I heard someone say Pete doesn't do windows!

What's the scoop?

我听到有人说皮特现在不加窗了！

有什么消息吗？

 

Well ... that's right. But you have to let me qualify that statement. Of course there are many data acquisition situations where it is a necessity to use windows. But almost all of the time when performing a modal test, the input excitation can be selected such that the use of windows can be eliminated. Let's first understand why acquisition of certain types of data can be distorted by the digitization and sampling process, what needs to be done to minimize the distortion, and how to work around the acquisition problem through the selection of specialized test excitation techniques.

嗯…情况就是那个样子。但是你得让我说明那个说法是对的。当然，有很多数据采集的情况必须要施加窗函数。但是，进行模态试验时，在几乎所有的时候都能够选择输入激励，这样就不用施加窗函数了。让我们首先理解为什么量化和采样过程会引起某些类型的数据采集发生失真，需要做些什么来将失真减至最小，并且如何通过选择专门的试验激励技术来解决这个采集的问题。

 

First let's remember that the Fourier Transform is defined from −∞ to +∞, but that we only acquire data over a very short time interval. As long as we can reconstruct the data, for all time, from the very small sample we measure, then there is no problem.

首先，我们记住傅里叶变换是在-∞到+∞上定义的，但是我们只能采集很短一段时间范围内的数据。从测量的这个非常短的样本中，只要我们都能够重构所有时间的数据，那就没有问题。

 

Figure 1 shows a simple sine wave, sampled for one time record, with the reconstruction of the time signal from the sample. Figure 1 also shows the FFT of this sampled signal. The time signal is expressed in the frequency domain as one discrete spectral line as expected. This happened because we captured an integer number of cycles of the sine wave in one record or sample of the data - in which case we say that the signal is periodic with respect to the sample interval.

图1显示了一个简单的正弦波、采样到的一段时间记录、从样本点中重建的时域信号。图1同时显示了采样到的信号的FFT结果。如同我们所期待的那样，时域信号在频域内表现为一根离散的谱线。出现这种情况是因为在一次数据记录或数据样本中，采集到了整数周期的正弦波形 — 在这种情况下，我们说信号相对于采样时间段是周期的。

But what if this is not the case. Figure 2 shows this situation. As before, we see the signal, the sample, the reconstructed signal and the FFT of the signal. Notice that the reconstructed signal contains a discontinuity that clearly did not exist in the original signal. The FFT of this signal is far from being a single spectral line as expected. Due to the sampling distortion, the frequency representation is smeared over the whole frequency bandwidth. This very serious error is called leakage and is by far the most serious digital signal processing error that is encountered.

但是如果情况并非如此，又会怎样呢？图2说明了这种情况。跟以前一样，我们观察这个信号、样本数据、重建的信号、和信号的FFT结果。注意到重建信号包含不连续点，显然它们在原始信号中并不存在。这个信号的FFT结果远不是我们所期望的那样为一根谱线。因为采样失真，频域表现形式在整个频带范围内发生了模糊。这个非常严重的误差称为泄漏，并且是目前为止我们遇到的最为严重的数字信号处理误差。

 

But why does this happen? The original signal was a simple sine wave. How did the frequency representation get so distorted? There's an easy explanation for this. The sampled data does not contain an integer number of cycles or repetitions of the signal.

但为什么会发生这种情况呢？原始信号原先是一个简单的正弦波。为什么频域表现形式变得如此失真？对此有一个简单的解释。采样到的数据没有包含整数周期的信号，或反复发生的信号。

 

Let's stop and recall some simple things we learned about Fourier series. If we start with a simple sine wave, we know that it is a trivial task to describe that signal with a Fourier series. It is basically just one term of the Fourier series which is a sine wave at ω with some amplitude A0. But do you remember what the series expansion was for a signal such as a rectangular series of pulses? Well, I don't want to expand on all of this right now but I think you would remember that it was a series of sinusoids at different frequencies with different amplitudes. In fact for the rectangular pulse, there were many terms in the series required in order to approximate that signal. That happened because the shape of the discontinuous rectangular pulse doesn't look like a nice smooth sine wave.

停下来，回忆一下我们学过的关于傅里叶级数的某些简单内容。如果从简单的正弦波开始，我们知道用傅里叶级数来描述这个信号是一个很简单的任务。大致说来，它只是傅里叶级数中的一项，是一个频率为ω、幅值为A0的正弦波。但是你还记得一个信号，如矩形脉冲序列的级数展开式是什么吗？好了，此刻我不想对这些内容进行全面展开，但是我认为你应该记得，它是一系列不同的频率上、具有不同幅值的正弦波。实际上对于矩形脉冲，为了逼近这个信号，在级数中需要有很多项。这种情况的发生是因为不连续的矩形脉冲的形状看起来不像一个很好的光滑的正弦波。

 

Now if I look back at the sampled sine wave in Figure 2, I can now see that by not capturing an integer number of cycles of the signal I have distorted the signal such that it appears to have a discontinuous nature at the end of the sample interval. This explains why the FFT is smeared over the frequency bandwidth. Basically, there are many terms needed in order to approximate this apparently discontinuous signal.

现在，如果我回过头来观察图2中采样到的正弦波，可以发现，由于没有采集到整数周期的信号，使得信号发生了失真，这样在信号采样时间段的末端，信号看上去具有不连续点。这解释了为什么FFT在整个频带范围内发生了模糊。从根本上讲，需要有很多项来逼近这个明显不连续的信号。

 

In order to minimize this error (and notice that I said minimize and not eliminate), we use weighting functions called windows. Basically we apply a weighting function to make the signal appear to better satisfy the periodicity requirements of the FFT process. Figure 3 shows a windowed time history.

为了将这个误差减至最小（并且注意我的说是减至最小，而不是消除），我们使用称为窗的加权函数。从根本上讲，我们施加一个加权函数来使信号看上去更好地满足FFT处理的周期性要求。图3显示了一个加窗后的时域波形。

The most common windows for modal testing today are the Rectangular window, the Hanning window, and the Flat Top window for shaker testing and the Force/Exponential window for impact testing.  The main thing to understand right from the start is that all windows distort data! Without going into all the detail, windows always distort the peak amplitude measured and always give the appearance of more damping than what actually exists in the measured FRF - two very important properties that we try to estimate from measured functions. The amplitudes are distorted as much as 36% for the Rectangular window and 16% for the Hanning window. The effects of these windows is best seen in the Frequency domain representation of the weighting function. All windows have a characteristic shape that identifies the amount of amplitude distortion possible, the damping effects introduced and the amount of smearing of information possible. Figure 4 shows the Rectangular, Hanning and Flat Top windows frequency representation. Sometime soon we will discuss what these curve more but for right now, I'm happy if you just understand that the windows, while a necessary evil in some measurement situations, distort data.

今天对于模态试验来讲，最常见的窗函数，对激振器试验，是矩形窗、汉宁窗、和平顶窗，而对锤击试验，是力窗/指数窗。在刚开始时，需要了解的主要一点是，所有的窗函数都会造成数据失真！不去深究所有细节，在测得的FRF中，窗函数总是使测得的峰的幅度发生失真，总是给出好像更大的阻尼，比实际存在的阻尼要大 — 我们试图从测得的函数中要估计出来的两个非常重要的特性。对矩形窗，最大失真可至36%，对汉宁窗，可至16%。最好在加权函数的频域表现形式中观察这些窗函数的影响。所有的窗函数都具有一个特征形状，它确定了可能的幅值失真的程度、带来的阻尼影响、以及可能的信息模糊的程度。图4显示出了矩形窗、汉宁窗、和平顶窗的频域表现形式。很快在某个时候我们会更深入地讨论这些曲线是什么意思，但是，目前你只要知道窗函数会造成数据失真，我就很高兴了，在某些测量情况下，我们不愿意加窗，却又不得不加窗。

 

So how do I get around not using windows on measured FRFs for a modal test? Basically, I try to satisfy Fourier's request - "either sample a repetition of the data or completely observe the signal in one sample of data". If you think about it, signals such as pseudo-random, burst random, sine chirp, and digital stepped sine all satisfy this requirement under most conditions and therefore are leakage free and do not require the use of a window. Maybe we can discuss the particulars about each of the windows another time, but this short explanation should suffice for now.

那么，对模态试验，我该怎么避免对测量的FRF加窗函数呢？从根本上讲，我要尽量满足傅里叶的要求 — “或者采样一段重复的信号，或者在一个数据样本中可以完整地观察到信号”。思考一下这点，在大多数条件下，诸如伪随机、猝发随机、正弦扫频、和数字步进正弦的信号都满足这个要求，因而没有泄露，不需要加窗。下一次，我们或许可以讨论每个窗函数的特点。但是目前来讲，这个非常简短的解释应该足够了。

 

Now I hope you understand why I don't like to use windows and I will avoid the use of windows at all costs - but every once and a while, I have no other choice. (Especially at home, where I can never get out of "doing windows"!)  If you have any other questions about modal analysis, just ask me.

现在，我希望你理解了我为什么不喜欢使用窗了，而且我将不惜一切代价地避免使用窗 — 但我时常又没有其他选择。（特别是在家里，我从来都逃不掉“擦窗”！）如果你有关于模态分析的任何其他问题，尽管问我好了。