Inverse Discrete Fourier Transform Calculator

IDFT Time-Domain Signal Reconstructor

Input Fields

Frequency domain values

X[k]

Comma-separated real or complex values (e.g. 1, 0, 0, 0 or 1+2j, 0, -1, 0)

CalculateReset

Calculate automaticallyIf enabled, the result will update automatically when you change any value.

Your Results

Time domain sequence x[n]

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IDFT Mathematical Definition

$$x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] \cdot e^{j 2\pi kn / N}$$

Where:

• $$x[n]$$ = output time-domain sample
• $$X[k]$$ = frequency-domain component
• $$N$$ = total number of samples
• $$j$$ = imaginary unit
• $$n$$ = sample index (0 to N-1)
• $$k$$ = frequency index (0 to N-1)

This is the standard definition of the IDFT, reconstructing the time-domain signal from its spectral representation.

Number of Turns – Calculation Example

Given:

• X[0] = 10, X[1] = -2 + 2j, X[2] = 0, X[3] = -2 – 2j
• N = 4

Calculation:

1. $$x[0] = \frac{1}{4} \cdot \left( X[0] + X[1] + X[2] + X[3] \right)$$
2. $$= \frac{1}{4} \cdot \left( 10 + (-2 + 2j) + 0 + (-2 – 2j) \right)$$
3. $$= \frac{1}{4} \cdot 6 = 1.5$$

The Inverse Discrete Fourier Transform (IDFT) is the fundamental tool in digital signal processing used to transform signals from the frequency domain back into the time domain. It’s especially valuable when analyzing or synthesizing signals in communication systems, filtering, and audio processing. This calculator takes in a list of complex-valued frequency components and computes each corresponding sample in the time domain, supporting educational use and quick verifications.