信号处理原理：第四章 非平稳信号与不确定性原理

1、信号处理原理Principle of Signal Processing.Stationary of Signal第四 章 非平稳信号与不确定性原理信号的平稳性短时傅立叶变换Heisenberg不确定性原理 .Stationary of SignalStationary of SignalStationary SignalSignals with frequency content unchanged in timeAll frequency components exist at all timesNon-stationary SignalFrequency changes in timeO

2、ne example: the “Chirp Signal”.Stationary of Signalx(t)=cos(2*pi*5*t)+cos(2*pi*10*t)+cos(2*pi*20*t)+cos(2*pi*50*t) .Stationary of Signalx(t)=cos(2*pi*5*t)+cos(2*pi*10*t)+cos(2*pi*20*t)+cos(2*pi*50*t) FT.Stationary of SignalTimeMagnitudeMagnitudeFrequency (Hz)TimeMagnitudeMagnitudeFrequency (Hz) Freq

3、uency: 2 Hz to 20 Hz Frequency: 20 Hz to 2 Hz.Stationary of SignalStationary of SignalNote that the exponential term in Eqn. (1) can also be written as:cos(2.pi.f.t)+j.sin(2.pi.f.t) . . . (3)The above expression has a real part of cosine of frequencyf, and an imaginary part of sine of frequencyf. So

4、 what we are actually doing is, multiplying the original signal with a complex expression which has sines and cosines of frequencyf. Then we integrate this product.Stationary of SignalStationary of SignalAt what time the frequency components occur? FT can not tell!FT only gives what frequency compon

5、ents exist in the signalThe time and frequency information can not be seen at the same timeMost of Transportation Signals are Non-stationary. (We need to know whether and also when an incident was happened.) .Stationary of Signal第x 章 非平稳信号与不确定性原理信号的平稳性短时傅立叶变换Heisenberg不确定性原理 STFTSHORT TERM FOURIER T

6、RANSFORMDennis Gabor (1946) Used STFT (Gabor Transform)To analyze only a small section of the signal at a time a technique called Windowing the Signal.The segment of signal is assumed Stationary x(t)is the signal itself,(t)is the window function, and*is the complex conjugate.STFTSHORT TERM FOURIER T

7、RANSFORMThe STFT of the signal is nothing but the FT of the signal multiplied by a window function.In the FT, the window is its kernel, theexpjwtfunction, which lasts at all times from minus infinity to plus infinity. Now, in STFT, the window is of finite length, thus it covers only a portion of the

8、 signal, which causes the frequency resolution to get poorer, while the FT gives the perfect frequency resolution STFTSHORT TERM FOURIER TRANSFORMA compromise between time-based and frequency-based views of a signal.both time and frequency are represented in limited precision.The precision is determ

9、ined by the size of the window.Once you choose a particular size for the time window - it will be the same for all frequencies.Many signals require a more flexible approach - so we can vary the window size to determine more accurately either time or frequency. - Wavelet TransformSTFTSHORT TERM FOURI

10、ER TRANSFORM300 Hz 200 Hz 100Hz 50HzSTFTSHORT TERM FOURIER TRANSFORM.Stationary of Signal第x 章 非平稳信号与不确定性原理信号的平稳性短时傅立叶变换Heisenberg不确定性原理 UncertaintyThe Heisenberg Uncertainty PrincipleSTFT is closely related to the choice of analysis windowNarrow window good time resolutionWide window (narrow band) g

11、ood frequency resolutionTwo extreme cases:(T)=(t) excellent time resolution, no frequency resolution(T)=1 excellent freq. resolution (FT), no time info!How to choose the window length?Window length defines the time and frequency resolutionsHeisenbergs inequalityCannot have arbitrarily good time and

12、frequency resolutions. One must trade one for the other. Their product is bounded from below.UncertaintyThe Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty PrincipleIn the context of harmonic analysis, a branch of mathematics, the uncertainty principle implies that one cannot at the same

13、time localize the value of a function and its Fourier transform. To wit, the following inequality holds,UncertaintyThe Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty PrincipleIn the context of signal processing, and in particular timefrequency analysis, uncertainty principles are referre

14、d to as the Gabor limit, after Dennis Gabor, or sometimes the HeisenbergGabor limit.One cannot simultaneously sharply localize a signal (function f ) in both the time domain and frequency domain (, its Fourier transform).UncertaintyThe Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Princ

15、ipleOne cannot know the exact time-frequency representation of a signal (instance of time);What one can know are the time intervals in which certain band of frequencies exist;This is a resolution problem, which has something to do with thewidthof the window function that is used. UncertaintyThe Heis

16、enberg Uncertainty PrincipleThe resolutions in FTIn the FT there is no resolution problem in the frequency domain, i.e., we know exactly what frequencies exist; similarly there is no time resolution problem in the time domain, since we know the value of the signal at every instant of time. Conversel

17、y, the time resolution in the FT, and the frequency resolution in the time domain are zero, since we have no information about them. UncertaintyThe Heisenberg Uncertainty PrincipleDilemmaIf we use a window of infinite length, we get the FT, which gives perfect frequency resolution, but no time infor

18、mation. in order to obtain the stationarity, we have to have a short enough window, in which the signal is stationary. The narrower we make the window, the better the time resolution, and better the assumption of stationarity, but poorer the frequency resolutionUncertaintyThe Heisenberg Uncertainty PrincipleNarrow window =good time resolution, poor frequency resolution.Wide window =good frequency resolution, poor time resolution.UncertaintyThe Heisenberg Uncertainty Princi