Cross Spectrogram — Control Systems/Time Frequency

Cross Spectrogram

Control Systems / Time Frequency

The cross spectrum of two streams, frame by frame – MATLAB's xspectrogram, one column at a time:

C_j[m] = |X_j[m]·Y_j*[m]| · scale[m]

where X_j and Y_j are the windowed one-sided transforms of the same L-sample frame of each input. A bin is large where the two signals share energy at that frequency in that frame, which is what separates a common component from two unrelated ones of the same size.

Ports

• x – the first signal, scalar.
• y – the second signal, scalar. The product conjugates y's transform; the published magnitude is the same either way round.
• C – the current frame, [L/2+1, 1], one-sided with DC first and Nyquist last. It changes only on a frame sample and holds in between.

Parameters

• Window Length – L, the frame and transform length: an even whole number from 4 to 32. Even because a one-sided spectrum needs a Nyquist bin; bounded above because the transform is unrolled at export and its cost grows as L².
• Hop Size – the samples from one frame to the next, 1 to L. MATLAB's noverlap is L − hop.
• Window – Hamming (MATLAB's default), Hann or Rectangular: the symmetric windows MATLAB's hamming() and hann() return, so hamming(8) starts at 0.08.
• Spectrum Type – the scaling of the cross spectrum before its magnitude is taken:
  • Power spectral density – ÷ (f_s·Σw²), every bin but DC and Nyquist doubled: xspectrogram's default.
  • Power spectrum – ÷ (Σw)², doubled the same way: 'power'.
• Bin Spacing – Δf, the frequency step between output bins in hertz; the sampling rate follows as f_s = L·Δf. It is read only by the power-spectral-density scaling, and it is the same parameter under the same name as Welch PSD's.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Frames and timing

A frame is the last L samples, oldest first. The first is taken when the window first fills – at sample L − 1, counting from 0 – and then every hop samples, so the output at sample L − 1 + j·hop is frame j: column j of MATLAB's xspectrogram(x, y, window, L−hop, L). Between frames the output holds the last one, because a wire holds; before the first frame it is all zeros.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. The window, the basis and the scale are inlined as numbers, so no generated core computes a sine, a cosine or a division; all five settings are structural, so re-export after changing one.

The three HDL targets form the transforms and the squares in genuine Q16.16; the magnitude of the cross product – a square root of a sum of squares of products – goes through real (simulation-only), exactly as Quaternion Modulus's root does. Scale the signal into roughly ±1 first: a frame sums L windowed samples and the block then squares or multiplies.

Simulink bridge

None (Support::None). xspectrogram 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.

Notes

• Stateful, and discrete by nature: two registers of L samples, a frame counter and the held frame.
• ⚠ It publishes the MAGNITUDE. xspectrogram returns a real matrix – |P| of the scaled complex cross spectrum – and this block matches it, so a negative product in the Nyquist bin comes out positive. The phase between the two signals is not on the port.
• Verified against R2026a: for two twenty-sample records at L = 8, hop = 3, f_s = 100, both scalings match MATLAB's to the last printed digit, and xspectrogram(x, x) is exactly Spectrogram's power spectral density.
• No state space. The product of two transforms is not linear.