I'm trying to generate a Simulink model which can output a sine wave which is dependant on the input frequency. It has to be time discrete with 1ms sample time.
The frequency input will be a positve ramp, then stable, then negative ramp.
This works for the fist part, the rising frequency. As the frequency becomes stable, it outputs zero. If the frequency decreases, it does strange things.
I come to that approach from the wikipedia article chirp signals, but do not know how to handle the stable and especially the falling frequency.
Rising frequency correct (10 Hz -> ~100m)s:
Stable frequency: output 0 (I know why, but what is the best approach to fix)
Decreasing frequency: Incorrect, 10Hz -> ~2ms is much too fast)
Related
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 want to select an optimal window for STFT for different audio signals. For a signal with frequency contents from 10 Hz to 300 Hz what will be the appropriate window size ? similarly for a signal with frequency contents 2000 Hz to 20000 Hz, what will be the optimal window size ?
I know that if a window size is 10 ms then this will give you a frequency resolution of about 100 Hz. But if the frequency contents in the signal lies from 100 Hz to 20000 HZ then 10 ms will be appropriate window size ? or we should go for some other window size because of 20000 Hz frequency content in a signal ?
I know the classic "uncertainty principle" of the Fourier Transform. You can either have high resolution in time or high resolution in frequency but not both at the same time. The window lengths allow you to trade off between the two.
Windowed analysis is designed for quasi-stationary signals. Quasi-stationary signals are signals which change over time but on some short period of time they might be considered stable.
One example of quasi-stationary signal is speech. Frequency components of this signal change over time when position of tongue and mouth changes, but on a short period of time approximately 0.01s they might be considered stable because tongue does not move this fast. The range of 0.01s is determined by our biology, we just can't move tongue faster than that.
Another example is music. When you touch the string you might consider it produces more or less stable sound for some short period of time. Usually 0.05 seconds. Within this period you might consider sound stable.
There might be other types of signals, for example, it might have frequency 10Ghz and be quasi-stationary of 1ms of time.
Windowed analysis allows to capture both stationary properties of signal and change of signal over time. Here it does not matter what sample rate does signal have, what frequency resolution do you need or what are the main harmonics. Are main harmonics near 100Hz or near 3000Hz. It is important on what period of time the signal is stationary and on what it can be considered as changing.
So for speech 25ms window is good just because speech is quasi-stationary on that range. For music you usually take longer windows because our fingers are moving slower than our mouth. You need to study your signal to decide optimal window length or you need to provide more information about it.
You need to specify your "optimality" criteria.
For a desired frequency resolution, you need a length or window size of roughly Fs/df (or from a fraction to twice that length or more, depending on S/N and window). However the length also needs to be similar to or shorter than the length of time during which your signal is stationary within your desired frequency resolution bounds. This may not be possible or known, thus requiring you to specify which criteria (df vs. dt) is more important for your desired "optimality".
If multiple window lengths meet your criteria, then the shortest length that is a multiple of very small primes is likely to be the most computationally efficient for the following FFTs within an STFT computational sequence.
Based on the sampling theorem, the sampling frequency needs to be larger than twice the highest frequency of the signal. And based on DFT (discrete Fourier Transform), we also know that the frequency resolution is the inverse of the entire signal duration, and the the entire frequency span is the inverse of the time resolution. Note that the frequency is simply the inverse of the period, thus the relationships go inversely with each other.
Frequency resolution = 1 / (overall time duration)
Frequency span = 1 / (time resolution)
Having said that, to process 20kHz audio signal, we need to sample in 40kHz. And if we want to get the frequency resolution down, say to 10Hz, we will need to sample the entire duration as long as 0.1Sec, which is 1/10Hz.
This is the reason we normally see that audio files are said to be 44k. Because the human hearing range is limited to 20kHz. To add some margin to it, we use 44k sampling frequency in stead of 40kHz.
I think the uncertainty principle goes with the fact that more localized signal in one domain, actually spread out on the other. For example, a pulse in time domain goes from negative infinity to positive infinite, i.e the entire stretch of the spectrum. And vice versa that the a single frequency signal in spectrum stretches from negative infinity to positive infinite in time domain. This is simply because we had to go forever in order to know if a signal could be a pure sinusoidal signal or not.
But for DFT, we can always get the frequency span if we sample twice the highest frequency of the signal, and the resolution we want if we sample the signal duration long enough. So, not so uncertain as the uncertainty principle says, as long as we know how many samples to take and how fast and how long to take them.
I am running some experiments on MATLAB, and I have noticed that, keeping the period fixed, increasing the sampling rate of a sine signal causes the different shifted waveforms in the Fourier transform to become more distinct. They get further apart, I think this makes sense because as the sampling rate increases, the difference between the Nyquist rate and the sampling rate increases too, which creates an effect opposed to aliasing. I have also noticed that the amplitude of the peaks of the transform also increase as the sampling rate increases. Even the DC component (frequency = 0) changes. It's shown as being 0 at some sampling rate, but when increasing the sampling rate it's not 0 anymore.
All the sampling rates are above the Nyquist rate. It seems odd to me that the Fourier transform changes its shape, since according to the sampling theorem, the original signal can be recovered if the sampling rate is above the Nyquist rate, no matter if it's 2 times the nyquist rate or 20 times. Wouldn't a different Fourier waveform mean a different recovered signal?
I am wondering, formally, what's the impact of the sampling rate
Thank you.
You're conflating conversion between time-discrete and time-continuous forms of a signal with reversibility of a transform.
The only guarantee is: For a given transform of some discrete signal, its inverse transform will yield the "same" discrete signal back. The discrete signal is abstract from any frequencies. All that the transform does is take some vector of complex values, and give the dimensionally matching vector of complex values back. You can then take this vector, run an inverse transform on it, and get the "original" vector. I use quotes since there may be some numerical errors that depend on the implementation. As you can see, nowhere does the word frequency appear because it's irrelevant.
So, your real question is then, how to get an FFT with values that are useful for something besides getting the original discrete signal back through an inverse transform. Say, how to get an FFT that will tell a human something nice about the frequency content of a signal. A transform "tweaked" for human usefulness, or for use in further signal processing such as automated music transcription, can't reproduce the original signal anymore after inversion. We're trading off veracity for usefulness. Detailed discussion of this can't really fit into one answer, and is off topic here anyway.
Another of your real questions is how to go between a continuous signal and a discrete signal - how to sample the continuous signal, and how to reconstruct it from its discrete representation. The reconstruction means a function (or process) that will yield the values the signal had at points in time between the samples. Again, this is a big topic.
You are seeing several things when you increase the sample rate:
most (forward) FFT implementations have an implicit scaling factor of N (sometimes sqrt(N)) - if you're increasing your FFT size as you increase the sample rate (i.e. keeping the time window constant) then the apparent magnitude of the peaks in the FFT will increase. When calculating absolute magnitude values you would normally need to take this scaling factor into account.
I'm guessing that you are not currently applying a window function prior to the FFT - this will result in "smearing" of the spectrum, due to spectral leakage, and the exact nature of this will be very dependent on the relationship between sample rate and the frequencies of the various components in your signal. Apply a window function and the spectrum should look a lot more consistent as you vary the sample rate.
Is there any formula to calculate the frequency (or frequencys) of a signal that is bad sampled?
For example, what's the output of an analog signal with F=22Khz when it's sampled at 25Khz, or 10Khz?
EDIT:
In this example, the sampled signal (on the right) have a different frequency than the original one, because it was bad sampled (Fs is minor than 2*F)
My question is: is there any formula to know what's the frequency of this 20kHz signal, sampled at 30kHz?
No any formula to know what's the frequency of 20kHz signal, sampled at 30kHz. But it is a fact that the frequency of undersampled signal will be reflected about Nyquist frequency. In your example 30 kHz means that Nyquist frequency is about 15 KHz, that is not enough to record original signal (20KHz) correctly, only 15 kHz of it distributed, another 5 KHz (reminder after distribution of 15 KHZ) during reflection about Nyquist frequency appear in position 15-5=10 KHz. This is final ansver. The frequency of sampled signal will be equal 10 kHz in your case
Unless the bandwidth of the signal is less than half the sampling rate, you lose information during sampling and generally can't distinguish frequencies after that due to aliasing.
See Undersampling for more details about sampling at rates lower than twice the maximum signal frequency.
There's no simple formula that can give you the spectral content of a signal or the main frequency. In general you need to calculate a Discrete Fourier Transform of the sampled signal to find that out. If you're interested in whether or not there's a specific frequency, or how strong it is, you can calculate DFT at that frequency. The Goertzel algorithm can be an option.
EDIT: a signal at frequency f such that fsample/2 <= f < fsample will alias to f* = fsample - f, hence a 20KHz sine wave sampled at 30KHz will appear as a 10KHz sine wave.
In general frequencies above the fsample/2 can be observed in the sampled signal, but their frequency is ambiguous. That is, a frequency component with frequency f cannot be distinguished from other components with frequencies N*fsample/2 + f and N*fsample/2 – f for nonzero integers N. This ambiguity is called aliasing*.
Assuming a constant sampling rate, any sampling will alias together spectral content from below and above the sampling rate. If you have frequency content on both sides of the sampling rate that you don't want combined, you will have to filter one or the other frequency band Out before the sampling, or you will have a problem. For instance a low-pass filter which only passes signals below Fs/2, or a bandpass filter that only passes signals strictly between n*Fs/2 and (n+1)*Fs/2 for some integer n, might be appropriate.
Note that the accuracy of the sampling rate must be higher (lower jitter) for n > 0. Lack of this lower jitter would be an example of bad sampling that would add random phase noise.