I'm trying to determine the best DSP method for what I'm trying to accomplish, which is the following:
In real-time, detect the presence of a frequency from a set of different predefined frequencies (no more than 40 different frequencies all within a 1000Hz range). I need to be able to do this even when there are other frequencies (outside of this set or range) that are more dominant.
It is my understanding that FFT might not be the best method for this, because it tells you the most dominant frequency (magnitude) at any given time. This seems like it wouldn't work because if I'm trying to detect say a frequency at 1650Hz (which is present), but there's also a frequency at 500Hz which is stronger, then it's not going to tell me the current frequency is 1650Hz.
I've heard that maybe the Goertzel algorithm might be better for what I'm trying to do, which is to detect single frequencies or a set of frequencies in real-time, even within sounds that have more dominant frequencies than the ones trying to be detected .
Any guidance is greatly appreciated and please correct me if I'm wrong on these assumptions. Thanks!
In vague and somewhat inaccurate terms, the output of the FFT is the magnitude and phase of all[1] frequencies. That is, your statement, "[The FFT] tells you the most dominant frequency (magnitude) at any given time" is incorrect. The FFT is often used as a first step to determine the most dominant frequency, but that's not what it does. In fact, if you are interested in the most dominant frequency, you need to take extra steps over and beyond the FFT: you take the magnitude of all frequencies output by the FFT, and then find the maximum. The corresponding frequency is the dominant frequency.
For your application as I understand it, the FFT is the correct algorithm.
The Goertzel algorithm is closely related to the FFT. It allows for some optimization over the FFT if you are only interested in the magnitude and/or phase of a small subset of frequencies. It might be the right choice for your application depending on the number of frequencies in question, but only as an optimization -- other than performance, it won't solve any problems the FFT won't solve. Because there is more written about the FFT, I suggest you start there and use the Goertzel algorithm only if the FFT proves to not be fast enough and you can establish the Goertzel will be faster in your case.
[1] For practical purposes, what's most inaccurate about this statement is that the frequencies are grouped together in "bins". There's a limited resolution to the analysis which depends on a variety of factors.
I am leaving my other answer as-is because I think it stands on it's own.
Based on your comments and private email, the problem you are facing is most likely this: sounds, like speech, that are principally in one frequency range, have harmonics that stretch into higher frequency ranges. This problem is exacerbated by low quality microphones and electronics, but it is not caused by them and wouldn't go away even with perfect equipment. Once your signal is cluttered with noise in the same band, you can't really distinguish on from off in a simple and reliable way, because on could be caused by the noise. You could try to do some adaptive thresholding based on noise in other bands, and you'll probably get somewhere, but that's no way to build a robust system.
There are a number of ways to solve this problem, but they all involve modulating your signal and using error detection and correction. Basically, you are building a modem and/or radio. Ultimately, what I'm saying is this: you can't solve your problem on the detector alone. You need to build some redundancy into your signal, and you may need to think about other methods of detection. I know of three methods of sending complex signals:
Amplitude modulation, which is what it sounds like you are doing now.
Frequency modulation, which tends to be more robust in the face of ambient noise. (compare FM and AM radio)
Phase modulation, which is more subtle and tricky.
These methods can be combined and multiplexed in various ways. Read about them on wikipedia. Moreover, once your base signal is transmitted, you can add error correction and detection on top.
I am not an expert in this area, but off the top of my head, I am not sure you'll be able to use PM silently, and AM is simply too sensitive to noise, as you've discovered, although it might work with the right kind of redundancy. FM is probably your best bet.
I am designing a RPG game like final fantasy.
I have the programming part done but what I lack is the maths. I am ok at maths but I am having trouble incorporating the players stas into mu sums.
How can I make an action timer that is based on the players speed?
How can I use attack and defence so that it is not always exactly the same damage?
How can I add randomness into the equations?
Can anyone point me to some resources that I can read to learn this sort of stuff.
EDIT: Clarification Of what I am looking for
for the damage I have (player attack x move strength) / enemy defence.
This works and scales well but i got a look at the algorithms from final fantasy 4 a while a got and this sum alone was over 15 steps. mine has only 2.
I am looking for real game examples if possible but would settle for papers or books that have sections that explain how they get these complex sums and why they don't use simple ones.
I eventually intent to implement but am looking for more academic knowledge at the moment.
Not knowing Final fantasy at all, here are some thoughts.
Attack/Defence could either be a 'chance to hit/block' or 'damage done/mitigated' (or, possibly, a blend of both). If you decide to go for 'damage done/mitigated', you'll probably want to do one of:
Generate a random number in a suitable range, added/subtracted from the base attack/defence value.
Generate a number in the range 0-1, multiplied by the attack/defence
Generate a number (with a Gaussian or Poisson distribution and a suitable standard deviation) in the range 0-2 (or so, to account for the occasional crit), multiplied by the attack/defence
For attack timers, decide what "double speed" and "triple speed" should do for the number of attacks in a given time. That should give you a decent lead for how to implement it. I can, off-hand, think of three methods.
Use N/speed as a base for the timer (that means double/triple speed gives 2/3 times the number of attacks in a given interval).
Use Basetime - Speed as the timer (requires a cap on speed, may not be an issue, most probably has an unintuitive relation between speed stat and timer, not much difference at low levels, a lot of difference at high levels).
Use Basetime - Sqrt(Speed) as the timer.
I doubt you'll find academic work on this. Determining formulae for damage, say, is heuristic. People just make stuff up based on their experience with various functions and then tweak the result based on gameplay.
It's important to have a good feel for what the function looks like when plotted on a graph. The best advice I can give for this is to study a course on sketching graphs of functions. A Google search on "sketching functions" will get you started.
Take a look at printed role playing games like Dungeons & Dragons and how they handle these issues. They are the inspiration for computer RPGs. I don't know of academic work
Some thoughts: you don't have to have an actual "formula". It can be rules like "roll a 20 sided die, weapon does 2 points of damage if the roll is <12 and 3 points of damage if the roll is >=12".
You might want to simplify continuous variables down to small ranges of integers for testing. That way you can calculate out tables with all the possible permutations and see if the results look reasonable. Once you have something good, you can interpolate the formulas for continuous inputs.
Another key issue is play balance. There aren't necessarily formulas for telling you whether your game mechanics are balanced, you have to test.
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For example, math logic, graph theory.
Everyone around tells me that math is necessary for programmer. I saw a lot of threads where people say that they used linear algebra and some other math, but no one described concrete cases when they used it.
I know that there are similar threads, but I couldn't see any description of such a case.
Computer graphics.
It's all matrix multiplication, vector spaces, affine spaces, projection, etc. Lots and lots of algebra.
For more information, here's the Wikipedia article on projection, along with the more specific case of 3D projection, with all of its various matrices. OpenGL, a common computer graphics library, is an example of applying affine matrix operations to transform and project objects onto a computer screen.
I think that a lot of programmers use more math than they think they do. It's just that it comes so intuitively to them that they don't even think about it. For instance, every time you write an if statement are you not using your Discrete Math knowledge?
In graphic world you need a lot of transformations.
In cryptography you need geometry and number theory.
In AI, you need algebra.
And statistics in financial environments.
Computer theory needs math theory: actually almost all the founders are from Maths.
Given a list of locations with latitudes and longitudes, sort the list in order from closest to farthest from a specific position.
All applications that deal with money need math.
I can't think of a single app that I have written that didn't require math at some point.
I wrote a parser compiler a few months back, and that's full of graph-theory. This was only designed to be slightly more powerful than regular expressions (in that multiple matches were allowed, and some other features were added), but even such a simple compiler requires loop detection, finite state automata, and tons more math.
Implementing the Advanced Encryption Standard (AES) algorithm required some basic understanding of finite field math. See act 4 of my blog post on it for details (code sample included).
I've used a lot of algebra when writing business apps.
Simple Examples
BMI = weight / (height * height);
compensation = 10 * hours * ((pratio * 2.3) + tratio);
A few years ago, I had a DSP project that had to compute a real radix-2 FFT of size N, in a given time. The vendor-supplied real radix-2 FFT wouldn't run in the allocated time, but their complex FFT of size N/2 would. It is easy to feed the real data into the complex FFT. Getting the answers out afterwards is not so easy: it is called post-weaving, or post-unweaving, or unweaving. Deriving the unweave equations from the FFT and complex number theory was not fun. Going from there to tightly-optimized DSP code was equally not fun.
Naturally, the signal I was measuring did not match the FFT sample size, which causes artifacts. The standard fix is to apply a Hanning window. This causes other artifacts. As part of understanding (and testing) that code, I had to understand the artifacts caused by the Hanning window, so I could interpret the results and decide whether the code was working or not.
I've used tons of math in various projects, including:
Graph theory for dealing with dependencies in large systems (e.g. a Makefile is a kind of directed graph)
Statistics and linear regression in profiling performance bottlenecks
Coordinate transformations in geospatial applications
In scientific computing, project requirements are often stated in algebraic form, especially for computationally intensive code
And that's just off the top of my head.
And of course, anything involving "pure" computer science (algorithms, computational complexity, lambda calculus) tends to look more and more like math the deeper you go.
In answering this image-comparison-algorithm question, I drew on lots of knowledge of math, some of it from other answers and web searches (where I had to apply my own knowledge to filter the information), and some from my own engineering training and lengthy programming background.
General Mindforming
Solving Problems - One fundamental method of math, independent of the area, is transofrming an unknown problem into a known one. Even if you don't have the same problems, you need the same skill. In math, as in programming, virtually everything has different representations. Understanding the equivalence between algorithms, problems or solutions that are completely different on the surface helps you avoid the hard parts.
(A similar thing happens in physics: to solve a kinematic problem, choice of the coordinate system is often the difference between one and ten pages full of formulas, even though problem and solution are identical.)
Precision of Language / Logical reasoning - Math has a very terse yet precise language. Learning to deal with that will prepare you for computers doing what you say, not what you meant. Also, the same precision is required to analyse if a specification is sufficient, to check a piece of code if it covers all possible cases, etc.
Beauty and elegance - This may be the argument that's hardest to grasp. I found the notion of "beauty" in code is very close to the one found in math. A beautiful proof is one whose idea is immediately convincing, and the proof itself is merely executing a sequence of executing the next obvious step.
The same goes for an elegant implementation.
(Most mathematicians I've encountered have a faible for putting the "Aha!" - effect at the end rather than at the beginning. As have most elite geeks).
You can learn these skills without one lesson of math, of course. But math ahs perfected this for centuries.
Applied Skills
Examples:
- Not having to run calc.exe for a quick estimation of memory requirements
- Some basic statistics to tell a valid performance measurement from a shot in the dark
- deducing a formula for a sequence of values, rather than hardcoding them
- Getting a feeling for what c*O(N log N) means.
- Recursion is the same as proof by inductance
(that list would probably go on if I'd actively watch myself for items for a day. This part is admittedly harder than I thought. Further suggestions welcome ;))
Where I use it
The company I work for does a lot of data acquisition, and our claim to fame (comapred to our competition) is the brain muscle that goes into extracting something useful out of the data. While I'm mostly unconcerned with that, I get enough math thrown my way. Before that, I've implemented and validated random number generators for statistical applications, implemented a differential equation solver, wrote simulations for selected laws of physics. And probably more.
I wrote some hash functions for mapping airline codes and flight numbers with good efficiency into a fairly limited number of data slots.
I went through a fair number of primes before finding numbers that worked well with my data. Testing required some statistics and estimates of probabilities.
In machine learning: we use Bayesian (and other probabilistic) models all the time, and we use quadratic programming in the form of Support Vector Machines, not to mention all kinds of mathematical transformations for the various kernel functions. Calculus (derivatives) factors into perceptron learning. Not to mention a whole theory of determining the accuracy of a machine learning classifier.
In artifical intelligence: constraint satisfaction, and logic weigh very heavily.
I was using co-ordinate geometry to solve a problem of finding the visible part of a stack of windows, not exactly overlapping on one another.
There are many other situations, but this is the one that I got from the top of my head. Inherently all operations that we do is mathematics or at least depends on/related to mathematics.
Thats why its important to know mathematics to have a more clearer understanding of things :)
Infact in some cases a lot of math has gone into our common sense that we don't notice that we are using math to solve a particular problem, since we have been using it for so long!
Thanks
-Graphics (matrices, translations, shaders, integral approximations, curves, etc, etc,...infinite dots)
-Algorithm Complexity calculations (specially in line of business' applications)
-Pointer Arithmetics
-Cryptographic under field arithmetics etc.
-GIS (triangles, squares algorithms like delone, bounding boxes, and many many etc)
-Performance monitor counters and the functions they describe
-Functional Programming (simply that, not saying more :))
-......
I used Combinatorials to stuff 20 bits of data into 14 bits of space.
Machine Vision or Computer Vision requires a thorough knowledge of probability and statistics. Object detection/recognition and many supervised segmentation techniques are based on Bayesian inference. Heavy on linear algebra too.
As an engineer, I'm trying really hard to think of an instance when I did not need math. Same story when I was a grad student. Granted, I'm not a programmer, but I use computers a lot.
Games and simulations need lots of maths - fluid dynamics, in particular, for things like flames, fog and smoke.
As an e-commerce developer, I have to use math every day for programming. At the very least, basic algebra.
There are other apps I've had to write for vector based image generation that require a strong knowledge of Geometry, Calculus and Trigonometry.
Then there is bit-masking...
Converting hexadecimal to base ten in your head...
Estimating load potential of an application...
Yep, if someone is no good with math, they're probably not a very good programmer.
Modern communications would completely collapse without math. If you want to make your head explode sometime, look up Galois fields, error correcting codes, and data compression. Then symbol constellations, band-limited interpolation functions (I'm talking about sinc and raised-cosine functions, not the simple linear and bicubic stuff), Fourier transforms, clock recovery, minimally-ambiguous symbol training sequences, Rayleigh and/or Ricean fading, and Kalman filtering. All of those involve math that makes my head hurt bad, and I got a Masters in Electrical Engineering. And that's just off the top of my head, from my wireless communications class.
The amount of math required to make your cell phone work is huge. To make a 3G cell phone with Internet access is staggering. To prove with sufficient confidence that an algorithm will work in most all cases sometimes takes people's careers.
But... if you're only ever going to work with this stuff as black boxes imported from a library (at their mercy, really), well, you might get away with just knowing enough algebra to debug mismatched parentheses. And there are a lot more of those jobs than the hard ones... but at the same time, the hard jobs are harder to find a replacement for.
Examples that I've personally coded:
wrote a simple video game where one spaceship shoots a laser at another ship. To know if the ship was in the laser's path, I used basic algebra y=mx+b to calculate if the paths intersect. (I was a child when I did this and was quite amazed that something that was taught on a chalkboard (algebra) could be applied to computer programming.)
calculating mortgage balances and repayment schedules with logarithms
analyzing consumer buying choices by calculating combinatorics
trigonometry to simulate camera lens behavior
Fourier Transform to analyze digital music files (WAV files)
stock market analysis with statistics (linear regressions)
using logarithms to understand binary search traversals and also disk space savings when using packing information into bit fields. (I don't calculate logarithms in actual code, but I figure them out during "design" to see if it's feasible to even bother coding it.)
None of my projects (so far) have required topics such as calculus, differential equations, or matrices. I didn't study mathematics in school but if a project requires math, I just reference my math books and if I'm stuck, I search google.
Edited to add: I think it's more realistic for some people to have a programming challenge motivate the learning of particular math subjects. For others, they enjoy math for its own sake and can learn it ahead of time to apply to future programming problems. I'm of the first type. For example, I studied logarithms in high school but didn't understand their power until I started doing programming and all of sudden, they seem to pop up all over the place.
The recurring theme I see from these responses is that this is clearly context-dependent.
If you're writing a 3D graphics engine then you'd be well advised to brush up on your vectors and matrices. If you're writing a simple e-commerce website then you'll get away with basic algebra.
So depending on what you want to do, you may not need any more math than you did to post your question(!), or you might conceivably need a PhD (i.e. you would like to write a custom geometry kernel for turbine fan blade design).
One time I was writing something for my Commodore 64 (I forget what, I must have been 6 years old) and I wanted to center some text horizontally on the screen.
I worked out the formula using a combination of math and trial-and-error; years later I would tackle such problems using actual algebra.
Drawing, moving, and guidance of missiles and guns and lasers and gravity bombs and whatnot in this little 2d video game I made: wordwarvi
Lots of uses sine/cosine, and their inverses, (via lookup tables... I'm old, ok?)
Any geo based site/app will need math. A simple example is "Show me all Bob's Pizzas within 10 miles of me" functionality on a website. You will need math to return lat/lons that occur within a 10 mile radius.
This is primarily a question whose answer will depend on the problem domain. Some problems require oodles of math and some require only addition and subtraction. Right now, I have a pet project which might require graph theory, not for the math so much as to get the basic vocabulary and concepts in my head.
If you're doing flight simulations and anything 3D, say hello to quaternions! If you're doing electrical engineering, you will be using trig and complex numbers. If you're doing a mortgage calculator, you will be doing discrete math. If you're doing an optimization problem, where you attempt to get the most profits from your widget factory, you will be doing what is called linear programming. If you are doing some operations involving, say, network addresses, welcome to the kind of bit-focused math that comes along with it. And that's just for the high-level languages.
If you are delving into highly-optimized data structures and implementing them yourself, you will probably do more math than if you were just grabbing a library.
Part of being a good programmer is being familiar with the domain in which you are programming. If you are working on software for Fidelity Mutual, you probably would need to know engineering economics. If you are developing software for Gallup, you probably need to know statistics. LucasArts... probably Linear Algebra. NASA... Differential Equations.
The thing about software engineering is you are almost always expected to wear many hats.
More or less anything having to do with finding the best layout, optimization, or object relationships is graph theory. You may not immediately think of it as such, but regardless - you're using math!
An explicit example: I wrote a node-based shader editor and optimizer, which took a set of linked nodes and converted them into shader code. Finding the correct order to output the code in such that all inputs for a certain node were available before that node needed them involved graph theory.
And like others have said, anything having to do with graphics implicitly requires knowledge of linear algebra, coordinate spaces transformations, and plenty of other subtopics of mathematics. Take a look at any recent graphics whitepaper, especially those involving lighting. Integrals? Infinite series?! Graph theory? Node traversal optimization? Yep, all of these are commonly used in graphics.
Also note that just because you don't realize that you're using some sort of mathematics when you're writing or designing software, doesn't mean that you aren't, and actually understanding the mathematics behind how and why algorithms and data structures work the way they do can often help you find elegant solutions to non-trivial problems.
In years of webapp development I didn't have much need with the Math API. As far as I can recall, I have ever only used the Math#min() and Math#max() of the Math API.
For example
if (i < 0) {
i = 0;
}
if (i > 10) {
i = 10;
}
can be done as
i = Math.max(0, Math.min(i, 10));