Calculator Inputs
Example Data Table
Use these values to test a typical finance cycle signal.
| Scenario | Amplitude | Period | Duty Cycle | Harmonics | Use Case |
|---|---|---|---|---|---|
| Quarterly liquidity pulse | 500 | 4 | 50% | 11 | Cash stress pattern |
| Annual revenue regime | 2500 | 12 | 60% | 21 | Seasonal sales switch |
| Credit spread state | 125 | 6 | 40% | 15 | Risk-on versus risk-off |
Formula Used
The calculator uses a generalized rectangular square wave. It supports amplitude, duty cycle, phase shift, vertical shift, and harmonic count.
Here, A is amplitude. T is period. d is duty cycle as a decimal. φ is phase shift. C is vertical shift. N is the number of harmonics. When duty cycle is 50%, even harmonics cancel. The formula becomes the classic odd harmonic square wave series.
How to Use This Calculator
- Enter a finance signal name, such as revenue cycle or market regime.
- Set the amplitude to represent the positive and negative swing.
- Enter the period of one full cycle.
- Choose the duty cycle for the positive part of the square wave.
- Adjust phase shift when the cycle starts later or earlier.
- Add a vertical shift when the signal has a nonzero baseline.
- Select the number of harmonics. Higher values improve edge sharpness.
- Press calculate. Review the result, graph, table, CSV, and PDF report.
Fourier Series Square Waves in Finance
Why square waves matter
Finance data often changes by regimes. A market can move from calm to stressed. Revenue can switch from low season to high season. Liquidity can tighten near a reporting date. These shifts may look like square waves. A square wave is not smooth. It jumps between two levels. That makes it useful for modeling on and off states.
How Fourier series helps
A Fourier series rebuilds a wave using sine and cosine terms. Each term is a harmonic. The first harmonic gives the broad cycle. Later harmonics add sharper turning points. This calculator lets you control the harmonic count. A low count gives a smooth estimate. A high count gives a sharper square shape. Near jump points, small overshoots may appear. This behavior is normal. It is related to the Gibbs effect.
Finance interpretation
The amplitude can represent cash movement, spread width, index points, or profit swing. The period can represent months, weeks, quarters, or trading sessions. The duty cycle shows how long the signal remains in the positive state. A 50 percent duty cycle means equal high and low phases. A 60 percent duty cycle means the positive state lasts longer.
Using the results
The approximation value estimates the wave at your selected x value. The ideal value shows the exact square wave state. The error shows the difference. RMS error summarizes accuracy over the full graph range. Mean absolute error gives a simpler average gap. The chart helps compare the smooth Fourier approximation against the ideal step signal. Export the table when you need spreadsheet review. Export the PDF when you need a quick report. This tool is best for scenario analysis. It does not predict prices by itself. Use it with sound market logic and tested assumptions.
FAQs
1. What does this calculator measure?
It estimates a square wave with a Fourier series. It also compares the approximation with the ideal square wave and reports error values.
2. Why is this useful in finance?
Many finance signals switch between regimes. Examples include risk-on states, cash stress windows, seasonal sales periods, and spread widening phases.
3. What is amplitude?
Amplitude is the size of the wave swing. A larger amplitude creates a larger positive and negative movement around the baseline.
4. What does duty cycle mean?
Duty cycle is the share of each period spent in the positive state. A 50 percent duty cycle creates a balanced square wave.
5. Should I use many harmonics?
More harmonics make the approximation sharper. They also increase oscillation near jumps. Test several values before using the result in analysis.
6. Why does overshoot appear near jumps?
Fourier approximations can overshoot near discontinuities. This is a known behavior. It becomes narrower, but it may not fully disappear.
7. What is RMS error?
RMS error measures the average size of errors across the graph range. It gives more weight to larger errors.
8. Can this forecast markets?
No. It models periodic regime behavior. Use it for analysis, stress testing, and signal design, not as a standalone forecast method.