Figure 7.1 (p. 554) Real and imaginary parts of the signal zn.

Figure 7.1 (p. 554) Real and imaginary parts of the signal zn.
Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.2 (p. 556) Illustration of a signal that has a z-transform, but does not have a DTFT. (a) An increasing exponential signal for which the DTFT does not exist. (b) The attenuating factor r–n associated with the z-transform. (c) The modified signal x[n]r–n is absolutely summable, provided that r > , and thus the z-transform of x[n] exists. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.3 (p. 557) The z-plane. A point z = rej is located at a distance r– from the origin and an angle  relative to the real axis. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.4 (p. 557) The unit circle, z = ej, in the z-plane.

Figure 7.5 (p. 559) Locations of poles and zeros of x[n] = nu[n] in the z-plane. The ROC is the shaded area. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.6 (p. 560) ROC and locations of poles and zeros of x[n] = –nu[–n – 1] in the z-plane.

Figure 7.7 (p. 560) ROC and locations of poles and zeros in the z-plane for Example 7.4.

Figure 7.8 (p. 564) The relationship between the ROC and the time extent of a signal. (a) A right-sided signal has an ROC of the form |z| > r+. (b) A left-sided signal has an ROC of the form |z| < r–. (c) A two-sided signal has an ROC of the form r+ < |z| < r–. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7. 9 (p. 565) ROCs for Example 7. 5
Figure 7.9 (p. 565) ROCs for Example (a) Two-sided signal x[n] has ROC in between the poles. (b) Right-sided signal y[n] has ROC outside of the circle containing the pole of largest magnitude. (c) Left-sided signal w[n] has ROC inside the circle containing the pole of smallest magnitude. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7. 10 (p. 567) ROCs for Example 7. 6
Figure (p. 567) ROCs for Example 7.6. (a) ROC and pole-zero plot for X(z). (b) ROC and pole-zero plot for Y(z). (c) ROC and pole-zero plot for a(X(z) and Y(z)). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 569) The effect of multiplication by n on the poles and zeros of a transfer function. (a) Locations of poles and zeros of X(z). (b) Locations of poles and zeros of X(z/). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.12 (p. 574) Locations of poles and ROC for Example 7.9.

Figure 7.13 (p. 575) Locations of poles and ROC for Example 7.10.

Figure (p. 583) The relationship between the location of a pole and the impulse response characteristics for a causal system. (a) A pole inside the unit circle contributes an exponentially decaying term to the impulse response. (b) A pole outside the unit circle contributes an exponentially increasing term to the impulse response. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 583) The relationship between the location of a pole and the impulse response characteristics for a stable system. (a) A pole inside the unit circle contributes a right-sided term to the impulse response. (b) A pole outside the unit circle contributes a left-sided term to the impulse response. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 584) A system that is both stable and causal must have all its poles inside the unit circle in the z-plane, as illustrated here. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.17 (p. 584) Locations of poles in the z-plane for the system in Example 7.16.

Figure (p. 586) A system that has a causal and stable inverse must have all its poles and zeros inside the unit circle, as illustrated here. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.19 (p. 589) Vector interpretation of e(j0 – g in the z-plane.

Figure (p. 589) The quantity |ej – g| is the length of a vector from g to ej in the z-plane. (a) Vectors from g to ej at several frequencies. (b) The function ej – g|. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7. 21 (p. 590) (a) Location of zero for multipath channel
Figure (p. 590) (a) Location of zero for multipath channel. (b) Location of pole for inverse of the multipath channel. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure (p. 591) Magnitude response of multipath channel (left panel) and inverse system (right panel). (a) a = 0.5ej/4. (b) a = 0.8ej/4. (c) a = 0.95ej/4. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7. 23a (p. 592) Solution for Example 7. 21
Figure 7.23a (p. 592) Solution for Example (a) Locations of poles and zeros in the z-plane. (b) The component of the magnitude response associated with a zero is given by the length of a vector from the zero to ej. (c) The component of the magnitude response associated with the pole at z = ej/4 is the inverse of the length of a vector from the pole to ej. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.23b (p. 593, continued) (d) The component of the magnitude response associated with the pole at is the inverse of the length of a vector from the pole to ej. (e) The system magnitude response is the product of the response in parts (b)–(d). Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.24 (p. 593) Solution to Problem 7.13.

Figure (p. 593) The quantity arg{ej – g} is the angle of the vector from g to ej with respect to a horizontal line through g, 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 7.26 (p. 595) Block diagram of the transfer function corresponding to Fig. 2.38.

Figure (p. 596) Development of the direct form II representation of an LTI system. (a) Representation of the transfer function H(z) as H2(z)H1(z). (b) Direct form II implementation of the transfer function H(z) obtained from (a) by collapsing the two sets of z–1 blocks. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.28 (p. 597) Cascade form of implementation for Example 7.22.

Figure 7. 29 (p. 597) Solution to Problem 7. 14
Figure (p. 597) Solution to Problem The number “4” near the zero at z = –1 indicates that there are 4 zeros at this location. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7. 30a (p. 601) Solution to Example 7
Figure 7.30a (p. 601) Solution to Example 7.23, depicted as a function of the month. (a) Account balance at the start of each month following possible withdrawal. Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 7.30b (p. 601) (b) Natural response.

Figure 7.30c (p. 602) Forced response.

Figure 7.31 (p. 604) Location of poles and zeros in the z-plane obtained by using MATLAB.

Figure 7.32 (p. 605) Magnitude response evaluated by using MATLAB.

Figure 7.1 (p. 554) Real and imaginary parts of the signal zn.

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Figure 7.1 (p. 554) Real and imaginary parts of the signal zn.

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