Transcription of Discrete Fourier Transform (DFT)
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Discrete Fourier Transform (DFT)
Discrete Fourier Transform (DFT). Recall the DTFT: . X. X( ) = x(n)e j n. n= . DTFT is not suitable for DSP applications because In DSP, we are able to compute the spectrum only at specific Discrete values of , Any signal in any DSP application can be measured only in a finite number of points. A finite signal measured at N points: . 0, n < 0, x(n) = y(n), 0 n (N 1), 0, n N, . where y(n) are the measurements taken at N points. EE 524, Fall 2004, # 5 1. Sample the spectrum X( ) in frequency so that 2 . X(k) = X(k ), = = . N. N 1. j2 kn X. X(k) = x(n)e N DFT. n=0. The inverse DFT is given by: N 1. 1 X kn x(n) = X(k)ej2 N . N. k=0. 1. N. (N 1 ). 1 X X. j2 km j2 kn x(n) = x(m)e N e N. N m=0. k=0. 1 1. N. ( N. ). X 1 X. j2 . k(m n). = x(m) e N = x(n). m=0. N. k=0. | {z }. (m n). EE 524, Fall 2004, # 5 2. The DFT pair: N 1. X kn X(k) = x(n)e j2 N analysis n=0. N 1. 1 X j2 kn x(n) = X(k)e N synthesis. N. k=0. Alternative formulation: N 1. X 2 . X(k) = x(n)W kn W = e j N. n=0. N 1. 1 X. x(n) = X(k)W kn.
Discrete Fourier Transform (DFT) Recall the DTFT: X(ω) = X∞ n=−∞ x(n)e−jωn. DTFT is not suitable for DSP applications because •In DSP, we are able to compute the spectrum only at specific discrete values of ω, •Any signal in any DSP application can be measured only in a finite number of points. A finite signal measured at N ...
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