I am attempting to calculate the MTF from a test target. I calculate the spread function easily enough, but the FFT results do not quite make sense to me. To summarize,the values seem to alternate giving me a reflection of what I would expect. To test, I used a simple square wave and numpy:
from numpy import fft
data = []
for x in range (0, 20):
data.append(0)
data[9] = 10
data[10] = 10
data[11] = 10
dataFFT = fft.fft(data)
The results look correct, with the exception of the sign... I am seeing the following for the first 4 values as an example:
30.00000000 +0.00000000e+00j
-29.02113033 +7.10542736e-15j
26.18033989 -1.24344979e-14j
-21.75570505 +1.24344979e-14j
So my question is why positive->negative->positive->negative in the real plane? This is not what I would expect... It I plot it, it almost appears that the correct function is mirrored around the x axis.
Note: I was expecting the following as an example:
This is what I am getting:
Your pulse is symmetric and positioned in the center of your FFT window (around N/2). Symmetric real data corresponds to only the cosine or "real" components of an FFT result. Note that the cosine function alternates between being -1 and 1 at the center of the FFT window, depending on the frequency bin index (representing cosine periods per FFT width). So the correlation of these FFT basis functions with a positive going pulse will also alternate as long as the pulse is narrower than half the cosine period.
If you want the largest FFT coefficients to be mostly positive, try centering your narrow rectangular pulse around time 0 (or circularly, time N), where the cosine function is always 1 for any frequency.
It works if you shift the data around 0 instead of half your array, with:
dataFFT = fft.fft(np.fftshift(data))
This isn't all that unexpected. If you want to check against conventional plots, make sure you convert that info to magnitude and phase before coming to any conclusions.
I did a quick check using your code and numpy.abs for mag, numpy,angle for phase. It sure looks like a sinc() function to me, which is what would be expected if the time-domain is a square pulse. If you do this, you'll find a pretty wide sinc, as would be expeceted for a short duration pulse on so few samples.
you forget to specify if your data is Real or Complex
not everyone code in python/numpy (including me) and if you do not know this then you probably handle data to/from FFT the wrong way.
FFT input can be both real or complex domain
FFT output is complex domain
so check the docs for your FFT implementation and specify it and also repair your data handling accordingly. Complex domain usually have first value Re and Second Im but that depends on FFT implementation/configuration.
signal
here is an example of impulse response from FFT
first is input Real domain signal (Im=0) single finite nonzero width pulse and second is the Re part of FFT output. The third is the Im part of FFT output. If you zoom it a bit then you will see amplitude range of y axis of each signal (on left).
Do not forget that different FFT implementations can have different normalization constants which will change the amplitude of signal. If you want magnitude and phase convert it like this:
mag=sqrt(Re*Re+Im*Im); // power
ang=atanxy(Re,Im); // phase angle
atanxy(dx,dy) is 4 quadrant arctan also called atan2 but be careful to get the operand order the same as your atanxy/atan2 implementation needs. Also can use mine C++ atanxy implementation
[Notes]
if your input signal is Real domain then FFT output is symmetric. Both Re and Im signals will be like:
{ a0,a1,a2,a3,...,a(n-1),a(n-1)...,a3,a2,a1,a0 }
exactly like on the image above. On the left are low frequencies and in the middle is the top frequency. If your input signal is Complex domain then the output can be anything.
Related
I am writing a python script for some geometrical data manipulation (calculating motion trajectories for a multi-drive industrial machine). Generally, the idea is that there is a given shape (let's say - an ellipse, but it general case it can be any convex shape, defined with a series of 2D points), which is rotated and it's uppermost tangent point must be followed. I don't have a problem with the latter part but I need a little hint with the 2D shape preparation.
Let's say that the ellipse was defined with too little points, for example - 25. (As I said, ultimately this can be any shape, for example a rounded hexagon). To maintain necessary precision I need far more points (let's say - 1000), preferably equally distributed over whole shape or with higher density of points near corners, sharp curves, etc.
I have a few things ringing in my head, I guess that DFT (FFT) would be a good starting point for this resampling, analyzing the scipy.signal.resample() I have found out that there are far more functions in the scipy.signal package which sound promising to me...
What I'm asking for is a suggestion which way I should follow, what tool I should try for this job, which may be the most suitable. Maybe there is a tool meant exactly for what I'm looking for or maybe I'm overthinking this and one of the implementations of FFT like resample() will work just fine (of course, after some adjustments at the starting and ending point of the shape to make sure it's closing without issues)?
Scipy.signal sounds promising, however, as far as I understand, it is meant to work with time series data, not geometrical data - I guess this may cause some problems as my data isn't a function (in a mathematical understanding).
Thanks and best regards!
As far as I understood, what you want is to get an interpolated version of your original data.
The DFT (or FFT) will not achieve this purpose, since it will perform an Fourier Transform (which is not what you want).
Talking theoretically, what you need to interpolate your data is to define a function to calculate the result in the new-data-points.
So, let's say your data contains 5 points, in which one you have a 1D (to simplify) number stored, representing your data, and you want a new array with 10 points, filled with the linear-interpolation of your original data.
Using numpy.interp:
import numpy as np
original_data = [2, 0, 3, 5, 1] # define your data in 1D
new_data_resolution = 0.5 # define new sampling distance (i.e, your x-axis resolution)
interp_data = np.interp(
x = np.arange(0, 5-1+new_data_resolution , new_data_resolution), # new sampling points (new axis)
xp = range(original_data),
fp = original_data
)
# now interp_data contains (5-1) / 0.5 + 1 = 9 points
After this, you will have a (5-1) / new_resolution (which is greater than 5, since new_resolution < 1)-length data, which values will be (in this case) a linear interpolation of your original data.
After you have achieved/understood this example, you can dive in the scipy.interpolate module to get a better understanding in the interpolation functions (my example uses a linear function to get the data in the missing points).
Applying this to n-D dimensional arrays is straight-forward, iterating over each dimension of your data.
I have implemented Demetri's Pitch Detector project for the iPhone and hitting up against two problems. 1) any sort of background noise sends the frequency reading bananas and 2) lower frequency sounds aren't being pitched correctly. I tried to tune my guitar and while the higher strings worked - the tuner could not correctly discern the low E.
The Pitch Detection code is located in RIOInterface.mm and goes something like this ...
// get the data
AudioUnitRender(...);
// convert int16 to float
Convert(...);
// divide the signal into even-odd configuration
vDSP_ctoz((COMPLEX*)outputBuffer, 2, &A, 1, nOver2);
// apply the fft
vDSP_fft_zrip(fftSetup, &A, stride, log2n, FFT_FORWARD);
// convert split real form to split vector
vDSP_ztoc(&A, 1, (COMPLEX *)outputBuffer, 2, nOver2);
Demetri then goes on to determine the 'dominant' frequency as follows:
float dominantFrequency = 0;
int bin = -1;
for (int i=0; i<n; i+=2) {
float curFreq = MagnitudeSquared(outputBuffer[i], outputBuffer[i+1]);
if (curFreq > dominantFrequency) {
dominantFrequency = curFreq;
bin = (i+1)/2;
}
}
memset(outputBuffer, 0, n*sizeof(SInt16));
// Update the UI with our newly acquired frequency value.
[THIS->listener frequencyChangedWithValue:bin*(THIS->sampleRate/bufferCapacity)];
To start with, I believe I need to apply a LOW PASS FILTER ... but I'm not an FFT expert and not sure exactly where or how to do that against the data returned from the vDSP functions. I'm also not sure how to improve the accuracy of the code in the lower frequencies. There seem to be other algorithms to determine the dominant frequency - but again, looking for a kick in the right direction when using the data returned by Apple's Accelerate framework.
UPDATE:
The accelerate framework actually has some windowing functions. I setup a basic window like this
windowSize = maxFrames;
transferBuffer = (float*)malloc(sizeof(float)*windowSize);
window = (float*)malloc(sizeof(float)*windowSize);
memset(window, 0, sizeof(float)*windowSize);
vDSP_hann_window(window, windowSize, vDSP_HANN_NORM);
which I then apply by inserting
vDSP_vmul(outputBuffer, 1, window, 1, transferBuffer, 1, windowSize);
before the vDSP_ctoz function. I then change the rest of the code to use 'transferBuffer' instead of outputBuffer ... but so far, haven't noticed any dramatic changes in the final pitch guess.
Pitch is not the same as peak magnitude frequency bin (which is what the FFT in the Accelerate framework might give you directly). So any peak frequency detector will not be reliable for pitch estimation. A low-pass filter will not help when the note has a missing or very weak fundamental (common in some voice, piano and guitar sounds) and/or lots of powerful overtones in its spectrum.
Look at a wide-band spectrum or spectrograph of your musical sounds and you will see the problem.
Other methods are usually needed for a more reliable estimate of musical pitch. Some of these include autocorrelation methods (AMDF, ASDF), Cepstrum/Cepstral analysis, harmonic product spectrum, phase vocoder, and/or composite algorithms such as RAPT (Robust Algorithm for Pitch Tracking) and YAAPT. An FFT is useful as only a sub-part of some of the above methods.
At the very least you need to apply a window function to your time domain data, prior to calculating the FFT. Without this step the power spectrum will contain artefacts (see: spectral leakage) which will interfere with your attempts at extracting pitch information.
A simple Hann (aka Hanning) window should suffice.
What is your sample frequency and blocksize? Low E is around 80 Hz, so you need to make sure your capture block is long enough to capture many cycles at this frequency. This is because the Fourier Transform divides the frequency spectrum into bins, each several Hz wide. If you sample at 44.1 kHz and have a 1024 point time domain sample, for instance, each bin will be 44100/1024 = 43.07 Hz wide. Thus a low E would be in the second bin. For a bunch of reasons (to do with spectral leakage and the nature of finite time blocks), practically speaking you should consider the first 3 or 4 bins of data in an FFT result with extreme suspicion.
If you drop the sample rate to 8 kHz, the same blocksize gives you bins that are 7.8125 Hz wide. Now low E will be in the 10th or 11th bin, which is much better. You could also use a longer blocksize.
And as Paul R points out, you MUST use a window to reduce spectral leakage.
The frequency response function of the iPhone drops off below 100 - 200 Hz (see http://blog.faberacoustical.com/2009/ios/iphone/iphone-microphone-frequency-response-comparison/ for an example).
If you are trying to detect the fundamental mode of a low guitar string, the microphone might be acting as a filter and suppressing the frequency you are interested in. There are a couple of options if you interested in using the fft data you can get - you can window the data in the frequency domain around the note you are trying to detect so that all you can see is the first mode even if it is of lower magnitude than higher modes(i.e. have a toggle to tune the first string and put it in this mode).
Or you can low pass filter the sound data - you can do this either in the time domain or even easier since you already have frequency domain data, in the frequency domain. A very simple time domain low pass filter is to do a time-moving average filter. A very simple frequency domain low pass filter is to multiply your fft magnitudes by a vector with 1's in the low frequency range and a linear (or even a step) ramp down in the higher frequencies.
I am analyzing a time-series dataset that I am pretty sure can be broken down using fft. I want to develop a model to estimate the data using a sum of sin/cos but I am having trouble with the syntax to find the frequencies in python
Here is a graph of the data
data graph
And here's a link to the original data: https://drive.google.com/open?id=1mqZtQ-txdd_AFbKGBlbSL6903CK-_kXl
Most of the examples I have seen have multiple samples per second/time period, however the data in this set represent by-minute observations of some metric. Because of this, I've had trouble translating the answers online to this problem
Here's my naive first approach
X = fftpack.fft(data)
freqs = fftpack.fftfreq(len(data))
plt.plot(freqs, np.abs(X))
plt.show()
Instead of peaking at the major frequencies, my plot only has one peak at 0.
result
The FFT you posted has been shifted so that 0 is at the center. Data to the left of the center represents negative frequencies and to the right represents positive frequencies. If you zoom in and look more closely, I think you will see that there are two peaks close to the center that you are interpreting as a single peak at 0. Just looking at the positive side, the location of this peak will tell you which frequency is contributing significant signal power.
Like you said, your x-axis is probably incorrect. scipy.fftpack.fftfreq needs to know the time between samples (in seconds, I think) of your time-domain signal to correctly determine the bandwidth and create the x-axis array in Hz. This should do it:
dt = 60 # 60 seconds between samples
freqs = fftpack.fftfreq(len(data),dt)
I'm trying to get the correct FFT bin index based on the given frequency. The audio is being sampled at 44.1k Hz and the FFT size is 1024. Given the signal is real (capture from PyAudio, decoded through numpy.fromstring, windowed by scipy.signal.hann), I then perform FFT through scipy.fftpack.rfft, and compute the decibel of the result, in whole, magnitude = 20 * scipy.log10(abs(rfft(audio_sample)))
Based on this, and this, I originally had my mapping from the FFT bin index, k, to any frequency, F, as:
F = k*Fs/N for k = 0 ... N/2-1 where Fs is the sampling rate, and N is the FFT bin size, in this case, 1024. And the reverse as:
k = F*N/Fs for F = 0Hz ... Fs/2-Fs/N
However, realizing that the rfft's result is no symmetric like fft, and provides the result, in an N size array. I now have some questions in regarding the mapping and the function. Documentation unfortunately did not provide much information as I'm novice in this area.
My questions:
To me, the result of rfft on an audio sample can be used directly from the first bin to the last bin, as no symmetry occurs in the output, is that correct?
Given the lack of symmetry from the above, the frequency resolution appears to have increased, is this interpretation correct?
Because of using rfft, my mapping function from bin index k to frequency F is now F = k*Fs/(2N) for k = 0 ... N-1 is this correct?
Conversely, the reverse mapping function from frequency F to bin index k now becomes k = 2*F*N/Fs for F = 0Hz ... Fs/2-(Fs/2/N), what about the correctness of this?
My general confusion arises from how rfft is related to fft, and how the mapping can be done correctly while using rfft. I believe my mapping is offset by a small amount, and that is crucial in my application. Please point out the mistake or advise on the matter if possible, thank you very much.
First to clear up a few things for you:
A quick reference to the fftpack documentation reveals that rfft only gives you an output vector from 0..512 (in your case). The reason for this is exactly because of the symmetry present when calculating the discrete Fourier transform of a real-valued input:
y[k] = y*[N-k] (see Wikipedia page on DFTs). Therefore, the rfft function only calculates and stores N/2+1 values since you can calculate the other half by just taking the complex conjugates (should you really want it for plotting (say)). The fft function makes no assumption on the input values (they can have both a real and imaginary part) and therefore no symmetry can be assumed in the output and it gives you a full output vector with N values. Admittedly, most applications use a real input, so people tend to assume the symmetry is always there. Note that the Fast Fourier Transform (FFT) is an (efficient) algorithm to calculate the Discrete Fourier Transform (DFT) and the rfft function also uses the FFT to do the calculation.
In light of the above, your indices for accessing the output vector are out of bounds, i.e. > 512. The reasons why/how you can do this depends on your code. You should clearly distinguish between the 'logical N' (that you use to map the bin frequencies, define the DFT etc.) and the 'computational N' (the actual number of values in your output vector), then all your problems should disappear.
To concretely answer your questions:
No. There is symmetry and you need to use this to calculate the last bins (but they give you no extra information).
No. The only way to increase resolution of a DFT is to increase your sample length.
No, but almost. F = k*Fs/N for k = 0..N/2
For an output vector with N bins you get frequencies from 0 to (N-1)/N*Fs. Using the rfft you will have an output vector with N/2+1 bins. You do the maths, but I get 0..Fs/2
Hope things are clearer now.
Audio processing is pretty new for me. And currently using Python Numpy for processing wave files. After calculating FFT matrix I am getting noisy power values for non-existent frequencies. I am interested in visualizing the data and accuracy is not a high priority. Is there a safe way to calculate the clipping value to remove these values, or should I use all FFT matrices for each sample set to come up with an average number ?
regards
Edit:
from numpy import *
import wave
import pymedia.audio.sound as sound
import time, struct
from pylab import ion, plot, draw, show
fp = wave.open("500-200f.wav", "rb")
sample_rate = fp.getframerate()
total_num_samps = fp.getnframes()
fft_length = 2048.
num_fft = (total_num_samps / fft_length ) - 2
temp = zeros((num_fft,fft_length), float)
for i in range(num_fft):
tempb = fp.readframes(fft_length);
data = struct.unpack("%dH"%(fft_length), tempb)
temp[i,:] = array(data, short)
pts = fft_length/2+1
data = (abs(fft.rfft(temp, fft_length)) / (pts))[:pts]
x_axis = arange(pts)*sample_rate*.5/pts
spec_range = pts
plot(x_axis, data[0])
show()
Here is the plot in non-logarithmic scale, for synthetic wave file containing 500hz(fading out) + 200hz sine wave created using Goldwave.
Simulated waveforms shouldn't show FFTs like your figure, so something is very wrong, and probably not with the FFT, but with the input waveform. The main problem in your plot is not the ripples, but the harmonics around 1000 Hz, and the subharmonic at 500 Hz. A simulated waveform shouldn't show any of this (for example, see my plot below).
First, you probably want to just try plotting out the raw waveform, and this will likely point to an obvious problem. Also, it seems odd to have a wave unpack to unsigned shorts, i.e. "H", and especially after this to not have a large zero-frequency component.
I was able to get a pretty close duplicate to your FFT by applying clipping to the waveform, as was suggested by both the subharmonic and higher harmonics (and Trevor). You could be introducing clipping either in the simulation or the unpacking. Either way, I bypassed this by creating the waveforms in numpy to start with.
Here's what the proper FFT should look like (i.e. basically perfect, except for the broadening of the peaks due to the windowing)
Here's one from a waveform that's been clipped (and is very similar to your FFT, from the subharmonic to the precise pattern of the three higher harmonics around 1000 Hz)
Here's the code I used to generate these
from numpy import *
from pylab import ion, plot, draw, show, xlabel, ylabel, figure
sample_rate = 20000.
times = arange(0, 10., 1./sample_rate)
wfm0 = sin(2*pi*200.*times)
wfm1 = sin(2*pi*500.*times) *(10.-times)/10.
wfm = wfm0+wfm1
# int test
#wfm *= 2**8
#wfm = wfm.astype(int16)
#wfm = wfm.astype(float)
# abs test
#wfm = abs(wfm)
# clip test
#wfm = clip(wfm, -1.2, 1.2)
fft_length = 5*2048.
total_num_samps = len(times)
num_fft = (total_num_samps / fft_length ) - 2
temp = zeros((num_fft,fft_length), float)
for i in range(num_fft):
temp[i,:] = wfm[i*fft_length:(i+1)*fft_length]
pts = fft_length/2+1
data = (abs(fft.rfft(temp, fft_length)) / (pts))[:pts]
x_axis = arange(pts)*sample_rate*.5/pts
spec_range = pts
plot(x_axis, data[2], linewidth=3)
xlabel("freq (Hz)")
ylabel('abs(FFT)')
show()
FFT's because they are windowed and sampled cause aliasing and sampling in the frequency domain as well. Filtering in the time domain is just multiplication in the frequency domain so you may want to just apply a filter which is just multiplying each frequency by a value for the function for the filter you are using. For example multiply by 1 in the passband and by zero every were else. The unexpected values are probably caused by aliasing where higher frequencies are being folded down to the ones you are seeing. The original signal needs to be band limited to half your sampling rate or you will get aliasing. Of more concern is aliasing that is distorting the area of interest because for this band of frequencies you want to know that the frequency is from the expected one.
The other thing to keep in mind is that when you grab a piece of data from a wave file you are mathmatically multiplying it by a square wave. This causes a sinx/x to be convolved with the frequency response to minimize this you can multiply the original windowed signal with something like a Hanning window.
It's worth mentioning for a 1D FFT that the first element (index [0]) contains the DC (zero-frequency) term, the elements [1:N/2] contain the positive frequencies and the elements [N/2+1:N-1] contain the negative frequencies. Since you didn't provide a code sample or additional information about the output of your FFT, I can't rule out the possibility that the "noisy power values at non-existent frequencies" aren't just the negative frequencies of your spectrum.
EDIT: Here is an example of a radix-2 FFT implemented in pure Python with a simple test routine that finds the FFT of a rectangular pulse, [1.,1.,1.,1.,0.,0.,0.,0.]. You can run the example on codepad and see that the FFT of that sequence is
[0j, Negative frequencies
(1+0.414213562373j), ^
0j, |
(1+2.41421356237j), |
(4+0j), <= DC term
(1-2.41421356237j), |
0j, v
(1-0.414213562373j)] Positive frequencies
Note that the code prints out the Fourier coefficients in order of ascending frequency, i.e. from the highest negative frequency up to DC, and then up to the highest positive frequency.
I don't know enough from your question to actually answer anything specific.
But here are a couple of things to try from my own experience writing FFTs:
Make sure you are following Nyquist rule
If you are viewing the linear output of the FFT... you will have trouble seeing your own signal and think everything is broken. Make sure you are looking at the dB of your FFT magnitude. (i.e. "plot(10*log10(abs(fft(x))))" )
Create a unitTest for your FFT() function by feeding generated data like a pure tone. Then feed the same generated data to Matlab's FFT(). Do a absolute value diff between the two output data series and make sure the max absolute value difference is something like 10^-6 (i.e. the only difference is caused by small floating point errors)
Make sure you are windowing your data
If all of those three things work, then your fft is fine. And your input data is probably the issue.
Check the input data to see if there is clipping http://www.users.globalnet.co.uk/~bunce/clip.gif
Time doamin clipping shows up as mirror images of the signal in the frequency domain at specific regular intervals with less amplitude.