FourierRelations - ( DTFT ) x [ n ] discrete aperiodic X (...

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ENEE630 ADVANCED DIGITAL SIGNAL PROCESSING Review: Fourier Relations The table reviews the Fourier transforming relationships between time and frequency domain representations of signals, including continuous and discrete signals, and periodic and aperiodic signals. TRANSFORM TIME-DOMAIN FREQUENCY-DOMAIN (Analysis) (Synthesis) Fourier Series ( FS ) x ( t ) continuous periodic X n discrete aperiodic X n = 1 T Z + T 2 - T 2 x ( t ) e - j 2 πnt/T dt x ( t ) = + X n = -∞ X n e j 2 πnt/T Fourier Transform ( FT ) x ( t ) continuous aperiodic X (Ω) continuous aperiodic X (Ω) = Z + -∞ x ( t ) e - j Ω t dt x ( t ) = 1 2 π Z + -∞ X (Ω) e j Ω t d Ω (or in f where Ω = 2 πf ) Discrete-Time Fourier Transform
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Unformatted text preview: ( DTFT ) x [ n ] discrete aperiodic X ( ω ) continuous periodic X ( ω ) = + ∞ X n =-∞ x [ n ] e-jωn x [ n ] = 1 2 π Z + π-π X ( ω ) e jωn dω Discrete Fourier Transform ( DFT ) x [ n ] discrete periodic X [ k ] discrete periodic X [ k ] = N-1 X n =0 x [ n ] W kn N x [ n ] = 1 N N-1 X k =0 X [ k ] W-kn N (where W kn N = e-j 2 πkn/N ) As a general rule, remember that continuous in one domain ↔ aperiodic in the other domain discrete in one domain ↔ periodic in the other domain 1...
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