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Unformatted text preview: Lecture 7: CT Convolution Tie Liu September 19, 2011 1 CT Convolution DT LTI systems: x [ n ] = ∞ summationdisplay k = −∞ x [ k ] δ [ n − k ] = ⇒ y [ n ] = ∞ summationdisplay k = −∞ x [ k ] h [ n − k ] CT LTI systems: Need to express x ( t ) as a linear combination of a “basic” signal and its time shifts. Can we use δ ( t ) = braceleftbigg 1 , t = 0 , t negationslash = 0 No, that would require uncountably infinite many time shifts of δ ( t ). And we do not know how to add uncountably infinite many numbers in general. Solution: Approximate! Let x Δ ( t ) = x ( k Δ) when t ∈ (( k − 1 / 2)Δ , ( k + 1 / 2)Δ]. When Δ → 0, we have x Δ ( t ) → x ( t ) for all t ∈ R . t x ( t ) x Δ ( t ) Δ- Δ 2 Δ- 2 Δ- 3 Δ 3 Δ 1 Can we express x Δ ( t ) as a linear combination of a “basic” signal and its time shifts? Yes, consider δ Δ ( t ) = braceleftbigg 1 Δ , t ∈ ( − Δ 2 , Δ 2 bracketrightbig , otherwise Note that δ Δ ( t − k Δ) are non-overlapping for different k and together they “fill up” the entire time axis. We thus have x Δ ( t ) = ∞ summationdisplay k = −∞ x ( k Δ) δ Δ ( t − k Δ)Δ where Δ here is used to compensate 1 / Δ in the definition of δ Δ ( t ). t δ Δ ( t ) Δ- Δ 2 Δ- 2 Δ- 3 Δ 3 Δ δ Δ ( t- Δ ) δ Δ ( t- 2 Δ ) δ Δ ( t- 3 Δ ) δ Δ ( t + Δ ) δ Δ ( t + 2Δ) δ Δ ( t + 3 Δ ) 1 Δ Let h Δ ( t ) be the response to δ Δ ( t ). By the LTI properties, the output signal)....
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- Fall '09
- Digital Signal Processing, LTI system theory, Impulse response, unit impulse response