Fourier Synchrosqueezed Transform — Control Systems/Time Frequency

Fourier Synchrosqueezed Transform

Control Systems / Time Frequency

The synchrosqueezed short-time transform of a stream – MATLAB's fsst, one column per sample. The last L samples are transformed twice, with the window and with the window's derivative; the ratio gives every bin its own instantaneous frequency and the bin's contribution is moved there:

ν[m] = m − Im(D[m] ÷ X[m]),   S[round(ν) mod L] += X[m]·(−1)^m

so a ridge an ordinary spectrogram spreads over several bins comes back on one. The same rows, summed over a band and scaled, reconstruct the signal – that is MATLAB's ifsst, and it is the fourth output.

Ports

• u – the sampled signal. Scalar: one channel and its own window.
• Mag – |S|, the synchrosqueezed column, [L/2+1, 1], one-sided with DC first and Nyquist last. Row m is the normalized frequency m/L in cycles per sample, or m·f_s/L in hertz for a signal sampled at f_s. This is the map to plot, and the map Time-Frequency Ridges reads.
• Re – the real part of that same column, [L/2+1, 1]: the synchrosqueezed transform is complex, and a wire carries a real number, so its two halves ride two ports.
• Im – its imaginary part, same size, same rows.
• xr – [1, 1], the signal reconstructed from the rows inside Reconstruction Band – the sample at the published column's own time, which is L/2 − 1 samples behind the input (see Frames and timing). Over the whole band it is the input itself, delayed by that much.

Parameters

• Window Length – L, the window and the transform length: an even whole number from 4 to 20. Even because a one-sided spectrum needs a Nyquist bin, and because the transform's phase shift onto the window centre is then exactly (−1)^m. Bounded above because the four transforms are unrolled at export: a column costs 4·L² inlined products.
• Window – the analysis window, the symmetric shapes MATLAB's kaiser(), hann() and hamming() return:
  • Kaiser – fsst's own default, shaped by Kaiser Beta.
  • Hann – two end taps of exactly zero.
  • Hamming – end taps of 0.08.
  The window's derivative comes from a not-a-knot cubic spline through its taps, which is what MATLAB differentiates, and it decides every instantaneous frequency the block computes.
• Kaiser Beta – β, the Kaiser shape parameter, a scalar of zero or more (MATLAB's fsst uses 10). Read only by the Kaiser window; a larger value means a wider main lobe and lower side lobes.
• Reconstruction Band – the rows xr is summed over, as [low high] in normalized frequency (cycles per sample, 0 to 0.5), so the default [0 0.5] is the whole spectrum whatever L is. Each edge snaps to the nearest row – MATLAB's rule – and a tie keeps the lower one. The band is read by xr only; Mag, Re and Im always carry every row.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Frames and timing

fsst fixes its overlap at L − 1 samples, so it has one column per input sample and pads the record with L/2 zeros in front. Column k therefore reads samples k − L/2 to k + L/2 − 1, and the newest column a stream can finish at sample n is k = n − L/2 + 1: the four outputs run L/2 − 1 samples behind the input, and they are all zero until the register first fills at sample L − 1. The leading zeros of the register are exactly MATLAB's leading zero padding, so the columns agree from the first one published.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The window, its spline derivative and the transform's basis are fixed by the configuration and inlined as numbers, so no generated core computes a sine, a cosine or a Bessel function; the reassignment's rounding is a bounded count and its range reduction a bounded halving loop, so no core needs a floor either (PLC Structured Text has none). All four settings are structural – re-export after changing one.

The three HDL targets carry the transform in real arithmetic and are therefore simulation-only: the instantaneous frequency is a division by the transform's own squared magnitude, which leaves Q16.16 in both directions. The window is held in the port's fixed-point format, so quantization happens only at the port boundary.

⚠ The destination row is a ROUNDING, so it is a discrete answer. A bin whose instantaneous frequency sits within a quantum of a half-row boundary can be reassigned one row apart in fixed point, and that row then differs by the whole of that bin's contribution. It is arithmetic and not a code-generation defect – the same algorithm on different numbers – and it is why a fixed-point comparison of this block should be read as a spread rather than a single figure. Feeding it a narrowband signal, whose bins all agree about one frequency, keeps every decision far from its boundary.

Simulink bridge

None (Support::None). fsst is a Signal Processing Toolbox function and that toolbox ships no Simulink library, so there is no path a diagram could name. The bridge reports this block rather than dropping it silently, and it therefore has no parity testbench; code export verification still covers it across all ten languages. Sampling Time (s) → SampleTime, as on every block, has nothing to cross to for the same reason.

Notes

• Stateful, and discrete by nature: a register of the last L samples and the column published from it.
• The reassignment runs over the whole two-sided spectrum before the one-sided half is taken, exactly as fsst does, so a bin from the negative half whose frequency wraps into the positive one still contributes. That is also why the rows are not doubled.
• Verified against R2026a: the whole complex matrix to 1.96e-14 on a scale of 10.68, the full-band reconstruction to 8.9e-16 and a band reconstruction to 4.4e-16 (the source banner carries what was checked).
• Not reproduced: a window vector given tap by tap, an odd window length, a complex input (fsst then returns a centred two-sided spectrum), and ifsst's reconstruction along a ridge with NumFrequencyBins – that one needs a ridge index per column, which is a second input this block does not have.
• It is not Spectrogram. Spectrogram publishes |X| every hop samples and holds in between; this block publishes a column every sample and moves each bin's energy to where its instantaneous frequency says it belongs, which is what makes a ridge sharp enough to follow.
• No state space. The instantaneous frequency is a ratio of two transforms, so the block is not linear in its input and model reduction correctly declines to merge it.