theory it could cause `b` to run backwards or at up to 130+16 = 146
counts per sample, thus about 3.5 samples per cycle or 2280 Hz.
(Actually, not even that much, because half the time you have to be
feeding it zeroes, so its maximum stable magnitude oscillates around
130560/2.)

(In practice I’ve never managed to get the above code
even up into the kHz range.)

`putchar()` then outputs the low 8 bits of `b`, forming a sawtooth
wave from it with twice the chopper frequency, around 500 Hz.

More conventionally formatted and without the uninitialized-read
undefined behavior, the above code reads as follows:

    int main()
    {
        int a = 0, b = 0;
        for (;;) {
            a += (b & 256 ? getchar() : -getchar());
            a -= a/512;
            b += 16;
            b += a/1024;
            putchar(b);
        }
    }

This kind of phase detector tries hard to keep the chopper in
quadrature with the detected signal.  If you want to know the power of
the detected signal, you can chop the input signal with a second
chopper in quadrature with the first and sum its (squared)
output.

Note that this kind of phase detector is optimized for detecting
square waves, and thus can lock onto odd harmonics of the signal it
thinks it’s detecting.  This may be an advantage or a disadvantage in
a particular application.

The 1/512 in the above code, which is a low-pass filter on the phase detector
output (and oscillator frequency input), directly limits how fast the
PLL can track a changing frequency; less apparent is that it also
limits how much noise immunity the PLL has, and how far the PLL’s
frequency can jump to achieve lock (the “capture range”).  The
proportionality factor by which the phase output adjusts the local
oscillator frequency (the 1/1024) limits how far the PLL can track
before losing lock (the “lock range”).  A couple of hacks to improve
these tradeoffs are to sweep the natural frequency of the LO when it
hasn’t yet achieved lock, to use several concurrent PLLs with
different natural frequencies, and to tighten the low-pass filter once
lock is achieved.

With a different kind of phase detector, a PLL is also useful for
things like beat detection and beat matching, which is in a sense its
primary use in hardware — generating clock signals with a
predetermined relationship to existing clock signals, including clock
and data recovery for asynchronous data transmission.

The one-line program is about the same length in machine code as in C.  An
amd64 (but LP64) assembly listing for the 63 bytes of this loop is as
follows, keeping `b`, the local oscillator state, in %ebp, and `a`,
the phase detector, in %ebx.  GCC -Os did not optimize the
multiplication and divisions into a conditional and bitshifts as you
might expect:

      40              	.L3:
      43 0011 89E8     		movl	%ebp, %eax
      44 0013 25000100 		andl	$256, %eax
      44      00
      45 0018 83F801   		cmpl	$1, %eax
      46 001b 4519ED   		sbbl	%r13d, %r13d
      47 001e 31C0     		xorl	%eax, %eax
      48 0020 4183CD01 		orl	$1, %r13d
      49 0024 E8000000 		call	getchar
      49      00
      51 0029 4489E9   		movl	%r13d, %ecx
      52 002c 0FAFC8   		imull	%eax, %ecx
      53 002f 89D8     		movl	%ebx, %eax
      54 0031 99       		cltd
      55 0032 41F7FC   		idivl	%r12d
      56 0035 29C1     		subl	%eax, %ecx
      57 0037 01CB     		addl	%ecx, %ebx
      59 0039 B9000400 		movl	$1024, %ecx
      59      00
      60 003e 89D8     		movl	%ebx, %eax
      61 0040 99       		cltd
      62 0041 F7F9     		idivl	%ecx
      63 0043 8D6C0510 		leal	16(%rbp,%rax), %ebp
      65 0047 89EF     		movl	%ebp, %edi
      66 0049 E8000000 		call	putchar
      66      00
      69 004e EBC1     		jmp	.L3

There are many ways to improve this simplified PLL.

A simple one is to use a simple-moving-average filter instead of an
exponential filter to smooth the phase detector; this gives you better
tracking of frequency changes for the same noise immunity, or better
noise immunity for the same tracking of frequency changes.

Different phase detectors are best for different kinds of signals.
For sinusoidal signals with no significant harmonics, like a person
whistling or an FM radio signal, you can get a better phase detector
by weighting samples toward the middle of the sample interval more
highly than samples toward its edges; multiplying by even a triangle
wave (a square wave convolved with a simple moving average) gets you
most of the way there.  For signals where only, say, leading edges are
significant, you can use a phase detector that only considers the few
samples near that leading edge.

The shower algorithm
--------------------

Let’s suppose you want to detect some fixed frequency f and you’ve
already resampled your signal to a sampling rate of 4f.  Two
orthogonal sinusoids at frequency f then are [-1, -1, 1, 1] and [1,
-1, -1, 1]; two others are [0, 1, 0, -1] and [1, 0, -1, 0].  If you can
find the dot product of your signal with a pair of these sinusoids,
then you can use their Pythagorean sum to precisely compute the energy
in that frequency component of the signal.

If you’re dumping the decimated samples into a four-sample circular
buffer x\[0], x\[1], x[2], x[3], with some incrementing pointer xp:

    x[xp++ & 3] = new_sample;

Then you could imagine using x[xp], x[xp-1], x[xp-2], and x[xp-3] with
the appropriate modulo math.  However, this is totally not necessary,
because you actually don’t care how these sinusoids are aligned with
the signal; you only care that they are orthogonal.  It’s totally
valid to compute one phase component as x\[0] - x[2] and the other as
x\[1] - x[3] and then compute their Pythagorean sum:

    return pythsum(x[0] - x[2], x[1] - x[3]);

So far, though, we’ve gotten away without requiring the potentially
large number of bit operations required to even square a number.  For
low-precision data types, like 8 bits, it would be totally valid to
use a lookup table of squares, then binary-search it to find the
square root.  However, there are more approximate alternatives that
may be good enough in many cases.

Specifically, max(|a|, |b|) and |a| + |b| are both approximations of the
Pythagorean sum that are never wrong by more than a factor of √2 and
can be computed in a small linear number of bit operations.  Both are
precisely accurate when a or b is 0 and have their worst case when |a|
== |b|; max(|a|, |b|) is low by a factor of √2 then, and |a|+|b| is
high by a factor of √2.  Both have level sets that are square, but
rotated 45°.  If we sum them, the resulting octagonal level sets have
a worst-case error of a bit under 12%, just under 1 dB.

This Pythagorean-sum-approximation algorithm looks like this:

    if (a < 0) a = -a;
    if (b < 0) b = -b;
    return ((a < b ? b : a) + a + b >> 1);

YMMV.  On something like an AVR ATMega, the necessary three
comparisons, three conditional branches, pair of additions, and right
shift are probably slower than just computing the Pythagorean sum with
the two-cycle multiplier.  In the other direction, the 3 dB worst-case
error of max(|a|, |b|) or |a| + |b| is insignificant in many
frequency-detection contexts, where you’re trying to determine whether
a signal is more like -15dB or more like your -50dB noise floor.

In many cases you might want to be integrating the wave over a
significant period of time.  In a case like that, there are a few
different cheap approaches you can take.  A circular buffer of four
simple-moving-average filters provides optimal noise immunity for a
given step response; you can do the same thing with the accumulators
for single-pole exponential filters; and a chain of two or three
moving-average filters inexpensively gives you something approaching a
Gaussian window.  Finally, you could simply use a buffer of four
accumulators to sum the corresponding samples across some rectangular
window, without attempting any kind of nonuniform weighting.

### The problem of frequency response in rectangular windows ###

The disadvantage of rectangular windows in general (whether simple
moving average filters or fixed windows of a few hundred samples or
whatever) is that their Fourier transform is sinc.  Implicitly
multiplying your signal by the rectangular function in this way
effectively convolves its frequency spectrum with sinc, which dies off
relatively slowly (1/n) as you get far away from its center.  Even a
simple triangular window, which can be achieved by convolving two
identical rectangular functions together (and thus squaring their
frequency response), dies off at a much more reasonable pace (1/n²).
And, in a sense, this is the basis for CIC decimation.

CIC decimation
--------------

CIC decimation does not itself detect signals; it just (linearly, with
linear phase, essentially with a convolution of simple moving
averages) low-pass filters them and reduces the sampling rate.
However, an Nth-order CIC decimation filter involves only N integer
additions (using N accumulators) per input sample plus N subtractions
per output sample (using N previous output samples).  As explained
above, though, there are very efficient ways to detect signals in such
decimated data.

Here’s a second-order CIC decimation filter (untested) that reduces
the sample rate by a factor of 73.  It uses unsigned math because in C
unsigned overflow is well-defined and guarantees the property that
(a+b) - b == a, regardless of overflow, and the CIC algorithm needs that.

    enum { decimation_factor = 73 };
    unsigned s1 = 0, s2 = 0, d1 = 0, d2 = 0, i = decimation_factor;
    for (;;) {
        s1 += getchar();  // integrator 1
        s2 += s1;         // integrator 2
        if (0 == --i) {
            i = decimation_factor;
            unsigned dx = s2 - d1; // differentiator (“comb”) 1
            d1 = s2;
            unsigned dx2 = dx - d2; // differentiator 2
            d2 = dx;
            putchar(dx2 / (decimation_factor * decimation_factor));
        }
    }

For each input sample, this involves two additions, a decrement (or
increment), and a jump conditional on zero.  Each output sample
additionally requires two subtractions; in this case, I’ve also
rescaled the output by dividing by a compile-time constant (which
usually costs a multiply), so that it will be in (I think precisely)
the same range as the original samples; however, this loses precision
and dynamic range, and in many cases, no such rescaling is needed.

A 440 Hz signal (A above middle C in A440 standard pitch) sampled at
8 ksps is 18.1̄8̄ samples per cycle.  The factor of 73 above resamples
an 8 ksps signal such that a signal of roughly 438.4 Hz, about 6 cents
flat, will occupy 4 output samples, as recommended above for the
shower algorithm.  How well an actual precise 440 Hz signal will be
detected depends on how long you integrate the results over.

The second-order nature of the above filter effectively windows each
sample with a triangular window.

Here’s an (n-1)th-order version (untested):

    enum { decimation_factor = 73, n = 4 };
    unsigned s[n] = {0}, d[n-1] = {0}, i = decimation_factor;
    for (;;) {
        s[0] = getchar();
        for (int j = 1; j < n; j++) s[j] += s[j-1];
        if (0 == --i) {
            i = decimation_factor;

            unsigned dj = s[n-1];
            for (int j = 0; j < n-1; j++) {
                unsigned djprime = dj - d[j];
                d[j] = dj;
                dj = djprime;
            }

            printf("%d\n", dj);
        }
    }

There’s a quasi-inverse of CIC decimation, which is CIC interpolation;
in essence, this amounts to using the Method of Finite Differences
that Babbage used to tabulate polynomials on the Difference Engine.  I
say it’s a quasi-inverse because it doesn’t undo the low-pass
filtering that CIC decimation does, I think not even the part that’s
below the Nyquist frequency of the decimated signal.

I think it’s feasible to control the CIC decimation rate using a phase
detector, like a PLL, and it may be possible to dither the decimation
rate with something like the Bresenham algorithm; however, since the
signal out of the comb filter is amplified by a linear factor of the
decimation rate, this may be somewhat tricky, as oscillations of the
decimation rate turn into oscillations of the output signal amplitude,
which needs to be controlled for.

If you are resampling to several different frequencies, the initial
per-sample integration steps, and even the counter increment, can be
shared between them.

The Minsky Circle Algorithm
---------------------------

In HAKMEM (MIT AI Lab memo 239), item 149 (p. 73) describes an
algorithm attributed to Minsky for drawing circles — or, more
precisely, very slightly eccentric ellipses:

    NEW X = OLD X - ε * OLD Y
    NEW Y = OLD Y + ε * NEW(!) X

Rendered into C:

    int x = 255, y = 0, n = 1000;
    while (--n) {
        x -= y * epsilon;
        y += x * epsilon;
        pset(x0 + x, y0 + y);
    }

As Minsky comments, “If ε is a power of 2, then we don’t even need
multiplication, let alone square roots, sines, and cosines!”

Here’s a version that outputs a sine wave in minimized C:

    main(x,y){for(y=100;1+putchar(x+128);x-=y/4,y+=x/4);}

Of course, division by 4 can be implemented by an arithmetic shift
right by 2 bits.

HAKMEM items 150–152 go into some more detailed analysis.

We can think of the epsilons as rotating the (x, y) phasor by some
angle (specifically, cos⁻¹ (1-½ε²), according to item 151;
note that Baker’s transcription contains an erroneous omitted superscript 2).
If we add input samples to x (or y) then it will sum whatever
frequency component is in sync with the rotation:

    for (;;) {
        x -= epsilon * y;
        y += epsilon * x + getchar();
        putchar(x * scale);
    }

Frequency components that are not in sync with the rotation will
average out to zero.  Frequency components that are in sync will grow
without limit.

You can use two different ε factors for the two multiplications, which
gives a more elliptical “circle” and a denser range of frequencies.
In particular, one of them can be 1, which means you can generate a
tone of arbitrary frequency with only a single scaling operation per
cycle, plus the addition and subtraction:

    main(x,y){for(y=100;1+putchar(x+128);y+=x-=y/8);}

In somewhat more standard formatting, although with a still somewhat
eccentric use of `for`:

    /* Output an audio sinusoid with Minsky’s ellipse algorithm */
    #include <stdio.h>

    int main()
    {
      for (int x=0, y=100; EOF != putchar(x+128); x -= y >> 3, y += x)
        ;
      return 0;
    }

The loop here compiles to this amd64 machine code:

      400450:	89 e8                	mov    %ebp,%eax
      400452:	c1 f8 03             	sar    $0x3,%eax
      400455:	29 c3                	sub    %eax,%ebx
      400457:	01 dd                	add    %ebx,%ebp
      400459:	48 8b 35 e0 04 20 00 	mov    0x2004e0(%rip),%rsi # 600940 <__bss_start>
      400460:	8d bb 80 00 00 00    	lea    0x80(%rbx),%edi
      400466:	e8 b5 ff ff ff       	callq  400420 <_IO_putc@plt>
      40046b:	83 f8 ff             	cmp    $0xffffffff,%eax
      40046e:	75 e0                	jne    400450 <main+0x10>

Unfortunately, this version spends almost all of its time calling and
returning from `_IO_putc`.  A version that writes into a 512-byte
buffer instead, achieving 850 megasamples per second on my laptop, is
as follows:

    #include <stdio.h>

    int main()
    {
      char buf[512];
      int x=0, y=100;
      for (;;) {
        for (int i = 0; i != sizeof(buf)/sizeof(buf[0]); i++) {
          x -= y >> 3;
          y += x;
          buf[i] = 128 + x;
        }
        if (fwrite(buf, sizeof(buf), 1, stdout) != 1) return -1;
      }
    }

Its inner loop compiles to the following nine instructions (again, on
amd64):

      400460:	89 ea                	mov    %ebp,%edx
      400462:	c1 fa 03             	sar    $0x3,%edx
      400465:	29 d3                	sub    %edx,%ebx
      400467:	48 63 d0             	movslq %eax,%rdx
      40046a:	83 c0 01             	add    $0x1,%eax
      40046d:	8d 4b 80             	lea    -0x80(%rbx),%ecx
      400470:	01 dd                	add    %ebx,%ebp
      400472:	3d 00 02 00 00       	cmp    $0x200,%eax
      400477:	88 0c 14             	mov    %cl,(%rsp,%rdx,1)
      40047a:	75 e4                	jne    400460 <main+0x20>

The book “[Minskys and Trinskys][0],” by Corey Ziegler Hunts, Julian
Ziegler Hunts, R.W. Gosper and Jack Holloway, explores the variations
of this algorithm in more detail; I don’t have the book, but according
to [Nick Bickford’s 2011 post on the subject][1], they prove that,
using δ for the ε factor that multiplies Y:

    Xₙ = X₀ cos(n ω)+(X₀/2-Y₀/ε) sin(n ω)/d
    Yₙ = Y₀ cos(n ω)+(X₀/δ-Y₀/2) sin(n ω)/d

where

    d = √(1/(δ ε)-¼)
    ω = 2 sin⁻¹(½√(δ ε))

If this is correct, Gosper’s earlier result in HAKMEM that ω =
cos⁻¹ (1-½ε²) should be a special case of it (where ε = δ); it isn’t
immediately obvious to me why this is so, but these do seem to be
consistent for a couple of trial values:

    cos  ω = 1 - ½ δ ε
    sin ½ω = ½ √(δ ε)

[0]: http://au.blurb.com/b/2172660-minskys-trinskys-3rd-edition
[1]: https://nbickford.wordpress.com/2011/04/03/the-minsky-circle-algorithm/

I hypothesize, but haven’t proven either experimentally or rigorously,
that if you start with 0 and add each new input sample to x, you will
accumulate a phasor in x and (possibly, depending on the algorithmic
variant, scaled) y of all of the samples encountered so far, rotated
by the appropriate angle.  This will give you the total you’ve
encountered so far of a given Fourier component.

The Goertzel Algorithm
----------------------

The Goertzel algorithm (sometimes more specifically
“the second-order Goertzel algorithm”)
is an optimized version of Minsky’s algorithm,
requiring only one multiply or quasi-multiply per input sample.  It’s
actually older than Minsky’s formulation, dating from 1958.  At its
heart is the oscillator s[n] = (2 cos ω) s[n-1] - s[n-2], which
oscillates with an angular frequency of ω per sample; to look at it
another way, it’s the state transition function (s, t) ← ((2 cos ω) s
- t, s).  To this you just add each input sample x: (s, t) ← (x + (2
cos ω) s - t, s); in this way, the energy in the desired frequency
accumulates in s and t, and at the end of the accumulation process,
you can measure it.

Since 2 cos ω is constant, this is just a constant multiplication
— and some values are inexpensive to multiply by.  For example, you
can multiply a value a by 1-2⁻⁴ as follows:

    a -= a >> 4;

This is what I mean by “quasi-multiply”.

cos⁻¹(1-2⁻⁴) ≈ 0.3554, and consequently this works out to an
oscillation period of 2π/0.3554 ≈ 17.68 samples.  Here’s a C version
that emits a sine wave that repeats precisely, due to roundoff error,
every 53 samples, for a period of 17.66̄ samples, 452.8 Hz (lamentably
about halfway between A and A# above middle C) in linear unsigned
8-bit samples at 8 ksps:

    /* Generate a 452.8 Hz sine wave. ./goertzel | aplay */
    #include <stdio.h>

    int main()
    {
      int s = 0, t = 32;
      for (; EOF != putchar(s + 128);) {
        int tmp = s;
        s += s;
        s -= s >> 4;
        s -= t;
        t = tmp;
      }
    }

Not counting the output bias addition, this requires an addition, two
subtractions, and a bit shift per sample.  Here’s a one-line version:

    main(s,t,u){for(t=32;u=s,1+putchar(128+(s-=t-s+s/8));t=u);}

(For the special case where ω = ⅓π, cos ω = ½, so 2 cos ω = 1; we can
thus get a 6-sample cycle with only a subtraction per sample; in that
case (s, t) ← (s-t, s).  At 8 ksps this is 1333.3̄ Hz, 19 cents sharp
of E6.)

In a sense, this oscillator works by obtaining the derivative
information — stored as an explicit second variable in Minsky’s
algorithm — from the difference between s[n-1] and s[n-2].  We can
rewrite the recurrence relation as follows:

    s[n] = (2 cos ω) s[n-1] - s[n-2]

Let’s suppose:

    s[n-2] = k cos θ₀
    s[n-1] = k cos (θ₀ + ω)

Presumably in this case s[n] should be identically k cos (θ₀ + 2ω).
Is it?

    s[n] = 2 cos ω s[n-1] - s[n-2]
         = 2 cos ω k cos (θ₀ + ω) - k cos θ₀
         = k (2 cos ω cos (θ₀ + ω) - cos θ₀)

As you would remember from high-school trigonometry if you were
smarter than I am,

    cos (t + h) = cos t cos h - sin t sin h
    sin (t + h) = sin t cos h + cos t sin h
    cos 2ω = cos² ω - sin² ω
    sin 2ω = 2 sin ω cos ω

So

    cos (θ₀ + 2ω) = cos θ₀ cos (2ω) - sin θ₀ sin (2ω)
                  = cos θ₀ cos² ω - cos θ₀ sin² ω - 2 sin θ₀ sin ω cos ω
    cos (θ₀ + ω) = cos θ₀ cos ω - sin θ₀ sin ω
    k cos (θ₀ + 2ω) = k (cos θ₀ cos² ω - cos θ₀ sin² ω - 2 sin θ₀ sin ω cos ω)
    s[n] = 2 cos ω s[n-1] - s[n-2]
    if
    s[n-2] = k cos θ₀
    s[n-1] = k cos (θ₀ + ω)
    then
    s[n] = 2 cos ω k cos (θ₀ + ω) - k cos θ₀
         = k (2 cos ω cos (θ₀ + ω) - cos θ₀)
         = k (2 cos θ₀ cos² ω - cos θ₀        - 2 sin θ₀ sin ω cos ω)
         = k (cos θ₀ (2 cos² ω - 1)           - 2 sin θ₀ sin ω cos ω)
         = k (cos θ₀ (cos² ω + cos² ω - 1)    - 2 sin θ₀ sin ω cos ω)
         = k (cos θ₀ (cos² ω - (1 - cos² ω))  - 2 sin θ₀ sin ω cos ω)
         = k (cos θ₀ (cos² ω - sin² ω)        - 2 sin θ₀ sin ω cos ω)
         = k (cos θ₀ cos² ω - cos θ₀ sin² ω   - 2 sin θ₀ sin ω cos ω)
         = k cos (θ₀ + 2ω)
    ∴ cos (θ₀ + 2ω) = (2 cos ω) cos (θ₀ + ω) - cos θ₀ QED

This shows that given two points on a sinusoid of the right angular
frequency and any amplitude or phase, the Goertzel algorithm will
continue extrapolating further points on it indefinitely.

Measuring the energy is inexpensive (requiring two real multiplies, a
subtraction, a couple of squares, and an addition) but not entirely
obvious if we only keep around the last two samples of s.  The
standard presentation is that we transform them into a complex number
encoding the magnitude and phase of the signal:

    y[n] = s[n] - exp(-i ω) s[n-1]

Which is to say:

    y[n] = (s[n] - (cos ω) s[n-1]) + i (sin ω) s[n-1]

Note that that, if ω is small, the exponential gives a value close to
1, so this is very nearly the difference between the two last values
of s.

The idea is that the basic recurrence relation given above for s will
rotate this resultant phasor around by an angle of ω without altering
its magnitude.  Does it?

In the case of an arbitrary multiplier, the Goertzel algorithm beats
Minsky’s circle algorithm by almost a factor of 2, since Goertzel
requires only a single multiply per sample, while Minsky requires two
multiplies per sample.  But in a case where the multiplication can be
achieved by a bit shift and is basically free, Minsky requires just

    s += t >> n;
    t -= s >> n;

while Goertzel requires

    u = s;
    s += s - t - (s >> n);
    t = u;

So you have three additions or subtractions instead of two, and maybe
a bit of shunting as well, but one less bit shift.

Another factor to consider is that, for a given multiplier precision,
the Minsky algorithm covers the frequency spectrum much more densely;
for a multiplier ε (2⁻ⁿ in the example above) the Minsky frequency is
cos⁻¹ (1-½ε²), while the Goertzel frequency is cos⁻¹ (1-½ε).  So, for
example, if we are multiplying x, y, and 2·s[n-1] by 1/256 to get the
value we subtract, Minsky gives us a 3.9-milliradian rotation and a
1608-sample period, while Goertzel gives us a 63-milliradian rotation
and a 100.5-sample period.  This is somewhat intuitive, in that the
Minsky algorithm applies the multiplier twice per iteration.

The single-scaling version of Minsky’s algorithm is presumably similar
to Goertzel in its angular resolution, but with two additions or
subtractions per sample instead of three; the results I mentioned
above for that algorithm should be sufficient to show whether that is
true.

You can think of the Goertzel algorithm (or the Minsky algorithm) as
integrating the complex phasor y at a particular frequency over the
input signal — that is, y is a sum table (or prefix sum or cumulative
sum) of the input signal at that frequency.  This suggests that if you
want to know the amount (and phase) of signal at that frequency
between two points in time, you can subtract the corresponding points
in y, just as with a first-order CIC filter, and thus inexpensively
apply a moving average filter to it.  (You will need to rotate the two
phasors into phase with a rotation appropriate to the window size,
costing two multiplies.)

### Goertzel and Minsky as complex integrators ###

This of course suffers from the rectangular-window problem I mentioned
in the shower-algorithm section.  You can apply the CIC approach,
making a second-order or third-order sum table and then infrequently
taking differences at some window width, giving you the amplitude and
phase of the chosen frequency windowed by a triangular or
near-Gaussian function.  However, I think computing these sum tables
will require accumulating them as complex numbers and doing the
rotation by the appropriate phase before adding each new
sample — which, in the general case, requires a complex multiplication
(four real multiplications).

However, in the case where we can do the multiplication inexpensively,
as in the examples above with a bit shift and a subtraction or
addition, this may be a reasonable approach.  (See also the section
below about avoiding multiplications.)

Karplus-Strong delay line filtering
-----------------------------------

The Karplus-Strong string synthesis algorithm consists of nothing more
than a recursive unity-gain comb filter with a little bit of low-pass
filtering.  Originally, the delay line was initialized with random
noise, but it’s the resonances of the filter that provide the envelope
and most of the frequency response.  Here’s a one-line C
implementation initialized with just an impulse:

    s[72]={512};main(i){for(;;i%=72)s[i]+=s[(i+1)%72],putchar(s[i++]/=2);}

This version does two indexed fetches, two indexed stores, three
divisions, an addition, and a couple of increments per sample, but the
divisions can be replaced by equality tests and conditional stores,
except for one which is a bit shift.  So you need an indexed fetch, an
addition, an indexed store, a index increment, a bit shift of 1, and
the compare-to-threshold-and-reset operation on the index to implement
the circular buffer.

Here’s the same algorithm written in a more reasonable way, with a
time limit:

    #include <stdio.h>

    enum { delay = 72 };

    int s[delay] = { 512 };

    int main()
    {
      int i = 0, i2 = 0;
      for (int n = 8000; n--; i = i2) {
        i2 = i + 1;
        if (i2 == delay) i2 = 0;

        s[i] += s[i2];
        s[i] >>= 1;

        putchar(s[i]);
      }

      return 0;
    }

As a recurrence relation, this is computing

    s[n] = ½s[n-71] + ½s[n-72]

Of course, you can start with an empty buffer and add input signal to it,
which will be convolved with the infinite impulse response of
the recurrent filter — which response is precisely the sound you hear
when running it with no input starting from the above buffer with just
an impulse in it.  That impulse response has every frequency that fits
an integer number of times into the input buffer — the fundamental of
72 samples (111.1̄ Hz), but also 36 samples (222.2̄ Hz), 24 samples
(333.3̄ Hz), and so on.  The averaging of adjacent samples attenuates
higher frequencies.

If you negate the recurrence, it will instead allow signals with an
odd number of half-cycles to resonate.  For example:

    s[n] = -½s[n-71] - ½s[n-72]

This will preserve signals with a period of 144 samples, 48 samples,
28.8 samples, 20.57 samples, 16 samples, and so forth.  This allows
you to shorten the buffer by half and eliminates all the even
harmonics, which may be a drawback or an advantage, depending on the
signal you’re trying to detect.

Autoregressive filtering
------------------------

CIC’s integration and comb filters, the Karplus-Strong comb filter,
and the Goertzel s recurrence are all special cases of autoregressive
filtering, as used in linear predictive coding for
speech — they “predict” each new sample as a linear combination of the
previous samples.  There exist efficient algorithms
for finding the optimal autoregressive filter to minimize the
(squared?) residual, such as the Yule–Walker equations.
The coefficients of this filter are the
coefficients of a polynomial in the z-domain whose zeroes are the
formants of, say, a speech signal.

Avoiding multiplication with single-addition and dual-addition multipliers
--------------------------------------------------------------------------

Several times in the above, we’ve referred to cases where it’s
possible to use just a bit shift instead of a constant multiplication,
or merely to subtract or add a shifted number.  This is interesting
because bit shifts are, in hardware and occasionally in software,
free — they require zero gates, zero bit operations, and zero time.
An interesting question, then, is what set of multipliers we can
achieve with a single addition (or subtraction, which requires the
same number of bit operations):

    >>> numpy.array(sorted(set(x for a in range(8)
                                 for b in range(8)
                                 for x in [(1 << a) + (1 << b),
                                           (1 << a) - (1 << b)])))
    array([-127, -126, -124, -120, -112,  -96,  -64,  -63,  -62,  -60,  -56,
            -48,  -32,  -31,  -30,  -28,  -24,  -16,  -15,  -14,  -12,   -8,
             -7,   -6,   -4,   -3,   -2,   -1,    0,    1,    2,    3,    4,
              5,    6,    7,    8,    9,   10,   12,   14,   15,   16,   17,
             18,   20,   24,   28,   30,   31,   32,   33,   34,   36,   40,
             48,   56,   60,   62,   63,   64,   65,   66,   68,   72,   80,
             96,  112,  120,  124,  126,  127,  128,  129,  130,  132,  136,
            144,  160,  192,  256])

So, for example, in the neighborhood of 127, we can compute
multiplications by 120, 124, 126, 127, 128, 129, 130, 132, and 136
with a single addition or subtraction plus some bit shifts.

Suppose we can manage two additions or subtractions, not just one.
Then we can, for example, multiply by 27 with two additions:

    x += x << 1;  // multiply by 3
    x += x << 3;  // multiply by 9

Multiplying by 27 in the usual way would have required three
additions: x + (x << 1) + (x << 4) + (x << 5).

This approach gives us 1052 separate multipliers with bit shifts of up to 8:

    >>> singles = set(x for a in range(8)
                        for b in range(8)
                        for x in [(1 << a) + (1 << b),
                                  (1 << a) - (1 << b)])
    >>> len(numpy.array(sorted(set(a*b for a in singles for b in singles))))
    1052

Additionally, though, we can add or subtract the original number,
possibly with a shift.  So, for example, to multiply by 59, although
its Hamming weight is 5, we can calculate (x << 6) - (x << 2) - x.
Combining this approach with the previous one gives us 1366
multipliers with bit shifts of up to 8:

    >>> len(sorted(set(x for a in singles
                         for b in singles
                         for x in [a*b, a+b, a-b])))
    1366

Allowing larger bit shifts actually extends our range further; with
shifts of up to 16, we can multiply by any constant integer factor in
(-256, 256) with two shifted additions or subtractions except for the
following handful:

    >>> singles = set(x for a in range(16)
                        for b in range(16)
                        for x in [(1 << a) + (1 << b),
                                  (1 << a) - (1 << b)])
    >>> doubles = set(x for a in singles
                        for b in singles
                        for x in [a*b, a+b, a-b])
    >>> numpy.array([x for x in range(-255, 256) if x not in doubles])
    array([-213, -211, -205, -203, -181, -179, -173, -171, -170, -169, -165,
           -149,  -85,  171,  173,  179,  181,  203,  205,  211,  213])

This implies that, given an appropriate constant scale factor, we can
always do an approximate multiplication with two shifted additions and
subtractions while introducing an error of one part in 2·171 = 342 or
less.  In fact, more than three-quarters of the multipliers in (-1024,
1024) are reachable:

    >>> len([x for x in range(-1023, 1024) if x not in doubles])
    499
    >>> len(doubles)
    24418

This is especially beneficial for higher-precision operands — for
example, 16-bit or 32-bit operands, for which the O(N²) bit operations
of a full-precision multiply could be prohibitive.

In cases like the Minsky algorithm and the Goertzel algorithm where
constant multiplication is being used to rotate a phasor progressively
over time by a constant angle, it may be reasonable to “dither” the angle
by alternating between two different inexpensive rotations; this
introduces phase noise or jitter, but when it amounts to a small
fraction of a cycle, it shouldn’t make much difference.

The extended zero-crossing approach
-----------------------------------

A problem shared by the Minsky algorithm, the Goertzel algorithm, and
the shower algorithm is that they are all ways to calculate or
approximate a component of the Fourier transform or the short-time
Fourier transform (which I think is the best any linear algorithm can
do), and as a result they are fundamentally limited by the same
uncertainty principle: to distinguish between, in my example, 110 Hz
and 106 Hz, they need at least 250 ms of data, regardless of the
sampling rate.

I think, however, that phase-locked loops and zero-crossing detection
can do better.  In the absence of noise, or with relatively low noise,
they can provide a very accurate measurement of frequency with very
little data; in the extreme, by measuring the time of a single
half-cycle.

However, even that is poor performance; in theory, in the absence of
noise, we need only three sequential samples of the sine wave!  Any
three samples, as long as they’re much less than half a wavelength
apart!  Their first differences give us two slopes, and their second
difference gives us the second derivative; the ratio between this
second derivative and the central value gives us the negated squared
angular frequency.  (Roughly; hmm, I should work out what it is for
real, because especially for high frequencies this is only
approximate.)

The problem is that the phase-locked loop with the square-wave
edge-detector that I showed is only getting information from the
samples immediately next to the zero crossing — as is the
zero-crossing detector.  But usually there is lots of useful
information further away from the zero crossing, too.  We should be
able to take advantage of that information to more precisely estimate
the phase of the wave at each point in time.

The phase of a pure sine wave of unity amplitude and angular frequency
is just atan2(x, dx/dt); for angular frequency ω and an arbitrary
amplitude, it is atan2(x, dx/dt/ω), as the amplitude is a common
factor of both arguments and thus cancels.

We don’t really need to know the precise derivative to uniquely
identify a part of the cycle, though.  We only need to know the value
at that sample and whether the derivative is positive or negative.
(Alternatively, we don’t really need to know the precise value; we
only need the derivative and whether the sample is positive or
negative.)

If we count the time interval since we last passed the current phase
angle, this should give us some kind of guess at the period of the
wave — potentially a new guess on every sample!  If we accumulate
these guesses over some period of time, we can take the median of the
accumulated guesses (rather than, say, the mean) as the period of our
wave.

This is like zero-crossing time measurement, but instead of measuring
time between crossings of just the X-axis, we’re counting the time
between the crossings of every single phase.  Some degree of
hysteresis is appropriate, but now the hysteresis threshold can be
more or less an angle rather than an amplitude.

If we’re concerned about computation time, though, we probably don’t
actually want to calculate the arctangent precisely, even if that were
possible for an unknown-frequency signal.  Instead we would like to
lump each sample into some kind of angle bin based on something that’s
cheap to compute about that sample.

For example, we could draw a square around the origin, with the sides
meeting where x[n] == x[n] - x[n-1] and where x[n] == x[n-1] - x[n].
On the right side of the origin, crossing the real axis, we have a
side where x[n] > 0; on the left side of the origin, crossing the real
axis, we have a side where x[n] < 0; on the top side, crossing the
imaginary axis, we have a side where x[n] - x[n-1] > 0; on the bottom
side, crossing the imaginary axis, we have a side where x[n] - x[n-1]
< 0.  Which side we put a given sample on depends on which of these is
greatest, subject to a somewhat arbitrary scaling decision that will
double performance for frequencies around some optimal frequency.  So
if we’re on the right side, that means that x[n] is higher than all of
0, x[n] - x[n-1], and x[n-1] - x[n]; if we’re on the left side, it
means it’s lower than all three; if we’re on the top side, x[n] -
x[n-1] is higher than 0 and higher than either x[n] or -x[n]; if
we’re on the bottom side, it’s lower than all three.

More on phase detection and frequency suppression
-------------------------------------------------

Frequecy detection is sort of intimately interwoven with frequency
suppression.  Consider the following example.

You want to suppress a 50Hz interfering signal introduced into
electrical measurements by a nearby 50Hz fluorescent tube.  The
waveform is periodic at 50Hz, and indeed symmetric, but very far from
sinusoidal.

The simplest approach is simply to apply a feedforward comb filter: by
adding the signal to itself as it appeared 10ms ago, you will
completely suppress the noise from the fluorescent tube, because that
comb filter has nulls at 50Hz, 150Hz, 250Hz, etc.  But it does some
violence to the remaining signal, since each impulse in the signal now
appears twice, 10ms apart, thus smearing things out in the time domain
and adding 6dB to components of 0Hz, 100Hz, 200Hz, etc.  And, of
course, if the lamp turns on or off suddenly, you’ll have half a cycle
of bleedthrough at the end, which is attenuated but not suppressed if
the turnoff is gradual.

Within the linear-filtering paradigm, you can trade some of these
undesirable characteristics against one another by using more than one
previous sample.  For example, instead of merely adding the signal
10ms ago, you can add half the signal 10ms ago and subtract half the
signal 20ms ago.  This results in further temporal smearing of
whatever gets through, but the echo signals are now 6dB quieter.  If
we carry this further and use the average of ten samples (the
corresponding points in the last ten half-cycles, half negated), the
added echo is now 20dB down.

However, I think we can do better with some nonlinearity.  For
example, if we use median filtering rather than mean filtering over
the corresponding points in the last ten half-cycles, random impulses
will not be echoed at all.  Or you can use a hybrid: instead of purely
the median, use the mean of the six medial values, discarding the two
highest and two lowest values as outliers; or use a weighted mean with
weights not restricted to 0 and 1.  And we could extrapolate an
expected amplitude for the waveform to suppress, allowing us to
completely suppress sufficiently gradual turn-ons and turn-offs.

Human-voice sounds are periodic, entirely asymmetric, and also very
far from sinusoidal due to their laryngetic origin as impulse trains.
It’s common for the second harmonic to be even stronger than the
fundamental!  For such signals we couldn’t subtract negative
half-cycles; we’d have to use the corresponding points in the last ten
cycles, instead.  And, since they’re not very stable in frequency,
we’d need to extrapolate frequency shifts as well.

How do you detect the frequency shifts, though?  You can look to see
if the waveform is to the left or the right of the expected waveform,
but of course it’s a question of how far to the left or right you’re
looking, which perhaps you can reformulate as a question about how to
approximate its derivative.  Or you could do a full cross-correlation
between signals.

One alternative I’ve been thinking about is to use the signal and a
high-pass-filtered version of its *integral* to do phase detection.
That way, you don’t have the extreme amplification of noise that
derivatives get you.

If we can have a prophecy budget, we can use subsequent cycles of the
signal as well as previous cycles to estimate the current cycle.