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www.jntuworld.com Short-Term Fourier Transform (STFT) Motivation • To analyze STFT from Joint Time-Freq Distribution Perspec- tive • To understand the limitations of STFT as a TFD Introduction • Most widely used method for studying non-stationary signals • Breakup given non-stationary signal into small segments and analyze each segment • What is the typical size of segment? – Approximately where stationary property holds www.jntuworld.com – 10-30 ms in case of speech • Can we continue breaking to achieve ﬁner time localization? • No, because after certain narrowing spectrum becomes mean- ingless and show no relation to original signal spectrum – Termed as windowing eﬀect – Finer time localization =⇒ small window in TD =⇒ larger window in FD =⇒ convolved with signal spectrum =⇒ more smoothing of true spectrum information • This should be attributed to limitation of STFT which makes short segments • It is not uncertainty principle as applied to signal that is limiting factor, it is uncertainty principle associated with small segments www.jntuworld.com • Distinction between the two uncertainty principles should be kept in mind and the two should not be confused • In STFT properties of signal are scrambled with the properties of window function • Unscrambling required for proper interpretation and estimation of original signal • In spite of these limitations STFT provides excellent time-freq str. for some signals under proper choice of window function and duration. • However, it may not be best tool for all non-stationary signals • Hence search for new TFA tools www.jntuworld.com The STFT and Spectrogram • To study properties of signal at time t, emphasize signal around t and suppress signals at other places. st (τ ) = s(τ )h(τ − t) • Modiﬁed signal is a fn. of two times, ﬁxed time t, and running time τ • Window fn. is chosen to leave the signal more or less unaltered around the time t, but to suppress the signal for times distant from t i.e., st(τ ) = s(τ ) for τ near t and 0 for τ away from t • Since modiﬁed signal emphasizes the signal around t, the FT will reﬂect the distribution of freq around that time 1 st(w) = √ e−jwτ st (τ )dτ 2π 1 =√ e−jwτ s(τ )h(τ − t)dτ 2π www.jntuworld.com • The energy density spectrum at time t is therefore, 1 PSP (t, w) = |St (w)|2 = | √ e−jwτ s(τ )h(τ − t)dτ |2 2π • For each diﬀerent time we get a diﬀerent spectrum and the totality of these spectra is the time-frequency distribution, PSP • More commonly termed as Spectrogram Short-Frequency Time Transform • In STFT we emphasize the desire to study freq. properties at time t • Conversely, we may wish to study time properties at a particular freq. • We then window spectrum, S(w) with a freq. window fn. H(w), and take the time transform, which, of course, is the IFT www.jntuworld.com • Termed as short-freq time transform and deﬁned as 1 ′ Sw (t) = √ ejw t S(w′ )H(w − w′ )dw′ 2π • If we relate h(t) and H(w) by H(w) = √1 2π h(t)e−jwtdt, then St (w) = e−jwt Sw (t) • The STFT is the same as short-freq. time tfm. except for the phase factor e−jwt • Since the distribution is the absolute square, the phase factor e−jwt does not enter into it • Either the STFT or short-freq time tfm can be used to deﬁne the joint distribution P (t, w) = |St (w)|2 = |Sw (t)|2 • This shows that the spectrogram can be used to study the www.jntuworld.com behavior of time properties at a particular freq. • This is done by choosing an H(w) that is narrow or equivalently by taking an h(t) that is broad Narrowband and Wideband Spectrogram • If time window h(t) is of short duration the freq. window H(w) is broad and in that case the spectrogram is termed as Wideband. • Alternatively, if time window h(t) is of long duration, the freq. window H(w) is narrow and we say we have a Narrowband spec- trogram. Characteristic Function MSP (θ, τ ) = |St (w)|2ejθt+jτ w dtdw = As(θ, τ )Ah (−θ, τ ) www.jntuworld.com where As(θ, τ ) = 1 s∗(t − 2 τ )s(t + 1 τ )ejθtdt 2 Ah(−θ, τ ) = 1 h∗(t − 2 τ )h(t + 1 τ )e−jθt dt 2 As() and Ah () are the termed as ambiguity fns. of the signal s(t) and window fn. h(t), respectively. Notation: The results we will obtain are revealing when expressed in terms of the phases and amplitudes of the signal and window and their transforms. s(t) = A(t)ejφ(t) and S(w) = B(w)ejψ(w) h(t) = Ah (t)ejφh(t) and H(w) = BH (w)ejψH (w) In the calculation of global averages (e.g., mean freq, bandwidth) we will have to indicate which density fn is being used. We will www.jntuworld.com use the superscript (s), to indicate the signal being used. < w >(SP )= w|St (w)|2dwdt < w >(s)= w|S(w)|2dw < w >(h) = w|H(w)|2dw Characterisitc Function: • The joint characteristic function of a time frequency density is M (θ, τ ) =< ejθt+jτ w >= P (t, w).ejθt+jτ w .dtdw www.jntuworld.com General Properties: Total Energy • Integrating over all time and freq we get total energy ESP = PSP (t, w)dwdt = MSP (0, 0) = As (0, 0)Ah(0, 0) = |s(t)|2dt |h(t)|2dt • If the energy of window is taken to be one, then energy of spectrogram is equal to total energy of the signal Marginals Time Marginal P (t) = |St (w)|2dw 1 ′ = 2π s(τ )h(τ − t)s∗(τ ′ )h∗(τ ′ − t)e−jw(τ −τ )dτ dτ ′dw www.jntuworld.com = s(τ )h(τ − t)s∗(τ ′)h∗(τ ′ − t)δ(τ − τ ′)dτ dτ ′ = |s(τ )|2|h(τ − t)|2dτ = A2(τ )A2 (τ − t)dτ h Freq. Marginal P (w) = 2 B 2(w′)Bh(w − w′)dw′ As can be seen from the equations, the marginals of the spec- trogram generally do not satisfy the correct marginals, namely, |s(t)|2 and |S(w)|2 P (t) = A2(t) = |s(t)|2 P (w) = B 2(w) = |S(w)|2 – The reason is that the spectrogram scrambles the energy distribution of the window with those of the signal – This introduces eﬀects unrelated to the properties of the www.jntuworld.com original signal – Notice that the time marginal of the spectrogram depends only on the magnitude of the signal and window and not on their phases. – Similarly, freq. marginal depends only on the amplitudes of the FT Averages of time and freq. fns: • Since the marginals are not satisﬁed, averages of time and freq. fns. will never be correctly given, < g1(t) + g2(w) >= {g1(t) + g2(w)}PSP (t, w)dωdt = g1(t)|s(t)|2dt + g2(w)|S(w)|2dw • This is in contrast to other distributions we will be studying where these types of averages are always correctly given www.jntuworld.com Finite Support • According to ﬁnite support property, the distribution should be zero before and after the signal ends • In case of spectrogram, if a time t is chosen before the signal starts, will the spectrogram be zero for that time? • Generally, no, because the modiﬁed signal as a function of t will not necessarily be zero since the window may pick up some of the signal • Even though s(t) may be zero for a time t, s(τ )h(τ − t) may not be zero for that time • similar condition applies to FD • Spectrogram does not possess ﬁnite support property in either time or freq. domains. www.jntuworld.com Localization Trade-Oﬀ • Good time localization =⇒ narrow window h(t) in TD • Good freq localization =⇒ narrow window H(w) in FD • But both h(t) and H(w) cannot be made arbitrarily narrow • This is the inherent trade oﬀ between time and freq localization in the spectrogram for a particular window • Degree of trade oﬀ depends on the window, signal, time and freq. • The uncertainty principle for the spectrogram quantiﬁes these trade oﬀ dependencies www.jntuworld.com Entanglement and Symmetry Between Window and Signal • Results obtained using the spectrogram generally do not give results regarding the signal solely, because the STFT entangles the signal and window • Therefore we must be cautious in interpreting the results and we must attempt to disentangle the window. This is not always easy • In fact, because of the basic symmetry in the defn. of the STFT between the window and signal, we have to be careful that we are not using the signal to study the window www.jntuworld.com Uncertainty Principle for the STFT: –A short duration signal obtained by windowing is given by st (τ ) = s(τ )h(t − τ ) – The normalized short duration signal at time t is given by s(τ )h(t − τ ) ηt (τ ) = | s(τ )h(t − τ ) |2 dτ – DR. is square root of total energy in windowed signal. – This normalization ensures that | ηt (τ ) |2 dτ = 1 i.e. total normalized energy will be unity for any t. – The STFT of ηt(τ ) is given by Ft(jw) = ηt (τ ).e−jwτ dτ. www.jntuworld.com – We can deﬁne all the relevant quantities such as mean time, duration, and bandwidth in the standard way, but they will be time dependent. Mean Time: < τ >t = τ | ηt(τ ) |2 dτ τ | s(τ )h(τ − t) |2 dτ < τ >t = | s(τ )h(τ − t) |2 dτ Duration: Tt2 = (τ − < τ >t)2. | ηt (τ ) |2 dτ (τ − < τ >t )2. | s(τ )h(τ − t) |2 dτ Tt2 = | s(τ )h(τ − t) |2 dτ Mean Frequency: < w >t= w | Ft(jw) |2 dw www.jntuworld.com Bandwidth: Bt2 = (w− < w >t )2 | Ft(jw) |2 dw Tt2 = (τ − < τ >t )2 | ηt (τ ) |2 dτ – Let < τ >t = 0 then, Tt2 = τ 2 | ηt (τ ) | dτ – Similarly, < w >t= 0 then, Bt2 = w2 | Ft (jw) | dw Bt2 = ′ | ηt(τ ) |2 dτ Tt2Bt2 = | τ ηt (τ ) | dτ. ′ | ηt (τ ) |2 dτ | f (x) |2 dx | g(x) |2 dx ≥ | f ∗(x)g(x)dx |2 www.jntuworld.com – Let, f = τ ηt & ′ g = ηt Tt2Bt2 ≥ | τ ηt (τ )ηt (τ )dτ |2 ∗ ′ – Substituting and simplifying we get 1 Tt2Bt2 ≥ 2 – This is the uncertainty principle for the STFT. – It is a function of time, the signal, and the window. – It should not be compared with the uncertainty principle ap- plied to the signal. – It is important to understand this uncertainty principle, be- cause it places limits on the technique of the STFT proce- dure. – However, it places no constraints on the original signal. www.jntuworld.com – It is true that if we modify the signal by the technique of STFT, we limit our abilities in terms of resolution. – Hence the search for new time-frequency analysis tools. Global Quantities Mean Time: < t >(SP )= t|St(w)|2dtdw Mean Freq: < w >(SP )= w|St (w)|2dtdw • Direct evaluation leads to < t >(SP )=< t >(s) − < t >(h) < w >(SP )=< w >(s) − < w >(h) • If the window is chosen so that its mean time and freq. are zero, which can be done by choosing a window symmetrical in time and whose spectrum is symmetrical in freq. domain, then the mean time and freq. of the spectrogram will be that of www.jntuworld.com the signal. • The second conditional moments are calculated to be < w2 >(SP )=< w2 >(s) + < w2 >(h) +2 < w >(s) < w >(h) < t2 >(SP )=< t2 >(s) + < t2 >(h) −2 < t >(s) < w >(h) by combining these with mean time and freq. defns., we ﬁnd that the durations and bandwidths are related by 2 2 T(SP ) = Ts2 + Th 2 2 2 B(SP ) = Bs + Bh which indicates how the duration of the windowed signal is related to the duration of the signal and window www.jntuworld.com Covariance and Correlation Coeﬃcient • First mixed moment < tw >(SP )= tw|St (w)|2dtdw =< tφ′ >(s) − < tφ′h >(h) − < t >(h) < φ′ >(s) + < t >(s)< φ′h >(h) subtracting < t >(SP )< w >(SP ), from both sides we have that the covariance of spectrogram is (SP ) (s) (h) Covtw =< tw >(SP ) − < t >(SP )< w >(SP )= Covtw − Covtw covariance of real signal is zero • If we take real window, the covariance of the spectrogram will be the covariance of the signal (SP ) (s) Covtw = Covtw for real windows www.jntuworld.com Local Averages Method of Calculation • STFT St (w) and windowed signal sh (t) = s(τ )h(τ − t) forms Fourier pair s(τ )h(τ − t) ↔ St (w) represents Fourier pair between τ and w • Modiﬁed signal expressed in terms of the phase and amplitude is st (τ ) = s(τ )h(τ − t) = A(τ )Ah(τ − t)ej[ψ(τ )+ψh(τ −t)] •A fruitful way to look at the situation is that we are dealing with a signal in the variable τ whose amplitude is A(τ )Ah(τ − t) and whose phase is ψ(τ )ψh (τ − t) www.jntuworld.com • The normalized modiﬁed signal is given by s(τ )h(τ − t) s(τ )h(τ − t) ηt (τ ) = = |s(τ )h(τ − t)|2dτ P (t) Conditional Average 1 < g(w) >t = P (t) g(w)|St (w)|2dw = d ηt (τ )g( 1 dτ )ηt (τ )dτ ∗ j Local Frequency Instantaneous Freq. • WKT < w >= ψ ′ (t)|s(t)|2dt = ψ ′ (t)A2(t)dt • Let A2 be replaced by A2 (τ )A2 (τ − t) h and ψ′ by ′ ψ ′ (τ ) + ψh(τ − t) to obtain 1 1 d < w >t = w|St(w)|2dw = ηt (τ ) ∗ ηt (τ )dτ (1) P (t) j dτ 1 = ′ A2(τ )A2 (τ − t){ψ ′(τ ) + ψh (τ − t)}dτ h (2) P (t) www.jntuworld.com Local Square Frequency ′ • WKT < w2 >= w2|S(w)|2dw = ( A (t) )2A2(t)dt + A(t) ψ ′2(t)A2(t)dt • < w2 >t= ηt (τ )( 1 dτ )2ηt (τ )dτ = ∗ j d d | dτ ηt (τ )|2dτ 1 • < w2 >t= P (t) d ( dτ A(τ )Ah (τ − t))2dτ + P 1 (t) ′ A2(τ )A2 (τ − t){ψ ′(τ ) + ψh(τ − t)}2dτ h Conditional or Instantaneous Bandwidth ′ • Bandwidth Eqn: B 2 = ( A (t) )2A2dt + (ψ ′ (t)− < w >)2A2(t)dt A(t) 1 2 • Bt2 = σw|t = P (t) (w− < w >t )2|St(w)|2dw 1 d Bt2 = ( A(τ )Ah (τ − t))2dτ (3) P (t) dτ 1 + 2 A2 (τ1)A2(τ2)A2 (τ1 − t)A2 (τ 2 − t) h h (4) 2P1 (t) ′ ′ ×[ψ ′(τ1) − ψ ′ (τ2) + ψh (τ1 − t) − ψh (τ2 − t)]2dτ1dτ2 (5) www.jntuworld.com • For convenience we deﬁne 1 d < w2 >0= t ( A(τ )Ah(τ − t))2dτ P (t) dτ Narrowing and Broadening Window • All the results obtained above using spectrogram suﬀer from windowing eﬀect • Thus these quantities are estimates and depends on the window function chosen • By narrowing window, we will get better temporal resolution and better estimates and we may be tempted to do so. But this will give very poor frequency resolution. • For instance let us consider the window such that A2 (t) → δ(t) h • In the limit < w >t → ψ ′ (t) www.jntuworld.com • If the window is narrowed to get increasing time resolution, the limiting value of the estimated inst. freq. is the derivative of the phase, which is the inst. freq. • This is a pleasing and impt. result. But a penalty to pay for this. • σw|t → ∞ Some Examples to illustrate Limitations of STFT Ex1: Sinusoid with Gaussian Window 2 /2 s(t) = ejω0 t and h(t) = (α/π)1/4e−αt 2 STFT: St (w) = 1 (απ)1 /2 e−j(w−w0 )texp[− (w−w0) ] 2a 2 TFD: PSP (t, w) = |St(w)|2 = 1 (απ)1/2 exp[− (w−w0) ] a Using it we have < w >t = w0 ; σw|t = 1/2a 2 www.jntuworld.com Avg. value of freq. for a given time is always w0 , but the width about that freq. is dependent on the window width. Ex2: Impulse For an impulse at t = t0 , with the same window as above √ 2 /2 s(t) = 2πδ(t − t0) and h(t) = (a/π)1/4e−at 2 /2 We have St (w) = (a/π)1/4e−jwt0 e−a(t−t0 ) 2 PSP (t, w) = |St (w)|2 = (a/π)1/2e−a(t−t0 ) Ex3: Sinusoid Plus Impulse Consider sum of a sinusoid and impulse √ 2 /2 s(t) = ejw0 t + 2πδ(t − t0) and h(t) = (a/π)1/4e−at (w−w0)2 2 /2 We have St (w) = 1 (aπ)1/4 e−j(w−w0 )t exp[− a ] + (a/π)1/4e−jwt0 e−a(t−t0 ) 2 2 and PSP (t, w) = |St (w)|2 = 1 (aπ)1/2 e−(w−w0 ) /a + (a/π)1/2e−a(t−t0 ) + 2 2 2 √ e−(w−w0 ) /a−a(t−t0 ) cos[w(t−t0 )−w0 t] π www.jntuworld.com This example illustrates one of the fundamental diﬃculties with the spectrogram. For one window we cannot have high resolution in time and frequency. The broadness of the self terms, the ﬁrst two terms of above eqn., depends on the window size in an inverse relation. If we try to make one narrow, then the other must be broad. That will not be the case with other distributions. In Fig. shown below we plot the spectrogram for diﬀ. window sizes. The cosine term is an example of the so called cross terms. Not that they essentially fall on the self terms in the spectrogram. www.jntuworld.com