minimum-phase filter kernels and partitioned convolution
Previous pages described parametric Fourier filters which can do radical and precise filtering, but which suffer from considerable latency, due to FFT length (25 - 50 ms typically). As it turns out, it is possible to reduce that latency to a few samples, using a combination of two techniques, both invented decades ago. The first one, homomorphic filtering, reorganizes the filterkernel, and the second, partitioned convolution, breaks up fast convolution in small blocks.
zero-phase filter kernel
When a filter kernel (impulse response) is created by the inverse
Fourier transform of a magnitude spectrum, that filter kernel is
symmetric by definition: a zero-phase filter. The above plot shows an
example. The anticausal coefficients introduce latency. The longer
filter, the more latency. Radical filters are long by definition.
is much harder to create a purely causal filter kernel from a given
spectrum. The causal filter is not simply the (scaled) right
half of the equivalent zero-phase
In the article 'Homomorphic
Prediction' by Alan Oppenheim &c. (1975), the following
description is given: signal
be considered the convolution of it's minimum- and maximum-phase
components. And in four simple equations, the
whole theory of 'homomorphic processing' is outlined. With help of this
developed a minimum-phase FIR generator in Pure Data, step by step. The
convolution of minimum- and maximum-phase
components, making up the zero-phase signal x[n], can be written like
If you manage to separate these components of a filter kernel x[n],
you have an anticausal and causal filter. But how can you separate two
things which are convolved, if you don't at least know one of the
originals? Seems impossible.
Now, think of this: convolution in signal domain is complex
multiplication in frequency domain. It happens to be the case that
positive and negative frequencies live separated here. So, couldn't we
perform complex division to get
min-phase and max-phase components? Ehhm.... that doesn't work.
The imaginary parts of the zero-phase spectrum all have value zero. To
split a zero-phase spectrum in minimum- and maximum-phase components,
is like splitting matter and antimatter out of the void.
Next step is to express the magnitude spectrum
from the phase spectrum. On the z-plane, where the spectrum is
initially expressed, this is not the case. The z-plane has rectangular
coordinates, cosine and sine components, which can be recalculated into
complex exponentials, 'polar coordinates'.
expressed as radius or amplitude, and the phase aspect as arc length on
circle. That is not orthogonal, as is illustrated below:
The logarithms of the complex exponentials must be taken, which transforms the z-plane to an s-plane by unwinding the unit circle.
On the s-plane, the natural logarithm of the amplitude is on the x-axis, and the natural logarithm of the z-plane's eiy, being the phase, is on the y-axis. Since ln(0) = -inf, the amplitudes' lower limit must be restricted to some small value, subliminal in the sense of audio perception. The number, regularly called 'epsilon', could have value 1/million for example. The natural logarithm of 1/million is about -13.81551. So radius lengths between 1/million and the unit circle would then translate to logarithmic magnitude values between -13.81551 and zero.
On the s-plane, minimum- and maximum-phase components are no longer
a product, but a
sum, because the complex logarithms of all data were taken. So the
phase-separation problem is
simplified from deconvolution to division
to subtraction. But we are still in frequency domain, where the
opposite phase components have cancelled each other, with no way to
split them into the constituent parts.
logarithms of amplitudes
On to the next domain. The spectral data will now be transformed to
'cepstrum', by applying
inverse Fourier transform. The fantasy term 'cepstrum' is an indication
of the domain's abstractness. What data is on the x and iy axes there?
For the purpose under concern, I find it useful to think of the
cepstrum as a signal, reorganized according to the geometry of the
s-plane instead of the z-plane. If the original signal was pure real,
the spectrum was symmetric, and the cepstral data is again pure real.
Also, the cepstral coefficients of a zero-phase spectrum
concentrate symmetrically around index zero, like it is the case for
It seems to me that the cepstrum represents information about the
signal in a negated form: components not or hardly present in the
signal will have most energy in the cepstrum. After all, a white
spectrum, if normalized, has all magnitudes zero on the s-plane so
there would be nothing in the cepstrum. Hmm, weird idea.
In cepstral domain, we can finally split the now reorganized signal
into it's phase components. All the maximum-phase coefficients are on
the left and all the minimum-phase components are on the right. In the
center is a point where they overlap, corresponding to the identity
index x in the zero-phase signal. The value in this point belongs
half to the maximum-phase and half to the minimum-phase part. Here is
how the cepstrum
coefficients belonging to the earlier plotted FIR are separated:
sum of cepstrum coefficients
maximum-phase cepstrum coefficients
minimum-phase cepstrum coefficients
In order to get a minimum-phase IR with the same magnitude as
before, the minimum-phase cepstrum coefficients are multiplied by two:
minimum-phase cepstrum coefficients multiplied by 2
The result of this action is a single-sided cepstrum. When this single-sided cepstrum is Fourier-transformed back to the s-plane, the s-plane coordinates will no longer be pure real. It is now a minimum-phase spectrum, with real and imaginary coefficients. The sequence of imaginary coefficients is orthogonal to the sequence of real coefficients. Much like an 'analytic signal', which is the transform of a single-sided spectrum.
logarithms of amplitudes
From the s-plane, it's now back to z-plane and from there to signal
domain, where the IR appears as a minimum-phase filter. It is a causal
sequence. For comparison, the zero-phase and minimum-phase equivalents
are shown together below. They both have the same magnitude spectrum.
zero-phase filter kernel
minimum-phase filter kernel
When FIR filter kernels are created directly from an arbitrary
magnitude spectrum, there is a risk that the spectrum sequence will
contain sharp edges. In a signal, sharp edges introduce alias
frequencies, they simply should not be there. In a spectrum, they
should not be there either because they introduce alias phases in the
signal. The easiest way to avoid them is to lo-pass-filter the
magnitude spectrum by default. This erases any possible sharp edge,
just to be sure. Alternatively,
the filter kernel
could be windowed in signal domain.
For the purpose of this discussion I'll use the Hann window as the
example. More sophisticated windows exist, but the Hann window is
it's Fourier transform has only three coefficients. Multiplication with
the Hann window in one domain is a three-point convolution in the other
Hann window Fourier transform
Even if you would accidentally define a brickwall spectrum, the
lo-pass filtering will remove the sharp edges.
For a zero-phase filter, it works straightforwardly. Sidelobes are
reduced to an acceptable level. Compare the two frequency response
plots below, for the case of an eight bin wide brickwall spectrum and
it's lo-pass-filtered version:
measured frequency response for brickwall spectrum
measured frequency response for lo-pass-filtered brickwall spectrum
Now comes the bad news. When the lo-pass-filtered brickwall is
recalculated into logarithms of amplitudes, the magnitude spectrum
appears on the s-plane with discontinuities again:
If this logarithmic spectrum is transformed to cepstrum rightaway,
there will still be alias phases in the cepstrum. In the sense of
lo-pass filtering, the z-plane and s-plane data are simply
incompatible! And look how the 'lo-pass-filtered' minimum-phase
brickwall performs in reality:
measured frequency response for lo-pass-filtered minimum-phase brickwall spectrum
In official methods, the problem with s-plane aliases is solved by upsampling the spectrum. Upsampling makes the spectrum smoother, and sharp edges disappear without data loss. It is 'most efficiently' done by zero-padding the filter kernel in time domain, then use very large FFT's to go to spectrum and cepstrum. The FFT sizes needed for this are discourageing. Is it possible to compute how many times the IR must be upsampled, in order to erase sharp edges in an s-plane spectrum? It depends on epsilon, for one thing. If epsilon is 1/million and it's natural log is -13.81551, the neighbour of epsilon must be exp(0.75 * -13.31551). Oops, that is about 0.00005. In the lo-pass-filtered brickwall spectrum, the neighbour of epsilon is 0.25. It is indeed not uncommon to zero-pad to a thousand times the IR length. For my 2048 pt Fourier filters, that would mean two-megapoint FFT's, to find a minimum-phase filter! Completely ridiculous and out of the question, for a realtime performance application.
Instead, I decided to just window the
cepstrum. This is equivalent to lo-pass filtering of the logaritmic
magnitudes. Of course, this is utterly incorrect from a theoretical
perspective. After all the operations, the doubly-filtered
brickwall amplitudes re-appear like this on the z-plane:
But hey, it helps. When the real frequency response associated with
spectrum coefficients is tested, it turns out to perform no better
than the zero-phase unfiltered
brickwall spectrum, but much better than with unwindowed cepstrum. Here
measured frequency response for lo-pass-filtered minimum-phase brickwall spectrum,
For radical filter settings, sidebands are very well perceivable in
test conditions with a sine sweep. I'll have to put up with that, till
I invent some decent alternative for the excessive upsampling.
By coincidence, my parametric Fourier filter (as described on previous pages) is to a
large extent immune to s-plane discontinuities. Look at the generic
spectrum shape below. The filter definition is based on a logarithmic
raised to a power and clipped at the top. The small numbers at the
'foot of the mountain' happen to translate well to a non-aliased
s-plane magnitude spectrum.
parametric Fourier filter lobe (linear scales)
Just like there's two ways to make an analytic signal via frequency
domain, there's also two ways to make a minimum-phase signal. In the
above description, the cepstrum was made single-sided, and an analytic
sequence resulted as a spectrum on the s-plane. Instead, Hilbert
Transform could be performed on the s-plane magnitudes, in order to
derive the phase spectrum as a -pi/2 radian rotated sequence. The two
methods were illustrated for signals on the page 'complexify
a real signal'. Phase separation by Hilbert transform is described
by Niranjan Damera-Venkata &c. in .
In order to rotate all phases of a sequence over -pi/2 radian, it's
Fourier transform must be multiplied with -i for the positive
frequencies and i for the negative frequencies. In the plot below, that
is done for the example cepstrum. The positive side is multiplied by
-1, and rotation by i is simply accomplished by feeding the cepstrum in
the imaginary input of the FFT, back on the way to s-plane. The
identity point, 'DC', is eliminated, like the Nyquist point should.
They wouldn't survive the imaginary side of Fourier transform anyway.
cepstrum made antisymmetric
The phase spectrum now appears at the real output of the FFT, and it must be combined with the original magnitude spectrum to compute the z-plane coordinates. The result is almost identical to the earlier described 'homomorphic' method. However, for the 'brickwall cases' the Hilbert transform method seems to be slightly more advantageous, as the original amplitudes are kept intact, whatever windowing is viciously done on the cepstrum.
measured frequency response for lo-pass-filtered minimum-phase brickwall spectrum,
cepstrum windowed, Hilbert transform method
When the cepstral multiplications of both methods (homomorphic and
Hilbert transform) are compared, a striking similarity appears. In both
cases, it is a (scaled / flipped / horizontally shifted) unit step
function with exceptions for DC and Nyquist point. In addition, the
Hilbert transform method requires multiplication by i.
homomorphic cepstral multiplication
Hilbert method cepstral multiplication
A long zero-phase FIR is in fact a weird filter solution. If you
think of a filter as a series of reflections, the zero-phase filter
produces reflections which precede the direct response, preflections
one could say. Even though the filter reflections are not audible as
echo's, the preflections are unnatural. If you listen well, they can be
perceived as a slurping sound, like a reversed playback, but very
short. I only became aware of them when comparing my filters, while
feeding an impulse for latency testing with
radical settings. In real
world musical situations, preflections won't be perceived as such, but
they can make the sound muddy. In contrast, reflections which follow
the direct response are natural and do not compromise the character of
zero-phase impulse response
minimum-phase impulse response
So, if you perceive a minimum-phase response as more direct and
present, it is not only due to reduced latency but also to the more
natural reflection pattern.
The minimum-phase IR can only reveal it's fast response when the input sound is not processed in filter-size blocks. It could be processed as plain time-domain convolution, but my favourite filters are up to a couple thousand coefficients, way too CPU-intensive for simple convolution. With 'partitioned convolution', a balance between efficiency and directness of response can be found. The technique uses FFT to perform fast convolution in small blocks, called partitions.
minimum-phase IR divided in partitions
Each partition of the IR is zero-padded and separately transformed
to frequency domain. The input signal is partitioned, zero-padded and
transformed likewise. Zero-padding is necessary to avoid circular
convolution. Complex multiplications in frequency domain replace time
domain convolution as usual. Overlap-add and overlap-save algorithms
seem to be used both in partitioned convolution. See page 'fast convolution'
partitions are processed and summed to the output
A more detailed scheme of operations can be found in this article by
Angelo Farina: REAL-TIME
PARTITIONED CONVOLUTION FOR AMBIOPHONICS SURROUND SOUND. I still
have to write my own partitioned convolution to learn more about it.
For the moment I'm happy to use Benjamin Saylor's [partconv~] object in
Remarkably, partitioned convolution works with unwindowed input and output signal, in contrast with regular fast convolution. The window is supposed to be implemented in the filter kernel. In itself, this is advantageous as you need not waste CPU time on windowed overlapping. But a welcome side-effect of windowed output is now sorely missed: the automatic gain cross-over at the moments of filter kernel switch. The sudden filter kernel replacement produces a nasty spike when an input signal is present. Even when the signal is in the filter's stop band!
A windowed overlap of the same signal with old and new filter is
just not available. The fact that partitioned convolution does not work
with block size equal to the filter size disables that option by
definition. So what to do about it then? The output gain could be
temporarily shut off to avoid the spike, but a short gain modification
sounds like a spike as well.
I found a practical second-best solution in a granular synthesis
equivalent. The output is shortly and smoothly reduced, and
filter-sized granule of delayed sound is copied over the location. This
gives the well-known amplitude modulation effect associated with cheap
pitch shifting (see illustrations on page 'pitch
shifting'), but it is at least much better than spikes.
The stuff described on this page is implemented in a Pure Data demo
patch. If you have Pd-extended 0.42.5 installed you can use it
rightaway. Otherwise download and install Pd-extended for your
platform, it's fairly easy (http://puredata.info). Get the zipped
demo patch from the link below the screenshot.
Pure Data patch for Pd-extended, 8 KB
Design of Optimal Minimum-Phase Digital FIR Filters Using Discrete Hilbert Transform -- Niranjan Damera-Venkata, Brian L. Evans, and Shawn R. McCaslin -- IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000