Last time, we talked about frequency spectra, and the purpose of determining them is to find out what frequency components are contained in a time-domain signal and at what magnitude. So, how is the magnitude (or strength) of a signal defined?
For periodic signals like sine waves, the amplitude (peak value) can be used to define the power. However, for signals that continuously contain various frequency components or random signals, amplitude cannot be used to define the power. Generally, the magnitude (strength) of a time-domain signal x(t) is defined by the mean square, and this is called the signal's power. Using the mean square, any time-domain signal can be defined. Power is fundamentally a quantity proportional to energy (energy per unit time) obtained by multiplying different physical quantities, such as current × voltage. While calling the squared value "power" is a convenient convention, it has become a widely used term in relation to FFT, such as "power spectrum."
Now, if it is a periodic signal, then let its period be T;
Power = 1/T ∫129_0^T??x^2 (t)?t?
This means that "the area of one period from interval 0 to T is 1/T (per unit time)." So, what about non-periodic signals like random signals? Strictly speaking, it is necessary to integrate over an infinite interval limit (T → ∽);
Power = lim┬(T→∞)┬ 1/T ∫12⁹⁹⁰⁻¹x²⁻¹ (t)t
This is the result. However, in reality, integration over an infinite interval is impossible, so the integral is limited to a finite time interval, and therefore the result is an estimate.
Now, in the previous issue, we learned that by performing a DFT on the time-domain signal x(t), we can obtain the power for each frequency component, which we call the power spectrum at a certain frequency in Hz. Looking at it from a different perspective, since the power spectrum is a decomposition of the entire power of the time-domain signal x(t) into its frequency components, the sum of the power spectra is the power of the time-domain signal x(t) itself. If we decompose it into L frequency components and let the resulting power spectrum be P(k):
Power = ∑129_(k=0)^L⁻P(k)
It is expressed as follows. In an FFT analyzer, the right-hand side of equation (3), which is the sum of the power for each frequency component, is specifically called the "overall" value. It is important to note here that the power of a signal is a squared value with the dimension of the square of the amplitude. Therefore, by taking its square root, we can obtain a value with dimensions equivalent to the amplitude. This value is called the root mean square (rms value). As a simple example, in the case of a sine wave, the power (squared value) B with amplitude A V (where V is volts) and its rms value C are:
B=A2/2 、C=√B=A/√2
The following relationship holds true. Equation (4) does not hold true for complex waveforms. In the power spectrum, the simplified relationship in equation (4) can be used because it can be decomposed into sinusoidal (cosine) components using FFT. So, what is the combined power of a sine wave with an amplitude of 2 V and a sine wave with an amplitude of 3 V? This will be homework for next time.
(Excerpt from the email newsletter issued on January 24, 2003)
