This time, I'll be discussing the delta function, an essential tool for extensions of the Fourier transform and the discrete Fourier transform. Please forgive any slight lack of mathematical rigor.
The definition of the Fourier transform is shown again.
Equation (1) is the formula for finding the frequency spectrum X(f) from a non-periodic time function x(t), and is called the Fourier transform. Equation (2) is the formula for finding the original time signal x(t) from the frequency spectrum X(f), and is called the inverse Fourier transform. For equations (1) and (2) to have finite values, x(t) must be:
It must be so.
While the Fourier transform can be calculated for single pulse waveforms and transient phenomenon waveforms, for infinitely continuing signals such as DC waveforms and sinusoidal signal waveforms, the above equation (3) is not satisfied, and therefore the usual integral calculation using equation (1) cannot be performed.
Therefore, we define a delta function such that the following equation holds.
Equation (4) means that everything is 0 except at one point (t = t0), and equation (5) means that the area under the curve representing the function is 1 (Figure 1). In reality, we can consider a rectangular pulse with width a and height 1/a as shown in Figure 2, and think of it as the limit waveform where the width a approaches 0 while the area remains 1.
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Figure 1 -
Figure 2
In simple terms, it takes the shape of a pulse waveform with a width of 0 and an infinite height.
This function was devised by Paul Dirac, a key figure in early quantum theory, and is called the Dirac delta function. However, in electrical circuits and control systems, it is also called the impulse function, as its shape suggests.
If we set t0 = 0 in equations (4) and (5);
δ(t)=0 t≠0
It will be.
Multiplying this delta function by an arbitrary differentiable continuous function x(t) and integrating over the range (-∞, ∞), from the definitions of equations (4) and (5):
We can derive this. Equation (8) means that the delta function extracts the value of any function x(t) at a given time, which is one of the important properties of the delta function.
Next, let's perform a Fourier transform on the delta function. Using equation (8):
In other words, the Fourier transform of the delta function is 1 (a constant). Intuitively speaking, the frequency spectrum of an ideal impulse waveform consists of infinitely large frequency components of constant magnitude.
Conversely, the inverse Fourier transform of 1 is:
Equation (11) cannot be calculated using ordinary integrals, but it can be defined in the theory of hyperfunctions, and the delta function and
It will be.
In other words;
Generally, the integral representation of the delta function is:
This expression is very important for calculating the Fourier transform.
This is a very intuitive explanation, but since the cosine waveform is an even function, if we ignore negative frequencies, equation (13) means that if we add the cosine waveform from low frequencies to high frequencies, it approaches an impulse waveform. This is illustrated in Figure 3.
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Figure 3 shows how a composite wave of cosine waveforms with varying frequencies f approaches an impulse waveform.
Next, let's consider the Fourier transform of the DC component.
x(t)=a
Cos is an even function, and Sin is an odd function;
In other words, the Fourier transform of the DC signal x(t) in equation (14) is a&(f). This can be considered as an impulse with a frequency of 0 (DC component) and a height of a.
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Figure 4. Fourier transform of a DC waveform
Next, let's consider the Fourier transforms of periodic signals, namely cosine waves and sine waves.
x(t)=a cos(2π f0 t)
The Fourier transform X(f) of is (transformed using Euler's formula);
The real part is then given as the sum of two impulses (Figure 5).
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Figure 5: Fourier transform of a cosine wave.
In exactly the same way;
x(t)=a sin(2π f0 t)
The Fourier transform X(f) of is:
The imaginary part is then given as the sum of two impulses (Figure 6).
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Figure 6: Fourier transform of a sine wave.
In digital signal processing, sampling is performed to discretize continuous signals for calculation. The delta function is particularly useful in formulating these calculations.
Finally, here's a summary.
- To enable Fourier transforms for waveforms that cannot be calculated using ordinary integrals, such as DC waveforms and periodic signals, we define a delta function, like that of an impulse waveform, in equations (4) and (5) (as a generalized function rather than a regular function).
- By using the delta function, the Fourier transform of impulse waveforms and DC signals can be obtained.
- The Fourier transform of a periodic signal, such as a sine wave (or cosine wave) signal, can be expressed as an impulse (delta function) at its frequency component.
【keyword】
Delta function, Dirac delta function, impulse function
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[Reference materials]
- "Spectral Analysis" by Mikio Hino, published by Asakura Shoten.
- "The Fast Fourier Transform" by E. Oran Brigham, published by Science and Technology Press.
- "Digital Fourier Analysis (I) - Fundamentals" by Kenichi Kido, published by Corona Publishing Co.
(Excerpt from the email newsletter issued on September 22, 2012)
