On the perils of cross-fading loops in organ samples

One common strategy when looping problematic organ samples is to employ a cross-fade. This is an irreversible audio modification that gradually transitions the samples leading up to the end of a loop to equal the samples that were leading into the start. The goal is to completely eliminate any sort of impulsive glitch and hopefully also create a “good spectral match”. While the ability to eliminate glitches is possible, creating a good spectral match might not be so simple.

We can think of an organ sample as being a linear combination of two signals:

• A voiced/correlated/predictable component (choose your word) which represents the tonal part of the sample. Strictly speaking, the tonal component is not entirely predictable – there are continuous subtle variations in the speech of a pipe… but they are typically small and we will assume predictability.
• An unvoiced/uncorrelated/unpredictable component which represents the pipe-noise of the sample.

Both of these components are necessary for realism in a sample.

The following image shows a cross-fade of two entirely correlated signals. The top picture contains the input signals and the cross-fade transition that will be used, the bottom contains the input signals with the cross-faded signal overlaid on top. There is nothing interesting about this picture: the two input signals were the same. The cross-fade transitions between two identical signals i.e. the output signal is equal to both of the input signals. Crossfade of correlated signal

The next image shows a cross-fade of the same correlated signal with some uniform white noise overlaid on top. What is going on in the middle of output signal? It looks like it’s been attenuated a little bit and doesn’t appear to have a uniform distribution anymore. Crossfade of correlated signal with uniform noise

This final image is a cross-fade purely consisting of two uniform random signals to add some more detail. Crossfade of uniform noise

It turns out that summing two uniformly distributed random variables yields a new random variable with a triangular distribution (this is how noise for triangular dither gets generated). During the cross-fade, the distribution of the uncorrelated components of the signal is actually changed. Not only that, but the power of the signal is reduced in the middle of the transition. Here is an audio sample of two white noise streams being cross-faded over 2 seconds:

Listen for the level drop in the middle.

If the cross-fade is too long when looping a sample, it could make the pipe wind-noise duck over the duration of the cross-fade. While the effect is subtle, I have heard it in some samples and it’s not particularly natural. I suppose the TLDR advice I can give is:

• If you can avoid cross-fading – avoid it.
• If you cannot avoid cross-fading – make the cross-fade as short as possible (milliseconds) to avoid an obvious level-drop of noise during the transition.

Arbitrary polynomial-segment signal generation (e.g. square, triangle, sawtooth waves)

There is a tool which is part of my Open Diapason project called “sampletune” which is designed to allow sample-set creators to tune samples by ear and save the tuning information back into the sampler chunk of the wave file. The utility plays back a fixed tone and there are keyboard controls to shift the pitch of the sample up and down until the beats disappear. I was hoping to prototype a sawtooth wave as the tuning signal, but realised during implementation that creating arbitrary frequency piecewise-linear waveforms is actually somewhat a difficult problem – fundamentally due to aliasing… so I parked that idea and ended up summing several harmonic sinusoids at some arbitrary levels.

Recently, I was thinking about this again and set about solving the problem. It was only in the last couple of weeks that I discovered tools like Band Limited Impulse Trains (see link at bottom of post) which seem to be the common way of generating these signals. These are fundamentally FIR techniques. I approached the problem using IIR filters and came up with a solution which I think is kind-of interesting – please let me know if you’ve seen it done this way so I can make some references!

The idea

If we generate a piecewise linear waveform wave at a frequency which is not a factor of the sampling rate, we will get aliasing. In fact, we can work out the number of shifted aliasing spectrum copies which we can expect to see by taking . If is irrational, we have an infinite number of copies. Let’s make the assumption that more-often than not, we are going to be generating a very large number of aliases. The next assumption we are going to make is that most of the signals we are interested in generating have a spectral envelope which more-or-less decays monotonically. With this assumption, we can say: if we generate our signal at a much higher sampling rate, we effectively push the level of the aliasing down in the section of the spectrum we care about (i.e. the audible part). Below is an image which hopefully illustrates this; the upper image shows the spectral envelope of the signal we are trying to create in blue, all of the black lines which follow are envelopes of the aliasing. The second image shows what would have happened if we created the same frequency signal at three times the sampling rate. The interesting content again is in blue and everything in black is either aliasing or components we don’t care about. What we can see is that by increasing the sample rate we generate our content at, we effectively push the aliasing components down in the frequency region we care about. So let’s assume that instead of generating our waveforms at , we will:

• Generate them at a sample rate of where is some large number.
• Filter out the high-frequency components above using some form of IIR filter structure.
• Decimate the output sequence by a factor of to get back to (which we can do because of the previous operation).

Now that we’ve postulated a method, let’s figure out if it makes any sense by answering the questions “does it have an efficient implementation?” and “does it actually produce reasonable output for typical signals we may want to generate?”.

Does it have an efficient implementation?

Denote a state-space representation of our SISO discrete system as: For clarity, let’s just consider a single update of the system state where represents the “previous state” and represents the first sample of new data we are pushing in. We can derive an expression to insert two samples in via substitution and from that derive expressions for inserting groups of samples.  We’re interested in inserting groups of samples which form linear segments into our filter. So let’s replace with some arbitrary polynomial. This gives us some flexibility; we need only a polynomial of zero degree (constant) for square waves, first degree for saw-tooth and triangle – but we could come up with more interesting waveforms which are defined by higher order polynomials. i.e. Let’s substitute this into our state expressions:  There are some interesting things to note here:

• The first output sample ( ) is only dependent on and the previous state. If we restrict our attention to always getting the value of the first sample, we don’t need to do any matrix operations to get an output sample – just vector operations.
• All of the parenthesised bits in the equations have no dependency on the coefficients of the polynomial we are inserting into the filter – they only depend on the number of samples we want to insert ( ), the system state matrices and/or the index of the polynomial coefficient. This means that we can build lookup tables for the parenthesised matrices/vectors that are indexed by and the polynomial coefficient index and effectively insert a large number of samples of any polynomial into the filter state with a single matrix-matrix multiply and a number of scalar-vector multiplies dependent on the order of the polynomial.

It’s looking like we have an efficient implementation – particularly if we are OK with allowing a tiny (almost one sample at ) amount of latency into the system.

Does it work?

Below is a screen capture of Adobe Audition showing the output of a small program which implemented the above scheme. The system is configured using an elliptic filter with 1 dB of passband ripple and a stop-band gain of -60 dB. Waveform with frequency of 600*pi Hz. Left channel generated using naive modulo, right generated using the proposed scheme. The spectrum of the left channel is green.

The waveform has a frequency of which is irrational and therefore has an infinite number of aliases when sampled (this is not strictly true in a digital implementation – but it’s still pretty bad). We can see that after the cutoff frequency of the filter, the aliasing components are forced to decay at the rate of the filter – but we predicted this should happen.

Below is another example using an 4th order Bessel filter instead of an elliptic filter. This preserves the shape of the time-domain signal much more closely than the elliptic system, but rolls off much slower. But the performance is still fairly good and might be useful. Waveform with frequency of 600*pi Hz. Left channel generated using naive modulo, right generated using the proposed scheme. The spectrum of the left channel is green.

Here is a link to a good paper describing how classic waveforms have been done previously “Alias-Free Digital Synthesis of Classic Analog Waveforms” [
T. Stilson and J. Smith, 1996]
.

Generating multiple low-quality random numbers… fast.

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,zero,zero,zero
sampletest[0x10000a9c1] <+3537>: mulss  %xmm1, %xmm2
sampletest[0x10000a9c5] <+3541>: movss  (%r10,%rdx,2), %xmm3      ; xmm3 = mem,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.

Given a vector: find N orthogonal vectors.

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); /* As m 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 = 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 = 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; float q2 = v; float q3 = v; 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 = q2; v = q2 * q2scale - sign; v = q3q2scale; v = q3; v = q3q2scale; v = q3 * q3scale - sign; }