DSP
FT, Hilbert transform, short-time Fourier transform, Wigner distributions , Randon transform,and featured transformation(wavelet transform)
FT,WT reversible transform(可逆)但是FT可以得到频率信息没有时间信息，在时间域没有频率的信息。如何同时得到。
对于平稳信号stationary，FT处理没有问题，因为所有的时间段都有所有的频率成分，但是如果是非stationary signal处理的结果就不对，因为不是所有时刻都有所有的成分只是某一个时间段有某一种时间成分。图1.5 1.6 1.7 1.8的区别。
At what time these frequency occur? For these signals the frequency componets do not appear all the times.But they have the same spectrum.
FT gives no information regarding where in time thoese spetral components appear.Only if we are interested in what spectral component exist in the signal without interests where these occur.
The Ultimate Solution:Wavelet Transform
provides time-frequency representation.
Then we take the lowpass portion again and pass it through low and high pass filters; we now have 4 sets of signals corresponding to 0-125 Hz, 125-250 Hz,250-500 Hz, and 500-1000 Hz. We continue like this until we have decomposed the signal to a pre-defined certain level. Then we have a bunch of signals, which actually represent the same signal, but all corresponding to different frequency bands. We know which signal corresponds to which frequency band, and if we put all of them together and plot them on a 3-D graph, we will have time in one axis, frequency in the second and amplitude in the third axis. This will show us which frequencies exist at which time ( there is an issue, called "uncertainty principle", which states that, we cannot exactly know what frequency exists at what time instance, but we can only know what frequency bands exist at what time intervals, more about this in the subsequent parts of this tutorial).
The uncertainty principle, originally found and formulated by Heisenberg, states that, the momentum and the position of a moving particle cannot be known simultaneously. This applies to our subject as follows:
The frequency and time information of a signal at some certain point in the time-frequency plane cannot be known. In other words: We cannot know what spectral component exists at any given time instant. The best we can do is to investigate what spectral components exist at any given interval of time. This is a problem of resolution, and it is the main reason why researchers have switched to WT from STFT. STFT gives a fixed resolution at all times, whereas WT gives a variable resolution as follows:(STFT给出的是fixed resolution，而WT给出的是可变的分辨率)
Higher frequencies are better resolved in time, and lower frequencies are better resolved in frequency. This means that, a certain high frequency component can be located better in time (with less relative error) than a low frequency component. On the contrary, a low frequency component can be located better in frequency compared to high frequency component.
Take a look at the following grid:
FT:signal x(t) 乘以指数项，在某个certain frequecy下，然后对所有的时间积分，指数项可以用欧拉展开。
$$exp(2pift)=cos(2pift)+jsin(2pift)$$
FT是一系列的sin和cos函数组成，频率是f，计算和x(t)的乘积然后积分，如果得到较大的值，则认为dominant spectral component at frequecy f.如果得到较小的值表示没有这个f。等式的右边是X(f)，也就是说得到的是某一个certain frequency在over time的积分 value。X是f的函数，通过积分可以得到所有的X(f)
注意：积分是minus infinity to plus infinity. 无论在时间域的任何时候有frequecy f，积分就能得到较大的值，也就是说FT对时间没有分辨率，也就是说他对于非平稳信号not suitable。如果f at all times，FT makes sense，It just tells a certain f exists or not.
在处理数据之前我们要知道whether a signal is stationary or not.
FT中的f axis 是采样频率的2倍,如果使用无限长的窗函数，得到就是FT，如果是有限长的窗函数得到的STFT，但是得到的频率的分辨率变poor了。
narrow window > good time resolution, poor frequency resolution
wide window > good frequency resolution ,poor time resolution
Mulitresolution analysis: good time resolution and poor frequency resolution at high frequency and ggod frequency resolution and poor time resolution at low frequency
WT作法类似于STFT，使信号乘以一个function，STFT是window function，resolution是固定的
wavelet：小波，refers to window function of finite length. translation(tau) is the same sense as used in STFT, it is the time information in transform domain.,sacle is S,it is defined as 1/frequency,是STFT中的倒数。在时间域中，high scale corresponding to a non-detailed view and low scales correspond to a detailed view.在频率域中，low frequency(high sacles) correspond to a a global information of signal,whereas high frequency correspond to a detailed information of a hidden pattern in the signal.
Scale: mathematical operation,拉伸或者压缩，大的scale值对应拉伸，小的值对应压缩。
in the case of WT, the scale change can be used to reduce the sampling rate