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**Unformatted text preview: **ECE-600 Phil Schniter October 18, 2010 The Discrete Fourier Transform (DFT): • The DFT is a computational tool used to analyze the frequency content of discrete-time 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 ECE-600 Phil Schniter October 18, 2010 DFT definitions: • For an N-length signal { x [ n ] } N − 1 n =0 , the N-point 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 ECE-600 Phil Schniter October 18, 2010 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 N-spaced 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 ECE-600 Phil Schniter October 18, 2010 The DFT and zero-padding: Again, say x [ n ] has duration N . The M-length “zero-padded” signal ˜ x [ n ] defines braceleftbigg x [ n ] n = 0 ...N − 1 n = N ...M − 1 has the M-point 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, zero-padding 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 ECE-600 Phil Schniter October 18, 2010 DTFT sampling and time-domain aliasing: • Previously we saw that, when the time-domain sampling rate (in samp/sec) is too small with respect to the signal bandwidth (in Hz), frequency-domain aliasing results. Wide bandwidth ⇔ quick time variations. • Now we will see that, when the frequency-domain sampling rate (in samp/Hz) is too slow with respect to the signal duration (in sec), time-domain aliasing results. Wide 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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