Enter Frequency-Domain Data
Use one row per bin in the format: k, real, imag. Missing bins can be zero-filled automatically.
Example Data Table
This example reconstructs the time sequence x[n] = [1, 0, -1, 0] using a four-point inverse transform.
| k | Real{X[k]} | Imag{X[k]} | Meaning |
|---|---|---|---|
| 0 | 0 | 0 | Zero DC contribution |
| 1 | 2 | 0 | Positive cosine content |
| 2 | 0 | 0 | No Nyquist component |
| 3 | 2 | 0 | Mirror of k = 1 |
Formula Used
The inverse discrete Fourier transform converts the frequency-domain sequence X[k] into the time-domain sequence x[n].
x[n] = (1 / N) Σk=0N-1 X[k] ej2πkn/N, for n = 0, 1, ..., N-1
When X[k] = ak + jbk, the calculator expands the complex exponential into cosine and sine terms.
Re{x[n]} = factor × Σ [ak cos(2πkn/N) - bk sin(2πkn/N)]
Im{x[n]} = factor × Σ [ak sin(2πkn/N) + bk cos(2πkn/N)]
The factor equals 1/N for the standard inverse transform. If you choose no scaling, the calculator uses factor = 1.
How to Use This Calculator
- Enter each frequency bin on its own line as k, real, imag.
- Set the sequence length N, or leave it blank for auto-detection.
- Optionally enter the sampling frequency to show a time axis in seconds.
- Choose the inverse scaling rule and missing-bin handling method.
- Click Compute IDFT to reconstruct x[n].
- Review the result summary, output table, and Plotly graph.
- Use the export buttons to download CSV or PDF files.
- Use the example loader to test the calculator instantly.
Frequently Asked Questions
1) What does this IDFT calculator compute?
It reconstructs a time-domain sequence from complex frequency-domain bins. The tool returns real and imaginary samples, magnitude, phase, summary metrics, and a graph for quick verification.
2) What input format should I use?
Use one line per bin with three values: k, real part, and imaginary part. Commas, spaces, semicolons, or tabs are accepted as separators.
3) What happens if some bins are missing?
You can either zero-fill them automatically or force the calculator to reject incomplete spectra. Zero-filling is useful when unspecified bins are known to be zero.
4) Why is the output sometimes complex?
If the input spectrum does not satisfy Hermitian symmetry, the reconstructed sequence can contain nonzero imaginary parts. Real signals usually require mirrored complex-conjugate bins.
5) What does the scaling option change?
Standard inverse scaling divides the summed result by N. Choosing no scaling leaves the sequence unscaled, which may match custom textbook or software conventions.
6) Why is sampling frequency optional?
The inverse transform only needs N and X[k]. Sampling frequency is optional because it only affects the horizontal axis label, converting sample index into physical time.
7) What does the energy check tell me?
It helps compare time-domain energy with the frequency-domain equivalent under the standard inverse scaling convention. Matching values can confirm consistent transform handling.
8) Can I export the reconstructed results?
Yes. After computing the transform, you can download a CSV table for spreadsheets or a PDF report for sharing, printing, or documentation.