Background

A while ago I was doing some profiling to find out ways to improve the load performance of my Open Diapason project. The load process is roughly as follows:

• Load wave file from disk into memory.
• Apply a linear phase filter to all loaded samples (this is to compensate for the response of the resampler which occurs during playback).
• Normalise and quantise the filtered audio using TPDF dither. The normalisation ensures that we use the full 16-bits of precision for each sample. Obviously, the scaling coefficient is stored along with the sample to ensure we play the content back at the intended level.

All of these steps are done across multiple threads to maximise performance (there is one thread that is reading audio files from disk ready to spawn threads which apply the filter and requantise the data)… this is one of those times where the speedup was roughly linear with the first couple of threads which I added.

Anyway, back to the profiling: I was expecting that the vast majority of the cycles would be spent in the application of the filter, but this actually turned out not to be the case. The dithering and re-quantisation of the audio – which I assumed would be cheap – turned out to be chewing cycles on the order of 30% of the load time (all of my profiling was done on a mid 2014 MacBook Pro with an i7 processor).

There were multiple things which I did to speed it up, but I think what I pushed in this change list was the most interesting one – so what follows in this blog is a bit of a write up.

The idea

Uniform random numbers are useful and often we need several of them. In my use-case (performing TPDF dither on a sample pair) I needed four random variables. Generating them needs to be fast but the “quality” of the randomness is not that important. The obvious way to do this is to use a Linear Congruential Generator which ends up forming a code pattern like this:

r0 = (state * A + C) % M;
r1 = (r0 * A + C) % M;
r2 = (r1 * A + C) % M;
r3 = (r2 * A + C) % M;
state = r3;

This forms a dependency chain where each random number requires knowledge of the previous one and could end up leading to pipeline issues (this is really dependent on how the compiler re-orders the code and the architecture of the platform). In the code generated on my machine, this turned out to be a bottleneck. We could get around the pipeline issues by creating four different LCGs:

r0 = (state0 * A0 + C0) % M;
r1 = (state1 * A1 + C1) % M;
r2 = (state2 * A2 + C2) % M;
r3 = (state3 * A3 + C3) % M;
state0 = r0;
state1 = r1;
state2 = r2;
state3 = r3;

But this will require more registers to store the different LCG states and possibly more registers or code memory to hold all the new coefficients. It would be nice to have something a bit simpler…

LCGs are really cool toys. We can choose our and values in such a way that the pseudo-random sequence has period . Given the algorithm generates the next value from the previous, this must mean that the random sequence contains every integer between 0 and . The first part of the idea, is to modify the code as:

r0 = (state0 * A0) % M;
r1 = (state1 * A1) % M;
r2 = (state2 * A2) % M;
r3 = (state3 * A3) % M;
state0 = (r0 + C0 % M) % M;
state1 = (r1 + C1 % M) % M;
state2 = (r2 + C2 % M) % M;
state3 = (r3 + C3 % M) % M;

This produces a different sequence in to because the accumulation now happens at the state assignment, but still has a period . Now for the dodgy part, we change the algorithm to this:

r0 = (state0 * A0) % M;
r1 = (state0 * A1) % M;
r2 = (state0 * A2) % M;
r3 = (state0 * A3) % M;
state0 = (r0 + C0 % M) % M;

Where , and to are all different but satisfy the required conditions for an LCG to work. It should not be difficult to infer that to still all have period – but produce different output sequences. I wrote a test program to look at the correlation of these sequences over their period which can be found on GitHub here – they are mostly independent over their period. I doubt they make great random numbers, but for performing dither, they are completely fine. The generated code seems pretty good also:

sampletest[0x10000a99f] <+3503>: imull  $0xd688014d, %r8d, %edi   ; imm = 0xD688014D 
sampletest[0x10000a9a6] <+3510>: imull  $0xdb71f7bd, %r8d, %r15d  ; imm = 0xDB71F7BD 
sampletest[0x10000a9ad] <+3517>: imull  $0xe05f354d, %r8d, %esi   ; imm = 0xE05F354D 
sampletest[0x10000a9b4] <+3524>: imull  $0xe54c7f35, %r8d, %ecx   ; imm = 0xE54C7F35 
sampletest[0x10000a9bb] <+3531>: movss  (%r11,%rdx,2), %xmm2      ; xmm2 = mem[0],zero,zero,zero 
sampletest[0x10000a9c1] <+3537>: mulss  %xmm1, %xmm2
sampletest[0x10000a9c5] <+3541>: movss  (%r10,%rdx,2), %xmm3      ; xmm3 = mem[0],zero,zero,zero 
sampletest[0x10000a9cb] <+3547>: mulss  %xmm1, %xmm3
sampletest[0x10000a9cf] <+3551>: leal   0x1(%rdi), %r8d
sampletest[0x10000a9d3] <+3555>: cvttss2si %xmm2, %r12
sampletest[0x10000a9d8] <+3560>: cvttss2si %xmm3, %r13
sampletest[0x10000a9dd] <+3565>: addq   %rdi, %rsi
sampletest[0x10000a9e0] <+3568>: addq   %r12, %rsi
sampletest[0x10000a9e3] <+3571>: addq   %r15, %rcx
sampletest[0x10000a9e6] <+3574>: addq   %r13, %rcx
sampletest[0x10000a9e9] <+3577>: shrq   $0x21, %rsi
sampletest[0x10000a9ed] <+3581>: shrq   $0x21, %rcx
sampletest[0x10000a9f1] <+3585>: movl   %edx, %edi
sampletest[0x10000a9f3] <+3587>: movw   %si, (%rax,%rdi,2)
sampletest[0x10000a9f7] <+3591>: leal   0x1(%rdx), %esi
sampletest[0x10000a9fa] <+3594>: movw   %cx, (%rax,%rsi,2)
sampletest[0x10000a9fe] <+3598>: addq   $0x2, %rdx
sampletest[0x10000aa02] <+3602>: decl   %ebx
sampletest[0x10000aa04] <+3604>: jne    0x10000a99f

See the code in the original commit for a better understanding of what’s going on. There is a hefty comment explaining the fixed-point dithering process.

When writing eigen solvers, it’s useful to be able to generate an orthonormal basis containing a particular vector. Another way of saying this is: given some fixed vector, find other vectors that are perpendicular to it and each other. I am under the impression from some websites that this is also useful in computer graphics (but I know nothing about that – let me know if you do!).

I’ve seen people ask how to do this on the internet (for example: here and here) and in general the recommended process is:

• If only one perpendicular vector is required, use a heuristic to come up with something perpendicular (see previous links).
• If two perpendicular vectors are required, create one perpendicular vector using the above heuristic then use the cross-product to come up with the final vector.
• For more than two perpendicular vectors, generate random vectors and use Gram Schmidt to (hopefully) orthogonalise them.
• For 3D graphics, see the “Update” heading below…

The first two usually require branching in the heuristic. The second may not work (if you’re really really unlucky) and requires multiple normalisation operations. Follows is an approach which I haven’t seen documented elsewhere which requires one square root for an arbitrary dimension vector set.

What we want is some technique which creates an orthogonal matrix from a vector. This is exactly what the Householder transformation matrix does:

Where is some unit vector. The matrix is hermitian and unitary and is hence orthonormal. If we can find a simple expression for which will force the first column of to be equal to some unit vector , we have all the information we need to create a basis containing . This turns out to be trivial:

Explicitly, this looks like:

If is real (i.e. ), we set:

Here is a very inefficient C implementation (the square root is not actually necessary – this can be seen in the fact that two entries are always multiplied together and the term in the square root is constant – also the most inner loop over j should start from 1 because when j==0, the value is simply ):

/* The first n values in m is a normalised vector which is desired to be
 * part of the basis set. Following this vector, n-1 new vectors of length n
 * will be added which are orthogonal to it. */
void house_ortho(double *m, unsigned n)
{
    const double x = sqrt(1.0 - m[0]);
    /* As m[0] approaches 1, x approaches 0. This detects that and will end
     * up forcing the result to be the identity matrix. */
    const double s = (x < 1e-20) ? 0.0 : (-1.0 / x);
    unsigned i, j;
    /* We first convert the m vector into the v vector. Note that we have
     * incorporated the multiply by 2 present in the Householder operation
     * into v by multiplying every element by sqrt(2). */
    for (m[0] = x, i = 1; i < n; i++)
        m[i] *= s;
    /* Create all of the orthogonal vectors using the definition of the
     * Householder transformation matrix: I - 2 v v*. */
    for (i = 1; i < n; i++) {
        for (j = 0; j < n; j++) {
            m[i*n+j] = ((i == j) ? 1.0 : 0.0) - m[i] * m[j];
        }
    }
    /* Convert the v vector back into m. Probably could have written this
     * code in a way where this didn't need to happen. */
    for (m[0] = 1.0 - x * x, j = 1; j < n; j++)
        m[j] = -m[j] * x;
}

The termination condition for the for loop (over i) in the middle of the code sample could be modified to generate a smaller number of vectors.

This process could be modified to work with a complex valued but is slightly more complicated because is hermitian (which implies that all values on the diagonal are real). The modification amounts to finding a unit vector which forces to be real, multiplying by this unit vector then dividing all resultant vectors by the factor. I haven’t tested this, but it should work.

Update

I posted a link to this blog on Twitter and reached out to see if anyone else had used this method. Warren Moore helpfully responded and pointed me at this paper “Building an Orthonormal Basis, Revisited” (Pixar) which describes numerical and implementation improvements to an algorithm proposed in this paper “Building an Orthonormal Basis from a 3D Unit Vector Without Normalization” (Frisvad, J. R.).

The algorithm which I’ve described in terms of finding the vector which creates the Householder matrix with the first column equal to some unit vector , ends up producing exactly the same equations (given a vector of dimension 3) given in Frisvad’s paper up to a few sign changes – albeit I’ve gone about defining the problem in a slightly different way. We can demonstrate the equivalence by expanding out the resulting householder matrix given the vector we derived earlier:

Note: we don’t actually need because the householder matrix is symmetric. The matrix can be expressed purely in terms of as:

The Pixar paper addresses the singularity which occurs as tends towards 1. We can do the same thing here by negating the definition of the Householder matrix which still produces a matrix with the same properties. i.e.

In this case, the vector is now defined as:

Which moves the singularity to occur as tends towards -1. The matrix still looks very similar:

Efficient implementation of 3-dimension basis finder

As shown in the Pixar paper, there is no need to test the sign of and pick a code path to follow (which would create a branch prediction performance hit) – we can simply extract the sign and use it as a multiplier to flip signs where they need to be flipped (if they need to be flipped).

Modifying some signs in the explicit matrices given previously, we can re-write the matrices as:

Or more simply:

Where:

And here’s the C implementation:

/* The first three values in v are a three point unit vector. The
 * function produces two more three point vectors of the same coordinate
 * space which are orthogonal. */
void ortho3(float *v)
{
    float q1        = v[0];
    float q2        = v[1];
    float q3        = v[2];
    float sign      = copysignf(1.0f, q1);
    float scale     = 1.0f / (q1 + sign);
    float q2scale   = q2 * scale;
    float q3scale   = q3 * scale;
    float q3q2scale = q2scale * q3;
    v[3]            = q2;
    v[4]            = q2 * q2scale - sign;
    v[5]            = q3q2scale;
    v[6]            = q3;
    v[7]            = q3q2scale;
    v[8]            = q3 * q3scale - sign;
}

This guide omits a fair amount of detail in order to provide a fairly quick guide.

1) Design in the s-domain

Butterworth filters have their poles evenly spaced on a circlular arc in the left hand plane of the [s/Laplace/analogue]-domain.

For a low-pass filter of order , the complex valued pole locations are:

Where and is the cutoff frequency in radians.

The transfer function in the s domain is:

Because the poles always occur in conjugate pairs for this kind of filter, it can always be expanded out into a polynomial with real-valued coefficients :

2) Convert the s-domain transfer function to the z-domain.

There are a few methods to convert an s-domain transfer function this into a [z/digital/discrete]-domain transfer function. The most common is the bilinear transform which makes the substitution:

Where:

Or if you’re performing “frequency warping”:

I won’t go into verbose details here, but you probably want frequency warping. The bilinear transform is a linear approximation to the function which maps the s-domain to the digital z-domain. This linear mapping is analogous to the first term of a series expansion of which is most accurate around z=1 (i.e. DC). The frequency warping operation moves the point where most accuracy is preserved to . For higher frequency cutoffs, this becomes important. can be set to the cutoff frequency of the filter to improve the frequency response near the transition.

For a 3-rd order system, the conversion yields (see notes at the end of this page about risks):

Which can be expanded into a rational polynomial transfer function in .

3) Convert from the z-domain to a difference equation.

First, you must make your z-domain expression causal. Positive powers of must be removed otherwise your filter would need to know the future! This is simple, just modify the numerator and denominator of the transfer function by the inverse of the highest power of . For example:

Multiply the numerator and denominator by .

Get as a function of .

The difference equation can be found by substituting z-powers as offsets to the data they are using. i.e. for the above case:

That is what you want to implement in your code.

Things to keep in mind

High-order systems can start behaving very poorly when designed in this way. Even on floating point, once a system reaches 4-th order or higher, if there are poles close to the real axis in the z-domain, the output can become severely degraded. There are ways to mitigate this: the simplest of which is to break the implementation into separate 2nd order filters (biquads).

This is a combination of information from the following two locations:

• Setting up CardDAV and CalDAV on Windows Phone 8.1
• Windows Phone 8: Solution how to Sync Calendar and Contacts

What is this?

This page gives an outline of how to get your Windows Phone to sync calendars and contacts with an ownCloud instance running on a server using a self-signed certificate. This whole process is a hack and I’m incredibly disappointed that Windows Phone does not support this natively – especially since they have a CalDAV and CardDAV implementation already available (used by iCloud and Google accounts). If this process stops working at some point, that should be expected.

I successfully got this working on a Lumia 735.

Process

I’m assuming that you’ve got ownCloud installed on a server using SSL. When the certificate was set up, the FQDN must really be the FDQN. i.e. if your ownCloud instance is hosted at “a.b.com”, the certificate must be for “a.b.com” – not “b.com”. I had this wrong on my home server and was getting the “80072F0D” error on the phone. If you have not yet set up ownCloud or have not set up SSL for your server yet, there are sites already documenting this process How to Create and Install an Apache Self Signed Certificate.

You then need to get the certificate installed on your phone. If you do not do this, you will get errors when the phone tries to start syncing. You can do this by opening the certificate in Internet Explorer on the phone (you can do this easily by copying the certificate to your ownCloud files, logging into your cloud in IE on the phone and opening the file.

You should then be able to follow the instructions here: Setting up CardDAV and CalDAV on Windows Phone 8.1.

Which roughly reads:

1. Create a fake iCloud account. Put garbage information in the fields and create it. The phone won’t check anything.
2. Modify the account, edit the advanced settings then change the servers for CalDAV and CardDAV (respectively) as:
   [domain]/remote.php/caldav/principals/[username]
   [domain]/remote.php/carddav/principals/[username]