Verification of FFT Properties

Verification of FFT conjugate symmetry and related properties. In signal processing, FFT (Fast Fourier Transform) serves as a critical mathematical tool for converting time-domain signals into frequency-domain representations. The conjugate symmetry property specifically refers to the characteristic where real-valued input signals produce FFT results with Hermitian symmetry (complex conjugate symmetry). Through property verification, we ensure FFT algorithm correctness and enable accurate processing of real-valued signals. Implementation typically involves: - Generating test signals (sine waves, chirp signals) as real-valued inputs - Computing FFT using functions like fft() in MATLAB or numpy.fft.fft() in Python - Verifying conjugate symmetry through real/imaginary component analysis - Comparing original and reconstructed signals via inverse FFT - Using assertions to validate symmetry conditions in automated testing Key algorithmic aspects include checking that for real input x[n], the FFT output satisfies X[k] = X*[N-k], where N is the FFT length and * denotes complex conjugation.