DFT Definition

Fourier Analysis · Axiom Academy

LESSON The Discrete Fourier Transform Understanding how the DFT extracts frequency content from discrete signals The DFT is defined as a sum over all N samples in the input signal. For each frequency bin k, we compute: This formula takes N time-domain samples x[n] and produces N frequency-domain coefficients X[k]. Each coefficient is a complex number representing the amplitude and phase of a specific frequency component. 2. Breaking Down the Components Let's understand each part of the DFT formula: x[n]: The input signal samples (n = 0, 1, ..., N-1) e^(-i2πkn/N): Complex exponential "probe" at frequency k k: Frequency bin index (k = 0, 1, ..., N-1) The complex exponential rotates at a rate determined by k. When this matches a frequency present in x[n], the sum produces a large magnitude. Otherwise, the contributions cancel out. 3. Interpreting X[k]: Frequency Content Each DFT coefficient X[k] is a complex number that tells us about a specific frequency: Magnitude |X[k]|: The strength or amplitude of frequency component k Phase ∠X[k]: The phase offset of that frequency component When we visualize |X[k]| across all k values, we get the frequency spectrum showing which frequencies are present in the signal and how strong they are. The DFT computes exactly N frequency coefficients, indexed by k = 0, 1, 2, ..., N-1: k = 0: DC component (average value, zero frequency) k = 1 to N/2-1: Positive frequencies k = N/2: Nyquist frequency (highest representable frequency)

This is the written version of the interactive lesson above. See the full Fourier Analysis course.