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Audio is made up of multiple frequencies occurring at any given time, and we can perform the FFT to get the Frequency bins, but what does the concept of Frequency mean when it comes to Sensor data?
For example, a Triaxial Accelerometer somehow converts a voltage signal and produces acceleration readings in ms^-2. Is the FFT performed with those X,Y,Z readings or the voltages sampled at Fs.
Am I overcomplicating things or is there a difference in mindset when performing DSP for Audio vs Sensor data?
A Fourier transform is tool to convert functions or signals into something that is easier to work with. It is a mathematical tool. The result can have an easy physical interpretation but that is not always the case.
Assume you have an object with constant mass and several periodic sin-like forces F_1*sin(c*t), F_2*sin(d*t), ... that act on the object. The total force is just the sum of those forces:
F(t) = F_1*sin(c*t) + F_2*sin(d*t) + ...
You get the particle's acceleration using Newton's 2nd law:
m * a(t) = F(t)
=> a(t) = F(t) / m = F_1/m * sin(c*t) + F_2/m * sin(d*t) + ...
Let's assume you measured a(t) but don't know the equation above. It you perform a Fourier transformation you can calculate the values of F_1/m, F_2/m, ... . That means your Fourier transform of the the acceleration is the amplitude of the force at the given frequency over the object's mass.
This interpretation works because the Fourier transform is linear and so is the adding of forces (See Newtons 2nd law). If you describe something non-linear chances are that there is no easy interpretation of the result of the transformation.
So when do you perform the FFT? It depends:
If you do it to improve you signal (remove noise) do it on the measured data.
If you want to analyse the physical object (resonances) do it on the acceleration.
(If the conversion is linear (ADC output to m/s^2 is a simple multiplication) it does not matter.)
I hope this makes things a bit clearer (at least from the physical point of view).
I have to reduce white noise from a sound record.Because of that i used fourier transform.But i dont know how to use the fft function's return values which is in frequincy domain.How can i use the fft data to reduce noise?
Here is my code
from scipy.io import wavfile
import matplotlib.pyplot as plt
import simpleaudio as sa
from numpy.fft import fft,fftfreq,ifft
#reading wav file
fs,data=wavfile.read("a.wav","wb")
n=len(data)
freqs=fftfreq(n)
mask=freqs>0
#calculating raw fft values
fft_vals=fft(data)
#calculating theorical fft values
fft_theo=2*np.abs(fft_vals/n)
#ploting
plt.plot(freqs[mask],fft_theo[mask])
plt.show()```
It is better for such questions to build a synthetic example, so you don't have to post a big datafile and people can still follow your question (MCVE).
It is also important to plot intermediate results since we are talking about operations on complex numbers, so we often have to take re, im parts, or absolutes and angles respectively.
The Fourier transform of a real function is complex but is symmetric for positive vs negative frequencies. One can also look at that from an information theoretical viewpoint: you wouldn't want N independent real numbers in time to result in 2N independent real numbers describing the spectrum.
While you normally plot the absolute or absolute squared (voltage vs. power) of the spectrum, you can leave it complex when you apply the filter. After back-conversion to time via the IFFT, to plot it, you'll have to convert it to a real number again, in this case by taking the absolute.
If you design the filter kernel in the time domain (FFT of a Gaussian will be a Gaussian), the IFFT of the product of the FFT of the filter and the spectrum will have only very small imaginary parts and you can then take the real part (which makes more sense from a physics viewpoint, you started with real part, end with real part).
import numpy as np
import matplotlib.pyplot as p
%matplotlib inline
T=3 # secs
d=0.04 # secs
n=int(T/d)
print(n)
t=np.arange(0,T,d)
fr=1 # Hz
y1= np.sin(2*np.pi*fr*t) +1 # dc offset helps with backconversion, try setting it to zero
y2= 1/5*np.sin(2*np.pi*7*fr*t+0.5)
y=y1+y2
f=np.fft.fftshift(np.fft.fft(y))
freq=np.fft.fftshift(np.fft.fftfreq(n,d))
filter=np.exp(- freq**2/6) # simple Gaussian filter in the frequency domain
filtered_spectrum=f*filter # apply the filter to the spectrum
filtered_data = np.fft.ifft(filtered_spectrum) # then backtransform to time domain
p.figure(figsize=(24,16))
p.subplot(321)
p.plot(t,y1,'.-',color='red', lw=0.5, ms=1, label='signal')
p.plot(t,y2,'.-',color='blue', lw=0.5,ms=1, label='noise')
p.plot(t,y,'.-',color='green', lw=4, ms=4, alpha=0.3, label='noisy signal')
p.xlabel('time (sec)')
p.ylabel('amplitude (Volt)')
p.legend()
p.subplot(322)
p.plot(freq,np.abs(f)/n, label='raw spectrum')
p.plot(freq,filter,label='filter')
p.xlabel(' freq (Hz)')
p.ylabel('amplitude (Volt)');
p.legend()
p.subplot(323)
p.plot(t, np.absolute(filtered_data),'.-',color='green', lw=4, ms=4, alpha=0.3, label='cleaned signal')
p.legend()
p.subplot(324)
p.plot(freq,np.abs(filtered_spectrum), label = 'filtered spectrum')
p.legend()
p.subplot(326)
p.plot(freq,np.log( np.abs(filtered_spectrum)), label = 'filtered spectrum')
p.legend()
p.title(' in the log plot the noise is still visible');
I have a dataset where a process is described as a time series made of ~2000 points and 1500 dimensions.
I would like to quantify how much each dimension is correlated with another time series measured by another method.
What is the appropriate way to do this (eventually done in python) ? I have heard that Pearson is not well suited for this task, at least without data preparation. What are your thoughts about that?
Many thanks!
A general good rule in data science is to first try the easy thing. Only when the easy thing fails should you move to something more complicated. With that in mind, here is how you would compute the Pearson correlation between each dimension and some other time series. The key function here being pearsonr:
import numpy as np
from scipy.stats import pearsonr
# Generate a random dataset using 2000 points and 1500 dimensions
n_times = 2000
n_dimensions = 1500
data = np.random.rand(n_times, n_dimensions)
# Generate another time series, also using 2000 points
other_time_series = np.random.rand(n_times)
# Compute correlation between each dimension and the other time series
correlations = np.zeros(n_dimensions)
for dimension in range(n_dimensions):
# The Pearson correlation function gives us both the correlation
# coefficient (r) and a p-value (p). Here, we only use the coefficient.
r, p = pearsonr(data[:, dimension], other_time_series)
correlations[dimension] = r
# Now we have, for each dimension, the Pearson correlation with the other time
# series!
len(correlations)
# Print the first 5 correlation coefficients
print(correlations[:5])
If Pearson correlation doesn't work well for you, you can try swapping out the pearsonr function with something else, like:
spearmanr Spearman rank-order correlation coefficient.
kendalltau Kendall’s tau, a correlation measure for ordinal data.
I'm working on advanced vision system which consist of two static cameras (used for obtaining accurate 3d object location) and some targeting device. Object detection and stereovision modules have been already done. Unfortunately, due to the delay of targeting system it is obligatory to develop a proper prediction module.
I did some tests using Kalman filter but it is working not accurate enough.
kalman = cv2.KalmanFilter(6,3,0)
...
kalman.statePre[0,0] = x
kalman.statePre[1,0] = y
kalman.statePre[2,0] = z
kalman.statePre[3,0] = 0
kalman.statePre[4,0] = 0
kalman.statePre[5,0] = 0
kalman.measurementMatrix = np.array([[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]],np.float32)
kalman.transitionMatrix = np.array([[1,0,0,1,0,0],[0,1,0,0,1,0],0,0,1,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],np.float32)
kalman.processNoiseCov = np.array([[1,0,0,0,0,0],[0,1,0,0,0,0],0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],np.float32) * 0.03
kalman.measurementNoiseCov = np.array([[1,0,0],[0,1,0],0,0,1]],np.float32) * 0.003
I noticed that time periods between two frames are different each time (due to the various detection time).
How could I use last timestamp diff as an input? (Transition matrices?, controlParam?)
I want to determine the prediction time e.g want to predict position of object in 0,5sec or 1,5sec
I could provide example input 3d points.
Thanks in advance
1. How could I use last timestamp diff as an input? (Transition matrices?, controlParam?)
Step size is controlled through prediction matrix. You also need to adjust process noise covariance matrix to control uncertainty growth.
You are using a constant speed prediction model, so that p_x(t+dt) = p_x(t) + v_x(t)·dt will predict position in X with a time step dt (and the same for coords. Y and Z). In that case, your prediction matrix should be:
kalman.transitionMatrix = np.array([[1,0,0,dt,0,0],[0,1,0,0,dt,0],0,0,1,0,0,dt],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]],np.float32)
I left the process noise cov. formulation as an exercise. Be careful with squaring or not squaring the dt term.
2. I want to determine the prediction time e.g want to predict position of object in 0,5sec or 1,5sec
You can follow two different approaches:
Use a small fixed dt (e.g. 0.02 sec for 50Hz) and calculate predictions in a loop until you reach your goal (e.g. get a new observation from your cameras).
Adjusting prediction and process noise matrices online to the desired dt (0,5 / 1,5 sec in your question) and execute a single prediction step.
If you are asking about how to anticipate the detection time of your cameras, that should be a different question and I am afraid I can't help you :-)
I'm not a DSP expert, but I understand that there are two ways that I can apply a discrete time-domain filter to a discrete time-domain waveform. The first is to convolve them in the time domain, and the second is to take the FFT of both, multiply both complex spectrums, and take IFFT of the result. One key difference in these methods is the second approach is subject to circular convolution.
As an example, if the filter and waveforms are both N points long, the first approach (i.e. convolution) produces a result that is N+N-1 points long, where the first half of this response is the filter filling up and the 2nd half is the filter emptying. To get a steady-state response, the filter needs to have fewer points than the waveform to be filtered.
Continuing this example with the second approach, and assuming the discrete time-domain waveform data is all real (not complex), the FFT of the filter and the waveform both produce FFTs of N points long. Multiplying both spectrums IFFT'ing the result produces a time-domain result also N points long. Here the response where the filter fills up and empties overlap each other in the time domain, and there's no steady state response. This is the effect of circular convolution. To avoid this, typically the filter size would be smaller than the waveform size and both would be zero-padded to allow space for the frequency convolution to expand in time after IFFT of the product of the two spectrums.
My question is, I often see work in the literature from well-established experts/companies where they have a discrete (real) time-domain waveform (N points), they FFT it, multiply it by some filter (also N points), and IFFT the result for subsequent processing. My naive thinking is this result should contain no steady-state response and thus should contain artifacts from the filter filling/emptying that would lead to errors in interpreting the resulting data, but I must be missing something. Under what circumstances can this be a valid approach?
Any insight would be greatly appreciated
The basic problem is not about zero padding vs the assumed periodicity, but that Fourier analysis decomposes the signal into sine waves which, at the most basic level, are assumed to be infinite in extent. Both approaches are correct in that the IFFT using the full FFT will return the exact input waveform, and both approaches are incorrect in that using less than the full spectrum can lead to effects at the edges (that usually extend a few wavelengths). The only difference is in the details of what you assume fills in the rest of infinity, not in whether you are making an assumption.
Back to your first paragraph: Usually, in DSP, the biggest problem I run into with FFTs is that they are non-causal, and for this reason I often prefer to stay in the time domain, using, for example, FIR and IIR filters.
Update:
In the question statement, the OP correctly points out some of the problems that can arise when using FFTs to filter signals, for example, edge effects, that can be particularly problematic when doing a convolution that is comparable in the length (in the time domain) to the sampled waveform. It's important to note though that not all filtering is done using FFTs, and in the paper cited by the OP, they are not using FFT filters, and the problems that would arise with an FFT filter implementation do not arise using their approach.
Consider, for example, a filter that implements a simple average over 128 sample points, using two different implementations.
FFT: In the FFT/convolution approach one would have a sample of, say, 256, points and convolve this with a wfm that is constant for the first half and goes to zero in the second half. The question here is (even after this system has run a few cycles), what determines the value of the first point of the result? The FFT assumes that the wfm is circular (i.e. infinitely periodic) so either: the first point of the result is determined by the last 127 (i.e. future) samples of the wfm (skipping over the middle of the wfm), or by 127 zeros if you zero-pad. Neither is correct.
FIR: Another approach is to implement the average with an FIR filter. For example, here one could use the average of the values in a 128 register FIFO queue. That is, as each sample point comes in, 1) put it in the queue, 2) dequeue the oldest item, 3) average all of the 128 items remaining in the queue; and this is your result for this sample point. This approach runs continuously, handling one point at a time, and returning the filtered result after each sample, and has none of the problems that occur from the FFT as it's applied to finite sample chunks. Each result is just the average of the current sample and the 127 samples that came before it.
The paper that OP cites takes an approach much more similar to the FIR filter than to the FFT filter (note though that the filter in the paper is more complicated, and the whole paper is basically an analysis of this filter.) See, for example, this free book which describes how to analyze and apply different filters, and note also that the Laplace approach to analysis of the FIR and IIR filters is quite similar what what's found in the cited paper.
Here's an example of convolution without zero padding for the DFT (circular convolution) vs linear convolution. This is the convolution of a length M=32 sequence with a length L=128 sequence (using Numpy/Matplotlib):
f = rand(32); g = rand(128)
h1 = convolve(f, g)
h2 = real(ifft(fft(f, 128)*fft(g)))
plot(h1); plot(h2,'r')
grid()
The first M-1 points are different, and it's short by M-1 points since it wasn't zero padded. These differences are a problem if you're doing block convolution, but techniques such as overlap and save or overlap and add are used to overcome this problem. Otherwise if you're just computing a one-off filtering operation, the valid result will start at index M-1 and end at index L-1, with a length of L-M+1.
As to the paper cited, I looked at their MATLAB code in appendix A. I think they made a mistake in applying the Hfinal transfer function to the negative frequencies without first conjugating it. Otherwise, you can see in their graphs that the clock jitter is a periodic signal, so using circular convolution is fine for a steady-state analysis.
Edit: Regarding conjugating the transfer function, the PLLs have a real-valued impulse response, and every real-valued signal has a conjugate symmetric spectrum. In the code you can see that they're just using Hfinal[N-i] to get the negative frequencies without taking the conjugate. I've plotted their transfer function from -50 MHz to 50 MHz:
N = 150000 # number of samples. Need >50k to get a good spectrum.
res = 100e6/N # resolution of single freq point
f = res * arange(-N/2, N/2) # set the frequency sweep [-50MHz,50MHz), N points
s = 2j*pi*f # set the xfer function to complex radians
f1 = 22e6 # define 3dB corner frequency for H1
zeta1 = 0.54 # define peaking for H1
f2 = 7e6 # define 3dB corner frequency for H2
zeta2 = 0.54 # define peaking for H2
f3 = 1.0e6 # define 3dB corner frequency for H3
# w1 = natural frequency
w1 = 2*pi*f1/((1 + 2*zeta1**2 + ((1 + 2*zeta1**2)**2 + 1)**0.5)**0.5)
# H1 transfer function
H1 = ((2*zeta1*w1*s + w1**2)/(s**2 + 2*zeta1*w1*s + w1**2))
# w2 = natural frequency
w2 = 2*pi*f2/((1 + 2*zeta2**2 + ((1 + 2*zeta2**2)**2 + 1)**0.5)**0.5)
# H2 transfer function
H2 = ((2*zeta2*w2*s + w2**2)/(s**2 + 2*zeta2*w2*s + w2**2))
w3 = 2*pi*f3 # w3 = 3dB point for a single pole high pass function.
H3 = s/(s+w3) # the H3 xfer function is a high pass
Ht = 2*(H1-H2)*H3 # Final transfer based on the difference functions
subplot(311); plot(f, abs(Ht)); ylabel("abs")
subplot(312); plot(f, real(Ht)); ylabel("real")
subplot(313); plot(f, imag(Ht)); ylabel("imag")
As you can see, the real component has even symmetry and the imaginary component has odd symmetry. In their code they only calculated the positive frequencies for a loglog plot (reasonable enough). However, for calculating the inverse transform they used the values for the positive frequencies for the negative frequencies by indexing Hfinal[N-i] but forgot to conjugate it.
I can shed some light to the reason why "windowing" is applied before FFT is applied.
As already pointed out the FFT assumes that we have a infinite signal. When we take a sample over a finite time T this is mathematically the equivalent of multiplying the signal with a rectangular function.
Multiplying in the time domain becomes convolution in the frequency domain. The frequency response of a rectangle is the sync function i.e. sin(x)/x. The x in the numerator is the kicker, because it dies down O(1/N).
If you have frequency components which are exactly multiples of 1/T this does not matter as the sync function is zero in all points except that frequency where it is 1.
However if you have a sine which fall between 2 points you will see the sync function sampled on the frequency point. It lloks like a magnified version of the sync function and the 'ghost' signals caused by the convolution die down with 1/N or 6dB/octave. If you have a signal 60db above the noise floor, you will not see the noise for 1000 frequencies left and right from your main signal, it will be swamped by the "skirts" of the sync function.
If you use a different time window you get a different frequency response, a cosine for example dies down with 1/x^2, there are specialized windows for different measurements. The Hanning window is often used as a general purpose window.
The point is that the rectangular window used when not applying any "windowing function" creates far worse artefacts than a well chosen window. i.e by "distorting" the time samples we get a much better picture in the frequency domain which closer resembles "reality", or rather the "reality" we expect and want to see.
Although there will be artifacts from assuming that a rectangular window of data is periodic at the FFT aperture width, which is one interpretation of what circular convolution does without sufficient zero padding, the differences may or may not be large enough to swamp the data analysis in question.