Spectrum, energy spectrum, power spectrum, octave spectrum, 1-/3 times-octave spectrum

In the process of acoustic signal processing, we often encounter the following concepts: energy spectrum, power spectrum, octave spectrum,1/3 frequency range spectrum. There are differences and links between these concepts. Some of these concepts are often mixed when people talk about problems. A little bit of time lately, I'm going to make a comb of these kinds of terms and record them here. Among them, the accurate description of the power spectral density needs to use a bit of the knowledge of a stationary stochastic process, considering that most people do not understand the random process, I try to use only the simplest Fourier analysis of the basic concepts to illustrate, although not very rigorous, but for our usual application is enough.

Time domain signals can be represented as function x (t)of time t. If the signal satisfies the absolute integrable condition, the requirement is:

Then this signal has Fourier transform:

where X (f) is called the frequency spectral density of the signal, referred to as the Spectrum (Spectrum). | X (f) |  called the amplitude spectrum (amplitude Spectrum),argx (f) is called the phase spectrum (phase Spectrum).

The energy of the signal is defined as:

For signals with infinite energy we can calculate their average power:

For signals with infinite energy but with limited average power, we call this a power signal. The following equation is known by the Parseval's (parseval) equation:

The first of these is the energy we usually call the signal, so | X (f) | 2 is known as the energy spectral density of the signal , which is referred to as Spectrum.

For power signals:

At this point we say:

For the power spectral density of the signal (power spectral density, PSD), or spectral power distribution (spectral power distribution, SPD), Referred to as power Spectrum (Power spectrum). The units of power spectral density are usually expressed in watts per hertz (w/hz).

The power spectral density of the signal is also the Fourier transform of the signal autocorrelation function. The power spectral density exists only if and only if the signal is a generalized stationary process. If the signal is not a stationary process, then the autocorrelation function must be a function of two variables, so there is no power spectral density, but a similar technique can be used to estimate the time-varying spectral density. We see a process of limiting the power spectral density when it is obtained. The actual calculation is usually taken for a limited period of time. and sampling the signal. Set the acquisition frequency to FS, the sampling time interval is δ. The relationship between the discrete signal X[n] and the continuous signal x (t) is as follows:

The average power at this time is:

The discrete Fourier transform expression is as follows:

Here the frequency of XM corresponds to:

The following equation is known by the Parseval's (parseval) equation:

Therefore, the estimated power spectral density is as follows:

The power spectral density is computed here by dividing fs/n because each XM is actually the power within the fs/n bandwidth.

frequency multiplier Power spectrum

in the field of audio analysis, often to analyze the frequency spectrum of audio signals, the most commonly used is the octave power spectrum and 1/3 octave power spectrum. The so-called octave power spectrum, is to divide the audio into a frequency band, and then calculate the power spectrum in each frequency band separately. The width of the adjacent bands is a 2:1 relationship. 1/3 Octave is the octave is subdivided into three segments. The following table shows the recommended frequency division methods for iec

band number	octave band center frequency	One-third octave band center frequency	band limits lower	upper
14	31.5	25	22	28
15		31.5	28	35
16		40	35	44
17	63	50	44	57
18		63	57	71
19		80	71	88
20	125	100	88	113
21st		125	113	141
22		160	141	176
23	250	200	176	225
24		250	225	283
25		315	283	353
26	500	400	353	440
27		500	440	565
28		630	565	707
29	1000	800	707	880
30		1000	880	1130
31		1250	1130	1414
32	2000	1600	1414	1760
33		2000	1760	2250
34		2500	2250	2825
35	4000	3150	2825	3530
36		4000	3530	4400
37		5000	4400	5650
38	8000	6300	5650	7070
39		8000	7070	8800
40		10000	8800	11300
41	1600	12500	11300	14140
42		16000	14140	17600
43		20000	17600	22500

Because the width of each frequency band is different, so the frequency spectrum of the plot with the normal power spectrum of the graph has a great difference. There is a historical reason for the power spectrum of octave. Before the popularization of digital signal processing technology, people used to design a series of filters and measure the power of the filter output to determine its spectrum. It is not possible to divide the frequency bands very thin, so people design the filter according to Octave (the Q value of the filter is the same according to the octave, so it can be realized with the same type and different parameters of the filter group).

However, use DFT The technology, can calculate the octave power spectrum. The method is simple, as long as a octave within the DFT  Compute the power overlay on each sub-band. In order to calculate the results accurately, at least ensure that there is a 3 4 lines. We know that the lower the frequency octave, the narrower the bandwidth, so as long as the minimum can guarantee that the octave can meet this condition, the other octave can certainly be satisfied. The 1/3

However, there are some differences between the computed octave power spectrum and the octave power spectrum obtained by the filter banks. The reason is that the cutoff characteristics of The filter banks are not as good as the DFT results. In addition, the signal before entering the filter group will be advanced a pre-filter, that is, the signal is weighted, usually have so-called a- weighted,B - weighted and C - weighted. Specifically, you can refer to the IEC 60651 "standard for sound levelMeters", which is clearly written.

Spectrum, energy spectrum, power spectrum, octave spectrum, 1-/3 times-octave spectrum