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I am trying to find a very fast and efficient Fourier transform (FFT). Does anyone know of any good ones. I need to run it on the iPhone so it must not be intensive. Instead, maybe you know of one that is wavelet like, i need frequency resolution but only a narrow band (vocal audio range up to 10khz max...even 10Khz might be too high). Im thinking also of truncating this FFT to keep the frequency resolution while eliminating the unwanted frequency band. This is for an iphone
...I have taken a look at the FFT in Aurio touch but it seems this is an int FFT but my app uses floats.....would it give a big performance increase to try and adapt program to an int FFT or not(which i really dont feel like doing...plus aurio touch uses a radix 2 FFT which is not that great).
The iPhone OS4 SDK will include the Accelerate framework, which will (finally) give us Apple-written FFT functions
Accelerate provides hundreds of
mathematical functions optimized for
iPhone and iPod touch, including
signal-processing routines, fast
Fourier transforms, basic vector and
matrix operations, and
industry-standard functions for
factoring matrices and solving systems
of linear equations.
I've wrapped Ooura's FFT library in Objective-C. Ooura's code is of comparable performance to FFTW, but totally and utterly free.
This code uses double-precision and has several built-in window types (rectangular, Blackwell, Triangle, Hamming). I use Ooura's FFT code to implement Welch's method, which will generate a much smoother spectra when viewed over time.
Check it out at:
http://github.com/alexbw/iPhoneFFT
Give the Fastest Fourier Transform in the West (FFTW) a go, The performance is good compared to others, but it is not completely free. See the details on commercial use here. Obviously being a c library you should have no problem linking it as a static library to your iphone app.
The performance of the FFTW sets the standard for arbitrary length FFT's - especially for non-power of 2 lengths in 2 and greater dimensions. The commercial license for FFTW is $5000, which may or may not fit in your budget.
However, it sounds like you have a 1D signal processing problem in which case you have a few more options - and if you can further either pad or sample your data to power-of-2 lengths, then many libraries will offer reasonable performance. Check out this list of FFT algorithms that FFTW used for comparison - many are free and some may be adequate. I'd probably start with good old numerical recipes which offers an easy power of 2, 1D FFT implementation for free and some typing - and would be very memory efficient.
BTW - for voice you probably only need to go to 3-4Khz....10Khz is way way up there for the
human voice.
Here is a primary source link to Ooura's numerical software:
http://www.kurims.kyoto-u.ac.jp/~ooura/
I have been using many of Ooura's FFTs over the years, I should send him a "domo" at the very least, and I use his real radix-4 in several iPad and iPhone applications under development. I did translate the code to operate with 32-bit single precision for performance on ARM. Looking at the assembly produced with XCode 3.2.2, it vectorizes with NEON SIMD instructions very nicely. I was half disappointed actually, as I was willing to vectorize the code a bit myself for even more performance. These optimizations cannot be had without first translating the FFT to single precision obviously.
While I have used Objective-C for many years, I actively develop using it, and even taught an object oriented programming course using it, I did not prepare such a wrapper (though I had done the same back in 1992 with a different FFT) for performance reasons.
I haven't tested FFTW against Ooura's FFT in at least 10 years, but when I did Ooura's library was faster for 1024 point real FFTs. However, it is quite possible that FFTW may do much better now -- but licensing it and cross-compiling it for ARM is inconvenient and I have always found FFTW to be far too bulky and obtrusive for my DSP needs. Apple's VecLib is very nice but unfortunately they have not ported it to iPhoneOS. I opened a feature request in BugReporter and you can too: https://bugreport.apple.com/
As answered before, the Accelerate Framework now provides some APIs that might help you.
Check:
Accelerate Framework Reference
vDSP Reference
Using Fourier Transforms
Related
I'm very new to coding, and need some help with a project of sorts. I have segments of video, and I need to be able to track the motion of an object(s) through these segments, and get data like a mapped path, average and instantaneous velocities, etc. I'm trying to do this in MATLAB, and have the 2016a version installed. Any help is greatly appreciated.
Without knowing exactly what project you're working on, I can guess that Matlab is the wrong tool for this job.
I've been using Matlab near-daily for about four years now, but when I want to track an object in a video, I use Tracker.
Matlab is a good language for a beginning programmer because you can start doing numerical calculations, and plotting the results, very quickly. More advanced programmers tend to use Matlab to process data (Mathworks has many useful libraries for things like Fourier Transforms); to do linear algebra; to do quick numerical analyses; and to build scientific models. These applications are mainly in math, science, and engineering.
If you want to learn Matlab, I recommend you find a project which plays to Matlab's strengths.
If you want to analyse images and videos, I recommend that you learn a language which is used professionally for this purpose, such as Python or Java.
I want to detect not the pitch, but the pitch class of a sung note.
So, whether it is C4 or C5 is not important: they must both be detected as C.
Imagine the 12 semitones arranged on a clock face, with the needle pointing to the pitch class. That's what I'm after! ideally I would like to be able to tell whether the sung note is spot-on or slightly off.
This is not a duplicate of previously asked questions, as it introduces the constraints that:
the sound source is a single human voice, hopefully with negligible background interference (although I may need to deal with this)
the octave is not important, only the pitch class
EDIT -- Links:
Real time pitch detection
Using the Apple FFT and Accelerate Framework
See my answer here for getting smooth FREQUENCY detection: https://stackoverflow.com/a/11042551/1457445
As far as snapping this frequency to the nearest note -- here is a method I created for my tuner app:
- (int) snapFreqToMIDI: (float) frequencyy {
int midiNote = (12*(log10(frequencyy/referenceA)/log10(2)) + 57) + 0.5;
return midiNote;
}
This will return the MIDI note value (http://www.phys.unsw.edu.au/jw/notes.html)
In order to get a string from this MIDI note value:
- (NSString*) midiToString: (int) midiNote {
NSArray *noteStrings = [[NSArray alloc] initWithObjects:#"C", #"C#", #"D", #"D#", #"E", #"F", #"F#", #"G", #"G#", #"A", #"A#", #"B", nil];
return [noteStrings objectAtIndex:midiNote%12];
}
For an example implementation of the pitch detection with output smoothing, look at musicianskit.com/developer.php
Pitch is a human psycho-perceptual phenomena. Peak frequency content is not the same as either pitch or pitch class. FFT and DFT methods will not directly provide pitch, only frequency. Neither will zero crossing measurements work well for human voice sources. Try AMDF, ASDF, autocorrelation or cepstral methods. There are also plenty of academic papers on the subject of pitch estimation.
There is another long list of pitch estimation algorithms here.
Edited addition: Apple's SpeakHere and aurioTouch sample apps (available from their iOS dev center) contain example source code for getting PCM sample blocks from the iPhone's mic.
Most of the frequency detection algorithms cited in other answers don't work well for voice. To see why this is so intuitively, consider that all the vowels in a language can be sung at one particular note. Even though all those vowels have very different frequency content, they would all have to be detected as the same note. Any note detection algorithm for voices must take this into account somehow. Furthermore, human speech and song contains many fricatives, many of which have no implicit pitch in them.
In the generic (non voice case) the feature you are looking for is called the chroma feature and there is a fairly large body of work on the subject. It is equivalently known as the harmonic pitch class profile. The original reference paper on the concept is Tayuka Fujishima's "Real-Time Chord Recognition of Musical Sound: A System Using Common Lisp Music". The Wikipedia entry has an overview of a more modern variant of the algorithm. There are a bunch of free papers and MATLAB implementations of chroma feature detection.
However, since you are focusing on the human voice only, and since the human voice naturally contains tons of overtones, what you are practically looking for in this specific scenario is a fundamental frequency detection algorithm, or f0 detection algorithm. There are several such algorithms explicitly tuned for voice. Also, here is a widely cited algorithm that works on multiple voices at once. You'd then check the detected frequency against the equal-tempered scale and then find the closest match.
Since I suspect that you're trying to build a pitch detector and/or corrector a la Autotune, you may want to use M. Morise's excellent WORLD implementation, which permits fast and good quality detection and modification of f0 on voice streams.
Lastly, be aware that there are only a few vocal pitch detectors that work well within the vocal fry register. Almost all of them, including WORLD, fail on vocal fry as well as very low voices. A number of papers refer to vocal fry as "creaky voice" and have developed specific algorithms to help with that type of voice input specifically.
If you are looking for the pitch class you should have a look at the chromagram (http://labrosa.ee.columbia.edu/matlab/chroma-ansyn/)
You can also simply dectect the f0 (using something like YIN algorithm) and return the appropriate semitone, most of fundamental frequency estimation algorithms suffer from octave error
Perform a Discrete Fourier Transform on samples from your input waveform, then sum values that correspond to equivalent notes in different octaves. Take the largest value as the dominant frequency.
You can likely find some existing DFT code in Objective C that suits your needs.
Putting up information as I find it...
Pitch detection algorithm on Wikipedia is a good place to start. It lists a few methods that fail for determining octave, which is okay for my purpose.
A good explanation of autocorrelation can be found here (why can't Wikipedia put things simply like that??).
Finally I have closure on this one, thanks to this article from DSP Dimension
The article contains source code.
Basically he performs an FFT. then he explains that frequencies that don't coincide spot on with the centre of the bin they fall in will smear over nearby bins in a sort of bell shaped curve. and he explains how to extract the exact frequency from this data in a second pass (FFT being the first pass).
the article then goes further to pitch shift; I can simply delete the code.
note that they supply a commercial library that does the same thing (and far more) only super optimised. there is a free version of the library that would probably do everything I need, although since I have worked through the iOS audio subsystem, I might as well just implement it myself.
for the record, I found an alternative way to extract the exact frequency by approximating a quadratic curve over the bin and its two neighbours here. I have no idea what is the relative accuracy between these two approaches.
As others have mentioned you should use a pitch detection algorithm. Since that ground is well-covered I will address a few particulars of your question. You said that you are looking for the pitch class of the note. However, the way to find this is to calculate the frequency of the note and then use a table to convert it to the pitch class, octave, and cents. I don't know of any way to obtain the pitch class without finding the fundamental frequency.
You will need a real-time pitch detection algorithm. In evaluating algorithms pay attention to the latency implied by each algorithm, compared with the accuracy you desire. Although some algorithms are better than others, fundamentally you must trade one for the other and cannot know both with certainty -- sort of like the Heisenberg uncertainty principle. (How can you know the note is C4 when only a fraction of a cycle has been heard?)
Your "smoothing" approach is equivalent to a digital filter, which will alter the frequency characteristics of the voice. In short, it may interfere with your attempts to estimate the pitch. If you have an interest in digital audio, digital filters are fundamental and useful tools in that field, and a fascinating subject besides. It helps to have a strong math background in understanding them, but you don't necessarily need that to get the basic idea.
Also, your zero crossing method is a basic technique to estimate the period of a waveform and thus the pitch. It can be done this way, but only with a lot of heuristics and fine-tuning. (Essentially, develop a number of "candidate" pitches and try to infer the dominant one. A lot of special cases will emerge that will confuse this. A quick one is the less 's'.) You'll find it much easier to begin with a frequency domain pitch detection algorithm.
if you re beginner this may be very helpful. It is available both on Java and IOS.
dywapitchtrack for ios
dywapitchtrack for java
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A professor asked me to help making a specification for a college project.
By the time the students should know the basics of programming.
The professor is a mathematician and has little experience in other programming languages, so it should really be in MATLAB.
I would like some projects ideas. The project should
last about 1 to 2 months
be done individually
have web interface would be great
doesn't necessary have to go deep in maths, but some would be great
use a database (or store data in files)
What kind of project would make the students excited?
If you have any other tips I'll appreciate.
UPDATE: The students are sophomores and have already studied vector calculus. This project is for an one year Discrete Mathematics course.
UPDATE 2: The topics covered in the course are
Formal Logic
Proofs, Recursion, and Analysis of Algorithms
Sets and Combinatorics
Relations, Functions, and Matrices
Graphs and Trees
Graph Algorithms
Boolean Algebra and Computer Logic
Modeling Arithmetic, Computation, and Languages
And it'll be based on this book Mathematical Structures for Computer Science: A Modern Approach to Discrete Mathematics by Judith L. Gersting
General Suggestions:
There are many teaching resources at The MathWorks that may give you some ideas for course projects. Some sample links:
The MATLAB Central blogs, specifically some posts by Loren that include using LEGO Mindstorms in teaching and a webinar about MATLAB for teaching (note: you will have to sign up to see the webinar)
The Curriculum Exchange: a repository of course materials
Teaching with MATLAB and Simulink: a number of other links you may find useful
Specific Suggestions:
One of my grad school projects in non-linear dynamics that I found interesting dealt with Lorenz oscillators. A Lorenz oscillator is a non-linear system of three variables that can exhibit chaotic behavior. Such a system would provide an opportunity to introduce the students to numerical computation (iterative methods for simulating systems of differential equations, stability and convergence, etc.).
The most interesting thing about this project was that we were using Lorenz oscillators to encode and decode signals. This "encrypted communication" aspect was really cool, and was based on the following journal article:
Kevin M. Cuomo and Alan V. Oppenheim,
Circuit Implementation of Synchronized Chaos with Applications
to Communications, Physical Review
Letters 71(1), 65-68 (1993)
The article addresses hardware implementations of a chaotic communication system, but the equivalent software implementation should be simple enough to derive (and much easier for the students to implement!).
Some other useful aspects of such a project:
The behavior of the system can be visualized in 2-D and 3-D plots, thus exposing the students to a number of graphing utilities in MATLAB (PLOT, PLOT3, COMET, COMET3, etc.).
Audio signals can be read from files, encrypted using the Lorenz equations, written out to a new file, and then decrypted once again. You could even have the students each encrypt a signal with their Lorenz oscillator code and give it to another student to decrypt. This would introduce them to various file operations (FREAD, FWRITE, SAVE, LOAD, etc.), and you could even introduce them to working with audio data file formats.
You can introduce the students to the use of the PUBLISH command in MATLAB, which allows you to format M-files and publish them to various output types (like HTML or Word documents). This will teach them techniques for making useful help documentation for their MATLAB code.
I have found that implementing and visualizing Dynamical systems is great
for giving an introduction to programming and to an interesting branch of
applied mathematics. Because one can see the 'life' in these systems,
our students really enjoy this practical module.
We usually start off by visualizing a 1D attractor, so that we can
overlay the evolution rule/rate of change with the current state of
the system. That way you can teach computational aspects (integrating the system) and
visualization, and the separation of both in implementation (on a simple level, refreshing
graphics at every n-th computation step, but in C++ leading to threads, unsure about MATLAB capabilities here).
Next we add noise, and then add a sigmoidal nonlinearity to the linear attractor. We combine this extension with an introduction to version control (we use a sandbox SVN repository for this): The
students first have to create branches, modify the evolution rule and then merge
it back into HEAD.
When going 2D you can simply start with a rotation and modify it to become a Hopf oscillator, and visualize either by morphing a grid over time or by going 3D when starting with a distinct point. You can also visualize the bifurcation diagram in 3D. So you again combine generic MATLAB skills like 3D plotting with the maths.
To link in other topics, browse around in wikipedia: you can bring in hunter/predator models, chaotic systems, physical systems, etc.etc.
We usually do not teach object-oriented-programming from within MATLAB, although it is possible and you can easily make up your own use cases in the dynamical systems setting.
When introducing inheritance, we will already have moved on to C++, and I'm again unaware of MATLAB's capabilities here.
Coming back to your five points:
Duration is easily adjusted, because the simple 1D attractor can be
done quickly and from then on, extensions are ample and modular.
We assign this as an individual task, but allow and encourage discussion among students.
About the web interface I'm at a loss: what exactly do you have in mind, why is it
important, what would it add to the assignment, how does it relate to learning MATLAB.
I would recommend dropping this.
Complexity: A simple attractor is easily understood, but the sky's the limit :)
Using a database really is a lot different from config files. As to the first, there
is a database toolbox for accessing databases from MATLAB. Few institutes have the license though, and apart from that: this IMHO does not belong into such a course. I suggest introducing to the concept of config files, e.g. for the location and strength of the attractor, and later for the system's respective properties.
All this said, I would at least also tell your professor (and your students!) that Python is rising up against MATLAB. We are in the progress of going Python with our tutorials, but I understand if someone wants to stick with what's familiar.
Also, we actually need the scientific content later on, so the usefulness for you will probably depend on which department your course will be related to.
A lot of things are possible.
The first example that comes in mind is to model a public transportation network (the network of your city, with underground, buses, tramways, ...). It is represented by a weighted directed graph (you can use sparse matrix to represent it, for example).
You may, for example, ask them to compute the shortest path from one station to another one (Moore-dijkistra algorithm, for example) and display it.
So, for the students, the several steps to do are:
choose an appropriate representation for the network (it could be some objects to represent the properties of the stations and the lines, and a sparse matrix for the network)
load all the data (you can provide them the data in an XML file)
be able to draw the network (since you will put the coordinates of the stations)
calculate the shortest path from one point to another and display it in a pretty way
create a fronted (with GUI)
Of course, this could be complicated by adding connection times (when you change from one line to another), asking for several options (shortest path with minimum connections, take in considerations the time you loose by waiting for a train/bus, ...)
The level of details will depend on the level of the students and the time they could spend on it (it could be very simple, or very realist)
You want to do a project with a web interface and a database, but not any serious math... and you're doing it in MATLAB? Do you understand that MATLAB is especially designed to be used for "deep math", and not for web interfaces or databases?
I think if this is an intro to a Discrete Mathematics course, you should probably do something involving Discrete Mathematics, and not waste the students' time as they learn a bunch of things in that language that they'll never actually use.
Why not do something involving audio? I did an undergraduate project in which we used MATLAB to automatically beat-match different tunes and DJ mix between them. The full program took all semester, but you could do a subset of it. wavread() and the like are built in and easy to use.
Or do some simple image processing like finding Waldo using cross-correlation.
Maybe do something involving cryptography, have them crack a simple encryption scheme and feel like hackers.
MATLAB started life as a MATrix LAB, so maybe concentrating on problems in linear algebra would be a natural fit.
Discrete math problems using matricies include:
Spanning trees and shortest paths
The marriage problem (bipartite graphs)
Matching algorithms
Maximal flow in a network
The transportation problem
See Gil Strang's "Intro to Applied Math" or Knuth's "Concrete Math" for ideas.
You might look here: http://www.mathworks.com/academia/student_center/tutorials/launchpad.html
on the MathWorks website. The interactive tutorial (second link) is quite popular.
--Loren
I always thought the one I was assigned in grad school was a good choice-a magnetic lens simulator. The math isn't completely overwhelming so you can focus more on learning the language, and it's a good intro to the graphical capabilities (e.g., animating the path of an off-axis electron going through the lens).
db I/O and fancy interfaces are out of place in a discrete math course.
my matlab labs were typically algorithm implementations, with charts as output, and simple file input.
how hard is the material? image processing is really easy in matlab, can you do some discrete 2D filtering? blurs and stuff. http://homepages.inf.ed.ac.uk/rbf/HIPR2/filtops.htm
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Would learning to program fractals help think clearly about certain set of programming problems?
Fractal programming would introduce you to recursion, iteration, graphics programming concepts, image processing, and user interface design. From a mathematics point of view, you would learn about geometry, complex numbers, Mobius transformations (Kleinian fractals), Affine transformation (IFS fractals), root-finding methods (Newton fractals).
And on top of all this, you get the reward of seeing your efforts result in strange and unusual images.
From what I remember you'll get a good handle on recursion if nothing else...maybe a little bitmap level programming as well...
If you are a beginner such activity will surely help you to improve your skills. Apart from that programming fractal visualizations depending on the fractal type and the goal you set may give you some specific skills or knowledge like:
working with graphics, image processing;
understanding recursion and recursive
structures;
optimization techniques;
low-level program optimizations;
understanding how computer operates
(e.g. why resolution would be
normally limited -> floating point
precision and error accumulation);
parallel programming;
some mathematical improvement and
extend your range of interest;
understanding various technologies
(e.g. you can code Mandelbrot
set in PixelBender which is
really fast since may be executed on
GPU);
understanding complex compression
algorithms (e.g. some kind of fractal
compression);
creativity (e.g. you invent your own
fractal set coloring algorithm);
much more else :)
It is indeed a versatile and interesting field, lots of things to explore and learn. I used to draw fractals a lot :)
Fractals got me thinking about complex numbers and branch-points. Whether that was a good thing is, I suppose, a matter of opinion. :-)
Any kind of programming experience is useful. So yes it is.
Especially for:
math problems
basic algorithms
and of course fractal programming
It'll maybe give you practice in implementing mathematical formulae.
Some fractals are good visual examples for explicit recursion; if you have a hard time with that concept, they might be good problems to work. You can start with "turtle graphics" style fractal paths like the Hilbert curve, or the classic "snowflake" fractal.
Many fractal-generation methods use heavy-duty number crunching (e.g., Mandelbrot and Julia sets). Number crunching is of course a field in itself, and tweaking your fractal generator to run as fast as possible can be a nice exercise in optimization.
I don't think programming fractals will teach you anything in particular. Depending on the fractal, I suppose it might teach you a bit about math or fractals in general.
However, I do think fractals are fun as an introduction to programming, and beginners/students are often fascinated by the result, be it more graphic fractals like mandelbrot or julia sets, or more easy to understand L-systems.
Of course, if you're new to programming, it'll also hopefully teach you a lot about programming in general. If nothing else, fractals are interesting to look at.
when I was an undergrad, we used fractal drawing to power our work in parellel processing. It gets fairly computationally intensive quickly, so having multiple CPUs available to do the work lets you see a visible increase in efficiency.
So, along with recursion, I'd say it helps with learning how to balance CPU load across parallel processors.
... or if the equipment isn't available, it probably teaches you Zen-like patience. :)
Great idea! I think coding up fractals makes a great "etude" (study) sized program. It has some nice features this way: generally you won't require much 3rd party code, they can be implemented in a reasonably short amount of time (and complexity) and you get something nice too look at in the end which also verifies your work.
Also there are loads of basic issues in both mathematics and the implementation of numerical algorithms that you will bump into if you do this.
From something as simple as a basic Mandelbrot set generator you can branch out into all sorts of issues as commenters have mentioned. Even sticking with just that, though, you can learn about optimization techniques (why is my generator so slow) and numerical issues (why can't I zoom past here), but also if you want to a bit of color theory (what's Lab* space anyway) and other bits and pieces.
Have fun!
Fractals is a very intellectually interesting topic and well even the simplest implementation will make you learn about complex number maths,graphics generation,scaling images, and general programming.
This question already has answers here:
Closed 14 years ago.
Duplicate of How to program a fractal
What are fractals?
Is this is one of the concepts that is brought over from Mathematics to programming to simplify or solve a particular set of problems?
I am closing this question and have posted a related question
If you want to know about fractals in a general non-programming way, I would suggest looking at a general non-programming site. Wikipedia has a good article on them. If you want to know about programming fractals, I would suggest looking at this already asked question:
How to program a fractal
It even has a fractal tag.
A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.
A fractal often has the following features:
It has a fine structure at arbitrarily small scales.
It is too irregular to be easily described in traditional Euclidean geometric language.
It is self-similar (at least approximately or stochastically).
It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
It has a simple and recursive definition.
http://en.wikipedia.org/wiki/Fractal
its a type of self-similar shape, often grounded in a repeated mathematical function (but not necessarily). It has nothing to do with programming technique, but the easiest way to view one is to write a program to draw it. (drawing a fractal with pen-and-paper is pretty time-consuming)
By 'self-similar' i mean, if you keep zooming in on different parts of the fractal, it doesn't get any "smoother" or more linear, as would happen with a non-fractal shape. It's degree of complexity is invariant of the zoom level.
the Wikipedia page is pretty useful
Look up Procedural Generation for one way of how fractals are used in programming. They are an excellent way of generating chaotic/seemingly complex data from a very simple source. The generated data often benefits from self-similarity and other bits of organzation that make the content make more sense to people.