An Introduction to Time-Frequency Analysis

An Introduction to Time-Frequency Analysis Outline . Introduction . STFT . Rectangular STFT . Gabor Transform . Wigner Distribution Function . Motions on the Time-Frequency Distribution . FRFT . LCT . Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design . Sampling Theory . Modulation and Multiplexing Introduction . Frequency? . Another way to consider things. . Frequency related applications . FDM . Sampling . Filter design , etc …. Introduction . Conventional Fourier transform . 1-D . Totally losing time information . Suitable for analyzing stationary signal ,i.e. frequency does not vary with time. [1] . Introduction . Time-frequency analysis . Mostly originated form FT . Implemented using FFT [1] Short Time Fourier Transform . Modification of Fourier Transform . Sliding window, mask function, weighting function w(t) . Mathematical expression . 2()()(,)(){()}jfXtfxedFxwtwt....... . . .. ..... . Reversing Shifting FT Short Time Fourier Transform . Requirements of the mask function . w(t) is an even function. i.e. w(t) w(-t). . max(w(t))=w(0),w(t1) w(t2) if |t1|<|t2|. . when |t| is large. . An example of window functions . t Window width K Short Time Fourier Transform . Requirements of the mask function . w(t) is an even function. i.e. w(t) w(-t). . max(w(t))=w(0),w(t1) w(t2) if |t1|<|t2|. . when |t| is large. . An illustration of evenness of mask functions . Signal Mask t0 . Short Time Fourier Transform . Effect of window width K . Controlling the time resolution and freq. resolution. . Small K . Better time resolution, but worse in freq. resolution . Large K . Better freq. resolution, but worse in time resolution Short Time Fourier Transform . The time-freq. area of STFTs are fixed K decreases tt f f Rectangular STFT .Rectangle as the mask function .Uniform weighting .Definition .Forward .Inverse where 2(,)() tjBBftXtfxed.... . . . ..21()(,)jftxtXtfedf. . .. .. 1tBttB.... 2B 1 B=0.25 t f 10 20 300 B=0.25 t f 10 20 300 f 0 Rectangular STFT . Examples of Rectangular STFTs . cos(4),010()cos(2),1020cos(),2030ttxttttt . . . ... . .... . ... 2,010()1,10200.5,2030itfttt ... . .... . ... B=0.5 2 2 1 1 0 0 t 10 2030 Rectangular STFT . Examples of Rectangular STFTs . cos(4),010()cos(2),1020cos(),2030ttxttttt . . . ... . .... . ... 2,010()1,10200.5,2030itfttt ... . .... . ... B=1 B=3 t f t f 0 10 20 30 0 10 20 30 0 1 22 1 0 Rectangular STFT . Properties of rec-STFTs . Linearity ()()() (,)(,)(,) htxtytHtfXtfYtf .. .. ....... . Shifting 00202()(,) tBjfjftBxedXtfe........ . .. . .... . Modulation 0202[()](,) tBjftjfBefxedXtf...... . . . ... Rectangular STFT . Properties of rec-STFTs . Integration 2(), (,) 0, jfvxvvBtvBXtfedfotherwise . . .. ..... ... . . Power integration **(,)()(,)() tBtBXtfdfxdYtfy... .. ... ... . Energy sum **((,)()(),)XtfdfdtBxfydYt... ... ...... .... Gabor Transform . Gaussian as the mask function . . Mathematical expression . 2221.9143()() 1.92143(,)()() tjftjfttxGtfexdedexe............ . .... . .. ... .... . Since where . GT’s time-freq area is the minimal against other STFTs! Gabor Transform . Compared with rec-STFTs . Window differences . Resolution . The GT has better clarity . Complexity Gabor Transform . Compared with rec-STFTs . Resolution . GT has better clarity . Example of 2()tyte... The rec-STFT The GT t f t f 00 00 Gabor Transform . Compared with the rec-STFTs . Window differences .Resolution .GT has better clarity .Example of 2()tyte... The rec-STFT The GT t f t 0 0 0 0 Gabor Transform . Properties of the GT . Linearity ()()() (,)(,)(,)zxyzxyGtfGtfGtf ..... .. ....... . Shifting 0002()(,)(,)jfxttxtGttfGtfe.. ... . Modulation 200() (,)(,)jtfxxteGfGtftf... Gabor Transform . Properties of the GT . Integration 222(1)(,)()kjtftkxGtfedfexkt.. . .. .. .. K=1-> recover original signal . Power integration . Energy sum . Power decayed Gabor Transform . Gaussian function centered at origin . 1.. . Generalization of the GT . Definition 221.91()2()2431.9143(,)()() ttjftjfxtGtfeexdeexd..... . .. . ....... . . ...... .. . .... 0.. Gabor Transform . . plays the same role as K,B.(window width) . increases -> window width decreases . .. decreases -> window width increases . Examples : Synthesized cosine wave 1..0.1.. t f t f 0 10 20 30 0 10 20 30 0 2 1 0 1 2 Gabor Transform . . plays the same role as K,B.(window width) . increases -> window width decreases . .. decreases -> window width increases . Examples : Synthesized cosine wave 1.5..5.. t f t f 0 10 20 30 0 10 20 30 0 2 1 0 1 2 Wigner Distribution Function . Definition . **2()()()() 2(,){ 2} 22jfxxtxtWtfedFxtxt... ..... . .. ....... . Auto correlated -> FT . Good mathematical properties . Autocorrelation . Higher clarity than GTs . But also introduce cross term problem! Wigner Distribution Function . Cross term problem . WDFs are not linear operations. . 22(,)||(,)||(,)hgsWtfWtfWtf.... *2***()()()[() 2] 222jfgtstgtsedt...... ..... . . .. ....... Wigner Distribution Function . An example of cross term problem . 22(4) 10(6) 91() 19tjtjttetxtet . .. ....... .... 11(4)9125() 1(26)192ittfttt . . . ........ ..... .. Without cross term With cross term t f f t 0 0 00 Wigner Distribution Function . Compared with the GT . Higher clarity . Higher complexity . An example . 44()cos(4) 2jtjteextt .. . .. .. WDF GT t f t f 00 0 0 Wigner Distribution Function . But clarity is not always better than GT . Due to cross term problem . Functions with phase degree higher than 2 WDF GT [1]t f t f 0 0 0 0 Wigner Distribution Function . Properties of WDFs . Shifting . Modulation 20e( 0) (,)(,)jftxxtWtfWtff... . Energy property 22(,)|()||()|xWtfdtdfxtdtXfdf .... ........ ...... Wigner Distribution Function . Properties of WDFs . Recovery property . is real . Energy property . Region property . Multiplication . Convolution . Correlation . Moment . Mean condition frequency and mean condition time Motions on the Time- Frequency Distribution . Operations on the time-frequency domain .Horizontal Shifting (Shifting on along the time axis) 0,00002()(,) ()(,) jSTFTGTxWDFxftxttSttfxttWttfe................... t f . Vertical Shifting (Shifting on along the freq. axis) 002,020()(,) ()(,) jftSTFTGTxjftWDFxextStffextWtff . . ................ f t Motions on the Time- Frequency Distribution . Operations on the time-frequency domain . Dilation ,1()(,) || 1()(,) || STFTGTxWDFxttxSafaaattxWafaaa . ......... ..... .. . Case 1 : a>1 . Case 2 : a<1 Motions on the Time- Frequency Distribution . Operations on the time-frequency domain .Shearing -Moving the side of signal on one direction .Case 1 : 2()()jatytext.. (,)(,) (,)(,) yxyxaStfStftWtfWttaf ....... t f Moving this side a>0 . Case 2 : 2()() tjaytext . .. (,)(,) (,)(,) yxyxStfStffWtfWtfafa ....... t f a>0 Moving this side Motions on the Time- Frequency Distribution . Rotations on the time-frequency domain . Clockwise 90 degrees . Using FTs t f 2|(,)||(,)| (,)(,) (,)(,) XjftXxxXxStfSftGtfGftWtfWfte.. .. . . . . . . ... Motions on the Time- Frequency Distribution . Rotations on the time-frequency domain . Generalized rotation with any angles . Using WDFs or GTs via the FRFT . Definition of the FRFT 22cot2csccot()[()]1cot()jujutjtFXuOxtjeeextdu....... .. . . .. .... . Additive property Motions on the Time- Frequency Distribution . Rotations on the time-frequency domain . [Theorem] The fractional Fourier transform (FRFT) with angle . is equivalent to the clockwise rotation operation with angle . for the WDF or GT. (,)(cossin,sincos) (,)(cossin,sincos) XXxxWuvWuvuvGuvGuvuv . . .... .... ......... Old New New Old ' ' cossinsincosuuvv .. .. ....... ......... ...... ' ' cossinsincosuuvv .. .. ...... ................ Clockwise rotation matrix Counterclockwise rotation matrix Motions on the Time- Frequency Distribution . Rotations on the time-frequency domain . [Theorem] The fractional Fourier transform (FRFT) with angle . is equivalent to the clockwise rotation operation with angle . for the WDF or GT. . Examples (Via GTs) [1] Motions on the Time- Frequency Distribution . Rotations on the time-frequency domain . [Theorem] The fractional Fourier transform (FRFT) with angle . is equivalent to the clockwise rotation operation with angle . for the WDF or GT. . Examples (Via GTs) [1] Motions on the Time- Frequency Distribution . Twisting operations on the time-frequency domain . LCT s Old New t t f f (,,,) (,,,) (,)(,) (,)(,) XxabcdXxabcdWuvWdubvcuavWaubvcudvWuv ....... ..... ' ' udbucavv ....... ................ Inverse exist since ad-bc=1 abcd .. .. .. LCT Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design . A signal has several components -> separable in time -> separable in freq. -> separable in time-freq. 41 t f t f Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design 42 .An example t f 1.2.1t0tRotation -> filtering in the FRFT domain Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design . An example [1] Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design .An example [1] 2220.230.38.50.469.6()0.50.50.5jtjtjtjtjtnteee..... Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design . An example [1] 2220.230.38.50.469.6()0.50.50.5jtjtjtjtjtnteee..... Applications on Time-Frequency Analysis . Signal Decomposition and Filter Design . An example [1] Applications on Time-Frequency Analysis . Sampling Theory . Nyquist theorem : , B . Adaptive sampling [1] Conclusions and Future work . Comparison among STFT,GT,WDF rec-STFT GT WDF Complexity正正正正正正 Clarity正正正正正正 . Time-frequency analysis apply to image processing?