This time, building on what we've discussed so far (such as the delta function and convolution), I'll talk about sampling time-series signals.
To process time signals from sensors and other sources into digital signals, it is necessary to convert these analog signals into digital values (numerical data) at regular intervals. This process is called sampling, and the interval is called the sampling period (its reciprocal is the sampling frequency).
-
Figure 1. Time signal and its sampling
Figure 1 shows an example where the analog time signal on the left is sampled with a sampling period τ (s), leaving only the numerical sequence at the dot (•) on the right. In this case, the sampling frequency is 1/τ (Hz).
So, what sampling frequency should we use to sample the original analog signal without losing any information? Intuitively, it might seem that a high sampling frequency (i.e., a finer sampling interval) would be best, but this would result in an unnecessarily large amount of data being processed, which is undesirable. This article will discuss how to determine the optimal sampling frequency.
As preparation for explaining the sampling of time signals, let's add one more property of the delta function (impulse function).
"The Fourier transform of an equally spaced impulse train is also an equally spaced impulse train."
That's the point.
An impulse train is formed by arranging unit impulse functions δ(t) at intervals of period τ;
.................................（1）
Then, its Fourier transform is:
.................................（2）
-
Figure 2. Impulse train with equal intervals τ and its Fourier transform
I will prove this property.
Equation (1) is a periodic function that repeats with period τ, and its complex Fourier coefficient c n is (due to the properties of the impulse function);
.................................（3）
Next;
.................................（4）
It can be rewritten as follows.
From this, the Fourier transform of equation (4) is:
And we can derive equation (2).
As shown in Figure 2, the Fourier transform of a time series of impulse trains with periods of τ intervals is an impulse train on the frequency axis with frequency intervals of 1/τ.
Figure 3 is an explanatory diagram showing how the Fourier transform of a sampled time series signal, generated by sampling an arbitrary time signal x (t) with a sampling period τ, is derived using the convolution method discussed previously. The left side of Figure 3 represents the time domain, and the right side represents the frequency domain.
-
Figure 3: Explanatory diagram for obtaining the Fourier transform of a time-series signal sampled with period τ using convolution.
The following is an explanation of this diagram.
Here, we assume that the frequency components of the time signal x(t) (Figure 3-(a)) are limited to a maximum of fm (Hz) (i.e., bandwidth-limited) and do not include any frequencies higher than that (Figure 3-(b)). This assumption is not theoretically necessary, but in practice, we can consider the time signal we are dealing with to be finite, so we make this assumption.
When a time signal x(t) is sampled with a sampling period τ, the time series is:
x(τ)、x(2τ)、x(3τ)、・・・・x(nτ)、・・・
This is the result. This time-series signal is:
.................................（5）
This can be expressed as follows: That is, the sample time series of x(t) is an infinitely continuing impulse train with equal intervals τ, and its amplitude is given by the value of x(t) at the same time. In Figure 3, this is (e), which is the product of (a) and (c).
Next, we move to the frequency domain.
X(f) is the Fourier transform of the time signal x(t) (Figure 3-(b)), and the Fourier transform of the impulse train (Figure 3-(c)) is also an impulse train on the frequency axis (Figure 3-(d)), as explained above.
As explained previously, a product on the time axis corresponds to a convolution operation (frequency convolution theorem) on the frequency axis. Therefore, the Fourier transform of a sample time-series signal is the convolution of X(f) and an impulse train on the frequency axis, as shown in Figure 3(f), and becomes a periodic function that repeats at a sampling frequency of 1/τ (Hz).
In the example in Figure 3, the sampling frequency (1/τ) was sufficiently large compared to the frequency bandwidth fm of the time signal (the sampling period τ was sufficiently small). However, let's consider the case where this is not the case. As the sampling period increases, the interval between frequency impulses becomes narrower, so a phenomenon occurs where convolutions overlap, as shown in Figure 4, and distortion occurs in the Fourier transform result. This phenomenon is called aliasing, and this distortion is called aliasing distortion.
From (f) in Figure 3, in order to avoid overlap, that is;
.................................（6）
You need to choose a sampling frequency of 1/τ such that this occurs.
In general, if any time function x(t) is band-limited by fm and sampled at a sampling frequency of 1/τ that satisfies equation (6), then the sampled time series signal can be uniquely determined using only equation (5) without any loss of information. Furthermore, the result of its Fourier transform, X(f), can also be accurately determined.
At this time, the time-continuous function x(t) can be obtained from the sampled numerical sequence x(nτ) using the following formula.
-
This text is a sample. It is included to check the size, amount, spacing, and line spacing of the characters.
.................................（7）
This relationship is called the sampling theorem.
Typically, when using an FFT analyzer, the time signal to be frequency-analyzed is an unknown signal, and its upper frequency limit fm is unknown. Therefore, the only option is to arbitrarily determine the sampling frequency fs. In this case, the maximum frequency that can be analyzed is fs/2, and this frequency is called the Nyquist frequency.
If the Nyquist frequency fs/2 is smaller than the upper frequency fm of the signal, aliasing distortion will occur. To prevent this distortion, FFT analyzers are equipped with a low-pass filter that limits the bandwidth at the Nyquist frequency fs/2. This is called an anti-aliasing filter (or aliasing prevention filter).
-
Figure 4: An example of aliasing occurring.
Finally, here's a summary.
- In order to process analog signals into digital signals, it is necessary to convert them into digital values at regular time intervals. This process is called sampling, and the regular interval is called the sampling period (the reciprocal of which is the sampling frequency).
- The Fourier transform of a time series of equally spaced impulses is a series of equally spaced impulses on the frequency axis.
- In order to correctly sample a time signal that is bandwidth-limited to a frequency band of fm, the sampling frequency must be at least 1/2 fm. This is called the sampling theorem.
- If sampling is not performed according to the sampling theorem, aliasing distortion will occur, making it impossible to correctly obtain the spectrum.
- To prevent this aliasing phenomenon, FFT analyzers are equipped with an anti-aliasing filter that is linked to the sampling frequency.
【keyword】
Sampling, sampling period, sampling frequency, delta function, impulse function, convolution, frequency convolution theorem, sampling theorem, Nyquist frequency, anti-aliasing filter, anti-aliasing filter
------------------------------------------
[Reference materials]
- "The Fast Fourier Transform" by E. Oran Brigham, published by Science and Technology Press.
- "Digital Fourier Analysis (I) - Fundamentals" by Kenichi Kido, Corona Publishing Co.
- "Spectral Analysis" by Mikio Hino, published by Asakura Shoten.
(Excerpt from the email newsletter issued on January 24, 2013)
