Frequency Analysis from the Basics (4) - "Fourier Transform"

Today, I'll be talking about the Fourier transform.

In our previous discussion on Fourier series expansion, we assumed that the time waveform was a periodic function with period T. So, how do we determine the frequency components of a continuously repeating or transient signal that has no periodicity at all?

Since there are no numerical restrictions on the period T in the Fourier series, we consider the case where there is no periodicity, i.e., when the period T approaches infinity.

Let's start with the complex Fourier series explained last time. If the period of the time function x(t) is T, then;

.................................(1)

.................................(2)

Since cn in equation (2) is discrete with a frequency interval of 1/T;

.................................(3)

Letting this, and substituting this symbol into equations (1) and (2), as we let T approach infinity, we get:

.................................(4)

Here, if we let T → ∞, then Δf → df and fn → f, and finally;

.................................(5)

This is how it works. Here;

.................................(6)

Equation (6) is the formula for obtaining the frequency spectrum X(f) from the non-periodic time function x(t), Fourier transformationConversion (or Fourier integral) It is called. Conversely, the formula for finding the original time signal x(t) from the spectrum X(f) is given by equation (5). Inverse Fourier Transform We call it that. Since X(f) is clearly a complex number, complex f-Lie spectrum It is called [this].

Equations (5) and (6) involve integrals of infinity, and the question is whether the integral value is finite. While we will omit the mathematically rigorous details, if the time signal x(t) satisfies the following condition (7), then both equations (5) and (6) will have finite values.

.................................(7)

This property is called the absolute integrability of x(t). Intuitively speaking, equation (7) does not hold true for continuous, infinitely continuing time waveforms like sine waves or random waves, but it does hold true for transient signals such as single pulse waves.

Now, comparing the Fourier series equation (2) and the Fourier transform equation (6), the cn in equation (2) is clearly a line spectrum with only wavenumber components of the fundamental frequency 1/T and its integer multiples, while the complex Fourier spectrum X(f) in equation (6) is a continuous spectrum.

As a concrete example, we will calculate the Fourier transform of the pulse waveform in Figure 1.

This time waveform clearly satisfies equation (7);

.........................(8)

This is the result (Figure 2).

In equation (8), the right-hand side is intentionally written in the form sin x/x, without canceling out the b in the numerator and denominator.
This is a function of this form. Sinc function It is called a signal processor and frequently appears in the field of signal processing.

Next, consider a period-time waveform in which the pulse waveform in Figure 1 repeats with period T, as shown in Figure 3.

Next, we calculate the Fourier series expansion of this waveform. If the fundamental frequency is f0, then f0 = 1/T

that's why;

.........................(9)

Now, comparing equations (8) and (9), we can clearly see that, excluding the constant 1/T, cn is equal to X(f) with nf 0 substituted. That is;

.......................(10)

This is the relationship.

Figure 4 is a graph of the complex Fourier coefficients cn with the complex Fourier spectrum X(f) superimposed on it, ignoring the coefficient 1/T.

Finally, here's a summary.

1. The Fourier transform is applied to non-periodic signals, such as transient signals.
2. The spectrum obtained by the Fourier transform is an infinitely continuous spectrum. In contrast, the Fourier series obtained by the Fourier series expansion is a line spectrum.
3. The vertical axis of the Fourier coefficients represents the physical quantity of the signal itself, while the vertical axis of the Fourier spectrum represents the product of the physical quantity of the signal and time.

【keyword】
Fourier transform, Fourier series expansion, Fourier integral, inverse Fourier transform, complex Fourier spectrum, absolutely integrable, line spectrum, continuous spectrum, Sinc function

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[Reference materials]

1. "Signal Processing," co-authored by Iwao Morishita and Hidefumi Obata, Society of Instrument and Control Engineers.
2. "Spectral Analysis" by Mikio Hino, published by Asakura Shoten.
3. "The Fast Fourier Transform" by E. Oran Brigham, published by Science and Technology Press.

(Excerpt from the email newsletter issued on July 20, 2012)