Fig. 2: The principle behind superlets and redundancy of spectral representation. | Nature Communications

Fig. 2: The principle behind superlets and redundancy of spectral representation.

From: Time-frequency super-resolution with superlets

Fig. 2

a Test setup where a target oscillation packet, T, is contaminated by frequency and time neighbors (NF and NT). Top: using a wavelet with small number of cycles enables good time separation but has poor frequency resolution (red), whereas wavelets with many cycles enable good frequency separation but suffer from temporal contamination (blue). Bottom: a particular instantiation with packets of seven cycles having: target frequency 50 Hz, neighbor frequency 70 Hz, neighbor time offset ten cycles. b Target contamination in frequency by NF (top) and in time by NT (bottom). Contamination is measured as the normalized response (magnitude) of a single wavelet (c = 3) or a multiplicative superlet (c1 = 3; o = 5) at the time-frequency location of the target (without the target being present) to NF with various frequencies (top) or NT with various time offsets (bottom). c Frequency (top) and time (bottom) superlet resolution measured as the half-width of the frequency and time peak in b, respectively, as a function of the order of a multiplicative superlet (line). The same is shown for the longest wavelet in the superlet set (dotted line). The frequency resolution limit is the Rayleigh frequency of T with Gaussian windowing. The temporal resolution limit is half the size of T (3.5 cycles). d A long signal composed of 3 summed unitary amplitude sine waves has an average power of 1.5 (green). Two superlet transforms (SLT) using multiplicative superlets with c1 = 3 (blue) and 5 (red) give an increasingly sharper representation of the higher frequencies, as their order is increased. Sharper representations signify less redundancy. Insets show the time-collapsed power spectra computed using the CWT (SLT of order 1), SLT, and Welch for the corresponding marked points on the average power traces.

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