I want to take an array with N number of audio data points and upsample it such that there are L*N points. I understand an accurate way to do this is to pad L-1 zero points between each original point and then to low pass the signal. According to this 4 minute video https://www.youtube.com/watch?v=sJslC6TuCoc I should lowpass at a frequency of Pi / L and then add a gain of L to the result to properly upsample my signal. I am having trouble with this low passing step and my result audio signal is not audible at all. Can anyone help me here? Is this "low pass" really more like a band reject filter or something?
My low pass algorithm is noted here (biquad transfer function with coefficients marked under "LPF"): http://music.columbia.edu/pipermail/music-dsp/1998-October/054185.html
You can interpolate all the added points using a high quality interpolation algorithm, such as a polyphase windowed Sinc FIR filter.
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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
I am trying to implement FFT, and I am OK with the code etc, but the general order of things is confusing me.
Am I right in thinking that this is the correct order of things to do?
Input -> Overlap input -> Windowing -> FFT -> Phase calculations/Overlap compensation -> Output
I'm getting results close to my input frequency, but they are consistently off by some factor that I can't work out, i.e. 440Hz is always 407Hz, 430Hz is always 420Hz.
The main bit that is confusing me is the initial overlap, as I have been looking at some open source FFT code and that is the part that I can never quite work out whats going on. I seem to be getting the idea from looking at those that overlapping is supposed to happen before windowing, but to me logically, wouldn't that mess with the windowing?
Any advice would be great
Thanks
The FFT is a discrete version of the continuous Fourier Transform.
The FFT produces a 1D vector of complex numbers. This complex vector is often used to calculate a 2D matrix of Frequency Magnitude versus Frequency, and represented as a 2D graph, like this one:
A single FFT is used when you want to understand the frequency spectrum of a signal. For example, from the above FFT graph we can say that most of the energy in this female soprano's G5 note is concentrated in the 784 Hz and 1572 Hz frequencies.
STFT or "Short-Time Fourier Transform" uses a sliding-frame FFT to produce a 2D matrix of Frequency versus Time, often represented as a graph called a Spectrogram, like this one:
The STFT is used when you want to know at what time a particular frequency event occurs in the signal. For example, from the above graph we can say that a large portion of the energy in this vocal phrase occurred between 0.05 and 0.15 seconds, in the frequency range of 100 Hz to 1500 Hz.
The workflow for the FFT is:
Sample the signal -> Window the entire sample frame -> FFT -> Calculate magnitude and phase -> Output something, usually a 2D graph
If your time-domain data is available in text form and if you can post it here, we can try to help you analyze it, or you can analyze it yourself with this online FFT: Sooeet FFT calculator
If you use window for FFT, your computation will be a kind of STFT.
There are some prepared codes of STFT like 'Spectrogram' etc.
To write the code by FFT, the overlapping is inevitable,but you can use some optimization methods to minimize ghost effects.Also, the practical way for windowing may be choosing the window's bandwidth according to frequency extension. It is clear that in high frequency data's you need to select small windows which is so time consuming.
I am not good enough in Matlab to write this code adhesively:)
Good Luck
I need to know a way to make FFT (DFT) work with just n points, where n is not a power of 2.
I want to analyze an modify the sound spectrum, in particular of Wave-Files, which have in common 44100 sampling points. But my FFT does not work, it only works with points which are in shape like 2^n.
So what can I do? Beside fill up the vector with zeros to the next power of 2 ?!
Any way to modify the FFT algorithm?
Thanks!
You can use the FFTW library or the code generators of the Spiral project. They implement FFT for numbers with small prime factors, break down large prime factors p by reducing it to a FFT of size (p-1) which is even, etc.
However, just for signal analysis it is questionable why you want to analyze exactly one second of sound and not smaller units. Also, you may want to use a windowing procedure to avoid the jumps at the ends of the segment.
Aside from padding the array as you suggest, or using some other library function, you can construct a Fourier transform with arbitrary length and spacing in the frequency domain (also for non-integer sample spacings).
This is a well know result and is based on the Chirp-z transform (or Bluestein's FFT). Another good reference is given by Rabiner and can be found at the above link.
In summary, with this approach you don't have to write the FFT yourself, you can simply use an existing high-performance FFT and then apply the convolution theorem to a suitably scaled and conditioned version of your signal.
The performance will still be, O(n*log n), multiplied by some implementation-dependent scaling factor.
The FFT is just a faster method of computing the DFT for certain length vectors; and a DFT can be computed for any length of input vector. You can also zero-pad your input vector to a length supported by your FFT library, which may be faster.
If you want to modify your sound file, you may need to use the overlap-add or overlap-save fast convolution filtering after determining the length of the impulse response of your frequency domain modification.
How do i get frequency using FFT? What's the right procedure and codes?
Pitch detection typically involves measuring the interval between harmonics in the power spectrum. The power spectrum is obtained form the FFT by taking the magnitude of the first N/2 bins (sqrt(re^2 + im^2)). However there are more sophisticated techniques for pitch detection, such as cepstral analysis, where we take the FFT of the log of the power spectrum, in order to identify periodicity in the spectral peaks.
A sustained note of a musical instrument is a periodic signal, and our friend Fourier (the second "F" in "FFT") tells us that any periodic signal can be constructed by adding a set of sine waves (generally with different amplitudes, frequencies, and phases). The fundamental is the lowest frequency component and it corresponds to pitch; the remaining components are overtones and are multiples of the fundamental's frequency. It is the relative mixture of fundamental and overtones that determines timbre, or the character of an instrument. A clarinet and a trumpet playing in unison sound "in tune" because they share the same fundamental frequency, however, they are individually identifiable because of their differing timbre (overtone mixture).
For your problem, you could sample the trumpet over a time window, calculate the FFT (which decomposes the sequence of samples into its constituent digital frequencies), and then assert that the pitch is the frequency of the bin with the greatest magnitude. If you desire, this could then be trivially quantized to the nearest musical half step, like E flat. (Lookup FFT on Wikipedia if you don't understand the relationship between the sampling frequency and the resultant frequency bins, or if you don't understand the detriment of having too low a sampling frequency.) This will probably meet your needs because the fundamental component usually has greater energy than any other component. The longer the window, the greater the pitch accuracy because the bin centers will become more closely spaced in frequency. However, if the window is so long that the trumpet is changing its pitch appreciably over the duration of the window, then the technique's effectiveness will break down considerably.
DansTuner is my open source project to solve this problem. I am in fact a trumpet player. It has pitch detection code lifted from Audacity.
ia added this org.apache.commons.math.transform.FastFourierTransforme package to the project and its works perfectly
Here is a short blog article on non-parametric techniques to estimating the PSD (power spectral density) along with some more detailed links. This might get you started in estimating the PSD - and then finding the pitch.