Signal processing refers to the process of filtering, transforming, analyzing, processing, and extracting characteristic parameters. In electronic instruments and measurements, the most typical is to use a spectrum analyzer to perform spectral analysis of the signal to understand and obtain the frequency (or spectral) characteristics of the signal. Until modern computers and related technologies were developed, this process could only be achieved with traditional spectrum analyzers consisting of hard-wired technology. As we all know, this traditional spectrum analyzer requires a high level in terms of design and manufacturing and components used. Especially in the wide frequency range and high index, it is more difficult to design and manufacture, and its price is very expensive. However, since the computer and the emerging digital signal processing (DSP) technology have matured and developed, the way to solve the signal spectrum analysis is gradually being replaced by DSP.
About discrete Fourier transform and digital filtering
As signal processing, the most directly related to spectral analysis is the Fourier transform, FT. It is well known that discrete Fourier transform (DFT) and digital filtering are the basic content of DSP. At present, DFT has many practical and effective fast DFT algorithms, namely FFT algorithm and software. Its performance is mainly determined by sampling (actually including analog-to-digital conversion) rate and CPU operation speed. The process of converting arbitrary signals (mainly reflecting various changes in the objective physical world, and most of them continuously changing analog quantities) into digital data that can be processed by the CPU is called "digitalization", which includes sampling and quantification. The step is to quantify what is commonly referred to as analog/digital conversion. The rate of sampling is related to the signal being processed. In order to ensure that the digitized signal data does not lose the characteristics of the original signal, the sampling frequency should be greater than or at least equal to twice the signal cutoff frequency. This is the famous Nyquist sampling theorem, or Nyquist sampling rate. The Nyquist sampling theorem is easy to prove. As for the computing speed of the CPU, it is well known that the current microcomputer has reached the level of hundreds or even gigahertz. In order to improve or realize the high-speed operation of FFT and other operations, Texas Instruments (IT) has been working on the development and production of dedicated DSP chips since the beginning. The famous TMS320 series chips are well known to the scientific community. According to recent reports, the new TMS320C64x runs at speeds up to 600MHz, and its eight functional units can simultaneously perform four 16-bit MAC operations or eight 8-bit MAC operations in each cycle. A single C64xDSP chip can simultaneously perform MPEG4 video encoding of one channel, MPEG4 video decoding of one channel, and an MPEG2 video decoding, and still have 50% of the margin reserved for multi-channel voice and data encoding, naturally, and other vendors. Developed and produced a number of special or general purpose DSP chips.
In the last century, the development of electrical filters has gone through the process from passive to active and from analog to digital. High-precision passive filters are very difficult technologies from design to manufacture. Although the active filter greatly improves the performance of the filter, it also reduces the difficulty of some manufacturing processes. However, from the great improvement of its performance, combined with other signal processing technologies, the means of implementation is convenient, and the number is still counted. The filter comes up later. Of course, this is also related to the development of EDA technology.
A digital filter is a discrete system whose characteristics or transfer functions are described by a difference equation based on a Z-transform. Digital filters fall into two broad categories, namely the IIR finite impulse response filter and the FIR infinite impulse response filter. The former is also known as the "recursive" filter, which is also known as the "non-recursive" filter. One can determine the difference equation describing the system according to the requirements of signal processing, and then design the filter according to the difference equation. There are also two ways to implement the filter. One is pure software, which is an algorithm software or software package. The other is hardware, which is designed as a specific hard-wire circuit, or even a dedicated or general-purpose chip. The design methods of digital filters and mature hardware and software products are not difficult to obtain. It will not be detailed here.
Other orthogonal transformation of the signal
It is known that Fourier transform or Fourier analysis implies such a meaning:
A signal of EP is synthesized by a sine wave represented by components of the spectrum obtained by its FT. In this sense, we refer to a set of orthogonal sinusoidal functions representing these sinusoids as the orthogonal basis functions of the Fourier transform (which can also be expressed in the form of complex functions). The research shows that not only the sinusoidal function can be used as the basis function of the orthogonal transform, but also the orthogonal perfect function system can be used as the basis function to perform orthogonal transform decomposition analysis on the signal (the sine function is naturally orthogonal complete function). system). Therefore, we refer to these transformations collectively as "orthogonal transformations." The most interesting non-sinusoidal orthogonal functions in practice are the Rademacher function, the Haar function, and the Wald function. For a period of time, the most used one is the Walsh function, which is a Redmeer function that Walsh completed in 1923. The Walsh function is a set of rectangular waves with values ​​of 1 and -1 that are very convenient for computer operations. Walsh functions are arranged or numbered in three ways, namely, column rate or Walsh arrangement, Paley arrangement, and Hadamard arrangement. Each of these three arrangements has its own characteristics. The Ardamar arrangement is the easiest to calculate quickly. The transformation using the Walsh function of the Hadamard arrangement is called the Walsh-Hadamard transform, or WHT or the straight-forward Hadamard transform. Since the operations of discrete orthogonal transforms are often done in the form of matrix multiplication, the matrix form of the Walsh-Hadamard function group has only two elements, 1 and -l, and the regularity of this Adama short matrix is ​​very strong. It is generated with a simple algorithm, so the fast algorithm of WHT is easy to implement. Now, this fast algorithm and its software already have mature products. Of course, when using this transformation we must remember that the spectrum it produces is based on short waves.
Another commonly used orthogonal transform is the discrete cosine transform DCT. It is known that the basis function of the Fourier transform is a sinusoidal function, that is, the number of times each component is a sine wave (or a complex vector) component determines the frequency of the sine wave, and the phase of each component constitutes the phase spectrum of the signal. That is to say, the Fourier spectrum of a signal consists of two parts, one is the amplitude characteristic, the other is the phase characteristic; or the real cosine component of the complex vector and the sinusoidal component as the imaginary part. In other words, only the amplitude characteristic spectrum does not fully represent the signal, and the phase characteristics must be complemented to be complete. This of course complicates both the representation and the arithmetic processing, and increases the amount of data representing the signal. Studies have shown that if the origin of the signal coordinates is appropriately offset, the transformed result can have only one of the sine or cosine components of the sine wave. This is a sine transform or a cosine transform. The discrete cosine transform DCT in signal processing is obtained by shifting the origin of the signal coordinates to the left by a half sampling interval. DCT has excellent information characteristics and has an efficient fast algorithm. Therefore, when the MPEG standard is developed, it is defined as a standard conversion of image compression coding.
At the end of this section, by the way, the discrete KL (KarhunenLover) transform is mentioned. KLT is often referred to as the best transform because the KLT filter and information compression coding distortion is minimal. However, since KLT's transformation basis function is indefinite, and there is no fast algorithm so far, it is only used in special occasions.
About wavelet analysis
We note all of these transformations or analyses, all of which are stationary or even periodic. In the case of Fourier analysis, its original starting point is the Fourier series, whose mathematical definition indicates that any non-sinusoidal periodic function (signal) can be decomposed into a sine wave with multiple frequencies of its fundamental frequency (and continuous current). The sum of the components). For the integral of the Fourier transform, the integration period is extended to infinity. In fact, the concept of frequency is exactly what Fouriye put forward in this work. Moreover, this kind of analysis method of transforming a thing from one "domain" to another "domain" and analyzing or expressing it from a new angle or scale has an epoch-making creation in the history of science. It is Fu Liye. from. However, it has long been discovered that transformation or analysis tools such as Fourier transform can only be used to process deterministic stationary signals, and that satisfactory analysis cannot be performed for mutated non-stationary signals; and that Fourier analysis yields The overall spectrum of the signal, but the local characteristics of the signal cannot be obtained. Therefore, the windowed Fourier transform appeared in the 1980s. Windowed Fourier transform is a localized time-frequency analysis method that combines the time domain (or spatial domain) to the frequency domain mapping analysis of the traditional Fourier transform in a windowed manner for local time periods (or The spatial interval is analyzed by frequency domain analysis, and the windowed Fourier transform partially solves the analysis problem of short-term signals. However, it has many inherent defects. For example, for short-term high-frequency signals, it is possible to adapt the frequency increase by narrowing the window width and sampling interval, but the window is too narrow to reduce the frequency resolution, and it is not suitable for low-frequency components. . Therefore, this leads to the search for new transformation (analysis) methods. Wavelet analysis emerged in this context and was quickly applied and developed.
Now briefly introduce the concept of wavelet analysis.
Let the continuous signal f(t) be given. Considering that the resolution of the actual signal is always limited, f(t) can be expressed as the following step function.
Where n is an integer representing the sample point, Cn0=f(n) is the sample value, and
Is its basis function or scale function. At this time, if the sampling interval is doubled, the number of samples is halved, and the signal is expressed as
As a result, the amount of data in the signal is compressed by half. This is the so-called dichotomy. Examine the deviation of the two signals before and after the two points
Is a wavelet function.
Some people explained that "wavelet" is a small waveform. "Small" means that it is attenuating, and "wave" means volatility, that is, an oscillation form whose amplitude is positive and negative.
The wavelet function ψ(t) can generate a set of orthogonal bases in L2(R) by translation and expansion:
{(ψ(2-kt-n),k,n is an integer}
Thus the given signal f(t) can be decomposed:
Usually, ψ(t) is also called a wavelet basis function. The wavelet basis function can have different formulas, and the aforementioned Haar function is a commonly used basis function. Of course, it can be used as a wavelet basis function, or it must be able to be expanded into a complete set of orthogonal function systems.
The development of wavelet analysis is very rapid. Although it can be traced back to Hilbert's argument in 1900 and the norm-orthogonal basis proposed by Hal in 1910, the actual main work should be that when Morlet of France analyzed the local properties of seismic waves in 1984, Because the Fourier transform is difficult to meet the requirements, it introduces the wavelet concept. Later, Grossman studied Morlet's signal by a certain function of the expansion and translation system, which opened the first step for the formation of wavelet analysis.
Among the many scientists who have made great contributions to wavelet analysis, the Mallat algorithm published by Maliat in 1987 has undoubtedly played a very important role in promoting the development of wavelet analysis. Naturally, in the development of wavelet analysis, many scientific and technological workers in China have also made great contributions.
Like the other analysis transformations described above, wavelet transforms are also available in both continuous and discrete forms. However, since wavelet functions are usually short-shaped pulse waves, discrete processing is relatively easy, and sometimes people ignore the difference.
In addition to being adapted to handle abrupt (or time-varying) non-stationary signals, wavelet transform has a very useful feature, multi-resolution characteristics. The so-called multi-resolution, that is, in wavelet analysis, the results of different resolutions can be easily obtained due to the different scaling functions used. This has been practically applied in the processing of image signals.
Wavelet analysis has evolved to the present and has yielded many proven results, including a common set of algorithms, software, and cured devices. For example, the ADV611 chip introduced by AD company, as the encoding/decoding and compression of video images, contains a wavelet filter, which can achieve a compression ratio of 7500:1 and good image quality. In instrumentation and measurement applications, there are also many achievements. For example, some people use it in the analysis of X-ray spectrum signals, and the quality of the spectral line signals processed by wavelet transform is greatly improved. It can be expected that this technology will be further developed and applied to a wider range of applications.
Conclusion
The above paper briefly introduces the current common signal processing, especially digital signal processing technology. However, they are basically only suitable for processing deterministic signals. There is a large class in signal processing technology called random signal processing or statistical signal processing. This type of processing technology is most widely used to combat noise and signal pollution operations. It is also known as signal estimation or signal recovery. The two most representative technologies are Weiner filtering and Kalman filtering. The former is also known as least squares filtering, which is very effective in recovering signals from noise. In fact, they have been proposed very early, but only in the development of modern computers and digital technology, only to get real practical applications. So we ended up simply mentioning this as the end of this article.
Stabilized Wood Vape devices are very hot in the market now, although it costs higher. We offer top-class stabilized wood as material, but with very competitive prices. Products include Priest 26650, Priest 21700, Magic Wand 18650, Dark Knight. Stabl wood.
Dongguan Marvec Electronic Technology Co.,Ltd, was established in 2012, is a comprehensive technology enterprise specializing in the r&d,production, sales and service of high-tech healthy e-cigarettes. Marvec always insists on the spirit of independent innovation and pursuit of excellence, and aims to product healthy e-cigarettes, become a first-class international manufacturing and service enterprise. The company has a strong technical research and development and product production capacity, with solid quality control and perfect after-sales service, we have a stable customer base and a good reputation in the domestic and international market. The company has a number of international certification and technical patents, products through CE, ROHS, FCC and other international tests, our products are exported to various countries and regions of the world. We are willing to work with our partners around the world to create a healthy and free new life!
Stabilized Wood Vape
Stabilized Wood Vape,E Cigarette Vape,Voltage Control Vape,Stabilized Wood Woody Vapes
Dongguan Marvec Electronic Technology Co.,Ltd , http://www.marvec-cn.com