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Discrete Time Fourier Transform

Discrete Time Fourier Transform
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Published Date:22-07-2017
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2. DSP Theory-cont. Rahil Mahdian 04.05.2015Discrete Time Fourier Transform - DTFT • Recall for continuous time Fourier transform, when the signal is sampled: Assuming Discrete-Time Fourier Transform (DTFT): Analysis: Synthesis: 2Properties of DTFT If xn is absolutely summable, it contains properties: • exists for each frequency, ω. Analysis Serie converges. is a continious function of frequency, ω. • Continuity of Stability of the frequency the system response Question. Is an ideal LPF implementable? Gibbs Phenomenon 3DTFT theories 4DTFT examples 5Ideal Window 6Ideal Window 7Discrete Fourier Transform basics (DFT) DTFT is still a continuous function of frequency We need to save good samples of DTFT in memory, instead. theorem: Lossless representation Signal is limited of signal by its samples in frequency Analog sampling: is possible spectrum Lossless representation Signal is of signal in frequency DFT domain limited in Time domain by its samples sampling: domain is possible 8DFT - intuition ௝ఠ 9DFT DFS relation 10DFT DFS relation Discrete Fourier series coefficients of are the Discrete Fourier Transform coefficients of We should take N samples from the DTFT of xn, , In a 2π period, to get the DFT coefficients of xn, named as By defining, DFS 11DFT 12DFT as a matrix operator 13Example • The DFT of a rectangular pulse • xn is of length 5 • We can consider xn of any length greater than 5 • Let’s pick N=5 • Calculate the DFS of the periodic form of xn 4 j2k /5n Xk e  n0 j2k 1 e  j2k /5 1 e 5 k 0,5,10,...    0 else  14Example Cont’d • If we consider xn of length 10 • We get a different set of DFT coefficients • Still samples of the DTFT but in different places 15Properties of DFT • Linearity DFT  x n X k 1 1 DFT xn Xk 2 2 DFT axn bxn aXk bXk 1 2 1 2 • Duality DFT xn Xk DFT Xn Nx k N • Circular Shift of a Sequence DFT xn Xk DFTj2k /Nm xn m0 n N-1 Xke N 16ADC and DAC 17ADC and DAC y(t) x (t) D.S.P. c xn yn Input Output ADC DAC Signal Signal Analogue Digital to to Digital Analogue Converter Converter ஶ ஶஶ ஶ ௝ఠି௝ఠ௡ ି௝ఠ௡௝ஐ௡்ି௝ఠ௡ ିஶ ௡ୀିஶ ௡ୀିஶ௡ୀିஶ ஶ ஶ ஶ 1 Poisson's formula 1 ௝௡(ஐ்ିఠ ) =න݆ܺܿΩ (෍݁ )݀Ω =෍ ఠାଶగ௞ 2ߨ ܶ ିஶ Ω = ௡ୀିஶ ௞ୀିஶ ் Question. What is the requirement to be able to recover the signal completely? “signal should be band-limited“ 18Nyquist rate ݏܰܰ ݏܰ ݏ 19Reconstruction Condition-DAC T 2ߨ Ω = ݏ ܶ ݎ r D/C xn x (t) r (t) = x (n) = x(nT) ݎ sݎ ݎ ܿ ݎ ݎ ܿ ܿ - ܰܿݏܰ 20