Introduction to Time-Frequency Analysis and Wavelets

Editor's Notes

• #3: Synchrosqueezing was introduced in the context of analyzing auditory signals [13]; it is a special case of reallocation methods [14–16], which aim to “sharpen” a time-frequency representation R(t,ω) by “allocating” its value to a different point (t,ω) in the time–frequency plane, determined by the local behavior of (time-freq function)R(t,ω) around (t,ω). In the case of synchrosqueezing, one starts from the continuous wavelet transform Ws of the signal s as defined. where ψ is an appropriately chosen wavelet, and reallocates the Ws(a, b) to get a concentrated time-frequency picture, from which instantaneous frequency lines can be extracted.
• #4: If ˆ ψ(ξ) is concentrated around ξ = ω0, then Ws(a, b) will be concentrated around a = ω0/ω. However, the wavelet transform Ws(a, b) will be spread out over a region around the horizontal line a = ω0/ω on the time–scale plane. although Ws(a, b) is spread out in a, its oscillatory behavior in b points to the original frequency ω, regardless of the value of a. For the purely harmonic signal s(t) = A cos(ωt), one obtains ωs(a, b) = ω, as desired, this is illustrated in Fig. Fig. 5. Left: the harmonic signal f (t) = sin(8t); middle: the continuous wavelet transform of f ; right: synchrosqueezed transform of f .
• #5: 5. For simplicity, we will suppress the dependence on s and denote ω(a, b) = ωs(a, b). In a next step, the information from the time–scale plane is transferred to the time–frequency plane, according to the map (b,a)→(b,ωs(a, b)), in an operation dubbed synchrosqueezing. the frequency variable ω and the scale variable a were “binned”, i.e. Ws(a, b) was computed only at discrete values ak. and its synchrosqueezed transform Ts(ω, b) was likewise determined only at the centers ω of the successive bins [ω − 12ω,ω + 12ω], withω −ω−1 = ω, by summing different contributions:
• #6: The following argument shows that the signal can still be reconstructed after the synchrosqueezing. We have Setting Cψ = 12 ∞0ˆ ψ(ξ)dξξ , we then obtain (assuming that s is real, so that ˆs (ξ ) = ˆs (−ξ), hence s(b) = π−1Re[∞0ˆs(ξ )×eibξ dξ ]) In the piecewise constant approximation corresponding to the binning in a, this becomes
• #7: Examples of linear time-frequency representations. Assume for example.
• #8: Left: the toy signal used for Figs. 1 and 2; middle: its instantaneous frequency; right: the result of synchrosqueezing for this signal. The “extra” component at very low frequency is due to the signal’s not being centered around 0.
• #12: Another example Far left: the instantaneous frequencies, ω(t) = 2t +1 −sin(t) and 8, of the two IMT components of the crossover signal f (t) = cos[t2 +t +cos(t)] +cos(8t); middle left: plot of f (t) with no noise added; middle: synchrosqueezed wavelet transforms of noiseless f (t); middle right: f (t) + noise (correspondingto SNR of 6.45 dB); far right: synchrosqueezed wavelet transforms of f (t) +noise.
• #13: where Cψ, C ψ are constants depending only on ψ.
• #14: Comparing the decomposition into components s1(t) and s2(t) of the crossover signal s(t) = s1(t) + s2(t) = cos(8t) + cos[t2 + t + cos(t)]. Rows: noise-free situation in the first two rows; In each case, s1 is in the top row, and s2 underneath. far left: true s j (t) j = 1, 2; middle left: zone marked on the synchrosqueezed transform for reconstruction of the component; center: part of the synchrosqueezed transform singled out for the reconstruction of a putative s j ; middle right: the corresponding candidate s j (t) according to the synchrosqueezed transform (plotted in blue over the original s j , in red);
• #15: Comparing the decomposition into components s1(t) and s2(t) of the crossover signal s(t) = s1(t) + s2(t) = cos(8t) + cos[t2 + t + cos(t)]. in the last two rows noise with SNR = 6.45 dB was added to the mixed signal. In each case, s1 is in the top row,and s2 underneath. far left: true s j (t) j = 1, 2; middle left: zone marked on the synchrosqueezed transform for reconstruction of the component; center: part of the synchrosqueezed transform singled out for the reconstruction of a putative s j ; middle right: the corresponding candidate s j (t) according to the synchrosqueezed transform (plotted in blue over the original s j , in red);