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The\ninstantaneous frequency can be obtained by differentiating the\ninstantaneous phase in respect to time. The instantaneous phase corresponds\nto the phase angle of the analytic signal.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "analytic_signal = hilbert(signal)\namplitude_envelope = np.abs(analytic_signal)\ninstantaneous_phase = np.unwrap(np.angle(analytic_signal))\ninstantaneous_frequency = np.diff(instantaneous_phase) / (2.0*np.pi) * fs\nfig, (ax0, ax1) = plt.subplots(nrows=2, sharex='all', tight_layout=True)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ax0.set_title(\"Amplitude-modulated Chirp Signal\")\nax0.set_ylabel(\"Amplitude\")\nax0.plot(t, signal, label='Signal')\nax0.plot(t, amplitude_envelope, label='Envelope')\nax0.legend()\nax1.set(xlabel=\"Time in seconds\", ylabel=\"Frequency in Hz\", ylim=(0, 120))\nax1.plot(t[1:], instantaneous_frequency, 'C2-',\n label='Instantaneous Frequency')\nax1.legend()\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-3caaca2beb3db00e.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "envelope", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.signal._signaltools:envelope" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Compute envelope of a real- or complex-valued signal." } ] } ], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "x", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "N", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "-1" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "hilbert" }, "references": [ ".. [1] Wikipedia, \"Analytic signal\".", " https://en.wikipedia.org/wiki/Analytic_signal", ".. [2] Wikipedia, \"Hilbert Transform\".", " https://en.wikipedia.org/wiki/Hilbert_transform", ".. [3] Leon Cohen, \"Time-Frequency Analysis\", 1995. Chapter 2.", ".. [4] Alan V. Oppenheim, Ronald W. Schafer. Discrete-Time Signal", " Processing, Third Edition, 2009. Chapter 12.", " ISBN 13: 978-1292-02572-8" ], "qa": "scipy.signal._signaltools:hilbert", "arbitrary": [], "local_refs": [ "N", "axis", "x", "xa" ] }