M. R. Portnoff. Time--frequency representation of digital signals and systems based on short--time fourier analysis. IEEE Trans. Acoust., Speech, Signal Processing., ASSP-28:55--69, Feb 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....may vary in time. Another important point is that their phases are recomputed from scratch and not given by analysis. The most famous analysis method for additive synthesis is probably the phase vocoder. Very good introductory texts on the phase vocoder can be found for example in [Moo78] [Por80], Dol86] or [Ser97a] It is mainly an implementation of the short time Fourier transform [All77] This section makes a short summary of the principles of this transformation and points out its main limitations. 3.1 Principles The Fourier transform converts the temporal signal (amplitude versus ....

M. R. Portnoff. Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Transform. IEEE Transactions on Acoustics, Speech, and Signal

....continuous Weyl Heisenberg group of translations in the phase plane, for what we have called Gabor systems. Gabor s 1946 paper certainly did not go unnoticed by the engineering community, but it lasted until 1980 that the attention for Gabor expansions was revived through the work of Portnoff [3], Bastiaans [4] and Janssen [5] This revival coincided, not entirely by accident, with the increasing interest in the electrical engineering community for time frequency tools, such as the Wigner Ville distribution and the short time Fourier transform. It should, however, be noted that as early ....

M.R. Portnoff, Time-frequency representation of digital signals and systems based on short-time Fourier analysis, IEEE Trans. ASSP, 28 (1980), pp. 55--69. 27

....the time dimension of a time feature representation of a signal. For its mathematical tractability and its practical importance in speech processing, the time feature representation that we investigate is the STFT. In contrast with the multiplicative modifications of the STFT investigated in [6] and [4] temporal processing is characterized by the convolution with respect to time of the STFT with a time frequency modification. The modified STFT can be written as Y (n; k ) 1 X r= Gamma1 F (n Gamma r; k )S(r; k ) 1) F (n; k ) n S(n; k ) where S(n; k ) is the STFT of the ....

....NY 1997. respect to the time index n. The modification F (n; k ) can be interpreted as a set of time trajectory filters, each operating on a different time trajectory indexed by the center frequency k . A modified time domain signal y(n) can be obtained by applying a synthesis formula to (1) [6]. It can beshown [3] that the time domain signal y(n) can also be obtained by directly filtering s(n) with an equivalent time domain linear time invariant filter. This equivalent filter dependson the inverse STFT of the modification F (n; k ) and the analysis and synthesis windows of the STFT. ....

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Portnoff M., "Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Analysis," IEEE Trans. on Acoustics, Speech and Signal Proc., Vol. 28, No. 1, February 1980, pp. 55-69.

....of a signal. As an example of a time frequency representation, the short time power spectrum is also depicted. If we describe s(n) by a two dimensional discrete time sequence as in (2.1) we can obtain a frequency representation with respect to each of the time indices m and n. As in [49], applying the Fourier transform (FT) in each dimension (with respect to both time indices) we obtain the two dimensional transform S( 1 X n= Gamma1 1 X m= Gamma1 s w (n; m)e Gammaj ( n m) 2.2) where we assumed that the infinite summations converge. Applying the double inverse ....

....= w(0)s(n) s(n) for w(0) 1: 2.9) It is evident that in order to recover the original signal s(n) from the two dimensional representations we need to impose a constraint on the analysis window, namely w(0) 6= 0. Equation (2. 9) is not the only way of recovering s(n) The reader is referred to [49] for alternative inversion formulas. 2.1.1 Relation to the Fourier Transform A relationship between the two dimensional transform (2.2) with the Fourier transforms of the signal and window function, S( and W ( respectively 1 , can be obtained. Substituting w(n Gamma m) by its Fourier ....

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Portnoff, M. Time-frequency representation of digital signals and systems based on short-time Fourier analysis. IEEE Trans. on Acoustics, Speech and Signal Processing ASSP-28, 1 (February 1980), 55--69.

....we wish to recover the harmonic component h. Clearly, this signal is instantaneously narrowband around each harmonic and therefore calls for a set of narrow bandwidth filters; however, time invariant filters would have to include the entire modulation bandwidth. Time frequency filtering methods [22 24] could be applied, but they are usually computationally expensive and moreover are not fully characterized in terms of their statistical performance. Demodulation methods are likewise unattractive for this application, because they do not preserve the scaling relationship of the harmonics. An ....

M. R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis," IEEE Trans. Acoust., Speech, Signal Processing, vol. 28, pp. 55--69, Jan. 1980.

....(t; f) 5) i.e. TH (t; f) is an appropriately defined time varying transfer function of H and TRx (t; f) accordingly a time varying power spectrum of the process x(t) 2. STFT BASED ANALYSIS AND FILTERING The short time Fourier transform (STFT) using an analysis window fl(t) is defined as [4] STFT (fl) x (t; f) Z s x(s)fl(s 0 t)e 0j2fs ds = hx; MfT t fli ; where we have introduced a time shift operator acting as (T t x) x( 0 t) and a frequency shift operator acting as (Mfx) t) x(t)e 0j2ft . The spectrogram is defined as the magnitude squared STFT, which can be ....

.... approaches the corresponding KL eigenvalue which can be defined by k;x = hRx ; uk Omega uk i : 8) STFT based signal processing is based on the well known reproducing formula: x( Z t Z f STFT (fl) x (t; f) MfT t g) dt df; hg; fli = 1; where g(t) is the synthesis window [4]. In a signal free notation, this can be written as a continuous (Weyl Heisenberg) resolution of the identity: I = Z t Z f (g Omega fl) t;f) dt df hg; fli = 1: A linear time varying filter Q (g;fl) M is obtained by inserting a multiplicative modification between the analysis and ....

M.R. Portnoff. Time--frequency representation of digital signals and systems based on short-time Fourier analysis. IEEE Trans. Sign. Proc., 28:55--69, 1980.

....intensive to apply, but also more immune to the effects of interference terms. 4.7.3 Linear time frequency filtering Time frequency filters based on linear TFRs offer better performance [78] The filtering mechanism is similar to that described above. In the case of the STFT based filter [98], the STFT of the signal is generated using an analysis window fl(t) then multiplied by the indicator function, and the output signal synthesised from this masked STFT using a synthesis filter g(t) The performance of the filter is highly dependent on the choice of analysis and synthesis windows, ....

M.R. Portnoff. Time-frequency representation of digital signals and systems based on short-time Fourier analysis. IEEE Trans. Acoust., Speech, Signal Proc., 28:55--69, 1980.

....a u (k) are any periodic functions of k with period N . A simple solution can be obtained when g(k) has finite length Q 1 and N (Q 1 Gamma1) 2 . Note that we have to consider the N Q 1 case only. The condition in Eq. 14 is similar to the biorthogonal like condition in [7] and on p. 63 in [11]: X n g(k Gamma nN)fl (k Gamma nN Gamma mM) 1 M ffi(m) 16) with M = 2N . If the length of g(k) Q 1 , is finite and M Q 1 , then it can be shown [7, 9] that fgmn g constitutes a frame, and that the dual frame (window) can be obtained as fl(k) g(k) MG(k) 17) where G(k) X ....

M. R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-28, No. 1, Feb. 1980, pp. 65-69.

....or FFT: Nkn i N n e n x N k X 2 1 0 1 p = 2.1) where x(n) is a set of N samples from the input sound and X(k) is a set of N complex valued samples, or bins, of the frequency spectrum. Sounds can be represented as a sequence of such spectra over successive time intervals [10]. OSE supports viewing of time varying spectra, as illustrated in Figure 7. The dimensions of the spectral viewer are time, amplitude and the frequency bins. Amplitude is the magnitude of the complex values in the spectrum, and is measured in on a logarithmic (i.e. decibel) scale. Frequencies can ....

M. R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis," IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-28, 55-69, 1980.

....although in general the inverse is defined over scale as in (5) in practice it often suffices to consider the recovery of x( k ) for each value of oe(n) ie from each level of the transform. In this case, the conditions for exact inversion are those corresponding to that for the discrete STFT [17]. Assuming a regular sampling of the and axes, these centre around the limit of the product between the sampling intervals in each domain, ie Delta (n) Delta (n) 2 (10) where Delta (n) i 1 (n) Gamma i (n) Delta (n) j 1 (n) Gamma j (n) 11) The required window functions then ....

M.R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis", IEEE Trans. Acous. Speech Sig. Proc., vol. ASSP-28, pp. 55-69, 1980.

....implemented by a scaling factor. The equation for reconstruction is then given by x(n) A A) Gamma1 A b = 1 Mkhk 2 1 X m= Gamma1 M Gamma1 X k=0 h (m Gamma n)e j 2 M nk b(m; k) 6) This synthesis equation is the same as that studied by Allen, Crochiere and Portnoff [6, 7, 9], with a particular choice of synthesis window. By moving the summation over index k inside we have x(n) 1 Mkhk 2 1 X m= Gamma1 h (m Gamma n) M Gamma1 X k=0 e j 2 M nk b(m; k) 7) This inner summation can then be implemented efficiently with inverse FFTs of the time slices ....

....and x2 . This is equivalent to minimizing the amount of energy of x1 smeared outside of R, as well as the amount of energy of component x2 smeared into R. We therefore seek fl opt(t) arg min fl(t) S(x1 ; R; fl) S(x2 ; R c ; fl) 10) By following an analysis similar to that of Vakman in [9] and Jones and Parks in [10] to study maximally localized signals, the problem of solving for fl opt in (10) simplifies by considering a basis function decomposition of the window: fl(t) 1 X k=0 ak fl k (t) 11) With this basis function decomposition and a complete set of basis elements, ....

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M. R. Portnoff, "Time-Frequency Representation of digital Signals and Systems Based on Short-Time Fourier Analysis," IEEE Trans. Acoust., Speech and Sig. Proc., vol. ASSP-28, pp. 55-68, February 1980.

....to circular correlation, and secondly it allows for the localisation of motion estimation. In order to incorporate this into our motion estimation algorithm, we can make use of the windowed Fourier transform corresponding to a local spectrum estimate centred around the spatial coordinate i [22], i.e. x( i ; j ) K 2 Gamma1 X k=0 w( i Gamma k )x( k )e Gamma 2 j Delta k =K (16) for a discrete window function w( of finite support and where K denotes the size of the window. By using (16) we can, in turn, define a local correlation field ae( ....

M. R. Portnoff, "Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Analysis," IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 28, Feb. 1980.

....that are based on input output observations. In this section we give a brief review of linear time frequency signal representations and their application to LTV system realization. Short Time Fourier Transform. The short time Fourier transform (STFT) of a signal x(t) is defined as [9, 3] (STFT fl x) t; f) Z x(s)fl(s 0 t)e 0j2 fs ds = hx; M f T t fli ; where fl(t) is the analysis window. The STFT depends significantly on the window and is highly redundant. Gabor Expansion. The Gabor expansion of a signal x(t) is defined as [3, 12] x(t) X k X l (G fl ....

....[2] HG = X k X l MG [k; l]P g;fl (kT; lF ) 20) where MG [k; l] is a multiplicator function that characterizes the time frequency filtering behaviour of HG in a manner similar to a sampled transfer function. Eq. 20) is just a discrete version of multiplicative modification of the STFT [9] where the system is given by H STFT = Z Z M(t; f)P g;fl (t; f)dt df: 21) Based upon appropriately selected analysis and synthesis windows one can show that any underspread system can be realized by such an STFT filter [6] This does not hold true for the Gabor filters as defined by (20) One ....

M.R. Portnoff. Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Analysis. IEEE Trans. Sign. Proc., 28:55--69, 1980.

....Group at the University of Warwick has developed a novel form of WT in which the link between scale and frequency is removed, and which provides a degree of shift invariance. Known as the Multiresolution Fourier Transform (MFT) 4] 5] this resembles a stack of windowed Fourier transforms (WFT) [6] in which the window size is varied systematically to give a multiresolution representation of the spacefrequency plane. As such, it constitutes a superset of the WT and WFT, providing a complete representation of the frequency domain at each scale and hence enabling regions to be analysed over a ....

....multiple scale approach [4] 5] When dealing with discrete signals, the three coordinate axes , and oe are sampled at appropriate intervals, the latter usually at powers of 2 in much the same way as the standard WT. The sampling in position and frequency follows that employed in a discrete WFT [6], with the intervals in each domain reflecting the extent of the window function and the requirement for a complete and invertible representation at each scale. For the discrete signal x( k ) its discrete MFT is therefore given by x( i (n) j (n) n) X k w n ( k Gamma i (n) x( k ) ....

[Article contains additional citation context not shown here]

M.R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis", IEEE Trans. Acous. Speech Sig. Proc., vol. ASSP-28, pp. 55-69, 1980.

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M. R. Portnoff. Time--frequency representation of digital signals and systems based on short--time fourier analysis. IEEE Trans. Acoust., Speech, Signal Processing., ASSP-28:55--69, Feb 1980.

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Portnoff, M. R. "Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Transform". IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-28:1, pp. 55--69, February 1980.

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M.R. Portnoff, "Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Analysis ", IEEE Transactions on ASSP, vol.ASSP-28(1): pp. 55-69, Feb 1980.

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M. R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-28, No. 1, Feb. 1980, pp.

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M. R. Portnoff, "Time-frequency representation of digital signals and systems based on short-time Fourier analysis," IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-28, No. 1, Feb. 1980, pp. 65-69.

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