Discrete-Time Fourier Transform 2.3 2002/01/17 2005/06/09 13:20:21.934 GMT-5 Adan Galvan jago@rice.edu Adan Galvan jago@rice.edu conjugate symmetry convolution discrete-time fourier transform DTFT frequency z-transform Review of the discrete time fourier transform.

Z-Transform Using x n to denote a discrete-time signal at index n ℤ , Z-Transform Xz n x n z n

Discrete-Time Fourier Transform Discrete-Time Fourier Transform X ω n x n ω n ω ω x n 1 2 ω X ω ω n n n Note that: X ω is the z-transfer evaluated on the unit circle in -plane: z ω . X ω is 2 -periodic in ω. x n implies X ω X ω , i.e., conjugate symmetry Other DTFT properties are: ↔ x n X ω ↔ x n X ω ↔ x n ℓ X ω ω ℓ ↔ x n ω 0 n X ω ω 0 ↔ x n y n 1 2 θ X θ Y ω θ ↔ x n y n X ω Y ω Linear Convolution x n y n m x m y n m