Signals and Systems, 2/E by Simon Haykin and Barry Van Veen

Figure 6.1 (p. 483) Real and imaginary parts of the complex exponential est, where s =  + j.
Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.2 (p. 485) The Laplace transform applies to more general signals than the Fourier transform does. (a) Signal for which the Fourier transform does not exist. (b) Attenuating factor associated with Laplace transform. (c) The modified signal x(t)e-t is absolutely integrable for  > 1. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.3 (p. 486) The s-plane. The horizontal axis is Re{s} and the vertical axis is Im{s}. Zeros are depicted at s = –1 and s = –4  2j, and poles are depicted at s = –3, s = 2  3j, and s = 4. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.4 (p. 487) The ROC for x(t) = eatu(t) is depicted by the shaded region. A pole is located at s = a. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.5 (p. 488) The ROC for y(t) = –eatu(–t) is depicted by the shaded region. A pole is located at s = a. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.6 (p. 492) Time shifts for which the unilateral Laplace transform time-shift property does not apply. (a) A nonzero portion of x(t) that occurs at times t  0 is shifted to times t < 0. (b) A nonzero portion x(t) that occurs at times t < 0 is shifted to times t  0. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 7 (p. 493) RC circuit for Examples 6. 4 and 6. 10
Figure 6.7 (p. 493) RC circuit for Examples 6.4 and Note that RC = 1/5. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 8 (p. 504) The solution to Example 6. 11
Figure 6.8 (p. 504) The solution to Example (a) Forced response of the system, y(f)(t). (b) Natural response of the system, y(n)(t). (c) System output. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 9 (p. 506) MEMS accelerometer responses for Example 6. 12
Figure 6.9 (p. 506) MEMS accelerometer responses for Example (a) Natural response y(n)(t). (b) Forced response y(f)(t). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 507) Laplace transform circuit models for use with Kirchhoff’s voltage law. (a) Resistor. (b) Inductor with initial current iL(0–). (c) Capacitor with initial voltage vc(0–). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 507) Laplace transform circuit models for use with Kirchhoff’s current law. (a) Resistor. (b) Inductor with initial current iL(0–). (c) Capacitor with initial voltage vc(0–). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 12 (p. 508) Electrical circuit for Example 6. 13
Figure (p. 508) Electrical circuit for Example (a) Original circuit. (b) Transformed circuit. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 510) The ROC (shaded region) of a sum of signals may be larger than the intersection of individual ROCs when pole-zero cancellation occurs. (a) ROC for x(t) = e–2tu(t); (b) ROC for y(t) = e–2tu(t) = e–3tu(t); (c) ROC for x(t) – y(t). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 514) Relationship between the time extent of a signal and the ROC, shown as the shaded region. (a) A left-sided signal has ROC to the left of a vertical line in the s-plane. (b) A right-sided signal has ROC to the right of a vertical line in the s-plane. (c) A two-sided signal has ROC given by a vertical strip of finite width in the s-plane. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 15 (p. 515) ROCs for signals in Example 6. 16
Figure (p. 515) ROCs for signals in Example (a) The shaded regions denote the ROCs of each individual term, e–2tu(t) and e–tu(–t). The doubly shaded region is the intersection of the individual ROCs and represents the ROC of the sum. (b) The shaded regions represent the individual ROCs of e–2tu(–t) and e–tu(t). In this case there is no intersection, and the Laplace transform of the sum does not converge for any value of s. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.16 (p. 517) Poles and ROC for Example 6.17.

Figure 6.17 (p. 518) Poles and ROC for Example 6.18.

Figure (p. 522) (a) Electromechanical system in which a motor is used to position a load. (b) Circuit diagram relating applied voltage to back electromotive force, armature resistance, and input current. Note that v(t) = K2dy(t)/dt. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 524) The relationship between the locations of poles and the impulse response in a causal system. (a) A pole in the left half of the s-plane corresponds to an exponentially decaying impulse response. (b) A pole in the right half of the s-plane corresponds to an exponentially increasing impulse response. The system is unstable in this case. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 524) The relationship between the locations of poles and the impulse response in a stable system. (a) A pole in the left half of the s-plane corresponds to a right-sided impulse response. (b) A pole in the right half of the s-plane corresponds to an left-sided impulse response. In this case, the system is noncausal. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 525) A system that is both stable and causal must have a transfer function with all of its poles in the left half of the s-plane, as shown here. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.22 (p. 529) The quantity j0 = g as a vector from g to j0 in the s-plane.

Figure (p. 529) The function |j – g| corresponds to the lengths of vectors from g to the j-axis in the s-plane. (a) Vectors from g to j for several frequencies. (b) |j – g| as a function of j. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 24 (p. 530) Components of the magnitude response
Figure (p. 530) Components of the magnitude response. (a) Magnitude response associated with a zero. (b) Magnitude response associated with a pole. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 530) The solution to Example (a) Pole-zero plot. (b) Approximate magnitude response. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 531) The quantity j0 – g as a vector from g to j0 in the s-plane. The phase angle of the vector is , its angle with respect to a horizontal line through g. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 27 (p. 532) The phase angle j – g
Figure (p. 532) The phase angle j – g. (a) Vectors from g to j for several different values of . (b) Plot of arg{j – g} as a continuous function of . Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 533) The phase response of the system in Example (a) Phase of the zero at s = (b) Phase of the pole at s = j5. (c) Phase of the pole at s = -0.1 – j5. (d) Phase response of the system. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.29 (p. 533) Solution to Problem 6.22.

Figure (p. 535) Bode diagram for first-order pole factor: 1/(1 + s/0). (a) Gain response. (b) Phase response. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6. 31a (p. 536) Bode diagram for Example 6. 25
Figure 6.31a (p. 536) Bode diagram for Example (a) Gain response of pole at s = –1 (solid line), zero at s = –10 (dashed line), and pole at s = –50 (dotted line). (b) Actual gain response (solid line) and asymptotic approximation (dashed line). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.31b (p. 536) (c) Phase response of pole at s = –1 (solid line), zero at s = –10 (dashed line), and pole at s = –50 (dotted line). (d) Actual phase response (solid line) and asymptotic approximation (dashed line). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.32 (p. 537) Asymptotic approximation to 20log10|Q(j)|, where

Figure (p. 538) Bode diagram of second-order pole factor for varying  (a) Gain response. (b) Phase response. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 539) Bode diagram for electromechanical system in Example (a) Actual magnitude response (solid line) and asymptotic approximation (dashed line). (b) Actual phase response (solid line) and asymptotic approximation (dashed line). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.35a (p. 540) Solution to Problem 6.23)

Figure 6.36 (p. 540) Solution to Problem 6.24.

Figure (p. 543) Locations of poles and zeros in the s-plane for a system obtained by using MATLAB. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 6.38 (p. 544) Magnitude response for a system obtained by using MATLAB.

Figure 6. 39 (p. 545) Bode diagram for the system in Example 6
Figure (p. 545) Bode diagram for the system in Example 6.25 obtained using MATLAB. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Signals and Systems, 2/E by Simon Haykin and Barry Van Veen

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