Math 414-501 — Spring 2010

Assignment 1

• Read sections 0.1-0.5
• Problems.
  1. Section 0.8: 2, 3, 4
  2. For the inner products and vectors below, verify Schwarz's inequality
     1. Use the inner product in Example 0.3, pg. 4, and the column vectors v and w given below:

        

     2. Use the L² inner product given in Definition 0.5, pg. 6, with a = 0 and b = 1, and the ``vectors'' (functions)

        f(x) = x² and g(x) = x+1.

  3. In my notes on Matlab M-files and Functions, we discussed anonymous functions, script m_files, and function m_files.
     1. Write a script m_file to plot y = x²sin(4x) on [−π,π].
     2. Write a function m_file to plot y = e^xcos(3x) on [−π,π].
     3. Write an anonymous function for f(x) = 3x²cos(2x). Use it to find f(π/5), f(π/3).
     4. Apply the function m_file inv_sq to the two matrices [1 1;2 1] and [1 1;1 1.001]. State your result for each matrix.

Due Friday, 1/29/2010.

Assignment 2

• Read sections 1.1, 1.2.1-1.2.4
• Problems.
  1. Section 0.8: 6, 7, 11, 12, 13. In 13, use Matlab to plot both cos(x) and its projection. (This requires plotting polynomials — more to come on how to do that efficiently!)
  2. Let V be a vector space with a complex inner product < ·,· >. Suppose that the set B = {u₁, u₂, ..., u_n} is an orthonomal basis for V. Show that if
     v = a₁u₁ + a₂u₂ + ... + a_nu_n and w = b₁u₁ + b₂u₂ + ... + b_nu_n,
     then <v, w > = ∑_j a_j b_j = b^T a.

Due Friday, 2/5/2010.

Assignment 3

• Read sections 1.25, 1.3.1, 1.3.2
• Problems.
  1. Section 0.8: 15, 22.
  2. Section 1.4: 1, 2, 7, 8, 9

Due Friday, 2/12/2010.

Assignment 4

• Read sections 1.25, 1.3.1, 1.3.2
• Problems.
  1. Section 1.4: 3, 4, 10, 11, 16, 20

Due Friday, 2/19/2010.

Assignment 5

• Read sections 1.3.4, 1.3.5, 2.1. Review sections 0.5.1 and 0.5.2.
• Problems.
  1. Chapter 1 exercises: 23(a,b,c,d), 32(c,d,e,f), 33.
  2. Suppose that f is a 2π-periodic function that has a continuous derivative f′. If the Fourier series for f and f′ are
     f(x) = a₀ + ∑_n a_ncos(nx) + b_nsin(nx) and   f′(x) = a′₀ + ∑_n a′_ncos(nx) + b′_nsin(nx),
     then show that the coefficients of the two series are related this way for n ≥ 1:
     a′_n = n b_n, b′_n = − n a_n and a′₀ = 0.
     (Integrate by parts. This was essentially done in Theorem 1.30 in the text. The result is also true if f′ is only piecewise continuous.)
  3. In the text (cf. Example 1.9), we derived the Fourier series for g(x) = x on − π < x < π.
     1. Use Example 1.9 and the previous problem to show that the Fourier series for f(x) = (π²x − x³)/12 is given by
        f(x) = ∑_n=1^∞ (−1)ⁿ⁺¹ sin(nx)/n³.
     2. Let S_N be the partial sum for the series for f. Following the example done in class on 2/26, show that truncating the series at n = N results in this estimate of the error:
        max_x|f(x) − S_N(x)| ≤ 1/(2N²)
     3. Use the series above and Parseval's theorem to find the sum of the series ∑_n=1^∞ n⁻⁶.
  4. Chapter 2 exericses: 2

Due Friday, 3/5/2010.

Assignment 6

• Read sections 2.2 and 2.3
• Problems.
  1. Chapter 2 exericses: 1, 4, 10
  2. Use exercise 10 above with a = 1 and the properties listed in Theorem 2.6 to find the Fourier transforms of the following functions:
     1. t e^-|t|
     2. e^-2|t-3|
     3. sign(t)e^-|t| (Hint : differentiate e^-|t| – the result is a piecewise continuous function.)
     4. 1/(1+t²)  (Hint: How are Fourier transforms and inverse Fourier transforms related?)

Due Friday, 3/12/2010.

Assignment 7

• Read sections 2.4 and 3.1.
• Problems.
  1. Chapter 2 exercises: 8, 13
  2. Let f(t) be a signal that is 0 when t < 0 or t > 1. Show that, for the Butterworth filter, one has

     L[f] = A e^{− α t} ∫ ₀^min(1,t) e^ατ f(τ)dτ, if t ≥ 0, and that L[f] = 0 if t < 0.

  3. Let h(t) be the impulse response function in exercise 12 of Chapter 2. For any signal f, show that

     L[f] = d⁻¹∫ _t−d^t f(τ)dτ

• Start putting together groups for projects.

Due Monday, 3/29/2010.

Assignment 8

• Read sections 3.1, 4.1 and 4.2
• Problems.
  1. Chapter 2 exercise: 14
  2. Chapter 3 exercise: 15
  3. Chapter 4 exercise: 2
  4. Let f(t) = t(1-t) for 0 ≤ t ≤ 1, and let f(t) = 0 otherwise. Write out the DFT approximation to the Fourier transformation for f. Give it explicitly in the n = 4 case.
• Start putting together groups for projects.

Due Monday, 4/4/2010.

Assignment 9

• Read sections 3.2.1, 4.3, 4.4, 5.1 and 5.2
• Problems.
  1. Chapter 4 exercises: 6, 7, 9 (Use the wavelet toobox for 9.)
  2. Chapter 5 exercises: 2, 8(a,b)
• Project proposals are due.

Due Monday, 4/19/2010.

Assignment 10

• Read sections 3.2.1, 4.3, 4.4, 5.1 and 5.2
• Problems.
  1. Chapter 5 exercises: 5, 8(c,d,e,f)
  2. Let the p_k's be the scaling coefficients in Example 5.8, p. 195. For these, the scaling and wavelet relations are
     φ(x) = p₀&phi(2x) + p₁&phi(2x−1) + p₂&phi(2x−2) + p₃&phi(2x−3),
     ψ(x) = p₃&phi(2x+2) − p₂&phi(2x+1) + p₁&phi(2x) − p₀&phi(2x−1).
     1. Show these p_k's satisfy the four properties in Theorem 5.9, p. 196.
     2. Find all four filters corresponding to these coefficients: low pass and high pass decomposition and reconstruction filters.
     3. Use Theorem 5.9 and the wavelet ψ(x) above to show that ∫ ψ(x)dx = 0.
     4. BONUS: Show that ∫ xψ(x)dx = 0.

Due Monday, 4/26/2010.

Assignment 11

• Read sections 5.3.3, 5.3.4, 6.1-6.3, 7.2 (We've been covering this material in class.)
• Problems.
  1. Chapter 5 exercises: 13
  2. Chapter 6 exercises; 1, 3
  3. Section 7.2: 1, 2
  4. Consider the samples of a function f(x) taken at the resolution 2⁻³ (j = 3) listed below. Suppose that you are using the db2 scaling function and wavelet.
     a³ = [2, −2, 0, 4, 5, 1, 2, 3],  k = −1 to 6. (I.e., a³₋₁ = 2, etc.)
     For each type of padding listed in section 6.3 show the padded signal required to obtain a²_k and b²_k for all possible k using at least one genuine sample

Not to be handed in.