Appendix E: Exercises

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 5 http://dx.doi.org/10.1190/1.9781560801610 ISBN 9781560801481 SEG Online Store
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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.]

4. If ${\displaystyle A\left(Z\right)=1+{\rm {Z}}+{\rm {Z}}^{2}+Z^{3}+Z^{4}}$, what are the coefficients of ${\displaystyle A^{2}\left(Z\right),A^{3}\left(Z\right),A^{4}\left({\rm {Z}}\right)}$?

Interpret these results in terms of convolution of waveforms.

(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 ${\displaystyle 5+2Z+Z^{2}}$. 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 + 2iZ)(–3–2i 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) ${\displaystyle \left({1,}{\ c,\ }{c}^{2},\ {c}^{3},\ {c}^{4},\ \dots \right)}$

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).