Cepstrum cepstrum Review

During the course of study, this parameter is often used. Later I worked mainly on audio/speech codec, but the concept of cepstrum was rarely used. I saw these words when I read a paper today, I think it is very practical. Let's review this signal feature.

Time-cepstrum analysis is very effective when extracting the signal characteristics transmitted by the carrier. Cepstrum is widely used in the extraction of Speech Recognition feature parameters. This is because the essence of speech is the vibration of vocal cords, and then the sound that we can hear or pick up is produced after the sound channels and oral modulation. Cepstrum can be used to analyze speech to extract this essential feature parameter.

1) mathematical description of inverted Spectrum

The mathematical expression of the inverted spectrum function CF (q) (Power cepstrum) is:

(2.6)

CF (q) is also called the power inverted spectrum or the power spectrum of the logarithm power spectrum. Commonly used in projects is the formula (2.6), that is:

(2.7)

C0 (q) is called the amplitude inverted spectrum, sometimes referred toInverted spectrum.

Physical Meaning of inverted spectrum variable q

To make its definition clearer, you can also define:

(2.8)

That is, the inverted spectrum is defined as the logarithm weighting of the Bilateral Power Spectrum of the signal, and then the Fourier inverse transformation is obtained. Contact the self-correlation function of the signal:

We can see that this definition method is very similar to the self-correlation function, and the Q and Tau variables are exactly the same in dimension.

In order to reflect the phase information, the original signal can be restored after the separation, and a method of complex spectrum operation is proposed. If the Fourier transformation of the signal x (t) is X (f ):

(2.9)

The X (t) Inverted spectrum is recorded:

(2.10)

Obviously, it retains the phase information.

The difference-only logarithm weighting function of the inverted spectrum is different from the related function. The objective is to enable the signal energy concentration after the second transformation, expand the spectrum range of the dynamic analysis, and improve the accuracy of the re-transformation. It can also be used to extract convolution components, making it easy to separate and recognize the original signal.

 

(2) Application of inverted Spectrum

Effect of Separating information channels on Signals

 

Figure 2.26 logarithm power spectrum diagram. In mechanical status monitoring and fault diagnosis, the measured signal is often a response from the fault source transmitted through the system path, that is, it is not the signal of the original fault point, to obtain the source signal, you must delete the impact of the transmission channel. For example, in noise measurement, the signal obtained is not only the source signal but also the echo signal reflected in different directions. to extract the source signal, the echo interference signal must also be deleted. If the input of the system is X (t), the output is Y (T), and the impulse response function is h (T), the temporal relationship between the two is Y (t) = x (t) * h (t)

Frequency: Y (f) = x (f) * H (F) or Sy (f) = SX (f) * | H (f) | 2

If the logarithm is obtained on both sides of the preceding formula, the following values are available:

(2.11)

As shown in formula (2.72) (2.26), the source signal is a signal with obvious periodic characteristics. After the influence of the System feature loggk (f) is corrected, the output signal loggy (f) is synthesized ).

For the (2.72) type for further Fourier transformation, the amplitude inverted spectrum can be obtained:

(2.12)

That is:

(2.13)

The above derivation shows that the signals can be output by convolution between x (t) and H (t) in the time domain, and the product relationship between X (F) and H (f) in the frequency domain; in the inverted frequency domain, the relationship between Cx (Q) and CH (q) is changed, which significantly distinguishes the system's special feature CH (q) from the signal feature Cx (q, this is very useful for clearing the impact of the transfer channel, and it is difficult to implement it using Power Spectrum Processing.

The image (2.26b) is the corresponding inverted spectrum. It is clearly shown from the figure that there are two components: one is the high inverted frequency Q2, which reflects the source signal features; the other is the low inverted frequency Q1, which reflects the system features. The two parts occupy different inverted frequency ranges on the inverted spectrum diagram. As needed, the signal and system impact can be separated and deleted to retain the source signal.

Use inverted spectrum to diagnose gear faults

For high-speed large-scale rotating machinery, its rotation condition is complex, especially when the equipment is faulty, bearing or gear defects, Oil Film rotation, friction, trapped flow and quality asymmetry, etc, the vibration is more complex, and it is difficult to identify using the general spectrum analysis method (to identify the frequency component that reflects the defect). However, using the inverted spectrum will enhance the recognition capability.

For example, a pair of working gears contains a certain number of cyclic components in the vibration or noise signals obtained from the test. If the gear has a defect, its vibration or noise signal will also greatly increase the harmonic component and the so-called side band frequency component.

What is the frequency of the Side Band? How is it produced?

There are two frequencies W1 and W2 in the rotating machine. Under these two frequencies, the response of the mechanical vibration shows a periodic pulse, that is, to present the signal whose amplitude is modulated by a difference frequency (W2-W1) with W2> W1) to form the waveform of the beat. This amplitude modulation signal is naturally generated. For example, the amplitude adjustment wave originated from the sine carrier of the Gear Meshing Frequency (number of teeth x axis rotation number) w0, and its amplitude became a function Sm (t) that changed with time due to the eccentric influence of the gear ), so:

(2.75)

If the rotation frequency of the gear shaft is WM, it can be written as follows:

(2.76)

As shown in figure 2.27a, it looks like a periodic function, but in fact it is not a periodic function, unless W0 and WM are in an integer relationship, which is used in actual applications, this is rare. Based on the triangular halfwidth relationship, the (2.76) formula can be written:

(2.77)

From the (2.77) formula is not difficult to see, it is by w0, (w0 + WM) and (w0-wm) three different sine wave Sum, with 2.27b) spectrum. Here, the difference frequency and frequency between (W0-WM) and (w0 + WM) are called the side band frequency.

In the above example, for a gear with 100 Teeth with four wheel frames, the quasi-rotation number of the shaft is 50 rpm, and Its Meshing Frequency is 5000Hz. The amplitude (the size of the bonding force) is modulated by the cycle of every four revolutions of Hz (because there are four wheel frames ). Therefore, in the measured Vibration Components, there are not only obvious shaft rotation numbers of 50Hz and Meshing Frequency (5000Hz), but also 4800hz and 5200hz side band frequencies.

In fact, if there are serious gear defects or multiple types of faults, and many mechanical causes such as non-aligning, looseness, and non-linear stiffness, or the occurrence of pattern truncation or other causes, the band frequency will increase significantly.

Too many frequency differences occur on a spectrum graph, making it difficult to identify, while a inverted spectrum graph is helpful for recognition, as shown in Figure 2.28. Figure (a) is a spectrum diagram of a speed reduction box, and figure (B) is its inverted spectrum diagram. It is clearly seen from the Cepstrum that there are two main frequency components: 117.6Hz (85 Ms) and 48.8Hz (20.5 ms ).