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Unformatted text preview: ECE600 Phil Schniter January 31, 2011 The Discrete Fourier Transform (DFT): • The DFT is a computational tool used to analyze the frequency content of finitelength discretetime signals. • Our DFT discussion will include: 1. definitions of the DFT and IDFT, 2. interpretations of the DFT: (a) sampled DTFT, (b) discrete Fourier series, 3. properties of the DFT, including circular convolution, 4. use of DFT for spectral analysis, 5. fast computation of the DFT via the FFT, 6. matrix/vector formulations. 1 ECE600 Phil Schniter January 31, 2011 DFT definitions: • For an Nlength signal { x [ n ] } N − 1 n =0 , the Npoint DFT is X [ k ] = N − 1 summationdisplay n =0 x [ n ] e − j 2 π N kn , k = 0 ...N − 1 and the corresponding IDFT is x [ n ] = 1 N N − 1 summationdisplay k =0 X [ k ] e j 2 π N kn , n = 0 ...N − 1 . • Note: – the signal has finite duration N , – there are only N DFT coefficients. 2 ECE600 Phil Schniter January 31, 2011 DFT Interpretation # 1 — sampled DTFT: Say signal x [ n ] has duration N . Then X ( e jω ) = ∞ summationdisplay n = −∞ x [ n ] e − jωn = N − 1 summationdisplay n =0 x [ n ] e − jωn . Meanwhile, X [ k ] = N − 1 summationdisplay n =0 x [ n ] e − j 2 π N kn = X ( e jω ) vextendsingle vextendsingle ω = 2 π N k for k = 0 ...N − 1 . Thus, the DFT returns 2 π Nspaced samples of the DTFT. Example for N = 8 : 2 4 6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 4 6 1 2 3 4 5 6 7 8 ω n x [ n ] X [ k ] X ( e jω ) 3 ECE600 Phil Schniter January 31, 2011 The DFT and zeropadding: Again, say x [ n ] has duration N . The Mlength “zeropadded” signal ˜ x [ n ] defines braceleftbigg x [ n ] n = 0 ...N − 1 n = N ...M − 1 has the Mpoint DFT ˜ X [ k ] = M − 1 summationdisplay n =0 ˜ x [ n ] e − j 2 π M kn = N − 1 summationdisplay n =0 x [ n ] e − j 2 π M kn = X ( e jω )  ω = 2 π M k . Thus, zeropadding increases the rate at which the DFT samples the DTFT! 1 2 3 4 5 6 7 1 2 1 2 3 4 5 6 2 4 6 8 5 10 15 20 25 30 1 2 1 2 3 4 5 6 2 4 6 8 ω ω n n x [ n ] ˜ x [ n ]  X ( e jω )   ˜ X ( e jω )  4 ECE600 Phil Schniter January 31, 2011 DTFT sampling and timedomain aliasing: • Previously we saw that, when the timedomain sampling rate (in samp/sec) is too small with respect to the signal bandwidth (in Hz), frequencydomain aliasing results. Wide bandwidth ⇔ quick time variations. • Now we will see that, when the frequencydomain sampling rate (in samp/Hz) is too slow with respect to the signal duration (in sec), timedomain aliasing results. Long duration ⇔ quick frequency variations. • Say we’re given Y [ k ] defines X ( e jω )  ω = 2 π N k for k = 0 ...N − 1 , where X ( e jω ) DTFT ←→ x [ n ] for some x [ n ] of generic length....
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 Fall '05
 UYSALBIYIKOGLU
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