I am trying to time align two signals. My problem is however, that they have been sampled at different rates, one has been sampled at 50 Hz the other at 100 Hz will my initial approach of cross correlation work or do I now need to either model these signals or interpolate the one sampled at 50 Hz. I feel this may be a hefty task as this is real-life data and my model will have a certain amount of error.
You can just re-sample the 50 Hz data to 100 Hz. There are plenty of libraries and sample code out there for doing this. The basic algorithm for 2x up-sampling is:
insert a 0 sample between each actual sample
apply a low pass filter (25 Hz cut-off)
Alternatively, if you're not interested in the higher frequency components then you can down-sample the 100 Hz data to 50 Hz:
apply a low pass filter (25 Hz cut-off)
delete every other sample
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I'm currently working on a program in C++ in which I am computing the time varying FFT of a wav file. I have a question regarding plotting the results of an FFT.
Say for example I have a 70 Hz signal that is produced by some instrument with certain harmonics. Even though I say this signal is 70 Hz, it's a real signal and I assume will have some randomness in which that 70Hz signal varies. Say I sample it for 1 second at a sample rate of 20kHz. I realize the sample period probably doesn't need to be 1 second, but bear with me.
Because I now have 20000 samples, when I compute the FFT. I will have 20000 or (19999) frequency bins. Let's also assume that my sample rate in conjunction some windowing techniques minimize spectral leakage.
My question then: Will the FFT still produce a relatively ideal impulse at 70Hz? Or will there 'appear to be' spectral leakage which is caused by the randomness the original signal? In otherwords, what does the FFT look like of a sinusoid whose frequency is a random variable?
Some of the more common modulation schemes will add sidebands that carry the information in the modulation. Depending on the amount and type of modulation with respect to the length of the FFT, the sidebands can either appear separate from the FFT peak, or just "fatten" a single peak.
Your spectrum will appear broadened and this happens in the real world. Look e.g for the Voight profile, which is a Lorentizan (the result of an ideal exponential decay) convolved with a Gaussian of a certain width, the width being determined by stochastic fluctuations, e.g. Doppler effect on molecules in a gas that is being probed by a narrow-band laser.
You will not get an 'ideal' frequency peak either way. The limit for the resolution of the FFT is one frequency bin, (frequency resolution being given by the inverse of the time vector length), but even that (as #xvan pointed out) is in general broadened by the window function. If your window is nonexistent, i.e. it is in fact a square window of the length of the time vector, then you'll get spectral peaks that are convolved with a sinc function, and thus broadened.
The best way to visualize this is to make a long vector and plot a spectrogram (often shown for audio signals) with enough resolution so you can see the individual variation. The FFT of the overall signal is then the projection of the moving peaks onto the vertical axis of the spectrogram. The FFT of a given time vector does not have any time resolution, but sums up all frequencies that happen during the time you FFT. So the spectrogram (often people simply use the STFT, short time fourier transform) has at any given time the 'full' resolution, i.e. narrow lineshape that you expect. The FFT of the full time vector shows the algebraic sum of all your lineshapes and therefore appears broadened.
To sum it up there are two separate effects:
a) broadening from the window function (as the commenters 1 and 2 pointed out)
b) broadening from the effect of frequency fluctuation that you are trying to simulate and that happens in real life (e.g. you sitting on a swing while receiving a radio signal).
Finally, note the significance of #xvan's comment : phi= phi(t). If the phase angle is time dependent then it has a derivative that is not zero. dphi/dt is a frequency shift, so your instantaneous frequency becomes f0 + dphi/dt.
I am running FFT algorithm to detect the music note played on a guitar.
The frequencies that I am interested are in the range 65.41Hz (C2) to 1864.7Hz (A#6).
If I set the sampling frequency of the input to 16KHz, the output of FFT would yield N points from 0Hz to 16KHz linearly. All the input I am interested would be in the first N/8 points approximately. The other N*7/8 points are of no use to me. They actually are decreasing my resolution.
From Nyquist's theory (https://en.wikipedia.org/wiki/Nyquist_frequency), the sampling frequency that is needed is just twice the maximum frequency one desires. In my case, this would be about 4KHz.
Is 4KHz really the ideal sampling frequency for a guitar tuning app?
Intuitively, one would feel a better sampling frequency would give you more accurate results. However, in this case, it seems having a lesser sampling frequency is better for improving the resolution. Regards.
You are confusing the pitch of a guitar note with spectral frequency. A guitar generates lots of overtones and harmonics at a much higher frequency than the pitch of a played note. Those higher harmonics and overtones, more than the possibly weak fundamental frequency in some cases, is what the human ear hears and interprets as the lower perceived pitch.
Any of the overtones and harmonics around or above 2 kHz that are not completely low pass filtered out before sampling at 4 kHz will cause aliasing and thus corruption of your sampled data and its spectrum.
If you want to create an accurate tuner, use a pitch estimation algorithm, not an FFT peak frequency bin estimator. And depending on which pitch estimation method you choose, a higher density of samples per unit time might allow finer accuracy or greater reliability under background noise or more prompt responsiveness.
Is 4KHz really the ideal sampling frequency for a guitar tuning app?
You've been mis-reading Nyquist's theorem if you ask it like that.
States that every sampling frequency above twice your maximum signal frequency will allow you to perfectly reconstruct your original signal. So there's no "ideal" frequency. Just a set of frequencies that are sufficient. What is ideal hence depends on a lot of other things: mainly, what your digitizer really supports (hint: most sound cards can do 44.1kHz, but not 4kHz), what kind of margin you want to have for filters etc to work on, and what kind of processing power you can spend (hint: modern smart phones, PCs and even pocket calculators don't really have a hard time processing a couple hundred kHz in real time).
Also note that #hotpaw2 is right, the harmonics are important, and are multiples of the base tone frequency.
However, in this case, it seems having a lesser sampling frequency is better for improving the resolution.
no. No matter where that comes from, it's wrong. Information theory's first and foremost result is that based upon more information, you can't make worse estimates. An oversampled signal is simply more information on the same signal.
Yes, if all you are interested in is frequencies up to 2 kHz then you only need a sampling frequency of 4 kHz. This should include an anti-aliasing filter in front of the ADC or any downconverter to prevent any higher frequency components from aliasing into a lower frequency.
If all you are interested in is specific frequencies (one or two) then you may want to look at the Goertzel algorithm which is more efficient than an FFT for a single frequency. Also, the chirp-Z transform can be used to effectively get a zoomed FFT (resulting in a higher resolution over a smaller bandwidth without the computational complexity of an FFT with the same resolution). You may want to check out this CZT tutorial
Let's say I have some data that corresponds to the average temperature in a city measured every minute for around 1 year. How can I determine if there's cyclical patterns from the data using an FFT?
I know how it works for sound... I do an FFT of a sound wave and now the magnitude is shown in the Y axis and the frequency in Hertz is shown in the X-axis because the sampling frequency is in Hertz. But in my previous example the sampling frequency would be... 1 sample every minute, right? So how should I change it to something meaningful? I would get cycles/minute instead of cycles per seconds? And what does cycles/minute would mean here?
I think your interpretation is correct - you are just scaling to different units. Once you've found the spectral peak you might find it more useful to take the reciprocal to express the value in minutes/cycle (ie the length of the periodic cycle). Effectively this is thinking in terms of wavelength rather than frequency.
I am trying to post-process pulse train data. It is a 0 to 5V square wave, where the frequency of pulses corresponds to a physical measurement. During measuring, I may see anywhere from 100 pulses/second to 10,000 pulses per second. The duty cycle changes.
I wrote a pulse counter function to analyze the pulse data in the time domain, but the result was very noisy. I suspect that an FFT may be appropriate, though I have never really done anything like this before.
Has anybody done anything similar? What would be the broad methodology behind analyzing the pulse train in the frequency domain? Would it be best to take an FFT at specific time intervals (for instance every seconds worth of data)?
An FFT might be useful if your pulse train were stationary over some interval (the length of an FFT) and embedded in noise. Otherwise, why not just use at the reciprocal of the time between rising edges?
I'm trying to analyze frequency detection algorithms on iOS platform. So I found several implementations using FFT and CoreAudio (example 1 and example 2). But in both cases there is some imprecision in frequency exists:
(1) For A4 (440Hz) shows 441.430664 Hz.
(1) For C6 (1046.5 Hz) shows 1518.09082 Hz.
(2) For A4 (440Hz) shows 440.72 Hz.
(2) For C6 (1046.5 Hz) shows 1042.396606 Hz.
Why this happens and how to avoid this problem and detect frequency in more accurate way?
Resolution in the frequency domain is inversely related to number of FFT bins. You need to either:
increase the size of your FFT to get finer resolution
use magnitude of adjacent bins to tweak the frequency estimate
use an alternative method for frequency estimation rather than FFT e.g. parametric model