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I am wondering if there is an established method for choosing the best filter order to use when performing linear predictive coding (such as that used in audio file formats like FLAC).
My current approach is:
Take a chunk of signal
Window the signal with a 0.5 tukey window
Get the auto-correlation coefficients
Calculate the LPC coefficients using the auto-correlation coefficients
Generate the predicted signal using a standard FIR filter with the LPC coefficients
Measure the error between the original and predicted signal
Goto step 1 and continue repeating with different filter orders
...
Choose best order based on lowest error
Is it possible to estimate the best filter order by looking at the error created from step 4 of the process? I would like to shortcut to step 8 from step 4 if possible.
Since you mentioned FLAC, I had a look at how they do it. It looks like in the process of calculating the LPC coefficients, they estimate the errors for all orders up to the max. The overall computation starts here, they use FLAC__lpc_compute_lp_coefficients to compute the coeffs and also estimate the error. This is then used in FLAC__lpc_compute_best_order to decide what coefficient to use (in the nonexhaustive case).
Another implementation to look at is libflake, which is choosing the highest order n with n-1's reflection coefficient > .10. This seems related to the approach described here (PDF) which chooses the lowest order n when n+1 and n+2's reflection coefficients are < .15. Both are looking for the point where the reflection coef blows up, but looking at figure 1 in the aforementioned PDF it looks like taking the search from above as Flake does makes more sense. Just another heuristic but might be interesting.
I have some vectors of experimental data that I need to massage, for example:
{
{0, 61237, 131895, 194760, 249935},
{0, 61939, 133775, 197516, 251018},
{0, 60919, 131391, 194112, 231930},
{0, 60735, 131015, 193584, 249607},
{0, 61919, 133631, 197186, 250526},
{0, 61557, 132847, 196143, 258687},
{0, 61643, 133011, 196516, 249891},
{0, 62137, 133947, 197848, 251106}
}
Each vector is the result of one run and consists of five numbers, which are times at which an object passes each of five sensors. Over the measurement interval the object's speed is constant (the sensor-to-sensor intervals are different because the sensor spacings are not uniform). From one run to the next the sensors' spacing remains the same, but the object's speed will vary a bit from one run to the next.
If the sensors were perfect, each vector ought to simply be a scalar multiple of any other vector (in proportion to the ratio of their speeds). But in reality each sensor will have some "jitter" and trigger early or late by some small random amount. I am trying to analyze how good the sensors themselves are, i.e. how much "jitter" is there in the measurements they give me?
So I think I need to do the following. To each vector I must scale it, and add then shift the vector a bit (adding or subtracting a fixed amount to each of its five elements). Then the StandardDeviation of each column will describe the amount of "noise" or "jitter" in that sensor. The amount that each vector is scaled, and the amount each vector is shifted, has to be chosen to minimize the standard deviation of columns.
It seemed to me that Mathematica probably has a good toolkit for getting this done, in fact I thought I might have found the answer with Standardize[] but it seems to be oriented towards processing a list of scalars, not a list of lists like I have (or at least I can't figure out to apply it to my case here).
So I am looking for some hints toward which library function(s) I might use to solve this problems, or perhaps the hint I might need to cleave the algorithm myself. Perhaps part of my problem is that I can't figure out where to look - is what I have here is a "signal processing" problem, or a data manipulation or data mining problem, or a minimization problem, or maybe it's a relatively standard statistical function that I simply haven't heard of before?
(As a bonus I would like to be able to control the weighting function used to optimize this scaling/shifting; e.g. in my data above I suspect that sensor#5 is having problems so I would like to fit the data to only consider the SDs of sensors 1-4 when doing the scaling/shifting)
I can't comment much on your algorithm itself, as data analysis is not my forte. However, from what I understand, you're trying to characterize the timing variations in each sensor. Since the data from a single sensor is in a single column of your matrix, I'd suggest transposing it and mapping Standardize on to each set of data. In other words,
dat = (* your data *)
Standardize /# Transpose[dat]
To put it back in columnar form, Transpose the result. To exclude you last sensor from this process, simply use Part ([ ]) and Span (;;)
Standardize /# Transpose[dat][[ ;; -2 ]]
Or, Most
Standardize /# Most[Transpose[dat]]
Thinking about it, I think you're going to have a hard time separating out the timing jitter from variation in velocity. Can you intentionally vary the velocity?
Hello kind people of the audio computing world,
I have an array of samples that respresent a recording. Let us say that it is 5 seconds at 44100Hz. How would I play this back at an increased pitch? And is it possible to increase and decrease the pitch dynamically? Like have the pitch slowly increase to double the speed and then back down.
In other words I want to take a recording and play it back as if it is being 'scratched' by a d.j.
Pseudocode is always welcomed. I will be writing this up in C.
Thanks,
EDIT 1
Allow me to clarify my intentions. I want to keep the playback at 44100Hz and so therefore I need to manipulate the samples before playback. This is also because I would want to mix the audio that has an increased pitch with audio that is running at a normal rate.
Expressed in another way, maybe I need to shrink the audio over the same number of samples somehow? That way when it is played back it will sound faster?
EDIT 2
Also, I would like to do this myself. No libraries please (unless you feel I could pick through the code and find something interesting).
EDIT 3
A sample piece of code written in C that takes 2 arguments (array of samples and pitch factor) and then returns an array of the new audio would be fantastic!
PS I've started a bounty on this not because I don't think the answers already given aren't valid. I just thought it would be good to get more feedback on the subject.
AWARD OF BOUNTY
Honestly I wish I could distribute the bounty over several different answers as they were quite a few that I thought were super helpful. Special shoutout to Daniel for passing me some code and AShelly and Hotpaw2 for putting in such detailed responses.
Ultimately though I used an answer from another SO question referenced by datageist and so the award goes to him.
Thanks again everyone!
Take a look at the "Elephant" paper in Nosredna's answer to this (very similar) SO question:
How do you do bicubic (or other non-linear) interpolation of re-sampled audio data?
Sample implementations are provided starting on page 37, and for reference, AShelly's answer corresponds to linear interpolation (on that same page). With a little tweaking, any of the other formulas in the paper could be plugged into that framework.
For evaluating the quality of a given interpolation method (and understanding the potential problems with using "cheaper" schemes), take a look at this page:
http://www.discodsp.com/highlife/aliasing/
For more theory than you probably want to deal with (with source code), this is a good reference as well:
https://ccrma.stanford.edu/~jos/resample/
One way is to keep a floating point index into the original wave, and mix interpolated samples into the output wave.
//Simulate scratching of `inwave`:
// `rate` is the speedup/slowdown factor.
// result mixed into `outwave`
// "Sample" is a typedef for the raw audio type.
void ScratchMix(Sample* outwave, Sample* inwave, float rate)
{
float index = 0;
while (index < inputLen)
{
int i = (int)index;
float frac = index-i; //will be between 0 and 1
Sample s1 = inwave[i];
Sample s2 = inwave[i+1];
*outwave++ += s1 + (s2-s1)*frac; //do clipping here if needed
index+=rate;
}
}
If you want to change rate on the fly, you can do that too.
If this creates noisy artifacts when rate > 1, try replacing *outwave++ += s1 + (s2-s1)*frac; with this technique (from this question)
*outwave++ = InterpolateHermite4pt3oX(inwave+i-1,frac);
where
public static float InterpolateHermite4pt3oX(Sample* x, float t)
{
float c0 = x[1];
float c1 = .5F * (x[2] - x[0]);
float c2 = x[0] - (2.5F * x[1]) + (2 * x[2]) - (.5F * x[3]);
float c3 = (.5F * (x[3] - x[0])) + (1.5F * (x[1] - x[2]));
return (((((c3 * t) + c2) * t) + c1) * t) + c0;
}
Example of using the linear interpolation technique on "Windows Startup.wav" with a factor of 1.1. The original is on top, the sped-up version is on the bottom:
It may not be mathematically perfect, but it sounds like it should, and ought to work fine for the OP's needs..
Yes, it is possible.
But this is not a small amount of pseudo code. You are asking for a time pitch modification algorithm, which is a fairly large and complicated amount of DSP code for decent results.
Here's a Time Pitch stretching overview from DSP Dimensions. You can also Google for phase vocoder algorithms.
ADDED:
If you want to "scratch", as a DJ might do with an LP on a physical turntable, you don't need time-pitch modification. Scratching changes the pitch and the speed of play by the same amount (not independently as would require time-pitch modification).
And the resulting array won't be of the same length, but will be shorter or longer by the amont of pitch/speed change.
You can change the pitch, as well as make the sound play faster or slower by the same ratio, by just resampling the signal using properly filtered interpolation. Just move each sample point, instead of by 1.0, by floating point addition by your desired rate change, then filter and interpolate the data at that point. Interpolation using a windowed Sinc interpolation kernel, with a low-pass filter transition frequency below the lower of the original and interpolated local sample rate, will work fairly well. Searching for "windowed Sinc interpolation" on the web returns lots of suitable result.
You need an interpolation method that includes a low-pass filter, or else you will hear horrible aliasing noise. (The exception to this might be if your original sound file is already severely low-pass filtered a decade or more below the sample rate.)
If you want this done easily, see AShelly's suggestion [edit: as a matter of fact, try it first anyway]. If you need good quality, you basically need a phase vocoder.
The very basic idea of a phase vocoder is to find the frequencies that the sound consists of, change those frequencies as needed and resynthesize the sound. So a brutal simplification would be:
run FFT
change all frequencies by a factor
run inverse FFT
If you're going to implement this yourself, you definitely should read a thorough explanation of how a phase vocoder works. The algorithm really needs many more considerations than the three-step simplification above.
Of course, ready-made implementations exist, but from the question I gather you want to do this yourself.
To decrease and increase the pitch is as simple as playing the sample back at a lower or higher rate than 44.1kHz. This will produce the slower/faster record sound but you'll need to add the 'scratchiness' of real records.
This helped me with resampling, which is same thing you need just looked from the opposite side.
If you can't find code, ping me, I have a nice C routine for this.
Since I have no idea about what I am doing right now, my wording may sound funny. But seriously, I need to learn.
The problem I'm facing is to come up with a method (model) to estimate how a software program works: namely running time and maximal memory usage. What I already have are a large amount of data. This data set gives an overview of how a program works under different conditions, e.g.
<code>
RUN Criterion_A Criterion_B Criterion_C Criterion_D Criterion_E <br>
------------------------------------------------------------------------
R0001 12 2 3556 27 9 <br>
R0002 2 5 2154 22 8 <br>
R0003 19 12 5556 37 9 <br>
R0004 10 3 1556 7 9 <br>
R0005 5 1 556 17 8 <br>
</code>
I have thousands of rows of such data. Now I need to know how I can estimate (forecast) the running time and maximal memory usage if I know all criteria in advance. What I need is an approximation that gives hints (upper limits, or ranges).
I have feeling that it is a typical ??? problem which I don't know. Could you guys show me some hints or give me some ideas (theories, explanations, webpages) or anything that may help. Thanks!
You want a new program that takes as input one or more criteria, then outputs an estimate of the running time or memory usage. This is a machine learning problem.
Your inputs can be listed as a vector of numbers, like this:
input = [ A, B, C, D, E ]
One of the simplest algorithms for this would be a K-nearest neighbor algorithm. The idea behind this is that you'll take your input vector of numbers, and find in your database the vector of numbers that is most similar to your input vector. For example, given this vector of inputs:
input = [ 11, 1.8, 3557, 29, 10 ]
You can assume that the running time and memory should be very similar to the values from this run (originally in your table listed above):
R0001 12 2 3556 27 9
There are several algorithms for calculating the similarity between these two vectors, one simple and intuitive such algorithm is the Euclidean distance. As an example, the Euclidean distance between the input vector and the vector from the table is this:
dist = sqrt( (11-12)^2 + (1.8-2)^2 + (3557-3556)^2 + (27-29)^2 + (9-10)^2 )
dist = 2.6533
It should be intuitively clear that points with lower distance should be better estimates for running time and memory usage, as the distance should describe the similarity between two sets of criteria. Assuming your criteria are informative and well-selected, points with similar criteria should have similar running time and memory usage.
Here's some example code of how to do this in R:
r1 = c(11,1.8,3557,29,10)
r2 = c(12,2.0,3556,27, 9)
print(r1)
print(r2)
dist_r1_r2 = sqrt( (11-12)^2 + (1.8-2)^2 + (3557-3556)^2 + (27-29)^2 + (9-10)^2 )
print(dist_r1_r2)
smarter_dist_r1_r2 = sqrt( sum( (r1 - r2)^2 ) )
print(smarter_dist_r1_r2)
Taking the running time and memory usage of your nearest row is the KNN algorithm for K=1. This approach can be extended to include data from multiple rows by taking a weighted combination of multiple rows from the database, with rows with lower distances to your input vector contributing more to the estimates. Read the Wikipedia page on KNN for more information, especially with regard to data normalization, including contributions from multiple points, and computing distances.
When calculating the difference between these lists of input vectors, you should consider normalizing your data. The rationale for doing this is that a difference of 1 unit between 3557 and 3556 for criteria C may not be equivalent to a difference of 1 between 11 and 12 for criteria A. If your data are normally distributed, you can convert them all to standard scores (or Z-scores) using this formula:
N_trans = (N - mean(N)) / sdev(N)
There is no general solution on the "right" way to normalize data as it depends on the type and range of data that you have, but Z-scores are easy to compute and a good method to try first.
There are many more sophisticated techniques for constructing estimates such as this, including linear regression, support vector regression, and non-linear modeling. The idea behind some of the more sophisticated methods is that you try and develop an equation that describes the relationship of your variables to running time or memory. For example, a simple application might just have one criterion and you can try and distinguish between models such as:
running_time = s1 * A + s0
running_time = s2 * A^2 + s1 * A + s0
running_time = s3 * log(A) + s2 * A^2 + s1 * A + s0
The idea is that A is your fixed criteria, and sN are a list of free parameters that you can tweak until you get a model that works well.
One problem with this approach is that there are many different possible models that have different numbers of parameters. Distinguishing between models that have different numbers of parameters is a difficult problem in statistics, and I don't recommend tackling it during your first foray into machine learning.
Some questions that you should ask yourself are:
Do all of my criteria affect both running time and memory usage? Do some affect only one or the other, and are some useless from a predictive point of view? Answering this question is called feature selection, and is an outstanding problem in machine learning.
Do you have any a priori estimates of how your variables should influence running time or memory usage? For example, you might know that your application uses a sorting algorithm that is N * log(N) in time, which means that you explicitly know the relationship between one criterion and your running time.
Do your rows of measured input criteria paired with running time and memory usage cover all of the plausible use cases for your application? If so, then your estimates will be much better, as machine learning can have a difficult time with data that it's unfamiliar with.
Do the running time and memory of your program depend on criteria that you don't input into your estimation strategy? For example, if you're depending on an external resource such as a web spider, problems with your network may influence running time and memory usage in ways that are difficult to predict. If this is the case, your estimates will have a lot more variance.
If the criterion you would be forecasting for lies within the range of currently known criteria then you should do some more research on the Interpolation process:
In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points
If it lies outside your currently known data range research Extrapolation which is less accurate:
In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points.
Methods
Interpolation methods for your browsing.
A powerpoint presentation detailing some methods used for Extrapolation.
I've got a 44Khz audio stream from a CD, represented as an array of 16 bit PCM samples. I'd like to cut it down to an 11KHz stream. How do I do that? From my days of engineering class many years ago, I know that the stream won't be able to describe anything over 5500Hz accurately anymore, so I assume I want to cut everything above that out too. Any ideas? Thanks.
Update: There is some code on this page that converts from 48KHz to 8KHz using a simple algorithm and a coefficient array that looks like { 1, 4, 12, 12, 4, 1 }. I think that is what I need, but I need it for a factor of 4x rather than 6x. Any idea how those constants are calculated? Also, I end up converting the 16 byte samples to floats anyway, so I can do the downsampling with floats rather than shorts, if that helps the quality at all.
Read on FIR and IIR filters. These are the filters that use a coefficent array.
If you do a google search on "FIR or IIR filter designer" you will find lots of software and online-applets that does the hard job (getting the coefficients) for you.
EDIT:
This page here ( http://www-users.cs.york.ac.uk/~fisher/mkfilter/ ) lets you enter the parameters of your filter and will spit out ready to use C-Code...
You're right in that you need apply lowpass filtering on your signal. Any signal over 5500 Hz will be present in your downsampled signal but 'aliased' as another frequency so you'll have to remove those before downsampling.
It's a good idea to do the filtering with floats. There are fixed point filter algorithms too but those generally have quality tradeoffs to work. If you've got floats then use them!
Using DFT's for filtering is generally overkill and it makes things more complicated because dft's are not a contiuous process but work on buffers.
Digital filters generally come in two tastes. FIR and IIR. The're generally the same idea but IIF filters use feedback loops to achieve a steeper response with far less coefficients. This might be a good idea for downsampling because you need a very steep filter slope there.
Downsampling is sort of a special case. Because you're going to throw away 3 out of 4 samples there's no need to calculate them. There is a special class of filters for this called polyphase filters.
Try googling for polyphase IIR or polyphase FIR for more information.
Notice (in additions to the other comments) that the simple-easy-intuitive approach "downsample by a factor of 4 by replacing each group of 4 consecutive samples by the average value", is not optimal but is nevertheless not wrong, nor practically nor conceptually. Because the averaging amounts precisely to a low pass filter (a rectangular window, which corresponds to a sinc in frequency). What would be conceptually wrong is to just downsample by taking one of each 4 samples: that would definitely introduce aliasing.
By the way: practically any software that does some resampling (audio, image or whatever; example for the audio case: sox) takes this into account, and frequently lets you choose the underlying low-pass filter.
You need to apply a lowpass filter before you downsample the signal to avoid "aliasing". The cutoff frequency of the lowpass filter should be less than the nyquist frequency, which is half the sample frequency.
The "best" solution possible is indeed a DFT, discarding the top 3/4 of the frequencies, and performing an inverse DFT, with the domain restricted to the bottom 1/4th. Discarding the top 3/4ths is a low-pass filter in this case. Padding to a power of 2 number of samples will probably give you a speed benefit. Be aware of how your FFT package stores samples though. If it's a complex FFT (which is much easier to analyze, and generally has nicer properties), the frequencies will either go from -22 to 22, or 0 to 44. In the first case, you want the middle 1/4th. In the latter, the outermost 1/4th.
You can do an adequate job by averaging sample values together. The naïve way of grabbing samples four by four and doing an equal weighted average works, but isn't too great. Instead you'll want to use a "kernel" function that averages them together in a non-intuitive way.
Mathwise, discarding everything outside the low-frequency band is multiplication by a box function in frequency space. The (inverse) Fourier transform turns pointwise multiplication into a convolution of the (inverse) Fourier transforms of the functions, and vice-versa. So, if we want to work in the time domain, we need to perform a convolution with the (inverse) Fourier transform of box function. This turns out to be proportional to the "sinc" function (sin at)/at, where a is the width of the box in the frequency space. So at every 4th location (since you're downsampling by a factor of 4) you can add up the points near it, multiplied by sin (a dt) / a dt, where dt is the distance in time to that location. How nearby? Well, that depends on how good you want it to sound. It's common to ignore everything outside the first zero, for instance, or just take the number of points to be the ratio by which you're downsampling.
Finally there's the piss-poor (but fast) way of just discarding the majority of the samples, keeping just the zeroth, the fourth, and so on.
Honestly, if it fits in memory, I'd recommend just going the DFT route. If it doesn't use one of the software filter packages that others have recommended to construct the filter for you.
The process you're after called "Decimation".
There are 2 steps:
Applying Low Pass Filter on the data (In your case LPF with Cut Off at Pi / 4).
Downsampling (In you case taking 1 out of 4 samples).
There are many methods to design and apply the Low Pass Filter.
You may start here:
http://en.wikipedia.org/wiki/Filter_design
You could make use of libsamplerate to do the heavy lifting. Libsamplerate is a C API, and takes care of calculating the filter coefficients. You to select from different quality filters so that you can trade off quality for speed.
If you would prefer not to write any code, you could just use Audacity to do the sample rate conversion. It offers a powerful GUI, and makes use of libsamplerate for it's sample rate conversion.
I would try applying DFT, chopping 3/4 of the result and applying inverse DFT. I can't tell if it will sound good without actually trying tough.
I recently came across BruteFIR which may already do some of what you're interested in?
You have to apply low-pass filter (removing frequencies above 5500 Hz) and then apply decimation (leave every Nth sample, every 4th in your case).
For decimation, FIR, not IIR filters are usually employed, because they don't depend on previous outputs and therefore you don't have to calculate anything for discarded samples. IIRs, generally, depends on both inputs and outputs, so, unless a specific type of IIR is used, you'd have to calculate every output sample before discarding 3/4 of them.
Just googled an intro-level article on the subject: https://www.dspguru.com/dsp/faqs/multirate/decimation