This time, we will discuss "convolution," one of the important properties of the Fourier transform that describes the relationship between time and frequency.
The convolution of time signals x(t) and h(t) is a sum-of-products operation of two functions and is defined by equation (1). It is also called a convolution integral because it is expressed as an integral.
=x(t)*h(t)
Equation (2) is a simplified representation of the integral in equation (1).
As a concrete example of convolution, in a linear system such as an electrical circuit like a filter (Figure 1), if x(t) is the input time signal and h(t) is the time function representing the characteristics of the system, then the output of the system is y(t) in equation (1).
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Figure 1 Linear system
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Figure 2: When a signal is input to a linear system
Here, we will investigate what happens to the output y(t) when the signal x(t) from step 1 is input to a system whose characteristic h(t) is a function as shown in Figure 2.
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Figure 3. Explanation of the convolution integral procedure (the diagram above shows the case when t = t').
The procedure for the convolution integral equation (1) involves folding the time axis of h(τ) (Figure a), shifting it in the positive direction of the time axis (Figure b), multiplying it by x(τ) (Figure c), and then performing the integral, which ultimately yields the output y(t) in Figure 3 (d).
If we actually calculate it from equation (1):
It can be seen that this is the case.
Also, although I will omit the explanation, the same result is obtained by swapping x(t) and h(t) in equation (1).
In other words;
Next, let's consider the case where a unit impulse function (the delta function we encountered previously) is input to the linear system in Figure 1.
In equation (4), by substituting the delta function δ(t) for x(t) and utilizing the properties of the delta function and its even function, we can:
Therefore, the output signal y(t) becomes h(t) itself.
In the linear system shown in Figure 1, h(t) is the output when a unit impulse function is input to the system, and is therefore called the impulse response.
I will now discuss the most important relationship between convolution and the Fourier transform.
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Figure 4. Time function in a linear system and its Fourier transform
As shown in Figure 4, let X(f), H(f), and Y(f) be the Fourier transforms of the time functions x(t), h(t), and y(t), respectively. Let's consider what kind of relationship functions that have a convolution relationship on the time axis have on the frequency axis.
Taking the Fourier transform of both sides of equation (1) gives:
The Fourier transform of the function obtained by shifting the time function h(t) by -τ is equal to the product of the Fourier transform of the original function and e-j2πft, so the expression in [ ] can be written as e-j2πft H(f), and therefore equation (8) is;
The Fourier transform of the convolution y(t) of two functions x(t) and h(t) is equal to the product of the Fourier transforms of the original functions. This relationship is called the convolution theorem. The converse is also true, so if we represent the Fourier transform and inverse Fourier transform with , we get the Fourier transform pair shown in equation (10).
x(t)*h(t)=X(f)H(f)
As will be explained later, FFT analyzers use this relationship to calculate frequency response functions, impulse responses, and other parameters.
In exactly the same way, the inverse Fourier transform of the convolution of two functions on the frequency axis is equal to the product of the original time functions. Using the same notation as in equation (10):
x(t)*h(t)=X(f)*H(f)
This is evident from the symmetry of the Fourier transform operation. Equation (11) is sometimes called the frequency convolution theorem.
This relationship is an important theorem in explaining window functions and other concepts in FFT analyzers.
Finally, in equation (11), if we let both time functions be x(t), the left side becomes the Fourier transform of x² (t), so if we let the new variable be k:
This is the result. Now, if we set k = 0, then ultimately;
The following relationship is obtained. This relationship is called Parseval's theorem. The physical meaning of equation (13) is that the total energy on the time axis is equal to the total energy on the frequency axis, which is an obvious statement. In that sense, |x(f)| 2 fX can be called the energy spectrum.
Finally, here's a summary.- The convolution integral of two time functions is defined by equation (1) or equation (3).
- When x(t) is input to a linear system of impulse responses h(t), the output is a convolution of h(t) and x(t).
- The Fourier transform of the convolution y(t) of two functions x(t) and h(t) is equal to the product of the Fourier transforms of the original functions. [Convolution Theorem (Equation (10))]
- The inverse Fourier transform of the convolution of two functions on the frequency axis is equal to the product of the original time functions. [Frequency Convolution Theorem (Equation (11))]
- The total energy on the time axis is equal to the total energy on the frequency axis. [Parseval's Theorem (Equation (13))]
【keyword】
Convolution, convolution integral, linear system, unit impulse function, delta function, impulse response, convolution theorem, frequency convolution theorem, Parseval's theorem, energy spectrum
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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.
- "Fourier Analysis" by H.P. Su, translated by Heihachi Sato, published by Morikita Publishing.
(Excerpt from the email newsletter issued on November 15, 2012)
