Engineering Tables/Fourier Transform Properties

Signal Fourier transform unitary, angular frequency Fourier transform unitary, ordinary frequency Remarks 1Linearity 2Shift in time domain 3Shift in frequency domain, dual of 2 4If is large, then is concentrated around 0 and spreads out and flattens 5Duality property of the Fourier transform. Results from swapping "dummy" variables of and . 6Generalized derivative property of the Fourier transform 7This is the dual to 6 8 denotes the convolution of and — this rule is the convolution theorem 9This is the dual of 810 For a purely real even function is a purely real even function is a purely real even function11 For a purely real odd function is a purely imaginary odd function is a purely imaginary odd function