# Hilbert Transform SineWave

`D09-F06-A04` · Statistical Time Series → Hilbert and Ehlers Cycle Analytics · archetype `record-transform` · difficulty 2/5 · verification **verified**

Full page: https://docs.thefintechbuilder.com/statistical-time-series/hilbert-and-ehlers-cycle-analytics/hilbert-transform-sinewave/
Agent skill: `npx skills add IslamBaraka90/Fintech-Algorithms-Library` — https://docs.thefintechbuilder.com/guides/agent-skill/

## Install and import

```bash
npm install fintech-algorithms
```

```ts
import { hilbertTransformSinewave } from "fintech-algorithms/statistical-time-series/hilbert-and-ehlers-cycle-analytics/hilbert-transform-sinewave";
```

## Signature

```ts
hilbertTransformSinewave(input)
```

Turns the Hilbert phase angle into the classic sinewave pair, the sine of the phase and a lead sine shifted a quarter turn ahead, whose crossings are the usual cycle-turn signal. Any contract failure is rethrown as an `Error` whose message begins `topic calculation failed: `.

## Parameters

| Name | Type | Required | Notes |
| --- | --- | --- | --- |
| `input` | `TopicInput` | yes | `bars` is a non-empty array of rows with `timestamp`, `open`, `high`, `low`, `close` and `volume`, strictly ordered by timestamp. The cycles family reads one key from `parameters`, `period`, defaulting to 20 and required to be an integer of at least 4; it is validated for this topic but its arithmetic never uses it. |

## Returns

`TopicResult`

Two series, `sine` and `lead_sine`, mirrored as `latest.sine` and `latest.lead_sine`. `sine` is the sine of the phase and `lead_sine` is the sine of the phase advanced by a quarter turn, so both stay within -1 to 1. `parameters` echoes the supplied parameters and `diagnostics` reports `causal` and `input_count`. Warm-up is six leading nulls in both series, so `ready_at` is 6.

## Warm-up

The first `6` positions are `null`. Both series are functions of the phase, which needs the 4-bar EMA of `close` to resolve at index 3 and three further bars for the quadrature filter, so they begin at index 6.

## Errors

- When `parameters.period` is present but is not an integer of at least 4 — throws Error
- When `bars` is empty, a `timestamp` is missing or not strictly increasing, an OHLCV field is not finite, `volume` is negative, or `high` and `low` do not bracket `open` and `close` — throws Error

## Complexity

Time `O(n)`, space `O(n)`.

## Worked example

Captured by running this function on the input its own test provides. Real output of real code — but not asserted against a published figure.

### Input

`input`:

```json
{
  "bars": [
    {
      "timestamp": "2024-01-02",
      "basis": "synthetic-unadjusted",
      "open": 100,
      "high": 101.45,
      "low": 98.695,
      "close": 100,
      "volume": 750000,
      "benchmark": 200
    },
    {
      "timestamp": "2024-01-03",
      "basis": "synthetic-unadjusted",
      "open": 101.49111452,
      "high": 103.38381693,
      "low": 100.05480022,
      "close": 101.78791214,
      "volume": 795117,
      "benchmark": 200.56326135
    },
    {
      "timestamp": "2024-01-04",
      "basis": "synthetic-unadjusted",
      "open": 102.45519048,
      "high": 104.6701838,
      "low": 100.91147007,
      "close": 102.9549389,
      "volume": 840234,
      "benchmark": 201.11020913
    }
  ],
  "parameters": {}
}
```

### Call

```ts
hilbertTransformSinewave(input)
```

### Returns

object with 9 fields: topic_id, title, state, ready, ready_at, series, latest, parameters, …

```json
{
  "topic_id": "D09-F06-A04",
  "title": "Hilbert Transform SineWave",
  "state": "calculated",
  "ready": true,
  "ready_at": 6,
  "series": {
    "sine": [null, null, null, null, null, null],
    "lead_sine": [null, null, null, null, null, null]
  },
  "latest": {
    "sine": 0.19131916990169168,
    "lead_sine": -0.5587619360262619
  },
  "parameters": {},
  "diagnostics": {
    "causal": true,
    "input_count": 96
  }
}
```

## Verification and provenance

Tier: **verified** (via E).

The worked example below is the figure published in this algorithm's article, replayed and asserted by the test suite on every build. The arithmetic cannot drift without the build failing.

Both tiers guarantee the signature. Full explanation: https://docs.thefintechbuilder.com/guides/verification/

Generated from the docs.json payload shipped inside fintech-algorithms@0.13.0.
The signature and parameter list are checked against the compiled implementation at build time,
so a description that contradicts the code fails the build rather than reaching this file.

## Links

- Article (how it works, step by step): https://thefintechbuilder.com/statistical-time-series/hilbert-and-ehlers-cycle-analytics/hilbert-transform-sinewave/
- Implementation source: https://github.com/IslamBaraka90/Fintech-Algorithms-Library/blob/main/src/statistical-time-series/hilbert-and-ehlers-cycle-analytics/hilbert-transform-sinewave/impl.ts
- Package on npm: https://www.npmjs.com/package/fintech-algorithms
- Domain index for agents: https://docs.thefintechbuilder.com/statistical-time-series/llms.txt