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  • compute sound FFT in Flex without playing it

    - by paleozogt
    Flex has the SoundMixer.computeSpectrum function that lets you compute an FFT from the currently playing sound. What I'd like to do is compute an FFT without playing the sound. Since Flash 10.1 lets us access the microphone bytes directly, it seems like I should be able to compute the FFT directly off of what the user is speaking.

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  • Cepstral Analysis for pitch detection

    - by Ohmu
    Hi! I'm looking to extract pitches from a sound signal. Someone on IRC just explain to me how taking a double FFT achieves this. Specifically: take FFT take log of square of absolute value (can be done with lookup table) take another FFT take absolute value I am attempting this using vDSP I can't understand how I didn't come across this technique earlier. I did a lot of hunting and asking questions; several weeks worth. More to the point, I can't understand why I didn't think of it. I am attempting to achieve this with vDSP library. it looks as though it has functions to handle all of these tasks. However, I'm wondering about the accuracy of the final result. I have previously used a technique which scours the frequency bins of a single FFT for local maxima. when it encounters one, it uses a cunning technique (the change in phase since the last FFT) to more accurately place the actual peak within the bin. I am worried that this precision will be lost with this technique I'm presenting here. I guess the technique could be used after the second FFT to get the fundamental accurately. But it kind of looks like the information is lost in step 2. as this is a potentially tricky process, could someone with some experience just look over what I'm doing and check it for sanity? also, I've heard there is an alternative technique involving fitting a quadratic over neighbouring bins. Is this of comparable accuracy? if so, I would favour it, as it doesn't involve remembering bin phases. so questions: does this approach makes sense? Can it be improved? I'm a bit worried about And the log square component; there seems to be a vDSP function to do exactly that: vDSP_vdbcon however, there is no indication it precalculates a log-table -- I assume it doesn't, as the FFT function requires an explicit pre-calculation function to be called and passed into it. and this function doesn't. Is there some danger of harmonics being picked up? is there any cunning way of making vDSP pull out the maxima, biggest first? Can anyone point me towards some research or literature on this technique? the main question: is it accurate enough? Can the accuracy be improved? I have just been told by an expert that the accuracy IS INDEED not sufficient. Is this the end of the line? Pi PS I get SO annoyed (npi) when I want to create tags, but cannot. :| I have suggested to the maintainers that SO keep track of attempted tags, but I'm sure I was ignored. we need tags for vDSP, accelerate framework, cepstral analysis

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  • How to remove the boundary effects arising due to zero padding in scipy/numpy fft?

    - by Omkar
    I have made a python code to smoothen a given signal using the Weierstrass transform, which is basically the convolution of a normalised gaussian with a signal. The code is as follows: #Importing relevant libraries from __future__ import division from scipy.signal import fftconvolve import numpy as np def smooth_func(sig, x, t= 0.002): N = len(x) x1 = x[-1] x0 = x[0] # defining a new array y which is symmetric around zero, to make the gaussian symmetric. y = np.linspace(-(x1-x0)/2, (x1-x0)/2, N) #gaussian centered around zero. gaus = np.exp(-y**(2)/t) #using fftconvolve to speed up the convolution; gaus.sum() is the normalization constant. return fftconvolve(sig, gaus/gaus.sum(), mode='same') If I run this code for say a step function, it smoothens the corner, but at the boundary it interprets another corner and smoothens that too, as a result giving unnecessary behaviour at the boundary. I explain this with a figure shown in the link below. Boundary effects This problem does not arise if we directly integrate to find convolution. Hence the problem is not in Weierstrass transform, and hence the problem is in the fftconvolve function of scipy. To understand why this problem arises we first need to understand the working of fftconvolve in scipy. The fftconvolve function basically uses the convolution theorem to speed up the computation. In short it says: convolution(int1,int2)=ifft(fft(int1)*fft(int2)) If we directly apply this theorem we dont get the desired result. To get the desired result we need to take the fft on a array double the size of max(int1,int2). But this leads to the undesired boundary effects. This is because in the fft code, if size(int) is greater than the size(over which to take fft) it zero pads the input and then takes the fft. This zero padding is exactly what is responsible for the undesired boundary effects. Can you suggest a way to remove this boundary effects? I have tried to remove it by a simple trick. After smoothening the function I am compairing the value of the smoothened signal with the original signal near the boundaries and if they dont match I replace the value of the smoothened func with the input signal at that point. It is as follows: i = 0 eps=1e-3 while abs(smooth[i]-sig[i])> eps: #compairing the signals on the left boundary smooth[i] = sig[i] i = i + 1 j = -1 while abs(smooth[j]-sig[j])> eps: # compairing on the right boundary. smooth[j] = sig[j] j = j - 1 There is a problem with this method, because of using an epsilon there are small jumps in the smoothened function, as shown below: jumps in the smooth func Can there be any changes made in the above method to solve this boundary problem?

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  • Using GNU Octave FFT functions

    - by CFP
    Hello world! I'm playing with octave's fft functions, and I can't really figure out how to scale their output: I use the following (very short) code to approximate a function: function y = f(x) y = x .^ 2; endfunction; X=[-4096:4095]/64; Y = f(X); # plot(X, Y); F = fft(Y); S = [0:2047]/2048; function points = approximate(input, count) size = size(input)(2); fourier = [fft(input)(1:count) zeros(1, size-count)]; points = ifft(fourier); endfunction; Y = f(X); plot(X, Y, X, approximate(Y, 10)); Basically, what it does is take a function, compute the image of an interval, fft-it, then keep a few harmonics, and ifft the result. Yet I get a plot that is vertically compressed (the vertical scale of the output is wrong). Any ideas? Thanks!

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  • DSP - Problems using the inverse Fast Fourier Transform

    - by Trap
    I've been playing around a little with the Exocortex implementation of the FFT, but I'm having some problems. First, after calculating the inverse FFT of an unchanged frequency spectrum obtained by a previous forward FFT, one would expect to get the original signal back, but this is not the case. I had to figure out that I needed to scale the FFT output to about 1 / fftLength to get the amplitudes ok. Why is this? Second, whenever I modify the amplitudes of the frequency bins before calling the iFFT the signal gets distorted at low frequencies. However, this does not happen if I attenuate all the bins by the same factor. Let me put a very simplified example of the output buffer of a 4-sample FFT: // Bin 0 (DC) FFTOut[0] = 0.0000610351563 FFTOut[1] = 0.0 // Bin 1 FFTOut[2] = 0.000331878662 FFTOut[3] = 0.000629425049 // Central bin FFTOut[4] = -0.0000381469727 FFTOut[5] = 0.0 // Bin 3, this is a negative frequency bin. FFTOut[6] = 0.000331878662 FFTOut[7] = -0.000629425049 The output is composed of pairs of floats, each representing the real and imaginay parts of a single bin. So, bin 0 (array indexes 0, 1) would represent the real and imaginary parts of the DC frequency. As you can see, bins 1 and 3 both have the same values, (except for the sign of the Im part), so I guess these are the negative frequency values, and finally indexes (4, 5) would be the central frequency bin. To attenuate the frequency bin 1 this is what I do: // Attenuate the 'positive' bin FFTOut[2] *= 0.5; FFTOut[3] *= 0.5; // Attenuate its corresponding negative bin. FFTOut[6] *= 0.5; FFTOut[7] *= 0.5; For the actual tests I'm using a 1024-length FFT and I always provide all the samples so no 0-padding is needed. // Attenuate var halfSize = fftWindowLength / 2; float leftFreq = 0f; float rightFreq = 22050f; for( var c = 1; c < halfSize; c++ ) { var freq = c * (44100d / halfSize); // Calc. positive and negative frequency locations. var k = c * 2; var nk = (fftWindowLength - c) * 2; // This kind of attenuation corresponds to a high-pass filter. // The attenuation at the transition band is linearly applied, could // this be the cause of the distortion of low frequencies? var attn = (freq < leftFreq) ? 0 : (freq < rightFreq) ? ((freq - leftFreq) / (rightFreq - leftFreq)) : 1; mFFTOut[ k ] *= (float)attn; mFFTOut[ k + 1 ] *= (float)attn; mFFTOut[ nk ] *= (float)attn; mFFTOut[ nk + 1 ] *= (float)attn; } Obviously I'm doing something wrong but can't figure out what or where.

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  • Android library to get pitch from WAV file

    - by Sakura
    I have a list of sampled data from the WAV file. I would like to pass in these values into a library and get the frequency of the music played in the WAV file. For now, I will have 1 frequency in the WAV file and I would like to find a library that is compatible with Android. I understand that I need to use FFT to get the frequency domain. Is there any good libraries for that? I found that [KissFFT][1] is quite popular but I am not very sure how compatible it is on Android. Is there an easier and good library that can perform the task I want? EDIT: I tried to use JTransforms to get the FFT of the WAV file but always failed at getting the correct frequency of the file. Currently, the WAV file contains sine curve of 440Hz, music note A4. However, I got the result as 441. Then I tried to get the frequency of G4, I got the result as 882Hz which is incorrect. The frequency of G4 is supposed to be 783Hz. Could it be due to not enough samples? If yes, how much samples should I take? //DFT DoubleFFT_1D fft = new DoubleFFT_1D(numOfFrames); double max_fftval = -1; int max_i = -1; double[] fftData = new double[numOfFrames * 2]; for (int i = 0; i < numOfFrames; i++) { // copying audio data to the fft data buffer, imaginary part is 0 fftData[2 * i] = buffer[i]; fftData[2 * i + 1] = 0; } fft.complexForward(fftData); for (int i = 0; i < fftData.length; i += 2) { // complex numbers -> vectors, so we compute the length of the vector, which is sqrt(realpart^2+imaginarypart^2) double vlen = Math.sqrt((fftData[i] * fftData[i]) + (fftData[i + 1] * fftData[i + 1])); //fd.append(Double.toString(vlen)); // fd.append(","); if (max_fftval < vlen) { // if this length is bigger than our stored biggest length max_fftval = vlen; max_i = i; } } //double dominantFreq = ((double)max_i / fftData.length) * sampleRate; double dominantFreq = (max_i/2.0) * sampleRate / numOfFrames; fd.append(Double.toString(dominantFreq)); Can someone help me out? EDIT2: I manage to fix the problem mentioned above by increasing the number of samples to 100000, however, sometimes I am getting the overtones as the frequency. Any idea how to fix it? Should I use Harmonic Product Frequency or Autocorrelation algorithms?

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  • Output from OouraFFT correct sometimes but completely false other times. Why ?

    - by Yan
    Hi I am using Ooura FFT to compute the FFT of the accelerometer data in windows of 1024 samples. The code works fine, but then for some reason it produces very strange outputs, i.e. continuous spectrum with amplitudes of the order of 10^200. Here is the code: OouraFFT *myFFT=[[OouraFFT alloc] initForSignalsOfLength:1024 NumWindows:10]; // had to allocate it UIAcceleration *tempAccel = nil; double *input=(double *)malloc(1024 * sizeof(double)); double *frequency=(double *)malloc(1024*sizeof(double)); if (input) { //NSLog(@"%d",[array count]); for (int u=0; u<[array count]; u++) { tempAccel = (UIAcceleration *)[array objectAtIndex:u]; input[u]=tempAccel.z; //NSLog(@"%g",input[u]); } } myFFT.inputData=input; // specifies input data to myFFT [myFFT calculateWelchPeriodogramWithNewSignalSegment]; // calculates FFT for (int i=0;i<myFFT.dataLength;i++) // loop to copy output of myFFT, length of spectrumData is half of input data, so copy twice { if (i<myFFT.numFrequencies) { frequency[i]=myFFT.spectrumData[i]; // } else { frequency[i]=myFFT.spectrumData[myFFT.dataLength-i]; // copy twice } } for (int i=0;i<[array count];i++) { TransformedAcceleration *NewAcceleration=[[TransformedAcceleration alloc]init]; tempAccel=(UIAcceleration*)[array objectAtIndex:i]; NewAcceleration.timestamp=tempAccel.timestamp; NewAcceleration.x=tempAccel.x; NewAcceleration.y=tempAccel.z; NewAcceleration.z=frequency[i]; [newcurrentarray addObject:NewAcceleration]; // this does not work //[self replaceAcceleration:NewAcceleration]; //[NewAcceleration release]; [NewAcceleration release]; } TransformedAcceleration *a=nil;//[[TransformedAcceleration alloc]init]; // object containing fft of x,y,z accelerations for(int i=0; i<[newcurrentarray count]; i++) { a=(TransformedAcceleration *)[newcurrentarray objectAtIndex:i]; //NSLog(@"%d,%@",i,[a printAcceleration]); fprintf(fp,[[a printAcceleration] UTF8String]); //this is going wrong somewhow } fclose(fp); [array release]; [myFFT release]; //[array removeAllObjects]; [newcurrentarray release]; free(input); free(frequency);

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  • fourier transform to transpose key of a wav file

    - by tbischel
    I want to write an app to transpose the key a wav file plays in (for fun, I know there are apps that already do this)... my main understanding of how this might be accomplished is to 1) chop the audio file into very small blocks (say 1/10 a second) 2) run an FFT on each block 3) phase shift the frequency space up or down depending on what key I want 4) use an inverse FFT to return each block to the time domain 5) glue all the blocks together But now I'm wondering if the transformed blocks would no longer be continuous when I try to glue them back together. Are there ideas how I should do this to guarantee continuity, or am I just worrying about nothing?

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  • Determining the magnitude of a certain frequency on the iPhone

    - by eagle
    I'm wondering what's the easiest/best way to determine the magnitude of a given frequency in a sound. It's my understanding that a FFT function will return the magnitudes of all frequencies in a signal. I'm wondering if there is any shortcut I could use if I'm only concerned about a specific frequency. I'll be using the iPhone mic to record the audio. My guess is that I'll be using the Audio Queue Services for recording since I don't need to record the audio to a file. I'm using SDK 4.0, so I can use any of the functions defined in the Accelerate framework (e.g. FFT functions) if needed. Update: I updated the question to be more clear as per Conrad's suggestion.

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  • Detecting periodic repetitions in the data stream

    - by pulegium
    Let's say I have an array of zeros: a = numpy.zeros(1000) I then introduce some repetitive 'events': a[range(0, 1000, 30)] = 1 Question is, how do I detect the 'signal' there? Because it's far from the ideal signal if I do the 'regular' FFT I don't get a clear indication of where my 'true' signal is: f = abs(numpy.fft.rfft(a)) Is there a method to detect these repetitions with some degree of certainty? Especially if I have few of those mixed in, for example here: a[range(0, 1000, 30)] = 1 a[range(0, 1000, 110)] = 1 a[range(0, 1000, 48)] = 1 I'd like to get three 'spikes' on the resulting data...

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  • android spectrum analysis of streaming input

    - by TheBeeKeeper
    for a school project I am trying to make an android application that, once started, will perform a spectrum analysis of live audio received from the microphone or a bluetooth headset. I know I should be using FFT, and have been looking at moonblink's open source audio analyzer ( http://code.google.com/p/moonblink/wiki/Audalyzer ) but am not familiar with android development, and his code is turning out to be too difficult for me to work with. So I suppose my questions are, are there any easier java based, or open source android apps that do spectrum analysis I can reference? Or is there any helpful information that can be given, such as; steps that need be taken to get the microphone input, put it into an fft algorithm, then display a graph of frequency and pitch over time from its output? Any help would be appreciated, thanks.

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  • Simple in-place discrete fourier transform ( DFT )

    - by Adam
    I'm writing a very simple in-place DFT. I am using the formula shown here: http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Definition along with Euler's formula to avoid having to use a complex number class just for this. So far I have this: private void fft(double[] data) { double[] real = new double[256]; double[] imag = new double[256]; double pi_div_128 = -1 * Math.PI / 128; for (int k = 0; k < 256; k++) { for (int n = 0; n < 256; n++) { real[k] += data[k] * Math.Cos(pi_div_128 * k * n); imag[k] += data[k] * Math.Sin(pi_div_128 * k * n); } data[k] = Math.Sqrt(real[k] * real[k] + imag[k] * imag[k]); } } But the Math.Cos and Math.Sin terms eventually go both positive and negative, so as I'm adding those terms multiplied with data[k], they cancel out and I just get some obscenely small value. I see how it is happening, but I can't make sense of how my code is perhaps mis-representing the mathematics. Any help is appreciated. FYI, I do have to write my own, I realize I can get off-the shelf FFT's.

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  • any good free C DSP library?

    - by Juan
    Hi everybody I am developing an application to process geophysical signals; Right now I have done everything in octave and its digital signal processing toolbox, speed is not bad, however the application specifications say I need to port to the final algorithm to C; I am doing lots of filtering, re-sampling and signal manipulation/characterization with FFTs and cepstrums. do you know a good free C library for DSP packaged with filter design, resampling, fft, etc? Thanks a lot for any suggestion

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  • Generating a scalogram of a signal

    - by Goz
    Hi there, I'm trying to build a scalogram view for my app to see whether there is relevant information we can retrieve from a wavelet transform as opposed to using a spectograms to see what can be retrieved via an FFT. So far I can take a wave form and I can perform the forward wavelet transform on it. However I am lost at the next step. How do I turn this information into power/energy information? I have a set of wave forms at different frequencies but I have, as I say, no frequency information. Can anyone tell me what the next step is for turning this transformed data into a scalogram? Any help would be much appreciated because my google skills are failing me!

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  • Converting excel files to python to frequency

    - by Jacob
    Essentially I've got an excel files with voltage in the first column, and time in the second. I want to find the period of the voltages, as it returns a graph of voltage in y axis and time in x axis with a periodicity, looking similar to a sine function. To find the frequency I have uploaded my excel file to python as I think this will make it easier- there may be something I've missed that will simplify this. So far in python I have: import xlrd import numpy as N import numpy.fft as F import matplotlib.pyplot as P wb = xlrd.open_workbook('temp7.xls') #LOADING EXCEL FILE wb.sheet_names() sh = wb.sheet_by_index(0) first_column = sh.col_values(1) #VALUES FROM EXCEL second_column = sh.col_values(2) #VALUES FROM EXCEL Now how do I find the frequency from this? Huge thanks to anyone who can help! Jacob

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  • Rapid spectral analysis of audio file using Python 2.6?

    - by Ephemeralis
    What I want to do is to have a subroutine that analyses every 200 milliseconds of a sound file which it is given and spits out the frequency intensity value (from 0 to 1 as a float) of a specific frequency range into an array which I later save. This value then goes on to be used as the opacity value for a graphic which is supposed to 'strobe' to the audio file. The problem is, I have never ventured into audio analysis before and have no clue where to start. I have looked pymedia and scipy/numpy thinking I would be able to use FFT in order to achieve this, but I am not really sure how I would manipulate this data to end up with the desired result. The documentation on the SpectrAnalyzer class of pymedia is virtually non-existant and the examples on the website do not actually work with the latest release of the library - which isn't exactly making my life easier. How would I go about starting this project? I am at a complete loss as to what libraries I should even be using.

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  • Python frequency detection

    - by Tsuki
    Ok what im trying to do is a kind of audio processing software that can detect a prevalent frequency an if the frequency is played for long enough (few ms) i know i got a positive match. i know i would need to use FFT or something simiral but in this field of math i suck, i did search the internet but didn not find a code that could do only this. the goal im trying to accieve is to make myself a custom protocol to send data trough sound, need very low bitrate per sec but im also very limited on the transmiting end so the recieving software will need to be able custom (cant use an actual hardware/software modem) also i want this to be software only (no additional hardware except soundcard) thanks alot for the help.

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  • Getting PCM values of WAV files

    - by user2431088
    I have a .wav mono file (16bit,44.1kHz) and im using this code below. If im not wrong, this would give me an output of values between -1 and 1 which i can apply FFT on ( to be converted to a spectrogram later on). However, my output is no where near -1 and 1. This is a portion of my output 7.01214599609375 17750.2552337646 8308.42733764648 0.000274658203125 1.00001525878906 0.67291259765625 1.3458251953125 16.0000305175781 24932 758.380676269531 0.0001068115234375 This is the code which i got from another post Edit 1: public static Double[] prepare(String wavePath, out int SampleRate) { Double[] data; byte[] wave; byte[] sR = new byte[4]; System.IO.FileStream WaveFile = System.IO.File.OpenRead(wavePath); wave = new byte[WaveFile.Length]; data = new Double[(wave.Length - 44) / 4];//shifting the headers out of the PCM data; WaveFile.Read(wave, 0, Convert.ToInt32(WaveFile.Length));//read the wave file into the wave variable /***********Converting and PCM accounting***************/ for (int i = 0; i < data.Length; i += 2) { data[i] = BitConverter.ToInt16(wave, i) / 32768.0; } /**************assigning sample rate**********************/ for (int i = 24; i < 28; i++) { sR[i - 24] = wave[i]; } SampleRate = BitConverter.ToInt16(sR, 0); return data; }

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  • How can I compare market data feed sources for quality and latency improvement?

    - by yves Baumes
    I am in the very first stages of implementing a tool to compare 2 market data feed sources in order to prove the quality of new developed sources to my boss ( meaning there are no regressions, no missed updates, or wrong ), and to prove latencies improvement. So the tool I need must be able to check updates differences as well as to tell which source is the best (in term of latency). Concrectly, reference source could be Reuters while the other one is a Feed handler we develop internally. People warned me that updates might not arrive in the same order as Reuters implementation could differs totally from ours. Therefore a simple algorithm based on the fact that updates could arrive in the same order is likely not to work. My very first idea would be to use fingerprint to compare feed sources, as Shazaam application does to find the title of the tube you are submitting. Google told me it is based on FFT. And I was wondering if signal processing theory could behaves well with market access applications. I wanted to know your own experience in that field, is that possible to develop a quite accurate algorithm to meet the needs? What was your own idea? What do you think about fingerprint based comparison?

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  • Determining what frequencies correspond to the x axis in aurioTouch sample application

    - by eagle
    I'm looking at the aurioTouch sample application for the iPhone SDK. It has a basic spectrum analyzer implemented when you choose the "FFT" option. One of the things the app is lacking is X axis labels (i.e. the frequency labels). In the aurioTouchAppDelegate.mm file, in the function - (void)drawOscilloscope at line 652, it has the following code: if (displayMode == aurioTouchDisplayModeOscilloscopeFFT) { if (fftBufferManager->HasNewAudioData()) { if (fftBufferManager->ComputeFFT(l_fftData)) [self setFFTData:l_fftData length:fftBufferManager->GetNumberFrames() / 2]; else hasNewFFTData = NO; } if (hasNewFFTData) { int y, maxY; maxY = drawBufferLen; for (y=0; y<maxY; y++) { CGFloat yFract = (CGFloat)y / (CGFloat)(maxY - 1); CGFloat fftIdx = yFract * ((CGFloat)fftLength); double fftIdx_i, fftIdx_f; fftIdx_f = modf(fftIdx, &fftIdx_i); SInt8 fft_l, fft_r; CGFloat fft_l_fl, fft_r_fl; CGFloat interpVal; fft_l = (fftData[(int)fftIdx_i] & 0xFF000000) >> 24; fft_r = (fftData[(int)fftIdx_i + 1] & 0xFF000000) >> 24; fft_l_fl = (CGFloat)(fft_l + 80) / 64.; fft_r_fl = (CGFloat)(fft_r + 80) / 64.; interpVal = fft_l_fl * (1. - fftIdx_f) + fft_r_fl * fftIdx_f; interpVal = CLAMP(0., interpVal, 1.); drawBuffers[0][y] = (interpVal * 120); } cycleOscilloscopeLines(); } } From my understanding, this part of the code is what is used to decide which magnitude to draw for each frequency in the UI. My question is how can I determine what frequency each iteration (or y value) represents inside the for loop. For example, if I want to know what the magnitude is for 6kHz, I'm thinking of adding a line similar to the following: if (yValueRepresentskHz(y, 6)) NSLog(@"The magnitude for 6kHz is %f", (interpVal * 120)); Please note that although they chose to use the variable name y, from what I understand, it actually represents the x-axis in the visual graph of the spectrum analyzer, and the value of the drawBuffers[0][y] represents the y-axis.

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  • STFT and ISTFT in Python

    - by endolith
    Is there any form of short-time Fourier transform with corresponding inverse transform built into SciPy or NumPy or whatever? There's the pyplot specgram function in matplotlib, which calls ax.specgram(), which calls mlab.specgram(), which calls _spectral_helper(): #The checks for if y is x are so that we can use the same function to #implement the core of psd(), csd(), and spectrogram() without doing #extra calculations. We return the unaveraged Pxy, freqs, and t. I'm not sure if this can be used to do an STFT and ISTFT, though. Is there anything else, or should I translate something like this?

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  • polynomial multiplication using fastfourier transform

    - by mawia
    i am going through the above topic from CLRS(CORMEN) (page 834) and I got stuck at this point. Can anybody please explain how the following expression, A(x)=A^{[0]}(x^2) +xA^{[1]}(x^2) follows from, n-1 ` S a_j x^j j=0 Where, A^{[0]} = a_0 + a_2x + a_4a^x ... a_{n-2}x^{\frac{n}{2-1}} A^{[1]} = a_1 + a_3x + a_5a^x ... a_{n-1}x^{\frac{n}{2-1}} WITH REGARDS THANKS

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