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4. ALTI CT system has transfer function H(s) = s² +1 (s + 1)(s² + 2s + 17) Compute the steady-state and transient responses resulting when the input x(t) is a) x(t) = u(t), with zero initial conditions b) x(t) = cost(t), t ≥ 0, with zero initial conditions

4. ALTI CT system has transfer function H(s) = s² +1 (s + 1)(s² + 2s + 17) Compute the steady-state and transient responses resulting when the input x(t) is a) x(t) = u(t), with zero initial conditions b) x(t) = cost(t), t ≥ 0, with zero initial conditions

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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4. ALTI CT system has transfer function H(s) = s² +1 (s + 1)(s² + 2s + 17) Compute the steady-state and transient responses resulting when the input x(t) is a) x(t) = u(t), with zero initial conditions b) x(t) = cost(t), t ≥ 0, with zero initial conditions
Transcribed Image Text:4. ALTI CT system has transfer function H(s) = s² +1 (s + 1)(s² + 2s + 17) Compute the steady-state and transient responses resulting when the input x(t) is a) x(t) = u(t), with zero initial conditions b) x(t) = cost(t), t ≥ 0, with zero initial conditions
TABLE 3.1 Properties of the Fourier Transform Property Linearity Right or left shift in time Time scaling Time reversal Multiplication by a power of t Multiplication by a complex exponential Multiplication by sin (wot) Multiplication by cos(wot) Differentiation in the time domain Integration in the time domain Convolution in the time domain Multiplication in the time domain Parseval's theorem Special case of Parseval's theorem Duality Transform Pair/Property ax(t) + bv(t)→aX(w) + bV(w) x(tc)X(w)e-jose x (at) < > ² x ( =) a > X a x(−1)→X(-w) = X(w) a>0 d" t"x(t) → j" don X (w) n = 1, 2, ... dw" x(t)ejat →X(w - wo) wo real x(t) sin(wt) ↔ { [X(w + w) − X(w − w)] x(t) cos(wit) → [X(w + wo) + X(w − w)] din X(t) → (jw)"X(w) n = 1, 2,... 1 [x(x) dx → — X (w) + 7X (0)8(w) jw x(t) *v(t) →X(w)V(w) x(t)v(t) → __X(w)*V(w) [x(1)v(1) dt = 1 [~_* x²(1) dt = 2 / 2TT X(t)→2πx(-w) [XV (oo) das 1X (w) ² do -xx TABLE 3.2 Common Fourier Transform Pairs 1, -∞ < t < ∞ → 2π8 (W) −0.5 + u(t) ↔ - jw 1 jw u(t) <> πδ(ω) + 8(t) →1 8(tc)e-jwc, c any real number 1 jw + b² e-btu(t). ejut → 2πd (w - wo), wo any real number τω 2π P:(t)→7 sinc- b>0 7 sinc→2πp, (w) 27 2|t| τω (1 - 2!!!) p.,(1)→ sinc²(e) T sinc²(# )+2π(1 - 2!!)P, (c) 2 T cos(wot) →T[8(w + wo) + d(w = wo)] cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)] sin(wot) → jπ[8(w + wo) − 8(w - wo)] sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]
Transcribed Image Text:TABLE 3.1 Properties of the Fourier Transform Property Linearity Right or left shift in time Time scaling Time reversal Multiplication by a power of t Multiplication by a complex exponential Multiplication by sin (wot) Multiplication by cos(wot) Differentiation in the time domain Integration in the time domain Convolution in the time domain Multiplication in the time domain Parseval's theorem Special case of Parseval's theorem Duality Transform Pair/Property ax(t) + bv(t)→aX(w) + bV(w) x(tc)X(w)e-jose x (at) < > ² x ( =) a > X a x(−1)→X(-w) = X(w) a>0 d" t"x(t) → j" don X (w) n = 1, 2, ... dw" x(t)ejat →X(w - wo) wo real x(t) sin(wt) ↔ { [X(w + w) − X(w − w)] x(t) cos(wit) → [X(w + wo) + X(w − w)] din X(t) → (jw)"X(w) n = 1, 2,... 1 [x(x) dx → — X (w) + 7X (0)8(w) jw x(t) *v(t) →X(w)V(w) x(t)v(t) → __X(w)*V(w) [x(1)v(1) dt = 1 [~_* x²(1) dt = 2 / 2TT X(t)→2πx(-w) [XV (oo) das 1X (w) ² do -xx TABLE 3.2 Common Fourier Transform Pairs 1, -∞ < t < ∞ → 2π8 (W) −0.5 + u(t) ↔ - jw 1 jw u(t) <> πδ(ω) + 8(t) →1 8(tc)e-jwc, c any real number 1 jw + b² e-btu(t). ejut → 2πd (w - wo), wo any real number τω 2π P:(t)→7 sinc- b>0 7 sinc→2πp, (w) 27 2|t| τω (1 - 2!!!) p.,(1)→ sinc²(e) T sinc²(# )+2π(1 - 2!!)P, (c) 2 T cos(wot) →T[8(w + wo) + d(w = wo)] cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)] sin(wot) → jπ[8(w + wo) − 8(w - wo)] sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]
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