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Unformatted text preview: EE 338 Discrete‐time Signals and Systems Part IV: Frequency Analysis of Signals and Systems 1. Discrete‐time Fourier transform (DTFT) • ∑ Frequency response of an LTI system: | | , the output is o Given an input o Given an input |cos , the output is | arg o Relationship between h(n), H(z), and H(ω) | , and H(ω) exists if and only if the system is BIBO stable (the ROC of H(z) includes the unit circle) LTI system representation in the time, frequency and transform domain: h(n), the different‐equation representation, H(ω), H(z), and the zero‐pole plot of H(z) Definition of DTFT: o o ∑ , X(ω) is a continuous function of ω X(ω) is periodic with period 2π: X(ω+2πr)=X(ω) for an integer r. 0, +/‐ 2π, +/‐ 4π, … correspond to the low frequency components, and +/‐ π, +/‐ 3π, +/‐ 5π correspond to the high frequency components | | ∞. X(ω) exists if and only if ∑ Properties of DTFT: linearity, shifting in the time/frequency domains, convolution property, symmetry property • o o 2. Discrete Fourier series (DFS) for period signals • For a periodic signal ∑ • • with period N, / ∑ , ∑ , the frequency 2 / . index k corresponds to frequency is periodic with period N. Properties of DFS: linearity (for two sequences of the SAME period), shifting property, symmetry ), the convolution property (periodic/circular convolution: property ( ∑ ) 3. Discrete Fourier Transform (DFT) for finite‐duration signals • For a finite‐duration signal x(n) where x(n)=0 for all n<0 and n≥N, ∑ • • 0 0 0 Properties of DFT: Linearity, circular shifting, circular convolution, symmetry property Implementation of linear convolution using DFT 1, ∑ 0 1 4. Relationship between DTFT, DFS and DFT • • ∑ Periodic extension of a finite‐duration signal x(n) with period N: DFS and DFT: 0 1 ∑ o , 0 0 1 ∑ o , 0 o Periodic extension of DFT in the freq. domain ↔ periodic extension of the signal in the time domain o Truncation of DFS in the freq. domain ↔ truncation of the signal in the time domain DTFT and DFS: | o DFS is the sampled version of DTFT separated by 2π/N: / Frequency‐domain sampling of X(ω) results in the periodic extension of x(n) with period N in the time domain: generate copies of x(n), shift the origin to integer multiples of N, and superimpose all replicas o If x(n)’s non‐zero duration is L, to avoid time‐domain sampling, we should sample at least L samples per 2π. Sampling X(ω) with N≥L samples/2π enables exact recovery of x(n) and X(ω). DTFT and DFT: o o o 0 Padding x(n) with more zeros (increasing N) does not change the frequency spectrum, but results in a denser sampling of X(ω). 0 1 • • ...
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- Spring '10