Reassignment method

The method of reassignment is a technique for sharpening a time-frequency representation (e.g. spectrogram or the short-time Fourier transform) by mapping the data to time-frequency coordinates that are nearer to the true region of support of the analyzed signal. The method has been independently introduced by several parties under various names, including method of reassignment, remapping, time-frequency reassignment, and modified moving-window method. The method of reassignment sharpens blurry time-frequency data by relocating the data according to local estimates of instantaneous frequency and group delay. This mapping to reassigned time-frequency coordinates is very precise for signals that are separable in time and frequency with respect to the analysis window.

The method of reassignment is a technique for sharpening a time-frequency representation (e.g. spectrogram or the short-time Fourier transform) by mapping the data to time-frequency coordinates that are nearer to the true region of support of the analyzed signal. The method has been independently introduced by several parties under various names, including method of reassignment, remapping, time-frequency reassignment, and modified moving-window method. The method of reassignment sharpens blurry time-frequency data by relocating the data according to local estimates of instantaneous frequency and group delay. This mapping to reassigned time-frequency coordinates is very precise for signals that are separable in time and frequency with respect to the analysis window. h_{\omega}(t) = h(t) e^{j \omega t} where h(t) is a (real-valued) lowpass kernel function, like the window function in the short-time Fourier transform. The coefficients in this decomposition are defined \begin{align} \epsilon( t, \omega ) &= \int x(\tau) h( t - \tau ) e^{ -j \omega \left[ \tau - t \right]} d\tau \\ &= e^{ j \omega t} \int x(\tau) h( t - \tau ) e^{ -j \omega \tau } d\tau \\ &= e^{ j \omega t} X(t, \omega) \\ &= X_{t}(\omega) \\ &= M_{t}(\omega) e^{j \phi_{\tau}(\omega)} \end{align} where M_{t}(\omega) is the magnitude, and \phi_{\tau}(\omega) the phase, of X_{t}(\omega) , the Fourier transform of the signal x(t) shifted in time by t and windowed by h(t) . x(t) can be reconstructed from the moving window coefficients by \begin{align} x(t) & = \iint X_{\tau}(\omega) h^{*}_{\omega}(\tau - t) d\omega d\tau \\ & = \iint X_{\tau}(\omega) h( \tau - t ) e^{ -j \omega \left[ \tau - t \right]} d\omega d\tau \\ &= \iint M_{\tau}(\omega) e^{j \phi_{\tau}(\omega)} h( \tau - t ) e^{ -j \omega \left[ \tau - t \right]} d\omega d\tau \\ &= \iint M_{\tau}(\omega) h( \tau - t ) e^{ j \left[ \phi_{\tau}(\omega) - \omega \tau+ \omega t \right] } d\omega d\tau \end{align} For signals having magnitude spectra, M(t,\omega) , whose time variation is slow relative to the phase variation, the maximum contribution to the reconstruction integral comes from the vicinity of the point t,\omega satisfying the phase stationarity condition \begin{align} \frac{\partial}{\partial \omega} \left[ \phi_{\tau}(\omega) - \omega \tau + \omega t\right] & = 0 \\ \frac{\partial}{\partial \tau} \left[ \phi_{\tau}(\omega) - \omega \tau + \omega t \right] & = 0 \end{align} or equivalently, around the point \hat{t}, \hat{\omega} defined by \begin{align} \hat{t}(\tau, \omega) & = \tau - \frac{\partial \phi_{\tau}(\omega)}{\partial \omega} = -\frac{\partial \phi(\tau, \omega)}{\partial \omega} \\ \hat{\omega}(\tau, \omega) & = \frac{\partial \phi_{\tau}(\omega)}{\partial \tau} = \omega + \frac{\partial \phi(\tau, \omega)}{\partial \tau} \end{align} This phenomenon is known in such fields as optics as the principle of stationary phase, which states that for periodic or quasi-periodic signals, the variation of the Fourier phase spectrum not attributable to periodic oscillation is slow with respect to time in the vicinity of the frequency of oscillation, and in surrounding regions the variation is relatively rapid. Analogously, for impulsive signals, that are concentrated in time, the variation of the phase spectrum is slow with respect to frequency near the time of the impulse, and in surrounding regions the variation is relatively rapid. In reconstruction, positive and negative contributions to the synthesized waveform cancel, due to destructive interference, in frequency regions of rapid phase variation. Only regions of slow phase variation (stationary phase) will contribute significantly to the reconstruction, and the maximum contribution (center of gravity) occurs at the point where the phase is changing most slowly with respect to time and frequency. The time-frequency coordinates thus computed are equal to the local group delay, \hat{t}_{g}(t,\omega), and local instantaneous frequency, \hat{\omega}_{i}(t,\omega), and are computed from the phase of the short-time Fourier transform, which is normally ignored when constructing the spectrogram. These quantities are local in the sense that they represent a windowed and filtered signal that is localized in time and frequency, and are not global properties of the signal under analysis. The modified moving window method, or method of reassignment, changes (reassigns) the point of attribution of \epsilon(t,\omega) to this point of maximum contribution \hat{t}(t,\omega), \hat{\omega}(t,\omega) , rather than to the point t,\omega at which it is computed. This point is sometimes called the center of gravity of the distribution, by way of analogy to a mass distribution. This analogy is a useful reminder that the attribution of spectral energy to the center of gravity of its distribution only makes sense when there is energy to attribute, so the method of reassignment has no meaning at points where the spectrogram is zero-valued. \hat{t} (t,\omega) &= t - \frac{\iint \tau \cdot W_{x}(t-\tau,\omega -\nu) \cdot \Phi(\tau,\nu) d\tau d\nu} {\iint W_{x} \left (t-\tau,\omega -\nu \right ) \cdot \Phi (\tau,\nu) d\tau d\nu } \\ \hat{\omega} (t,\omega) & = \omega - \frac{\iint \nu \cdot W_{x}(t-\tau,\omega -\nu) \cdot \Phi(\tau,\nu) d\tau d\nu} {\iint W_{x}(t-\tau,\omega -\nu) \cdot \Phi(\tau,\nu) d\tau d\nu} \end{align} where W_{x}(t,\omega) is the Wigner–Ville distribution of x(t) , and \Phi(t,\omega) is the kernel function that defines the distribution. They further described an efficient method for computing the times and frequencies for the reassigned spectrogram efficiently and accurately without explicitly computing the partial derivatives of phase. In the case of the spectrogram, the reassignment operations can be computed by \begin{align} \hat{t} (t,\omega) & = t - \Re \left \{ \frac{ X_{\mathcal{T}h}(t,\omega) \cdot X^*(t,\omega) }{ | X(t,\omega) |^2 } \right \} \\ \hat{\omega}(t,\omega) & = \omega + \Im \left \{ \frac{ X_{\mathcal{D}h}(t,\omega) \cdot X^*(t,\omega) }{ | X(t,\omega) |^2 } \right \} \end{align} where X(t,\omega) is the short-time Fourier transform computed using an analysis window h(t), X_{\mathcal{T}h}(t,\omega) is the short-time Fourier transform computed using a time-weighted analysis window h_{\mathcal{T}}(t) = t \cdot h(t) and X_{\mathcal{D}h}(t,\omega) is the short-time Fourier transform computed using a time-derivative analysis window h_{\mathcal{D}}(t) = \tfrac{d}{dt}h(t) . Using the auxiliary window functions h_{\mathcal{T}}(t) and h_{\mathcal{D}}(t) , the reassignment operations can be computed at any time-frequency coordinate t,\omega from an algebraic combination of three Fourier transforms evaluated at t,\omega . Since these algorithms operate only on short-time spectral data evaluated at a single time and frequency, and do not explicitly compute any derivatives, this gives an efficient method of computing the reassigned discrete short-time Fourier transform. One constraint in this method of computation is that the | X(t,\omega) |^2 must be non-zero. This is not much of a restriction, since the reassignment operation itself implies that there is some energy to reassign, and has no meaning when the distribution is zero-valued. ==Separability== The short-time Fourier transform can often be used to estimate the amplitudes and phases of the individual components in a multi-component signal, such as a quasi-harmonic musical instrument tone. Moreover, the time and frequency reassignment operations can be used to sharpen the representation by attributing the spectral energy reported by the short-time Fourier transform to the point that is the local center of gravity of the complex energy distribution. For a signal consisting of a single component, the instantaneous frequency can be estimated from the partial derivatives of phase of any short-time Fourier transform channel that passes the component. If the signal is to be decomposed into many components, x(t) = \sum_{n} A_{n}(t) e^{j \theta_{n}(t)} and the instantaneous frequency of each component is defined as the derivative of its phase with respect to time, that is, \omega_{n}(t) = \frac{d \theta_{n}(t)}{d t}, then the instantaneous frequency of each individual component can be computed from the phase of the response of a filter that passes that component, provided that no more than one component lies in the passband of the filter. This is the property, in the frequency domain, that Nelson called separability and is required of all signals so analyzed. If this property is not met, then the desired multi-component decomposition cannot be achieved, because the parameters of individual components cannot be estimated from the short-time Fourier transform. In such cases, a different analysis window must be chosen so that the separability criterion is satisfied. If the components of a signal are separable in frequency with respect to a particular short-time spectral analysis window, then the output of each short-time Fourier transform filter is a filtered version of, at most, a single dominant (having significant energy) component, and so the derivative, with respect to time, of the phase of the X(t,\omega_0) is equal to the derivative with respect to time, of the phase of the dominant component at \omega_0. Therefore, if a component, x_n(t), having instantaneous frequency \omega_{n}(t) is the dominant component in the vicinity of \omega_0, then the instantaneous frequency of that component can be computed from the phase of the short-time Fourier transform evaluated at \omega_0. That is, \begin{align} \omega_{n}(t) &= \frac{\partial}{\partial t} \arg\{ x_{n}(t) \} \\ &= \frac{\partial }{\partial t} \arg\{ X(t,\omega_{0}) \} \end{align} Just as each bandpass filter in the short-time Fourier transform filterbank may pass at most a single complex exponential component, two temporal events must be sufficiently separated in time that they do not lie in the same windowed segment of the input signal. This is the property of separability in the time domain, and is equivalent to requiring that the time between two events be greater than the length of the impulse response of the short-time Fourier transform filters, the span of non-zero samples in h(t). Long-window reassigned spectrogram of speech.png|Long-window reassigned spectrogram of the word "open", computed using a 54.4 ms Kaiser window with a shaping parameter of 9, emphasizing harmonics. Short-window reassigned spectrogram of speech.png|Short-window reassigned spectrogram of the word "open", computed using a 13.6 ms Kaiser window with a shaping parameter of 9, emphasizing formants and glottal pulses. In general, there is an infinite number of equally valid decompositions for a multi-component signal. The separability property must be considered in the context of the desired decomposition. For example, in the analysis of a speech signal, an analysis window that is long relative to the time between glottal pulses is sufficient to separate harmonics, but the individual glottal pulses will be smeared, because many pulses are covered by each window (that is, the individual pulses are not separable, in time, by the chosen analysis window). An analysis window that is much shorter than the time between glottal pulses may resolve the glottal pulses, because no window spans more than one pulse, but the harmonic frequencies are smeared together, because the main lobe of the analysis window spectrum is wider than the spacing between the harmonics (that is, the harmonics are not separable, in frequency, by the chosen analysis window).

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