Frequency Analysis from the Basics (12) - "Power Spectrum (Part 1)"

Following on from our previous discussion of Fourier spectra, this time we'll talk about power spectra.

The primary purpose of spectral analysis is to measure the power spectrum, which is the most fundamental and important function of an FFT analyzer as an electrical measuring instrument. Therefore, we will discuss the power spectrum in two parts. This is part 1.

The power spectrum is a function of frequency and represents the energy per unit time, or power (signal strength), for each frequency contained in the original time signal.

In the field of signal processing, such as spectral analysis, "power" refers to the mean square of the time signal.
For example, the mean square of a time signal x(t) with period T, -x², is:

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

Alternatively, the mean squared value -x² of a time signal x(t) without periodicity is;

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

It can be expressed as follows.

Generally, time signals can be broadly classified into deterministic signals and non-deterministic signals. Deterministic signals are signals that can be described as a function of time, such as periodic signals, while non-deterministic signals are probabilistic signals that cannot be clearly described by mathematical formulas, such as random signals (irregular signals).

First, let's consider a periodic signal x(t) with period T. By complex Fourier series expansion, we can express it as follows:

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

This can be expressed as follows. Here, ck is the complex Fourier coefficient that appeared in the column two issues ago (Newsletter No. 144). (The argument k is used as frequency data.)

Next, squaring both sides of equation (3) and taking the average over period T, we obtain equation (4) below due to the orthogonality of the complex exponential function.

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

Alternatively, since the magnitudes of the positive and negative frequency components are equal;

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

Here,

Let P(0) = c0 ², then equation (5) becomes;

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

This is the result. Equation (6) shows that the sum of the components P(k) for each frequency is the power of the time signal x(t)​ ​(left side), so the frequency function P(k) is called the power spectrum. Unlike the Fourier spectrum (or complex Fourier coefficients), the power spectrum loses phase information.

We will now explain the procedure for finding P(k) using the FFT operation.

Let's take equation (7) below, which was presented in the previous measurement column "Fourier Spectra," as an example.

n＝0、1、2、……、N-1

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

Substituting N=64 and k=4 into equation (7) above and performing an N-point FFT operation, the Fourier spectrum is calculated as shown in Table 1.

Table 1: Calculation results when N = 64 and k = 4 are substituted into equation (7).

Frequency point (k)	Real part	Imaginary part
k=4 (positive frequency)
Nk = 60 (negative frequency)

From this result, divide by the time window length T (T=Nτ, where τ is omitted, so N=64);

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

This can be calculated as follows. This is equal to the mean square of x(t) in equation (7), thus confirming equation (6).

In actual FFT analyzers, since the Fourier transform is finite and discrete, theoretically, as explained two lessons ago, it is possible to obtain the power spectrum up to N/2 points from N points of time data. However, due to the anti-aliasing filter, the analyzer is designed to measure up to a slightly smaller number of points, N/2.56. For example, when N=64, it obtains up to 64/2.56 = 25 points. (Actually, there are DC components, so it's 26 points...) Please refer to Figures 1 and 2 for these relationships.

Here, we will summarize the relationship between time waveforms and power spectra.

In Figure 3, assume that the time waveforms of N points sampled at a sampling frequency fs (upper figure) are subjected to an N-point FFT to obtain the power spectrum shown in the lower figure.

Time axis

1. time resolution
2. Number of sampling points N
3. Time window length

frequency axis

1. frequency range
2. Number of analysis lines
3. frequency resolution

From the last equation, we can see that in order to increase (decrease) the frequency resolution Δf of the power spectrum, we can either decrease the sampling frequency fs or increase the number of sampling points N.

Furthermore, regarding why the constant "2.56" is used in the above relationship, it is because the number of FFT points is usually a power of 2.
This is because we want the frequency resolution to be a nice, even number that divides it evenly, as we will be running the program using this method.

The number of sampling points N and the corresponding number of analysis lines L that can be used with Ono Sokki 's FFT analyzer.
This is shown in Table 2 below.

Table 2 Sample points and number of analysis lines

Next time, in Power Spectrum (Part 2), we will discuss PSD and ESD.

Finally, here's a summary.

1. The main purpose of spectral analysis is to determine the power spectrum.
2. The power spectrum represents the energy per unit time, or power (signal strength), for each frequency contained in the original time signal. It is a positive real value that does not carry phase information.
3. Generally speaking, power is the mean square of a time signal (the square of its RMS value), and the power spectrum can be said to be a decomposition of that power by frequency.
4. In principle, power spectra up to N/2 points can be obtained from time waveforms at N points, but in actual FFT analyzers, the number of analysis lines is smaller, down to N/2.56.
5. The power spectrum is analyzed with a certain frequency resolution Δf, but to increase (improve or decrease) that resolution, the sampling time window length T must be increased. In other words, either the sampling frequency fs is lowered, or the number of sampling points N is increased.

【keyword】
Spectral analysis, Fourier spectrum, power spectrum, power, signal strength, mean square, deterministic signal, non-deterministic signal, complex Fourier coefficients, anti-aliasing low-pass filter, frequency resolution

[Reference materials]

1. "Spectral Analysis," by Mikio Hino, Asakura Shoten (1977)
2. "Digital Fourier Analysis (I) - Fundamentals -" by Kenichi Kido, Corona Publishing Co., Ltd. (2007)

(Excerpt from the email newsletter issued on November 21, 2013)