Advanced Digital Signal Processing. Stationary/nonstationary signals. Time-Frequency Analysis... Some nonstationary signals. Time-Frequency Analysis

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1 Advanced Digital ignal Proceing Prof. Nizamettin AYDIN Time-Frequency Analyi 2 tationary/nontationary ignal Time-Frequency Analyi Fourier Tranform indow function indow ize Overlap ratio In real world mot ignal are highly nontationary and ometime lat only for a hort time. ignal analyi method which aume that the ignal i tationary are not appropriate. Therefore time-frequency analyi of uch ignal i neceary. 3 4 ome nontationary ignal Time-Frequency Analyi The time repreentation i uually the firt decription of a ignal ( obtained by a receiver recording variation with time. The frequency repreentation, which i obtained by the well known Fourier tranform (FT), highlight the exitence of periodicity, and i alo a ueful way to decribe a ignal Forward (red) and revere (blue) flow component are hown (after Hilbert tranform proce) 5 6

2 Time-Frequency Analyi... The relationhip between frequency and time repreentation of a ignal can be defined a ( ω) jωt jωt e dt, ( = ( ω) e dω no frequency information Time axi no time information Frequency axi Time-Frequency Analyi A joint time-frequency repreentation i neceary to oberve evolution of the ignal both in time and frequency.. Linear method (indowed Fourier Tranform (FT), and avelet Tranform (T)) Decompoe a ignal into time-frequency atom. Computationally efficient Time-frequency reolution trade-off 2. Bilinear (quadratic) method (igner-ville ditribution) Baed upon etimating an intantaneou energy ditribution uing a bilinear operation on the ignal. Computationally intene Arbitrarily high reolution in time-and frequency Cro term interference 7 8 Harmonic Fourier Analyi + jωt t e dt F ( ω, ) freq time Bai function are mooth, analytic olution to natural differential equation Aume ignal i tationary 9 indowed Fourier Tranform......indowed Fourier Tranform... + jωt g ( t τ e dt F ( ω, ) freq g(: hort time analyi window localied around t= and ω= Alo called hort-time Fourier Tranform time indow of fixed ize but varying hape Fixed ize implie fixed time and frequency reolution Ueful for analyi of narrowband procee 2 2

3 e g...indowed Fourier Tranform... for each Frequency for each Time end end Coefficient (F,T) = ignal window (F,T) all time...indowed Fourier Tranform... egmenting a long ignal into maller ection with 28 point and 52 point Hanning window (criticaly ampled) Coefficient Frequency.5 Time Number of ample indowed Fourier Tranform Three important FT parameter for analyi of a particular ignal need to be determined: indow type, indow ize, indow type... The FT make an implicit aumption that the ignal within the meaured time i repetitive. Mot real ignal will have dicontinuitie at the end of the meaured time, and when the FFT aume the ignal repeat, it will alo aume dicontinuitie that are not there. Required overlap ratio 5 6 indow type......indow type ome window type and correponding power pectra Rec tangl Hammin Bartlett Blackman Dicontinuitie can be eliminated by multiplying the ignal with a window function. Hanning.5 5 ample Normalied frequency Gauian.5 5 ample Normalied frequency 7 8 3

4 indow ize... In a FFT proce, there i a well known trade-off between frequency reolution ( ω) and time reolution (, which can be expreed a t =, ω ω ω = where i window length and ω i ampling frequency. To ue the FT, one ha to make a trade-off between time-reolution and frequency reolution....indow ize If no overlap i employed, proceing N length data by uing length analyi window will reult in a time-frequency ditribution having a dimenion that almot equal to the dimenion of the original ignal pace (critically ampled FT). The actual dimenion of the time-frequency ditribution i N M = [ +] The bet combination of t and f depend on the ignal being proceed and bet time-frequency reolution trade-off need to be determined empirically. 9 2 indow overlap ratio... A hort duration ignal may be lot when a windowing function i ued prior to the FFT. In thi cae an overlap ratio to ome degree mut be employed. In overlapped FT, the data frame of length are proceed equentially by liding the window -O L time at each proceing tage, where O L i the number of overlapped ample. Conequently, overlapping FFT window produce higher dimenional FT. In an overlapped FT proce, the dimenion of the reultant time-frequency An example of poible embolic ditribution i ignal at the edge of two N OL conecutive frame and related 3d M = [ +] pectrum without a window and OL with a Hannig window function 2...indow overlap ratio... The overlapping proce introduce a predictable time hift on the actual location of a tranient event on the time-frequency plane of the FFT. The duration of the time hift depend on the overlap ratio ued, while the direction of the time hift i dictated by the way that the data are arranged prior to the FFT. Duration of the time hift can be etimated a (number of overlapped ample/2) ampling time. The time hift can be adjuted by adding zero equally at both end of the original data array. In thi cae the dimenion of the overlapped FT i = N M O L [ + ] 22...indow overlap ratio egmenting a long ignal into maller ection with 52 point Hanning window (different overlap trategie) Number o ample TFD with 6, 32, 64,28, 256, 52 point windowing Normalied IP and energy with 6(black), 32(red), 64(green), 28(blue), 256(magenta), 52(cyan) point windowing Normalied IP Normalied energy

5 TFD with 6, 32, 64,28, 256, 256, 52 point Hanning windowing Normalied IP Normalied IP with 6(black), 32(red), 64(green), 28(blue), 256(magenta), 52(cyan) point windowing Normalied energy with 6(black), 32(red), 64(green), 28(blue), 256(magenta), 52(cyan) point windowing onogram.9.8 Normalied energy onogram + frequencie forward flow - frequencie revere flow Fat Fourier Tranform pectrum