{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:3UOQQW3LLSXRTWZPAEHCM4RHQJ","short_pith_number":"pith:3UOQQW3L","schema_version":"1.0","canonical_sha256":"dd1d085b6b5caf19db2f010e2672278271b0fe15795246749c716cce08062e25","source":{"kind":"arxiv","id":"1208.1363","version":1},"attestation_state":"computed","paper":{"title":"Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Nicolas Le Bihan, Stephen J. Sangwine, Todd A. Ell","submitted_at":"2012-08-07T08:32:56Z","abstract_excerpt":"The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1208.1363","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-08-07T08:32:56Z","cross_cats_sorted":[],"title_canon_sha256":"26f1a18183b0c256dc2bf1085126cbb3fc500cc31a36df23020ebd5b5ddbdee5","abstract_canon_sha256":"d6a1d6c63c40cb118766897a45eef8f1ca68ad7aa0b6f884ff6474c30782db71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:24.265994Z","signature_b64":"1NjQ7qG65RxzmWveQjStZ6Yern3yDYJWNHGbXjooEhut5Rjuxka3JUFYtDzAxUjzO4hYnXo8w8E75OrQJWySDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd1d085b6b5caf19db2f010e2672278271b0fe15795246749c716cce08062e25","last_reissued_at":"2026-05-18T02:57:24.265280Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:24.265280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Nicolas Le Bihan, Stephen J. Sangwine, Todd A. Ell","submitted_at":"2012-08-07T08:32:56Z","abstract_excerpt":"The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1363","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1208.1363","created_at":"2026-05-18T02:57:24.265417+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.1363v1","created_at":"2026-05-18T02:57:24.265417+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.1363","created_at":"2026-05-18T02:57:24.265417+00:00"},{"alias_kind":"pith_short_12","alias_value":"3UOQQW3LLSXR","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"3UOQQW3LLSXRTWZP","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"3UOQQW3L","created_at":"2026-05-18T12:26:53.410803+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ","json":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ.json","graph_json":"https://pith.science/api/pith-number/3UOQQW3LLSXRTWZPAEHCM4RHQJ/graph.json","events_json":"https://pith.science/api/pith-number/3UOQQW3LLSXRTWZPAEHCM4RHQJ/events.json","paper":"https://pith.science/paper/3UOQQW3L"},"agent_actions":{"view_html":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ","download_json":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ.json","view_paper":"https://pith.science/paper/3UOQQW3L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.1363&json=true","fetch_graph":"https://pith.science/api/pith-number/3UOQQW3LLSXRTWZPAEHCM4RHQJ/graph.json","fetch_events":"https://pith.science/api/pith-number/3UOQQW3LLSXRTWZPAEHCM4RHQJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ/action/storage_attestation","attest_author":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ/action/author_attestation","sign_citation":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ/action/citation_signature","submit_replication":"https://pith.science/pith/3UOQQW3LLSXRTWZPAEHCM4RHQJ/action/replication_record"}},"created_at":"2026-05-18T02:57:24.265417+00:00","updated_at":"2026-05-18T02:57:24.265417+00:00"}