DFT Calculation Examples

Fourier Analysis · Axiom Academy

EXAMPLE DFT Calculation Examples Computing the DFT of a 4-point sequence step by step Excellent work! You've completed this DFT calculation. Here's what we learned: DFT Formula: For an N-point sequence, X[k] = sum of x[n] times e^(-i2πkn/N) for n from 0 to N-1 Twiddle Factors: W_N = e^(-i2π/N) represents the fundamental rotation. For N=4, W_4 = e^(-iπ/2) = -i DC Component: X[0] is always the sum of all time-domain samples Symmetry: For real-valued sequences, X[N-k] = X*[k] (complex conjugate symmetry) Physical Meaning: Each X[k] represents the amplitude and phase of a complex exponential at frequency k/N Calculation Strategy: Powers of twiddle factors cycle: W_4^0=1, W_4^1=-i, W_4^2=-1, W_4^3=i, W_4^4=1 Practice computing DFTs for other simple sequences to build intuition. Notice how the sequence [1, 0, -1, 0] produces only imaginary frequency components due to its odd symmetry!

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