Spectrum Analysis Using the Discrete Fourier Transform ∗

The discrete Fourier transform (DFT) maps a nite number of discrete time-domain samples to the same number of discrete Fourier-domain samples. Being practical to compute, it is the primary transform applied to real-world sampled data in digital signal processing. The DFT has special relationships with the discrete-time Fourier transform and the continuous-time Fourier transform that let it be used as a practical approximation of them through truncation and windowing of an in nite-length signal. Di erent window functions make various tradeo s in the spectral distortions and artifacts introduced by DFTbased spectrum analysis. 1 Discrete-Time Fourier Transform The Discrete-Time Fourier Transform (DTFT) is the primary theoretical tool for understanding the frequency content of a discrete-time (sampled) signal. The DTFT is de ned as X (ω) = ∞ ∑ n=−∞ ( x (n) e−(iωn) ) (1) The inverse DTFT (IDTFT) is de ned by an integral formula, because it operates on a continuous-frequency DTFT spectrum: x (n) = 1 2π ∫ π −π X (k) edω (2) The DTFT is very useful for theory and analysis, but is not practical for numerically computing a spectrum digitally, because 1. in nite time samples means • in nite computation • in nite delay 2. The transform is continuous in the discrete-time frequency, ω For practical computation of the frequency content of real-world signals, the Discrete Fourier Transform (DFT) is used. ∗Version 1.6: Sep 7, 2006 2:10 pm GMT-5 †http://creativecommons.org/licenses/by/1.0 "Discrete-Time Fourier Transform (DTFT)" "Discrete-Time Fourier Transform (DTFT)" http://cnx.org/content/m12032/1.6/ Connexions module: m12032 2 2 Discrete Fourier Transform The DFT transforms N samples of a discrete-time signal to the same number of discrete frequency samples, and is de ned as X (k) = N−1 ∑ n=0 ( x (n) e−( i2πnk N ) ) (3) The DFT is invertible by the inverse discrete Fourier transform (IDFT):