# Appendix E: Exercises

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 5 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | ISBN 9781560801481 |

Store | SEG Online Store |

Dost thou love life? Then do not squander time, for that is the stuff life is made of. - Benjamin Franklin

1. The causal FIR filter 2 + *Z* is a minimum-delay filter. Show that the causal feedback filter 1/(2 + *Z*) is also a minimum-delay filter.

2. Find the autocorrelations of each of the following wavelets by multiplying the wavelet’s *Z*-transform by the *Z*-transform of its time reverse: (1, –4, 3, 2), (5, –3, 2, –1), (6, 2, –1, 1). Next, find the crosscorrelation between each pair of wavelets by means of the transform.

3. Find the following convolutions by multiplying *Z*-transforms:

(–6,5) * (3, –2,1). [Answer: (–18,27, –16,5).]

(2,01,1,3, –2) * (1,–1). [Answer: (2, –2., 1,0,2, –5,2).]

(4,3,2,1) * (2, –1). Answer: [8,2,1,0, –1.]

(6,3) * (1,2,3,4). Answer: [6,15,24,33,12.]

4. If , what are the coefficients of ?

Interpret these results in terms of convolution of waveforms.

[Respective answers:

(1,2,3,4,5,4,3,2,1)

(1,3,6,10,15,18,19,18,15,10,6,3,1)

(1,4,10,20,35,52,68,80,85,80,85,80,68,52,35,20,10,4,1).]

5. The wavelet (5, –2, 1) has the *Z*-transform . Factor this polynomial.

[Answer: (*Z* + 1 + *2i*)(*Z +* 1 – *2i*). Note that the *Z*-transform is the product of two conjugate factors.]

6. Factor the *Z*-transform of the wavelet (864, –144,186, –55, –79, 4,4). [Answer: (4 + *Z*) (4 – *Z*)(–3 + 2*iZ*)(–3–2*i* *Z*)(2 – *Z*)(3 + *Z*). Note the conjugate factors.]

7. Find the convolution of the following wavelets by means of the *Z*-transform, and another method of your choice. *a* = (2, –1, 3) and *b* = (–1, 0, 1). [Answer: (–2, 1, 5, –1, –3).]

8. Find the *Z*-transforms of the following wavelets:

(a) (1, 0, 3, 0, 0, –1, 0, 2) (b)

Here, give a closed-form expression for the *Z*-transform. [Answer: 1/(1 – *cZ*).]

(c) (1, 0, –1, 0, 1, 0, –1, 0, 1, 0, –1). (d) (2, 1 ) * (–1, 3, 4), where * indicates convolution. (e) (–l, 1, 2) * (3, 4).

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