數(shù)控車床自動(dòng)回轉(zhuǎn)刀架設(shè)計(jì)-電氣含開題及7張CAD圖
數(shù)控車床自動(dòng)回轉(zhuǎn)刀架設(shè)計(jì)-電氣含開題及7張CAD圖,數(shù)控車床,自動(dòng),回轉(zhuǎn),刀架,設(shè)計(jì),電氣,開題,cad
XXXXX學(xué)
XX設(shè)計(jì)（XX）
中期檢查報(bào)告
題目 數(shù)控車床自動(dòng)回轉(zhuǎn)刀架設(shè)計(jì)
XXXXXXXXX院XXXXXXXXX專業(yè)XXXXX班
學(xué)生姓名 XXXX
學(xué) 號 XXXXXXX
指導(dǎo)教師 XXX
20XX 年 5 月 10日
自 檢 報(bào) 告
(要點(diǎn):1.畢業(yè)設(shè)計(jì)(論文)工作任務(wù)的進(jìn)展情況 2.未按計(jì)劃完成工作任務(wù)的原因 3.工作中所遇到的問題4.下一步工作打算)
:
1.畢業(yè)設(shè)計(jì)(論文)工作任務(wù)的進(jìn)展情況
我的畢業(yè)設(shè)計(jì)題目是數(shù)控車床自動(dòng)回轉(zhuǎn)刀架設(shè)計(jì)，現(xiàn)在畢業(yè)設(shè)計(jì)已經(jīng)進(jìn)行了10周左右，在老師的精心的指導(dǎo)下，我已經(jīng)完成對刀架傳動(dòng)系統(tǒng)、液壓缸的選擇計(jì)算；液壓傳動(dòng)系統(tǒng)原理圖、零件圖、液壓缸部裝圖和部分機(jī)械總成圖的繪制；雖然在有些地方還達(dá)不到老師的要求，但是在剩下的時(shí)間里我會加快進(jìn)度，爭取早日完成任務(wù)。
2.未按計(jì)劃完成工作任務(wù)的原因
在機(jī)械圖這部分，我還沒有達(dá)到老師所要求的進(jìn)度。原因有幾個(gè)方面：關(guān)于數(shù)控車床的基礎(chǔ)知識還存在一些問題，液壓傳動(dòng)系統(tǒng)的知識掌握的還不是很牢固；在查閱書籍這方面還有些不足，導(dǎo)致這一部分沒有達(dá)到進(jìn)度。
3.工作中所遇到的問題
在畢業(yè)設(shè)計(jì)的幾周中，在自己的實(shí)際設(shè)計(jì)過程中，發(fā)現(xiàn)了許多問題：首先對軸上零件的固定方法掌握的不很全面，在卡具的設(shè)計(jì)過程中，對關(guān)于夾緊力大小的計(jì)算時(shí)，對于取安全系數(shù)不是很清楚，這些問題在老師的幫助下，我自己又查閱了一些資料，最后這些問題都得到了很好地解決；其次，在畫液壓傳動(dòng)系統(tǒng)原理圖時(shí)，對液壓缸的內(nèi)部結(jié)構(gòu)不是十分了解；通過老師的進(jìn)一步講解，我圓滿地解決了這一問題并加以應(yīng)用。通過這一段時(shí)間的畢業(yè)設(shè)計(jì)，我感到自己所學(xué)的知識還有很大的不足。因此，我要在以后更加努力地學(xué)習(xí)，有不明白的時(shí)候，要把它徹底弄懂。
4.下一步工作打算
第十周 繪制控制系統(tǒng)程序流程圖
第十一周 控制系統(tǒng)硬件電路設(shè)計(jì)
第十二周 控制系統(tǒng)軟件編程
第十三周 程序調(diào)試
第十四周 撰寫15000字畢業(yè)設(shè)計(jì)論文
第十五周 打印輸出畢業(yè)設(shè)計(jì)資料，準(zhǔn)備畢業(yè)答辯
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4
附錄Ⅱ
Proceedings of the 7th ICFDM2006
International Conference on Frontiers of Design and Manufacturing
June 19-22, 2006, Guangzhou, China
Pages 255-260
A STUDY ON VIBRATION SIGNAL-BASED FEATURE EXTRACTION
FOR GRINDING SPINDLE-TYPED ROTOR-BEARING SYSTEM
DURING ACCELERATION
Jong-Kweon Park , Bong-Suk Kim , Soo-Hun Lee and Jun-Yeob Song
Intelligence and Precision Machinery Research Division, KIMM, 305 343, Rep. of Korea
School of Mechanical Engineering, Ajou University, 443 749, Rep. of Korea
Abstract: The goal of system monitoring is to minimize economic loss, to increase reliability, to maximize productivity, and to maintain product quality in manufacturing. Since vibration signals sufficiently contain the abundant
running information of the real system and the hidden fault symptoms, the feature extraction through those signals is widely applied for performance evaluation fault diagnostics of rotating machineries.
This paper shows feature extraction from vibration signals gathered in the grinding spindle-typed rotor-bearing system during acceleration in order to monitor an abnormal condition of current system like shaft crack by using various kinds of signal processing methods such as the Fast Fourier Transform, Short-Time Fourier Transform, Wigner-Ville Distribution, and Discrete Wavelet Transform. As well, the result of feature extraction in shaft crack condition was compared with that in normal condition.
Keywords: Feature extraction, Grinding spindle-typed rotor-bearing system, Non-stationary signal processing method, Accelerating process, Wavelet transform
1. Introduction
The condition monitoring or fault diagnosis in rotating machineries and machining process is a crucial requirement in order to maintain reliability, safety, and product quality and to prevent failures or damages .Compared with other machining methods, high- performance grinding process is one of the most complicated and important cutting processes as final machining stage; consequently, the monitoring of grinding process and machine is much more necessary in order to supervise the process and machine and also detect abnormalities . Among various kinds of approaches, vibration signal analysis method for feature extraction and nondestructive damage identification has been widely utilized due to capability to carry the abundant dynamic information and to indicate detailed motion of mechanical systems and to describe simultaneously when a fault occurs or what is its frequency . However, since most of the vibration signals sampled on mechanical systems are non-stationary or transient signals which sufficiently contain additional information or abnormal symptom, which can not be revealed from stationary signal, it is the key how to accurately draw dominant feature components from vibration signals because non-stationary signal is more complex than stationary signal. Up to date, for feature extraction of rotating machinery, many kinds of research results have mainly been focused on the stationary signal process; on the other hand, little research has been accomplished for the non-stationary signal process such as speed-up process; especially, there is almost no feature extraction using vibration signal of speed-up condition in the field of grinding process.
.This paper was about a study to extract the dominant features from vibration signals acquired in a laboratory grinding spindle-typed rotor-bearing system during acceleration by using several signal processing methods such as Time Domain Analysis (TDA), Frequency
Domain Analysis (FDA), and the Time-Frequency Analysis Method (TFAM). Modal testing, which detects dynamic characteristics of the system like natural frequency, was performed for the purpose of determining operating range for acceleration in test setup. Vibration data from the bearing housing passing through the distinctive resonance frequencies and frequency band in speed-up process were gathered through the experiments with normal and crack shaft condition. To get prominent signals of abnormality from acquired time data as a fundamental stage for diagnosis or monitoring technology, the Fast Fourier Transform (FFT), Short-Time Fourier Transform (STFT), Wigner-Ville distribution (WVD), and Wavelet Transform (WT) using commercial software were carried out and compared with each result
2. Theoretical Background
2.1. Review of Signal Analysis Methods
There are two major types of signal in the first natural division category: the stationary signal and non-stationary signal. Stationary signals are constant in their statistical parameters over time. Moreover, stationary signals are further divided into deterministic and random signals.
Random signals are unpredictable in their frequency content and their amplitude level, but they still have relatively uniform statistical characteristics over time.
On the other hand, non-stationary signals are divided into continuous and transient types. Transient signals are defined as signals which start and end at zero level and last a finite amount of time. In the stationary signal analyses, there are the RMS, Peak Value, Average/Distribution, and dynamic time models such as AR model and ARMA model in time
domain analysis as well as the Fourier Transform (FT) in frequency domain analysis. However, most signals of mechanical system like vibrations, noises, and sounds, generate non-stationary signals that data are intricate and irregular, so it is certainly necessary to use non-stationary signal processing methods to extract valid information from those signals. The FT is mainly used in frequency analysis of stationary signals, while frequency analysis
tends to accompany errors with regard to non-stationary signals. Since the frequency analysis simply shows only frequency components when non-stationary transient signals occur, it is difficult to find the time information. However, the TFAMs such as the STFT, WVD, and WT,
which perform frequency analysis at the time when failure occurs, compensate defects of time and frequency domain analysis, so that, it is considerably useful for applications to a number of fields.
2.2. Fast Fourier Transform
An infinite-range FT of a real-valued or a complex-valued record x(t) is defined by the complex-valued quantity.
（2.1）
Theoretically, as noted previously, this transform X(f) will not exist for an x(t) that is a representative member of a stationary random process when the infinite limits are used. However, by restricting the limits to a finite time interval of x(t), say in the range (0,T), then the finite-range the FT will exist, as defined by
（2.2）
Assume now that this x(t) is sampled at N equally t apart, where t has been spaced points a distance selected to produce a sufficiently high Nyquist frequency. As before, the sampling times are t=nt. However, it is convenient here to start with n=0 like Eq. (3). Let
（2.3）
Then, for arbitrary f, the discrete version of Eq. (2) is
（2.4）
The usual selection of discrete frequency values for the computation of X(f,T) is
（2.5）
At these frequencies, the transformed values give the Fourier components defined by
（2.6）
Where t has been included with X(f ) to have a scale kfactor of unity before the summation. Note that results are unique only out to k=N/2 since the Nyquist frequency occurs at this point. The function defined in Eq. (6) is often referred to as the Discrete Fourier Transform (DFT). A drawback in utilizing the DFT Eq. (6) for discrete signal processing is the time it takes to compute the transform values at all frequencies for large arrays of discrete data. More efficient numerical algorithms were developed to improve computational speed, and this has resulted in numerous the FFT algorithms.
2.3. Short-time Fourier Transform
The STFT is a Fourier Transform performed in successive time frames :
（2.7）
Where t is time, w is frequency, and h(u) is a temporal window function such as Rectangular, Gaussian, Blackman, Hanning, Hamming, etc. The STFT characterizes the temporal signal f(t) in both time and frequency domains. The main limitation of this TFAM is caused by the fixed temporal window applied, whose size (number of sampling points) defines the time and frequency resolutions; a large frame size improves the frequency resolution, while decreasing the time resolution and vice versa. Since fixed window is unable to adjust time and frequency resolution by signals, time-frequency resolution about non-stationary signals analysis needs to make mutual balance.
2.4. Wigner-ville Distribution
The WVD provides a relationship between time and frequency during the period of the time data window that is not present in standard Fourier Spectral Analysis. It is capable of displaying any phase and magnitude changes presented. The WVD can be defined in both the time and frequency domains. Due to computational efficiency and good time resolution, the time domain definition is used. The definition of the WVD is
（2.8）
Where x(t) is the time signal, conjugate, t is the time domain variable, and frequency. Any real signal x(t) is not only contaminated by the noise, but also by the interference terms. Suppression of both of these requires a combination of time and frequency windowing with the WVD. This approach is called the Smoothed Pseudo-Wigner-Ville Distribution (SPWVD), and expressed as follows:
（2.9）
In this Eq. (9), two independent windows exist. The function W() is a result of the application of a truncating window to the original time data, and determines the frequency resolution of the WVD. The g(t) is a smoothing window, which determine the times resolution. The independence of the two window functions enables them to be applied individually, or in combination, so that the desired degree of interference suppression can be achieved. Discrete smoothed version of the SPWVD can be expressed as
（2.10）
Where the n and m denote the time and frequency indices, respectively, and S is analytic signal obtained form Hilbert transform of the original windowed signal. N is the half-size of the FFT window w(k) and Q is the half-size of the post smoothing window g(p). The SPWVD is achieved by convolving the WVD with smoothing window.
2.5. Wavelet Transform
The WT, a popular tool for studying intermittent and localized phenomena in various signals, is efficient signal processing method to obtain time information lost in the FT, partly irregular fluctuation, time change changing, and discontinuous point. Due to the multi-resolution ability of the WT, the noised signals can be separated into several approximation signals and detail signals. The WT of a signal f(t) is defined as the sum of all of the time of the signal f(t) multiplied by a scaled and shifted version of the wavelet function (t). The coefficients C(a,b) of the WT of the signal f(t) can be expressed as follows
（2.11）
Where a and b are the scaling and shifting parameters in the WT. Basically, a small scaling parameter corresponds to a compressed wavelet function. As a result, the rapidly changing features in the signal f(t), i.e. high frequency components, can be obtained from the WT by using a small scaling parameter. On the other hand, the low frequency features in the signal f(t) can be extracted by using a large scaling parameter with a stretched wavelet function. For the sampled signal, a digital signal, f(k), k=0, 1, 2, !-, the Discrete Wavelet Transform (DWT) should be considered. The most commonly used DWT is the scaling and shifting of parameters with powers of two. That is:
（2.12）
（2.13）
Where j is the number of levels in the DWT. The coefficients C(a,b) of the DWT can be divided into two parts: that is, one is the approximation coefficients and the other is the detailed coefficients. The approximation coefficients are the high scale and the low frequency components of the signal f(t), while the detail coefficients are the low scale and the high frequency components of the signal f(t). The approximation of the DWT for the sampled signal f(t) at Coefficients level j can be expressed as Eq. (14).
（2.14）
Where (n) is the scaling function associated with j,k (n). Similarly, the detail the wavelet function coefficients (D) of the DWT for the sampled signal f(t) at level j can be expressed as follows.
英文翻譯中文
提取磨削主軸型轉(zhuǎn)子軸承系統(tǒng)在加速度期間振動(dòng)信號的特征研究
摘要:
目標(biāo)監(jiān)控系統(tǒng)，是為了最大限度地減少經(jīng)濟(jì)損失, 以增加可靠性, 最大限度地提高工作效率，從而保持制造業(yè)產(chǎn)品質(zhì)量。由于振動(dòng)信號含有真實(shí)系統(tǒng)和故障患癥狀的相當(dāng)豐富的運(yùn)動(dòng)信息的，因此這些信號被廣泛應(yīng)用于評價(jià)故障診斷旋轉(zhuǎn)機(jī)械。本文主要講述提取聚集在磨削轉(zhuǎn)子系統(tǒng)加速度振動(dòng)信號并監(jiān)測異常的情況，目前可以利用各種方法對信號進(jìn)行處理，如軸斷裂的處理方法有傅立葉變換、短時(shí)傅里葉變換、魏格納分布,以及離散小波變換，同時(shí)將常狀態(tài)與軸斷裂提取情況特征相比。
關(guān)鍵詞:
特征提取 磨梭型轉(zhuǎn)子系統(tǒng) 非平穩(wěn)信號處理的方法 加速進(jìn)程 小波變換
1簡介
狀態(tài)監(jiān)測和故障診斷在旋轉(zhuǎn)機(jī)械的加工過程中是一個(gè)重要要求，其目的為了保持可靠性和安全性，以及產(chǎn)品質(zhì)量，防止故障或損壞。與其他加工方法相比，高性能研磨過程是切削過程最復(fù)雜和最重要的最后加工階段；因此，磨削加工工藝及設(shè)備，更需要利用監(jiān)督過程發(fā)現(xiàn)機(jī)器異常。在各種方式中由于有能力進(jìn)行豐富的動(dòng)態(tài)查詢，并進(jìn)行詳細(xì)的分析，機(jī)械系統(tǒng)在什么頻率發(fā)生故障，振動(dòng)信號分析方法的特征提取和無損鑒定已被廣泛使用，不過，由于大部份的振動(dòng)信號為非靜態(tài)或暫態(tài)信號中含有大量的干擾或異常的癥狀，這些并不能顯示平穩(wěn)信號，它的關(guān)鍵是如何準(zhǔn)確地繪制振動(dòng)信號的主成分，因?yàn)榉瞧椒€(wěn)信號較平穩(wěn)的信號更復(fù)雜。截至目前為止，對于旋轉(zhuǎn)機(jī)械特征提取，多種研究成果主要集中在平穩(wěn)信號的研究過程。在另一方面，對加速進(jìn)程非平穩(wěn)信號研究卻很少， 因此幾乎沒有任何研磨過程加速狀態(tài)領(lǐng)域的振動(dòng)信號特征提取。
本文是關(guān)于研究從動(dòng)態(tài)信號中提煉的有用信號，利用一個(gè)實(shí)驗(yàn)室紡錘型轉(zhuǎn)子系統(tǒng)在加速狀態(tài)下振動(dòng)，用幾個(gè)信號處理方法如時(shí)間域分析(tda)和頻域分析(FDA)、時(shí)間-頻率分析法(tfam)測試模型，從而展現(xiàn)其中的動(dòng)態(tài)系統(tǒng)的特性，在正常及裂紋軸條件下振動(dòng)數(shù)據(jù)從軸承箱通過獨(dú)特的共振頻率和頻段混合。取得異常段時(shí)間數(shù)據(jù)作為一個(gè)階段的診斷或監(jiān)測技術(shù)，用(快速傅立葉變換),短時(shí)傅里葉(Fourier),魏格納分布(WVD), 和小波變換(WT)用商業(yè)軟件進(jìn)行比較每個(gè)結(jié)果。
2理論背景
2.1回顧信號分析方法
第一部自然分法中主要有兩種類型的信號: 穩(wěn)態(tài)信號和非平穩(wěn)信號。 穩(wěn)態(tài)信號在整個(gè)過程中是不變的。此外，穩(wěn)態(tài)信號又進(jìn)一步分為確定性信號和隨機(jī)信號，隨機(jī)信號是意想不到的頻率內(nèi)容及其振幅的大小，但他們?nèi)匀挥邢鄬y(tǒng)一的統(tǒng)計(jì)特征隨著時(shí)間變化。
在另一方面，非穩(wěn)態(tài)信號分為連續(xù)型和瞬態(tài)型。暫態(tài)信號是指信號的開始和結(jié)束時(shí)為零級，只是持續(xù)在一定的時(shí)間內(nèi)的信號。
對靜態(tài)的信號分析量有有效值，峰值，平均值與動(dòng)態(tài)的時(shí)間模式，如AR模型和ARMA模型在時(shí)間域分析以及傅立葉變換的頻域分析。然而，大多數(shù)信號的機(jī)械系統(tǒng)，如振動(dòng)、噪音和聲音都生非平穩(wěn)信號，數(shù)據(jù)是錯(cuò)綜復(fù)雜和不規(guī)則的，因此絕對有必要使用非平穩(wěn)信號處理方法，從這些信號提取有效信息。傅立葉變換主要用于穩(wěn)態(tài)信號頻率分析，雖然對非穩(wěn)態(tài)信號頻率分析往往伴隨著錯(cuò)誤,既然率分析只是傾向于頻率元件非穩(wěn)態(tài)信號發(fā)生時(shí)間，其中很難找到時(shí)間相關(guān)信息。不過，時(shí)域分析如短時(shí)短時(shí)傅立葉分析、魏格納分布、和小波分析其中推演出失效時(shí)頻率分析，從而彌補(bǔ)缺陷，這是一個(gè)在很多領(lǐng)域都相當(dāng)有用的方法。
2.2傅立葉變換
一個(gè)實(shí)值或復(fù)值函數(shù)的無窮傅立葉變換由x(t)得到
（2.1）
理論上,如前所述, 這個(gè)變換X(f)將不會存在,對于一個(gè)X(t)這是一個(gè)典型無限極限應(yīng)用在固頂過程的例子。不過這里限制了極限一個(gè)有限的時(shí)間間隔的x(t)即(0,t), 然后在有限范圍內(nèi)的傅立葉變換是永遠(yuǎn)存在的，正如下面公式所示:
（2.2）
現(xiàn)在假定x(t)在同一個(gè)t時(shí)刻采樣,這里t間隔被用于產(chǎn)生足夠高的奈奎斯特頻率。一如以往，采樣時(shí)間為t=nt。 然而,在這里以N=0開始很方便，象公式(2.3)。讓
（2.3）
那么，對任意f，公式（2.2）的離散表達(dá)為：
（2.4）
通常選擇的離散頻率值計(jì)算x(f,t)
（2.5）
在這些頻率中，轉(zhuǎn)化的傅里葉成分?jǐn)?shù)值有以下公式定義：
（2.6）
這里t已被包含在X(f)中從而在統(tǒng)一之前成為一個(gè)規(guī)模因子?？吹降慕Y(jié)果是獨(dú)一無二的,只有當(dāng)在K=n/2這一點(diǎn)奈奎斯特頻率才發(fā)生。在公式2.6中定義的函數(shù)，往往被稱為離散傅立葉變換(DFT)。
利用傅立葉變換公式(2.6)的缺點(diǎn)為處理離散信號所花費(fèi)的大量的時(shí)間，計(jì)算的價(jià)值轉(zhuǎn)化所有頻率離散數(shù)據(jù)。更有效的數(shù)值算法由此產(chǎn)生，分別制定了改進(jìn)的計(jì)算速度,而這已經(jīng)導(dǎo)致了大量的傅立葉算法。
2.3短時(shí)傅里葉變換
傅立葉短時(shí)變換是在連續(xù)的時(shí)間進(jìn)行傅立葉變換
（2.7）
這里t是時(shí)間w是頻率h(u)是窗口功能函數(shù)，如矩形、高斯、布拉克曼、漢寧等，
短時(shí)特征時(shí)間信號f(t)在時(shí)間和頻率范圍內(nèi)。主要限制短時(shí)傅立葉變換的是固定時(shí)間窗，其大小(多少個(gè)采樣點(diǎn))確定的時(shí)間和頻率決定；大窗提高頻率分辨率，同時(shí)減少了時(shí)間分辨率，反之亦然。由于固定窗口是無法調(diào)整的時(shí)間和頻率分辨率的信號為適應(yīng)分析非平穩(wěn)信號，時(shí)間和頻率分辨率需要彼此的平衡。
2.4魏格納分布
魏格納分布提供了實(shí)時(shí)數(shù)據(jù)窗口而不是標(biāo)準(zhǔn)的傅立葉譜分析時(shí)間和頻率的關(guān)系, 它能夠展示任何相位和幅度的變化。該魏格納分布可以定義在時(shí)間和頻率范圍。由于計(jì)算效率和及時(shí)解決，這種方法被采用。魏格納分布的定義式：
（2.8）
這里X(t)是時(shí)間信號X* (t)是他的復(fù)頻率T是時(shí)間域變量w是頻率任何真正的信號x(t) 不僅受到噪音干擾，而且也受到外界系統(tǒng)的干擾。去掉這兩方面干擾需要利用魏格納分布結(jié)合時(shí)間和頻率這樣計(jì)算x(f,t)的方法是所謂的平滑魏格納分布并表示如下:
（2.9）
在這一公式(2.9)中，存在兩個(gè)獨(dú)立的窗口。函數(shù)w(w)是由于采用了漢寧窗口原有的實(shí)時(shí)數(shù)據(jù)，并決定了頻率分辨率的魏格納分布。G(T)是平滑的窗口，它決定著時(shí)間的定義。 獨(dú)立的兩個(gè)窗口功能，使他們能夠應(yīng)用單獨(dú)或合并因此，理想的程度的抑制干擾才能實(shí)現(xiàn)。離散魏格納分布的觀測可表達(dá)為：
（2.10）
這里N和M分別表示的是時(shí)間和頻率指數(shù)，s是從希爾伯特變換獲得的解析信號，N是傅立葉窗的一半w（k）和Q是平滑窗g（p）的一半平滑魏格納分布由通過濾波窗口魏格納分布得到。
2.5小波變換
小波變換是一個(gè)學(xué)習(xí)各種信號的間歇和局部現(xiàn)象受歡迎的工具，是有效的獲得傅立葉變換中丟失時(shí)間信息的信號處理方法，包括部分不規(guī)則的波動(dòng)，隨著時(shí)間變化的變化， 以及間斷點(diǎn)。由于多分辨能力的小波變換，噪聲信號可分成幾個(gè)逼近信號和細(xì)節(jié)信號。 小波的一個(gè)信號f(t)的定義為所有的時(shí)間信號f(t)乘以一個(gè)規(guī)模和轉(zhuǎn)換的小波函數(shù)。t 系數(shù)c(a,b)。小波變的信號f(t)可表達(dá)如下：
（2.11）
A和B屬于小波變換中的比例變換因子。基本上，一個(gè)小規(guī)模的參數(shù)對應(yīng)著一個(gè)小波壓縮功能。因此迅速變化的信號f(t)，即高頻成分，可從小波變換用小比例參數(shù)獲得。 在另一方面，低頻信號f(t)可以提取采用大尺度參數(shù)與比例因子由小波函數(shù)獲得。
為采集信號，采用數(shù)字信號，f(k),k=0,1,2! ---，應(yīng)該考慮離散小波變換。最常用離散小波變換是將比例變換因子定位2的冪，即:
（2.12）
（2.13）
其中j是離散小波變換的級別。系數(shù)c(a,b)可分為兩部分。即: 一是逼近系數(shù)，另一種是詳盡的系數(shù)。逼近系數(shù)是高幅值和低頻成分的信號f(t)，詳盡系數(shù)是低幅值高頻率的部分，離散小波變換的逼近系數(shù)Aj可以定義為
（2.14）
Q(N)為與小波函數(shù)相關(guān)的比例縮放函數(shù)。同樣，抽樣信號f ( t )的離散小波變換的詳盡系數(shù)(d)在j級可表述如下：
15
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