《动态信号处理》PPT课件.ppt

Dynamic Signal Processing 动态信号处理,FFT Properties FFT特性,The Fast Fourier Transform (FFT) is an algorithm for transforming the signal from the time domain to the frequency domain. However, the signal cannot be transformed to the frequency domain in a continuous manner.It must first be sampled and digitized.This means that samples from the time domain are digitized to samples in the frequency domain, as shown in figure on next slide. Because of sampling, there is no longer an exact representation in either domain. However, a sampled representation can be closer to the ideal if the samples are placed closer together. 快速傅立叶变换(FFT)是一种将信号从时域转换到频域的运算方法。但是，信号不能以连续的方法转换到频域，必须先进行采样和数字化处理。也就是说将时域的样本通过数字化处理转换为频域的样本。请见下一张幻灯片图示。由于采样原因，两个域都不会有准确的图示。但是如果样本能放得更近一些，样本图示能更贴近理想图示。,FFT Samples in Time and Frequency Domains 时域与频域内的FFT 样本,Time Record 时间记录,A time record is defined to be finite number of consecutive, equally-spaced samples of the input signal. Because it makes the transform algorithm simpler and much faster, the finite number of samples is restricted to a multiple of 2. 1024 intervals or equally-spaced samples equal 210 intervals。 时间记录是指连续的、间隔相等的有限数目的输入信号样本。为了使转换方法变得更加简单和迅速，样本数目限制为2的幂，如1024（2的10次方）个样本。 This digitized time record is transformed as a complete block into a complete block of frequency lines. All the samples (of the input signal) of the time record are needed to compute each and every line in the frequency domain. This does not mean a single time domain sample transforms to exactly one frequency domain line. 这一数字化的时间记录以整体的方式转换成频域的一组记录。计算频域内的每根线时需要用到时间记录的所有（输入信号）样本。但这并不是指一个时域样本转换成一个频域线。,How Many Spectral Lines Are There? 有多少频谱线？,The FFT algorithm is a “complex-valued” operation; that is, it produces a “real” and an “imaginary” result, and the components of the frequency domain will appear at both positive and negative frequencies. FFT运算法则是一种“复数”运算：它会产生一个“实部”的和一个“虚部”的结果，使频域的构成部分会出现在正的和负的频率上。 Each of these components in the frequency domain is complex-valued; that is, they each have both amplitude and phase. Thus, the FFT transforms a finite number of equally spaced samples from the time domain to only half as many lines in the frequency domain 频域内的每个组成部分都是复数值，即都有幅值和相位。因此， FFT将一组数目有限的且间隔相等的样本从时域转换成只有一半线数的频域。,How Many Spectral Lines Are There? 有多少频谱线？,Thus, for a real-valued signal (the numbers are not complex), 800 data points would be required to compute a 400 line spectrum; 1600 data points for an 800 line spectrum, etc. However, some of the high frequency data is “discarded” to compensate for the roll-off of the anti-aliasing filters. In the “real world”, 1024 data points are actually required for a 400-line transform; 2048 points for an 800-line transform, etc. 因此，对于一个真实值信号来说（不是复数），需要800个数据点来计算一个400线的频谱；需要1600个数据点来计算一个800线的频谱，依此类推。然而，一些高频数据会被“舍弃”来补偿抗混滤波器的衰减。在实际情况下，需要1024个数据点来转换400线的频谱；需要2048个数据点来转换800线的频谱，依此类推。,What Is the Spacing of the Lines? 频谱线的间距是多少？,The lowest frequency that can be resolved must be based on the length of the time record. If the period (T) of the input signal is longer than the time record, there is no way of determining the period. Therefore, the lowest frequency line of the FFT must occur at a frequency equal to the reciprocal of the time record length. Delta F = 1/T_测不准原理 能被分解的最低频率取决于时间记录的长度。如果输入信号的周期(T) 比时间记录长，就无法计算其周期。因此， FFT 最低频率线的频率与时间记录长度互为倒数。,What Is the Frequency Range of the FFT FFT的频率范围,Fmax = (Sample Size)/2(Period of Time Record) 频率量程 = (样本大小)/2.56(时间记录周期) The number of time samples (sample size) is fixed by the implementation of the FFT algorithm. Therefore, the period of the time record (sec/cycle) must be varied to change Fmax (cycles/sec). To do this, the sample rate must be varied so that there always is the chosen, fixed number of time samples in the variable time record period. To cover high frequencies, the time record period must be shorter so that sampling is very fast. FFT运算法则确定了时间样本的数量(样本大小) 。因此，时间记录的长度(秒/转)必须是可变的，以此获得不同的频率量程(转/秒)。为做到这一点，采样率必须是可变得，从而保证在可变的时间记录长度内得到选定的、固定数量的时间样本。为了测量高频率，时间记录周期要更短些，这样就要求采样速度更快。,Sampling and Digitizing 采样与数字化,The input is a continuous analog voltage coming from the accelerometer and is proportional to the acceleration. Since the FFT requires digitized samples of the input for its digital calculations, a “sampler” and an “analog to digital converter” (A/D) needs to be added to the FFT processor to create a spectrum analyzer. 输入信号是一个连续的模拟电压，它从加速计输出并与加速度成正比。由于FFT需要输入数字化样本来进行数字计算，因此需要将一个“采样器”、一个“模拟数字转换器” (A/D) 和FFT处理器安装在一起，组成一个频谱分析仪。,Aliasing 混淆,Typically, the A/D converter must be able to acquire at least a hundred thousand readings per second. 一般来说，模拟/数字转换器每秒需要采集至少十万个读数。 The reason an FFT spectrum analyzer needs so many samples per second is to avoid “aliasing”. FFT频谱分析仪每秒需要采集如此多的样本（高采样率）是为了避免产生“混淆”。,Aliasing 混淆,Sampling Frequency = 2x Signal Frequency 采样频率= 2x 信号频率,Sampling Freq. = Signal Freq. 采样频率 = 信号频率,Sampling Freq. 2x Signal Freq. 采样频率 2x 信号频率,Aliasing in Frequency Domain 频域内的混淆,It is easy to see that a sampling frequency that is exactly twice the input frequency would not always be enough in the time domain. If the sampling rate is low (2x signal freq.), the alias products will fall in the frequency range of the input (“creating” fictitious frequencies) and no amount of filtering will be able to remove them from the signal. 如果采样频率只是输入频率的两倍，那么在时域中是不足够的。如果采样率很低（2x 信号频率），混淆就会发生在输入的频率量程内（“造成”虚假的频率），而且再多的滤波量也不能将混淆从信号中去除。,Aliasing in Frequency Domain 频域内的混淆,Anti-Alias Filter 抗混滤波器,The only way to be certain that the input frequency range is limited is to add a low-pass filter before the sampler and the A/D converter. Such a filter is called an anti-alias filter. The gradual roll-off area of the filter is known as the transition band. Large input signals are not well attenuated in the transition band; they can still alias. To avoid this, the sampling “frequency” is raised to twice the highest frequency of the transition band. Typically, this means that to do this the sample rate actually is now 2.5 to 4 times the maximum desired input frequency. 要确保有限过渡频率的输入频率量程，唯一的方法就是在采样器和模拟/数字转换器前先安装一个低通滤波器。这种滤波器被称为抗混滤波器。滤波器的逐渐截至区域就是过渡频段。一些大的输入信号并没有在过渡频段内得到良好衰减，因此仍然会产生混淆。为避免此类情况发生，采样频率应增大至转换频段最高频率的两倍，实际上现在采样率一般是需要的最高输入频率的2.5 至 4 倍。,Calculate tMAX and FMAX 计算时间记录长度和频率量程,Nyquists Sampling Theorem, briefly stated, is sampling must be done at a frequency rate of at least twice the highest frequency component of interest in order to not lose any information contained in the sampled signal. Thus, to satisfy Nyquists Theorem, select a sampling rate that is slightly higher than twice FMAX (2.56 X Fmax is typical). Nyquist采样法则，简单说来，就是必须以至少信号最高频率两倍的频率来进行采样，这样就不会丢失被测信号所包含的信息。因此，为了满足Nyquist采样法则，就要选择一个略高于两倍频率量程的采样率。(通常2.56 X频率量程).,Calculate tMAX and FMAX 计算 tMAX 和 FMAX,Tmax= # FFT lines/ Freq. Span = Sample Size/(2.56Freq. Span) Tmax= FFT频谱线数/频率量程(Fmax) = 样本大小/(2.56频率量程Fmax ) Fmax= # FFT lines/ Tmax = Sample Size/(2.56 Tmax) Fmax= FFT 频谱线数/ Tmax = 样本大小/(2.56 Tmax) Tmax in s, Fmax or Freq. Span in Hz Tmax 单位是秒, 频率量程单位是赫兹,Energy Leakage 能量泄漏,Time record is not periodical thought raw signal is. 时间记录是非周期性的，而原始信号是周期性的。 FFT of raw signal should be ONE FFT line, but now energy of this line leaks into others. 原始信号的FFT 应是一个FFT谱线，但现在这一频谱线的能量泄漏至其他线中。 It is obvious here that the leakage problem can entirely mask small signals located close to the sine wave(s). Most of the problem seems to be at the edges of the time records.The solution to this problem is known as “windowing”. 很明显，泄漏问题会完全掩盖住位于正弦波频率附近的小信号。问题主要出现在时间记录的边缘部位。解决这一问题的方法就是“窗函数”。,FFT of non-periodical time record/非周期性时间记录的FFT,Windowing 窗函数,Windowing forces the sampled data at the beginning and at the end of the sampling period to be equal to zero, thereby minimizing leakage. 加窗函数能使采样周期开始与结束时的样本数据等于零，从而将泄漏最小化。,Window reduces leakage, but not eliminate it 加窗函数能减少泄漏，不能消除泄漏,Sine wave not periodic in time record 时间记录中非周期性的正弦波,FFT results without window function 无窗函数的FFT结果,Leakage free Periodic in time record 无泄漏时间记录中呈周期性,Windowed Not periodic in time record 加窗函数时间记录中呈非周期性,Hanning Window 汉宁窗,The most commonly used window in vibration analysis is the Hanning window. The Hanning window does a good job with sine waves, both periodic and non-periodic. However, even with the Hanning window, some leakage will be present when the signal in the time record is not periodic. 在振动分析中最常用的窗函数就是汉宁窗，它能用于周期性和非周期性的正弦波。然而，即使加了汉宁窗，如果时间记录中的信号是非周期性的，仍然会出现一些泄漏。,Hanning Window 汉宁窗,While the Hanning window provides improved frequency resolution, it sacrifices amplitude accuracy(maximum -16%). 虽然汉宁窗能改善频率分辨率，却放弃了幅值的精确度（最多-16%）。,Wrong application of Hanning Window 汉宁窗的错误应用,Unwindowed transient 未加窗的瞬态信号,Hanning windowed transient 加汉宁窗后的瞬态信号,Usually Rectangular Window is used for transient test such as bump test. 通常情况下，瞬态测试，如敲击测试 使用矩形窗。,Uniform (Rectangular) Window 矩形窗_力窗,Since Hanning windows do not work well with transient vibrations, a uniform window (no window) could be used because it weights all of the time record uniformly. 由于汉宁窗不能很好得运用于瞬态振动，因此需要使用矩形窗（无窗），它能统一评估所有的时间记录。 In addition, amplitude variation for a uniform window can be up to 36%. 此外，矩形窗窗的幅值偏差能达到36%。,UNIFORM (RECTANGULAR) WINDOW (No Window) 矩形窗（无窗）,Flat Top Window 平顶窗,The Hanning function gives the filter a very rounded top. Although this characteristic is desirable to clearly identify frequency peaks, it is unacceptable if the signals amplitude needs to be accurately measured. The solution is to provide a window function that gives the filter a flatter passband. 汉宁窗函数使滤波器有一个圆形顶部。虽然这一特征能清楚识别频率峰值，但如果信号幅值需要精确测量时，这一特征是难以接受的。解决方法就是提供一个窗函数，它能使滤波器有一个更平坦的通频带。,Flat Top Window 平顶窗,The amplitude error from this window function does not exceed .1 dB (1%). Some of the ability to resolve a small component, closely spaced to a large one, is lost. 这一窗函数的幅值误差不超过0.1 dB (1%)，而辨别靠得较近的小幅值的频率峰与大幅值的频谱峰的能力部分丧失了。,Summary of Windows 窗函数小结,Many other window functions are possible but the three are by far the most common for general measurements. In summary, the Flat Top window provides the best amplitude measurement, the Kaiser-Bessell window provides the best frequency separation, whereas the Hanning window provides the best compromise. 也可使用许多其他窗函数，有三种函数是目前为止最普遍运用于一般测量的窗函数。平顶窗能提供最精确的幅值测量， 卡塞-贝塞尔窗能提供最好的频率分辨力，而汉宁窗提供最好的折衷方法。,Averaging 平均,In many cases, the amplitudes of fundamental and passing frequencies are fairly constant, but is accompanied by some random amplitudes and frequencies. Even in cases where the amplitudes are low, the discrete frequency amplitudes remain fairly constant, while the random signal amplitudes will fluctuate between zero and some peak value. As a result, the average amplitudes of the random signals will be significantly less than their peak values. In both cases, the amplitudes that are stable remain close to the same value, regardless of the number of averages taken. On the other hand, random or transient amplitudes tend to approach zero as more and more averages are taken. 在许多情况下，基频和通过频率的幅值是不变的，但也伴随有一些随机的幅值和频率。有时当幅值很低时，离散的频率幅值仍保持不变，而随机信号幅值会在零值和某个峰值间波动。因此，随机信号的平均幅值会大大小于它们的峰值。在以上两种情况下，无论取多少次平均值，稳定的幅值都保持在一个相同值附近。另一方面，平均值取得越多，随机或瞬态幅值就越趋近于零。,Overlap - No Overlap 重叠无重叠(1),The time record is not constant but is deliberately varied to change the frequency span of the analyzer. For wide frequency spans the time record is shorter. And no devices can reduce time for a specified frequency span. 时间记录长度是不固定的，而是通过其变化来改变分析仪的频带宽度。宽频带的时间记录较短。对于一个指定的频带宽度，没有仪器能减少其时间记录的长度。,Overlap - No Overlap 重叠无重叠(2),After capturing the time record, it will then require a finite time compute the FFT (this is where the high speed processors have an effect). 在捕捉到时间记录之后，会需要一段时间来计算FFT（高速处理器会对其产生影响）,Overlap Overlapped 重叠有重叠(1),In essence, a full length time record is first captured. Then portions of new time records are stored and added to portions of the old data. 首先采集到的是一个全长度的时间记录。然后，部分新的时间记录会被存储起来，加入到部分旧数据中，形成一个完整的时间记录。,Overlap Overlapped 重叠有重叠(2),This can greatly speed up the measurement process, especially when capturing 8 or more averages and is acceptable for most condition monitoring measurements - as long as the vibration is periodic (that is, repeating, predictable and without pronounced transients). 这能大大加快测量进程，尤其是当捕捉到8个或更多平均值，并能应用于大多数的状态监测测量时只要振动是周期性的（即反复的，可预测的，无明显瞬变现象）,Example of Sampling Time 采样时间举例,For an example, if it was necessary to capture a 400-line FFT, with 8 averages and a frequency range from 0 - 25 Hz, it would require 128 seconds just to sample the data without overlap (not including transducer and data collector system settling time). 如果需要捕捉一个400线的FFT，且要8次平均和0 - 25 Hz的频率量程，则需要花费128秒去进行无重叠的数据采样（不包括传感器和数据采集系统的初始化时间）。,Example of Sampling Time 采样时间举例,Data Sampling Time = (400 lines)(8 ave.) /25= 128 seconds 数据采样时间= (400 线)(8 平均值) /25= 128 秒,Example of Sampling Time 采样时间举例,Sampling with 50% overlap 有50% 重叠的采样,Frequency Resolution in Spectrum 频谱频率分辨率,FREQUENCY RESOLUTION = FREQ. SPAN / # FFT LINES 频率分辨率= 频带宽度 / FFT 频谱线数 The higher the number of lines of resolution that are specified for a spectrum, the higher will be the precision of the frequency read by the analyzer and displayed. It is important to note that the precision (accuracy range) of any displayed frequency will be the frequency reading plus or minus one-half the resolution. 频谱分辨率越高，分析仪所读取并显示