I'm writing a tone generator program for a microcontroller.
I use an hardware timer to trigger an interrupt and check if I need to set the signal to high or low in a particular moment for a given note.
I'm using pretty limited hardware, so the slower I run the timer the more time I have to do other stuff (serial communication, loading the next notes to generate, etc.).
I need to find the frequency at which I should run the timer to have an optimal result, which is, generate a frequency that is accurate enough and still have time to compute the other stuff.
To achieve this, I need to find an approximate (within some percent value, as the higher are the frequencies the more they need to be imprecise in value for a human ear to notice the error) LCM of all the frequencies I need to play: this value will be the frequency at which to run the hardware timer.
Is there a simple enough algorithm to compute such number? (EDIT, I shall clarify "simple enough": fast enough to run in a time t << 1 sec. for less than 50 values on a 8 bit AVR microcontroller and implementable in a few dozens of lines at worst.)
LCM(a,b,c) = LCM(LCM(a,b),c)
Thus you can compute LCMs in a loop, bringing in frequencies one at a time.
Furthermore,
LCM(a,b) = a*b/GCD(a,b)
and GCDs are easily computed without any factoring by using the Euclidean algorithm.
To make this an algorithm for approximate LCMs, do something like round lower frequencies to multiples of 10 Hz and higher frequencies to multiples of 50 Hz. Another idea that is a bit more principled would be to first convert the frequency to an octave (I think that the formula is f maps to log(f/16)/log(2)) This will give you a number between 0 and 10 (or slightly higher --but anything above 10 is almost beyond human hearing so you could perhaps round down). You could break 0-10 into say 50 intervals 0.0, 0.2, 0.4, ... and for each number compute ahead of time the frequency corresponding to that octave (which would be f = 16*2^o where o is the octave). For each of these -- go through by hand once and for all and find a nearby round number that has a number of smallish prime factors. For example, if o = 5.4 then f = 675.58 -- round to 675; if o = 5.8 then f = 891.44 -- round to 890. Assemble these 50 numbers into a sorted array, using binary search to replace each of your frequencies by the closest frequency in the array.
An idea:
project the frequency range to a smaller interval
Let's say your frequency range is from 20 to 20000 and you aim for a 2% accurary, you'll calculate for a 1-50 range. It has to be a non-linear transformation to keep the accurary for lower frequencies. The goal is both to compute the result faster and to have a smaller LCM.
Use a prime factors table to easily compute the LCM on that reduced range
Store the pre-calculated prime factors powers in an array (size about 50x7 for range 1-50), and then use it for the LCM: the LCM of a number is the product of multiplying the highest power of each prime factor of the number together. It's easy to code and blazingly fast to run.
Do the first step in reverse to get the final number.
Related
I need to calculate the FFT of audiodata in an Android Project and I use jTransforms to achieve this.
The samples of the audiodata are a few seconds long and are recorded with a samplerate of 11025 Hertz.
I am not sure how to set the length of the FFT in jTransforms.
I do not really need high frequency resolution, so a size of 1024 would be enough.
But from what I have understood learning about the FFT, if I decrease the FFT size F and use a sample with a lenght of N > F, only the first F values of the original sample are transformed.
Is that true or did I understand something wrong?
If it is true, is there an efficient way to tranform the whole signal and decreasing the FFT-Size afterwards?
I need this to classify different signals using Support Vector Machines, and FFT-Sizes > 1024 would give me too much features as output, so I would have to reduce the result of the FFT to a more compact vector.
If you only want the FFT magnitude results, then use the FFT repeatedly on successive 1024 chunk lengths of data, and vector sum all the successive magnitude results to get an estimate for the entire much longer signal.
See Welch's Method on estimating spectral density for an explanation of why this might be a useful technique.
Im not familiar with the jTransform library, but do you really set the size of the transform before calculating it? Amplitude values of the time-domain signal and the sampling frequency (11.025 kHz) is enough to calculate the FFT (note that the FFT assumes constant sampling rate)
The resolution in frequency domain will be determined by Nyquist's theorem; the maximum resolvable frequency in your signal will be equal to half your sampling rate. In other words, sampling your signal with 11.025 kHz, you can expect your frequency graph to contain frequency values (and corresponding amplitudes) between 0 Hz - 5.5125 kHz.
UPDATE:
The resolution of the FFT (the narrowness of the frequency bins) will increase/improve if your input signal is longer, thus 1024 samples might not be a long sequence enough if you need to distinguish between very small changes in frequency. If thats not a problem for you application, and the nature of your data is not variying quickly, and you have the processing time, then taking an average of 3-4 FFT estimates will greatly reduce noise and improve estimates.
I'v got some problem to understand the difference between Logarithmic(Lcc) and Uniform(Ucc) cost criteria and also how to use it in calculations.
Could someone please explain the difference between the two and perhaps show how to calculate the complexity for a problem like A+B*C
(Yes this is part of an assignment =) )
Thx for any help!
/Marthin
Uniform Cost Criteria assigns a constant cost to every machine operation regardless of the number of bits involved WHILE Logarithm Cost Criteria assigns a cost to every machine operation proportional to the number of bits involved
Problem size influence complexity
Since complexity depends on the size of the
problem we define complexity to be a function
of problem size
Definition: Let T(n) denote the complexity for
an algorithm that is applied to a problem of
size n.
The size (n) of a problem instance (I) is the
number of (binary) bits used to represent the
instance. So problem size is the length of the
binary description of the instance.
This is called Logarithmic cost criteria
Unit Cost Criteria
If you assume that:
- every computer instruction takes one time
unit,
- every register is one storage unit
- and that a number always fits in a register
then you can use the number of inputs as
problem size since the length of input (in bits)
will be a constant times the number of inputs.
Uniform cost criteria assume that every instruction takes a single unit of time and that every register requires a single unit of space.
Logarithmic cost criteria assume that every instruction takes a logarithmic number of time units (with respect to the length of the operands) and that every register requires a logarithmic number of units of space.
In simpler terms, what this means is that uniform cost criteria count the number of operations, and logarithmic cost criteria count the number of bit operations.
For example, suppose we have an 8-bit adder.
If we're using uniform cost criteria to analyze the run-time of the adder, we would say that addition takes a single time unit; i.e., T(N)=1.
If we're using logarithmic cost criteria to analyze the run-time of the adder, we would say that addition takes lgn time units; i.e., T(N)=lgn, where n is the worst case number you would have to add in terms of time complexity (in this example, n would be 256). Thus, T(N)=8.
More specifically, say we're adding 256 to 32. To perform the addition, we have to add the binary bits together in the 1s column, the 2s column, the 4s column, etc (columns meaning the bit locations). The number 256 requires 8 bits. This is where logarithms come into our analysis. lg256=8. So to add the two numbers, we have to perform addition on 8 columns. Logarithmic cost criteria say that each of these 8 addition calculations takes a single unit of time. Uniform cost criteria say that the entire set of 8 addition calculations takes a single unit of time.
Similar analysis can be made in terms of space as well. Registers either take up a constant amount of space (under uniform cost criteria) or a logarithmic amount of space (under uniform cost criteria).
I think you should do some research on Big O notation... http://en.wikipedia.org/wiki/Big_O_notation#Orders_of_common_functions
If there is a part of the description you find difficult edit your question.
I searched the site but did not find exactly what I was looking for... I wanted to generate a discrete random number from normal distribution.
For example, if I have a range from a minimum of 4 and a maximum of 10 and an average of 7. What code or function call ( Objective C preferred ) would I need to return a number in that range. Naturally, due to normal distribution more numbers returned would center round the average of 7.
As a second example, can the bell curve/distribution be skewed toward one end of the other? Lets say I need to generate a random number with a range of minimum of 4 and maximum of 10, and I want the majority of the numbers returned to center around the number 8 with a natural fall of based on a skewed bell curve.
Any help is greatly appreciated....
Anthony
What do you need this for? Can you do it the craps player's way?
Generate two random integers in the range of 2 to 5 (inclusive, of course) and add them together. Or flip a coin (0,1) six times and add 4 to the result.
Summing multiple dice produces a normal distribution (a "bell curve"), while eliminating high or low throws can be used to skew the distribution in various ways.
The key is you are going for discrete numbers (and I hope you mean integers by that). Multiple dice throws famously generate a normal distribution. In fact, I think that's how we were first introduced to the Gaussian curve in school.
Of course the more throws, the more closely you approximate the bell curve. Rolling a single die gives a flat line. Rolling two dice just creates a ramp up and down that isn't terribly close to a bell. Six coin flips gets you closer.
So consider this...
If I understand your question correctly, you only have seven possible outcomes--the integers (4,5,6,7,8,9,10). You can set up an array of seven probabilities to approximate any distribution you like.
Many frameworks and libraries have this built-in.
Also, just like TokenMacGuy said a normal distribution isn't characterized by the interval it's defined on, but rather by two parameters: Mean μ and standard deviation σ. With both these parameters you can confine a certain quantile of the distribution to a certain interval, so that 95 % of all points fall in that interval. But resticting it completely to any interval other than (−∞, ∞) is impossible.
There are several methods to generate normal-distributed values from uniform random values (which is what most random or pseudorandom number generators are generating:
The Box-Muller transform is probably the easiest although not exactly fast to compute. Depending on the number of numbers you need, it should be sufficient, though and definitely very easy to write.
Another option is Marsaglia's Polar method which is usually faster1.
A third method is the Ziggurat algorithm which is considerably faster to compute but much more complex to program. In applications that really use a lot of random numbers it may be the best choice, though.
As a general advice, though: Don't write it yourself if you have access to a library that generates normal-distributed random numbers for you already.
For skewing your distribution I'd just use a regular normal distribution, choosing μ and σ appropriately for one side of your curve and then determine on which side of your wanted mean a point fell, stretching it appropriately to fit your desired distribution.
For generating only integers I'd suggest you just round towards the nearest integer when the random number happens to fall within your desired interval and reject it if it doesn't (drawing a new random number then). This way you won't artificially skew the distribution (such as you would if you were clamping the values at 4 or 10, respectively).
1 In testing with deliberately bad random number generators (yes, worse than RANDU) I've noticed that the polar method results in an endless loop, rejecting every sample. Won't happen with random numbers that fulfill the usual statistic expectations to them, though.
Yes, there are sophisticated mathematical solutions, but for "simple but practical" I'd go with Nosredna's comment. For a simple Java solution:
Random random=new Random();
public int bell7()
{
int n=4;
for (int x=0;x<6;++x)
n+=random.nextInt(2);
return n;
}
If you're not a Java person, Random.nextInt(n) returns a random integer between 0 and n-1. I think the rest should be similar to what you'd see in any programming language.
If the range was large, then instead of nextInt(2)'s I'd use a bigger number in there so there would be fewer iterations through the loop, depending on frequency of call and performance requirements.
Dan Dyer and Jay are exactly right. What you really want is a binomial distribution, not a normal distribution. The shape of a binomial distribution looks a lot like a normal distribution, but it is discrete and bounded whereas a normal distribution is continuous and unbounded.
Jay's code generates a binomial distribution with 6 trials and a 50% probability of success on each trial. If you want to "skew" your distribution, simply change the line that decides whether to add 1 to n so that the probability is something other than 50%.
The normal distribution is not described by its endpoints. Normally it's described by it's mean (which you have given to be 7) and its standard deviation. An important feature of this is that it is possible to get a value far outside the expected range from this distribution, although that will be vanishingly rare, the further you get from the mean.
The usual means for getting a value from a distribution is to generate a random value from a uniform distribution, which is quite easily done with, for example, rand(), and then use that as an argument to a cumulative distribution function, which maps probabilities to upper bounds. For the standard distribution, this function is
F(x) = 0.5 - 0.5*erf( (x-μ)/(σ * sqrt(2.0)))
where erf() is the error function which may be described by a taylor series:
erf(z) = 2.0/sqrt(2.0) * Σ∞n=0 ((-1)nz2n + 1)/(n!(2n + 1))
I'll leave it as an excercise to translate this into C.
If you prefer not to engage in the exercise, you might consider using the Gnu Scientific Library, which among many other features, has a technique to generate random numbers in one of many common distributions, of which the Gaussian Distribution (hint) is one.
Obviously, all of these functions return floating point values. You will have to use some rounding strategy to convert to a discrete value. A useful (but naive) approach is to simply downcast to integer.
I've tried the typical physics equations for this but none of them really work because the equations deal with constant acceleration and mine will need to change to work correctly. Basically I have a car that can be going at a large range of speeds and needs to slow down and stop over a given distance and time as it reaches the end of its path.
So, I have:
V0, or the current speed
Vf, or the speed I want to reach (typically 0)
t, or the amount of time I want to take to reach the end of my path
d, or the distance I want to go as I change from V0 to Vf
I want to calculate
a, or the acceleration needed to go from V0 to Vf
The reason this becomes a programming-specific question is because a needs to be recalculated every single timestep as the car keeps stopping. So, V0 constantly is changed to be V0 from last timestep plus the a that was calculated last timestep. So essentially it will start stopping slowly then will eventually stop more abruptly, sort of like a car in real life.
EDITS:
All right, thanks for the great responses. A lot of what I needed was just some help thinking about this. Let me be more specific now that I've got some more ideas from you all:
I have a car c that is 64 pixels from its destination, so d=64. It is driving at 2 pixels per timestep, where a timestep is 1/60 of a second. I want to find the acceleration a that will bring it to a speed of 0.2 pixels per timestep by the time it has traveled d.
d = 64 //distance
V0 = 2 //initial velocity (in ppt)
Vf = 0.2 //final velocity (in ppt)
Also because this happens in a game loop, a variable delta is passed through to each action, which is the multiple of 1/60s that the last timestep took. In other words, if it took 1/60s, then delta is 1.0, if it took 1/30s, then delta is 0.5. Before acceleration is actually applied, it is multiplied by this delta value. Similarly, before the car moves again its velocity is multiplied by the delta value. This is pretty standard stuff, but it might be what is causing problems with my calculations.
Linear acceleration a for a distance d going from a starting speed Vi to a final speed Vf:
a = (Vf*Vf - Vi*Vi)/(2 * d)
EDIT:
After your edit, let me try and gauge what you need...
If you take this formula and insert your numbers, you get a constant acceleration of -0,0309375. Now, let's keep calling this result 'a'.
What you need between timestamps (frames?) is not actually the acceleration, but new location of the vehicle, right? So you use the following formula:
Sd = Vi * t + 0.5 * t * t * a
where Sd is the current distance from the start position at current frame/moment/sum_of_deltas, Vi is the starting speed, and t is the time since the start.
With this, your decceleration is constant, but even if it is linear, your speed will accomodate to your constraints.
If you want a non-linear decceleration, you could find some non-linear interpolation method, and interpolate not acceleration, but simply position between two points.
location = non_linear_function(time);
The four constraints you give are one too many for a linear system (one with constant acceleration), where any three of the variables would suffice to compute the acceleration and thereby determine the fourth variables. However, the system is way under-specified for a completely general nonlinear system -- there may be uncountably infinite ways to change acceleration over time while satisfying all the constraints as given. Can you perhaps specify better along what kind of curve acceleration should change over time?
Using 0 index to mean "at the start", 1 to mean "at the end", and D for Delta to mean "variation", given a linearly changing acceleration
a(t) = a0 + t * (a1-a0)/Dt
where a0 and a1 are the two parameters we want to compute to satisfy all the various constraints, I compute (if there's been no misstep, as I did it all by hand):
DV = Dt * (a0+a1)/2
Ds = Dt * (V0 + ((a1-a0)/6 + a0/2) * Dt)
Given DV, Dt and Ds are all given, this leaves 2 linear equations in the unknowns a0 and a1 so you can solve for these (but I'm leaving things in this form to make it easier to double check on my derivations!!!).
If you're applying the proper formulas at every step to compute changes in space and velocity, it should make no difference whether you compute a0 and a1 once and for all or recompute them at every step based on the remaining Dt, Ds and DV.
If you're trying to simulate a time-dependent acceleration in your equations, it just means that you should assume that. You have to integrate F = ma along with the acceleration equations, that's all. If acceleration isn't constant, you just have to solve a system of equations instead of just one.
So now it's really three vector equations that you have to integrate simultaneously: one for each component of displacement, velocity, and acceleration, or nine equations in total. The force as a function of time will be an input for your problem.
If you're assuming 1D motion you're down to three simultaneous equations. The ones for velocity and displacement are both pretty easy.
In real life, a car's stopping ability depends on the pressure on the brake pedal, any engine braking that's going on, surface conditions, and such: also, there's that "grab" at the end when the car really stops. Modeling that is complicated, and you're unlikely to find good answers on a programming website. Find some automotive engineers.
Aside from that, I don't know what you're asking for. Are you trying to determine a braking schedule? As in there's a certain amount of deceleration while coasting, and then applying the brake? In real driving, the time is not usually considered in these maneuvers, but rather the distance.
As far as I can tell, your problem is that you aren't asking for anything specific, which suggests that you really haven't figured out what you actually want. If you'd provide a sample use for this, we could probably help you. As it is, you've provided the bare bones of a problem that is either overdetermined or way underconstrained, and there's really nothing we can do with that.
if you need to go from 10m/s to 0m/s in 1m with linear acceleration you need 2 equations.
first find the time (t) it takes to stop.
v0 = initial velocity
vf = final velocity
x0 = initial displacement
xf = final displacement
a = constant linear acceleration
(xf-x0)=.5*(v0-vf)*t
t=2*(xf-x0)/(v0-vf)
t=2*(1m-0m)/(10m/s-0m/s)
t=.2seconds
next to calculate the linear acceleration between x0 & xf
(xf-x0)=(v0-vf)*t+.5*a*t^2
(1m-0m)=(10m/s-0m/s)*(.2s)+.5*a*((.2s)^2)
1m=(10m/s)*(.2s)+.5*a*(.04s^2)
1m=2m+a*(.02s^2)
-1m=a*(.02s^2)
a=-1m/(.02s^2)
a=-50m/s^2
in terms of gravity (g's)
a=(-50m/s^2)/(9.8m/s^2)
a=5.1g over the .2 seconds from 0m to 10m
Problem is either overconstrained or underconstrained (a is not constant? is there a maximum a?) or ambiguous.
Simplest formula would be a=(Vf-V0)/t
Edit: if time is not constrained, and distance s is constrained, and acceleration is constant, then the relevant formulae are s = (Vf+V0)/2 * t, t=(Vf-V0)/a which simplifies to a = (Vf2 - V02) / (2s).
Trying to understand an fft (Fast Fourier Transform) routine I'm using (stealing)(recycling)
Input is an array of 512 data points which are a sample waveform.
Test data is generated into this array. fft transforms this array into frequency domain.
Trying to understand relationship between freq, period, sample rate and position in fft array. I'll illustrate with examples:
========================================
Sample rate is 1000 samples/s.
Generate a set of samples at 10Hz.
Input array has peak values at arr(28), arr(128), arr(228) ...
period = 100 sample points
peak value in fft array is at index 6 (excluding a huge value at 0)
========================================
Sample rate is 8000 samples/s
Generate set of samples at 440Hz
Input array peak values include arr(7), arr(25), arr(43), arr(61) ...
period = 18 sample points
peak value in fft array is at index 29 (excluding a huge value at 0)
========================================
How do I relate the index of the peak in the fft array to frequency ?
If you ignore the imaginary part, the frequency distribution is linear across bins:
Frequency#i = (Sampling rate/2)*(i/Nbins).
So for your first example, assumming you had 256 bins, the largest bin corresponds to a frequency of 1000/2 * 6/256 = 11.7 Hz.
Since your input was 10Hz, I'd guess that bin 5 (9.7Hz) also had a big component.
To get better accuracy, you need to take more samples, to get smaller bins.
Your second example gives 8000/2*29/256 = 453Hz. Again, close, but you need more bins.
Your resolution here is only 4000/256 = 15.6Hz.
It would be helpful if you were to provide your sample dataset.
My guess would be that you have what are called sampling artifacts. The strong signal at DC ( frequency 0 ) suggests that this is the case.
You should always ensure that the average value in your input data is zero - find the average and subtract it from each sample point before invoking the fft is good practice.
Along the same lines, you have to be careful about the sampling window artifact. It is important that the first and last data point are close to zero because otherwise the "step" from outside to inside the sampling window has the effect of injecting a whole lot of energy at different frequencies.
The bottom line is that doing an fft analysis requires more care than simply recycling a fft routine found somewhere.
Here are the first 100 sample points of a 10Hz signal as described in the question, massaged to avoid sampling artifacts
> sinx[1:100]
[1] 0.000000e+00 6.279052e-02 1.253332e-01 1.873813e-01 2.486899e-01 3.090170e-01 3.681246e-01 4.257793e-01 4.817537e-01 5.358268e-01
[11] 5.877853e-01 6.374240e-01 6.845471e-01 7.289686e-01 7.705132e-01 8.090170e-01 8.443279e-01 8.763067e-01 9.048271e-01 9.297765e-01
[21] 9.510565e-01 9.685832e-01 9.822873e-01 9.921147e-01 9.980267e-01 1.000000e+00 9.980267e-01 9.921147e-01 9.822873e-01 9.685832e-01
[31] 9.510565e-01 9.297765e-01 9.048271e-01 8.763067e-01 8.443279e-01 8.090170e-01 7.705132e-01 7.289686e-01 6.845471e-01 6.374240e-01
[41] 5.877853e-01 5.358268e-01 4.817537e-01 4.257793e-01 3.681246e-01 3.090170e-01 2.486899e-01 1.873813e-01 1.253332e-01 6.279052e-02
[51] -2.542075e-15 -6.279052e-02 -1.253332e-01 -1.873813e-01 -2.486899e-01 -3.090170e-01 -3.681246e-01 -4.257793e-01 -4.817537e-01 -5.358268e-01
[61] -5.877853e-01 -6.374240e-01 -6.845471e-01 -7.289686e-01 -7.705132e-01 -8.090170e-01 -8.443279e-01 -8.763067e-01 -9.048271e-01 -9.297765e-01
[71] -9.510565e-01 -9.685832e-01 -9.822873e-01 -9.921147e-01 -9.980267e-01 -1.000000e+00 -9.980267e-01 -9.921147e-01 -9.822873e-01 -9.685832e-01
[81] -9.510565e-01 -9.297765e-01 -9.048271e-01 -8.763067e-01 -8.443279e-01 -8.090170e-01 -7.705132e-01 -7.289686e-01 -6.845471e-01 -6.374240e-01
[91] -5.877853e-01 -5.358268e-01 -4.817537e-01 -4.257793e-01 -3.681246e-01 -3.090170e-01 -2.486899e-01 -1.873813e-01 -1.253332e-01 -6.279052e-02
And here is the resulting absolute values of the fft frequency domain
[1] 7.160038e-13 1.008741e-01 2.080408e-01 3.291725e-01 4.753899e-01 6.653660e-01 9.352601e-01 1.368212e+00 2.211653e+00 4.691243e+00 5.001674e+02
[12] 5.293086e+00 2.742218e+00 1.891330e+00 1.462830e+00 1.203175e+00 1.028079e+00 9.014559e-01 8.052577e-01 7.294489e-01
I'm a little rusty too on math and signal processing but with the additional info I can give it a shot.
If you want to know the signal energy per bin you need the magnitude of the complex output. So just looking at the real output is not enough. Even when the input is only real numbers. For every bin the magnitude of the output is sqrt(real^2 + imag^2), just like pythagoras :-)
bins 0 to 449 are positive frequencies from 0 Hz to 500 Hz. bins 500 to 1000 are negative frequencies and should be the same as the positive for a real signal. If you process one buffer every second frequencies and array indices line up nicely. So the peak at index 6 corresponds with 6Hz so that's a bit strange. This might be because you're only looking at the real output data and the real and imaginary data combine to give an expected peak at index 10. The frequencies should map linearly to the bins.
The peaks at 0 indicates a DC offset.
It's been some time since I've done FFT's but here's what I remember
FFT usually takes complex numbers as input and output. So I'm not really sure how the real and imaginary part of the input and output map to the arrays.
I don't really understand what you're doing. In the first example you say you process sample buffers at 10Hz for a sample rate of 1000 Hz? So you should have 10 buffers per second with 100 samples each. I don't get how your input array can be at least 228 samples long.
Usually the first half of the output buffer are frequency bins from 0 frequency (=dc offset) to 1/2 sample rate. and the 2nd half are negative frequencies. if your input is only real data with 0 for the imaginary signal positive and negative frequencies are the same. The relationship of real/imaginary signal on the output contains phase information from your input signal.
The frequency for bin i is i * (samplerate / n), where n is the number of samples in the FFT's input window.
If you're handling audio, since pitch is proportional to log of frequency, the pitch resolution of the bins increases as the frequency does -- it's hard to resolve low frequency signals accurately. To do so you need to use larger FFT windows, which reduces time resolution. There is a tradeoff of frequency against time resolution for a given sample rate.
You mention a bin with a large value at 0 -- this is the bin with frequency 0, i.e. the DC component. If this is large, then presumably your values are generally positive. Bin n/2 (in your case 256) is the Nyquist frequency, half the sample rate, which is the highest frequency that can be resolved in the sampled signal at this rate.
If the signal is real, then bins n/2+1 to n-1 will contain the complex conjugates of bins n/2-1 to 1, respectively. The DC value only appears once.
The samples are, as others have said, equally spaced in the frequency domain (not logarithmic).
For example 1, you should get this:
alt text http://home.comcast.net/~kootsoop/images/SINE1.jpg
For the other example you should get
alt text http://home.comcast.net/~kootsoop/images/SINE2.jpg
So your answers both appear to be correct regarding the peak location.
What I'm not getting is the large DC component. Are you sure you are generating a sine wave as the input? Does the input go negative? For a sinewave, the DC should be close to zero provided you get enough cycles.
Another avenue is to craft a Goertzel's Algorithm of each note center frequency you are looking for.
Once you get one implementation of the algorithm working you can make it such that it takes parameters to set it's center frequency. With that you could easily run 88 of them or what ever you need in a collection and scan for the peak value.
The Goertzel Algorithm is basically a single bin FFT. Using this method you can place your bins logarithmically as musical notes naturally go.
Some pseudo code from Wikipedia:
s_prev = 0
s_prev2 = 0
coeff = 2*cos(2*PI*normalized_frequency);
for each sample, x[n],
s = x[n] + coeff*s_prev - s_prev2;
s_prev2 = s_prev;
s_prev = s;
end
power = s_prev2*s_prev2 + s_prev*s_prev - coeff*s_prev2*s_prev;
The two variables representing the previous two samples are maintained for the next iteration. This can be then used in a streaming application. I thinks perhaps the power calculation should be inside the loop as well. (However it is not depicted as such in the Wiki article.)
In the tone detection case there would be 88 different coeficients, 88 pairs of previous samples and would result in 88 power output samples indicating the relative level in that frequency bin.
WaveyDavey says that he's capturing sound from a mic, thru the audio hardware of his computer, BUT that his results are not zero-centered. This sounds like a problem with the hardware. It SHOULD BE zero-centered.
When the room is quiet, the stream of values coming from the sound API should be very close to 0 amplitude, with slight +- variations for ambient noise. If a vibratory sound is present in the room (e.g. a piano, a flute, a voice) the data stream should show a fundamentally sinusoidal-based wave that goes both positive and negative, and averages near zero. If this is not the case, the system has some funk going on!
-Rick