Discrete Fourier lab · RecallWeave

Offline · No automatic saving A lab plus a 12-question lesson.

1 Choose a sample grid

2 Keep the pieces you want

Frequency pairs included in reconstruction

Interior p keeps p and N−p together. DC and Nyquist each keep one bin.

Raw coefficient magnitudes |X[k]|. Green bins are retained; gray bins are omitted. A bin magnitude is not a sample amplitude.

3 Compare the reconstructed grid

RMS residual
Maximum real residual
Input energy Σx²
Full coefficient energy Σ|X|²/N

Entered samplesReconstruction

Only the N sample positions are shown. No connecting curve or unknown values between samples are inferred.

Selected reconstruction energy: . Maximum imaginary residual: . Error = reconstructed real value − entered sample.

Inspect every sample and residual

Original and reconstructed values on the finite grid
n	Entered	Reconstructed real	Error	Imaginary

Inspect coefficients and phase

Unscaled forward coefficients — displayed numbers are rounded
Bin k	Kept	Real	Imaginary	Magnitude	Phase (radians)

Includes the numerical samples, selected pairs, raw coefficients, residuals and energy values. Download only when you want a record.

Predict. Change. Explain.

Find the mean. Choose Constant. Predict DC, then remove it. Why does the reconstruction change from 3 to approximately 0?
Keep a pair. Enter 1, 0, -1, 0. Keep pair 1. How do two magnitudes of 2 rebuild samples of amplitude 1?
Find a singleton. Choose Alternating. Keep only Nyquist. Why does it need no second bin?
Explain the loss. Return to the mixture. Drop a pair and use the RMS residual to describe what the reconstruction leaves out.

One convention throughout

X[k] = Σ x[n] e^−i2πkn/N
y[n] = (1/N) Σ_{selected k} X[k] e^+i2πkn/N

The forward sum is unscaled. The inverse divides by N. For real samples, partners k and N−k are complex conjugates, apart from numerical roundoff.

What this can tell you

It compares a finite array with a reconstruction on that same grid. Bin k represents k/N cycles per sample before interpreting negative-frequency partners. No sampling rate is supplied, so the lab does not assign hertz or infer a continuous signal.

Continue in RecallWeave

Download the lesson deck, open the learner, and choose the JSON file in its lesson selector. The four concepts and 12 questions use the existing learning flow.