Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing

SOLUTIONS

Chapter 1

1.1 Color JPEG compressed images are typically 5 to 50 times smaller than they would be if stored “naively,” so the ratio of naively stored to JPEG-stored might range from a low of 0.02 to 0.2.

1.3 Work out e^2πik/8 = cos(kπ/4) + i sin(kπ/4).

1.4 Use equation (1.23).

1.5 Apply Euler’s identity to cos(ωt) and sin(ωt), group the e^iωt and e^−iωt terms.

1.8 Check closure under addition.

1.9 Consider whether cx is in Rⁿ for a complex constant c.

1.11 Use (p + q)² ≤ 2p² + 2q² to show that

and also that x + y is in L²(N) if x and y are in L²(N), and thus obtain closure under addition.

To show that L²(N) L^∞(N) argue that Σ_kx²_k < ∞ implies the existence of some bound M so that |x_k| ≤ M for all k.

1.13 Use equation (1.22).

1.14 Use |e^iωt| = 1 if ω and t are real, and e^ae_b = e^a+b.

1.15 Use Euler’s identity on each piece, e.g., cos(αx) = (e^iαx + e^−iαx)/2.

1.18 Recall that u · v = ||u|| ||v|| cos(θ), and that vectors are orthogonal when u · v = 0. The peak-to-peak distance in part (e) should be .

1.19 Use equation (1.22). Examine E_6,1 versus E_6,5 and E_6,2 versus E_6,4.

1.22 Use Theorem 1.8.3 for parts (b) and (d).

1.23 Use Theorem 1.8.3 for parts (a) and (c), and don’t forget ...