II-A. DT LTI System Analysis: Impulse Response & Convolution
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چکیده
k x[k]δ[n − k] : convolution of x[n] and δ[n] = · · · + x[−2]δ[n + 2] + x[−1]δ[n + 1] + x[0]δ[n] + x[1]δ[n − 1] + · · · : any sequence can be decomposed as a sum of scaled and shifted impulses. (a) Scale (by x[k]) and shift (from δ[n] to δ[n− k]) imply linearity and time-invariance, respectively. (b) Envision x[n] : {..., x[−1], x[0], x[1], ...} as a signal vector, and δ[n − k] : {..., 0, δ[n − k], 0, ...} is itself a signal vector, too. (c) The sifting property is used to prove the convolution thm of an LTI system. 2. Impulse response is the system’s response(output) to an impulse input: h[n] ≡ y[n]|when input x[n]=δ[n] δ[n] → LTI → h[n] [Ex] y[n] = 0.9y[n− 1]+ x[n]. The impulse response h[n] can be computed recursively from y[n] by setting x[n] = δ[n]. If the input is an impulse x[n] = δ[n] and y[−1] = 0(zero-state), then y[0] = x[0] = 1, y[1] = 0.9y[0] + 0 = 0.9, y[2] = 0.9y[1] = 0.92, y[3] = 0.93, . . .. Thus the impulse response is h[n] = 0.9u[n]. 3. [Convolution Sum Thm] The output of any LTI system is the discrete convolution of the input x[n] with the system’s impulse response h[n]:
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II-B. CT LTI System Analysis: Impulse Response & Convolution
τ :−∞ x(τ)h(t− τ)dτ = x(t) = y(t) Note that i. Linearity justifies superposition principle; ii. Time-Invariance justifies the system’s response to δ(t − n∆τ) is h(t− n∆τ). iii. For time-varying systems, the system response to δ(t − n∆τ) becomes h(t, n∆τ), and y(t) = ∫ ∞ τ :−∞ x(τ)h(t, τ)dτ , where h(t, τ) is the system response at instant t to a unit impulse input located at τ . 3. [Convolution...
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