Math 658 - Fall 2008

Assignment 1 - Due Wednesday, September 3, 2008

• Read sections 1.3, 2.1-2.3.
• Problems
  1. Verify the Fourier series for the functions in entries 1, 5, 6, 13, 18, 19 of Table 1, p. 26-28.
  2. Consider the trig polynomial T(x) = ∑_|n|≤N d_n e^inx.
     1. Show that the Fourier coefficents of T are just the d_n's for |n|≤ N and 0 for |n|> N.
     2. Show that if f has the Fourier series ∑_n c_n e^inx, then
        (2π)^-1 ∫₀^2π f(x)T(x)dx = ∑_|n|≤N c_nd_n.
        
     3. Show that (2π)^-1 ∫₀^2π |T(x)|²dx = ∑_|n|≤N |d_n |², so Bessel's inequality becomes an equation for trig polynomials.
     4. Show that if f has the Fourier series ∑_n c_n e^inx, then
        (2π)^-1 ∫₀^2π|f(x) − T(x)|²dx = (2π)^-1 ∫₀^2π|f(x)|²dx − ∑_|n|≤N |c_n|² + ∑_|n|≤N |d_n − c_n|²

Assignment 2 - Due Friday, September 12, 2008

• Read sections 2.4-2.6.
• Problems
  1. Section 2.2: 2, 7
  2. Section 2.3: 2(a), 5, 7(b,c)
  3. Consider the Fourier series in #17 in Table 1 on p. 28. Estimate the number of terms your would need to take to get the (unifrom) error below 10⁻⁵.
  4. Let f and g be piecewise continuous, 2π periodic functions. Define the convolution
     f∗g(θ) := (2π)^-1 ∫₀^2πf(θ -φ)g(φ)dφ.
     Let f and g have the Fourier series ∑_n c_n e^inθ and ∑_n d_n e^inθ, respectively.
     1. Show that the Fourier series for f∗g is ∑_n c_n d_n e^inθ.
     2. Show that this series is uniformly convergent. (Hint. You will need the M-test, Schwarz's inequality and Bessel's inequality.)
     3. Verify that the Fourier series for f(−θ) is ∑_n c_n e^inθ. Using this series and assuming that f∗g(θ) = ∑_n c_n d_n e^inθ, show that Parseval's (Plancheral's) equation holds for piecewise continuous f:
        (2π)^-1 ∫₀^2π|f(φ)|²dφ = ∑_n |c_n|²
  5. Use the same notation as in the previous problem. Define the function H(θ) := f(θ)g(θ). Let ∑_n C_n e^inθ be the Fourier series for H. Show that C_n = ∑_k c_n−k d_k . In addition, show that for n fixed the series ∑_k c_n−k d_k is absolutely convergent. (If we consider the sequences c = {c_n} and d = {d_n}, the sequence C ={C_n} is called the convolution of c and d. We write C=c∗d. Notice the similarity to f∗g.)

Assignment 3 - Due Wednesday, October 1, 2008

• Read sections 3.1-3.3, 7.1.
• Problems
  1. Section 2.6: 1
  2. Let g_N(θ) be the function defined in the problem above on p. 61; also, let a, b be fixed and satisfy 0 < a < b < 2π. Show that g_N converges to 0 uniformly on the interval [a,b].
  3. Let Σ_n(u) be the Cesàro mean of the Dirichlet kernels D_k(u). (See pgs. 60 and 34 for definitions.) Show that
     Σ_n(u) = [2π(n+1) sin²(½u)]^{− 1} sin²(½(n+1)u).
     
  4. Let σ_n(θ) be the Cesàro mean of the partial sums s_n(θ) of a Fourier series for a PC function f.
     1. Show that σ_n(θ) = ∑_{|k|≤ n}(1-|k| ∕(n+1)) c_ke^ikθ.
     2. Show that σ_n(θ) is also given by the convolution (see Assignment 2)
        σ_n(θ) = 2πΣ_n ∗f(θ).
        
     3. Bonus: show that if f is continuous, then σ_n converges uniformly to f.
  5. Let K_a(θ) be the 2π periodic extension of the function defined in entry #12 in Table 1 on p. 28, and let f be a 2π periodic PS function.
     1. For θ fixed, show that K_a∗f(θ) is, in the sense of integrals, the average of f over the interval [θ - a, θ + a].
     2. Given the complex form of the Fourier series for f, use the results from Assignment 2 to find the complex form of the Fourier series for K_a∗f.
     3. Let f be the 2π-periodic extension of the function f(θ) = 1, 0 < θ < π, and f(θ) = − 1, − π < θ < 0. At the discontinuities, just take f to be 0. Let the parameter a satisfy 0 < a < π/8. Find the Fourier series of K_a∗f. By doing the integrals involved, directly calculate the function K_a∗f. Make a sketch of several periods of K_a∗f for different values of a.
  6. Show that ℓ₂, the set of bi-infinite sequences having finite energy, is a Hilbert space. The inner product for this space is
     < x, y > = ∑_k x_k y_k .

Assignment 4 - Due Friday, October 24, 2008

• Read sections 7.5, 7.6 (DFT), and 9.1.
• Problems
  1. Section 7.1: 2, 3(a), 5 (You may use the FT and convolution theorem.)
  2. Section 7.2: 4, 6, 13(a,b)
  3. Section 7.3: 3, 8, 10
  4. Show that if a linear filter L is causal (cf. Linear Filters) and if the signal f(t) = 0 for t < 0, then
     L[f](t) = ∫₀^t f(t - τ)h(τ)dτ.

Assignment 5 - Due Friday, November 7, 2008

• Read sections 9.1 and 9.4.
• Problems
  1. Let f(t) be band-limited, with its Fourier transform f^(ω) (f "hat") having support in [−2π, 2π]. The Nyquist rate for f is 2π/π = 2, and the Nyquist frequency is ½·2 = 1.
     1. Show that if f is sampled at the Nyquist frequency, instead of the Nyquist rate, then the sampling theorem doesn't recover f. Instead, it gives another band-limited function, f_alias. Show that on [−π, π],
        f^_alias(ω) = f^(ω − 2π) + f^(ω) + f^(ω + 2π),
        and that f^_alias = 0 otherwise.
     2. Referring to the previous part, consider the (single-frequency) band-limited function
        f(t) = sin(5πt/3+π/6),
        whose angular frequency 5π/3 is in the band [−2π, 2π]. What is f_alias(t) in this case? Use your favorite software to make a plot showing f(t), f_alias(t) on the interval − 6 ≤ t ≤ 6, and also use circles or x's to indicate samples of these two functions at integer values of t.
  2. Section 7.5: 2, 3, 6(a,b).
  3. Find the n-dimensional Fourier transform of the Gaussian, g(r) =(2π)^{− n/2} exp(− r²/2), where r = |x|.
  4. Prove the three formulas listed at the end of section 2 on p. 3 of the Notes on the Discrete Fourier Transform.
  5. Section 9.4: 3.

Updated 10/29/08 (fjn)