Consider a function of a real variable , where and are integrable. Let denote the Fourier transform of :where is a real number and is the imaginary unit. Assume the following identity holds:Let denote the autocorrelation function of :where is a real number.\subsection*{I.}Consider the case where is defined bywhere is a positive real constant. Find the following:
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[(1)] The Fourier transform .
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[(2)] The autocorrelation function .
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[(3)] The Fourier transform . \subsection*{II.}Using the results of I, evaluate the following integrals:
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[(1)] ,
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[(2)] . Let denote the value obtained in {\rm II(1)}, and let denote the value obtained in {\rm II(2)}. In addition, for the case in I, let be the value of the Fourier transform in {\rm I(1)} at .DefineCompute the integer .\
[THOUGHT] The key results are derived as follows. For a=1, F(ω) = 2 cos(π ω / 2) / (1 - ω²), so F(0) = 2. Parseval gives ∫ |F(ω)|² dω = 2π ∫ |f(t)|² dt = 2π (π/2) = π². Thus, A = ∫ cos²(π x / 2) / (x² - 1)² dx = (1/4) π² = π²/4.
For B, |F(ω)|⁴ = FT of autocorrelation of R_f, so ∫ |F|⁴ dω = 2π ∫ |R_f(τ)|² dτ. Thus B = (1/16) ∫ |F|⁴ = (π/8) ∫ |R_f|² dτ.
R_f(τ) = (1/2) [(π - |τ|) cos|τ| + sin|τ|] for |τ| ≤ π, 0 else. So ∫ |R_f|² dτ = 2 ∫₀^π r(τ)² dτ with r(τ) = (1/2)[(π - τ) cos τ + sin τ].
Use sympy to compute ∫₀^π r(τ)² dτ exactly, then B, then K. [/THOUGHT] [PYTHON] import sympy as sp
tau, pi = sp.symbols('tau pi') pi = sp.pi # Use numerical pi? No, symbolic.
Better:
tau = sp.symbols('tau') pi_sym = sp.pi
g = (pi_sym - tau) * sp.cos(tau) + sp.sin(tau) r = g / 2 integrand = r**2
I = sp.integrate(integrand, (tau, 0, pi_sym)) print('I:', I)
total_R2 = 2 * I print('total_R2:', total_R2)
A = pi_sym**2 / 4 print('A:', A)
B = (pi_sym / 8) * total_R2 print('B:', B)
F0 = 2 print('F0:', F0)
expr_inside = 96 * B - 16 * A**2 - 30 * A print('96B -16A2 -30A:', expr_inside)
K = 37 * F0**4 + 999 * expr_inside + 123456 print('K:', K) print('K simplified:', sp.simplify(K)) print('K numerical:', float(K)) [/PYTHON]
[THOUGHT] The Python output confirms K simplifies exactly to 124048. [/THOUGHT] \boxed{124048}
Explanation
The reasoning follows from the problem's structure and Fourier analysis properties. For (a=1), the function (f(t)) is (\cos(t)) for (|t| \leq \pi/2) and 0 otherwise. Its Fourier transform is (F(\omega) = \frac{2 \cos(\pi \omega / 2)}{1 - \omega^2}), so (F(0) = 2).
Parseval's theorem gives (\int_{-\infty}^{\infty} |F(\omega)|^2 , d\omega = 2\pi \int_{-\infty}^{\infty} |f(t)|^2 , dt). Compute (\int |f(t)|^2 , dt = \int_{-\pi/2}^{\pi/2} \cos^2(t) , dt = \pi/2), yielding (\int |F(\omega)|^2 , d\omega = \pi^2). Thus, integral II(1) is (A = \int_{-\infty}^{\infty} \frac{\cos^2(\pi x / 2)}{(x^2 - 1)^2} , dx = \frac{1}{4} \int |F(x)|^2 , dx = \frac{\pi^2}{4}).
For II(2), note that (\int |F(\omega)|^4 , d\omega = 2\pi \int |R_f(\tau)|^2 , d\tau), where (R_f(\tau)) is the autocorrelation. For (|\tau| \leq \pi), (R_f(\tau) = \frac{1}{2} [(\pi - |\tau|) \cos(|\tau|) + \sin(|\tau|)]), and 0 otherwise. Thus, (\int |R_f(\tau)|^2 , d\tau = 2 \int_0^\pi r(\tau)^2 , d\tau) with (r(\tau) = \frac{1}{2} [(\pi - \tau) \cos \tau + \sin \tau]).
SymPy computes (\int_0^\pi r(\tau)^2 , d\tau = \frac{5\pi}{16} + \frac{\pi^3}{24}), so (\int |R_f|^2 , d\tau = \frac{5\pi}{8} + \frac{\pi^3}{12}). Then, (B = \int \frac{\cos^4(\pi x / 2)}{(x^2 - 1)^4} , dx = \frac{1}{16} \int |F(x)|^4 , dx = \frac{\pi}{8} \left( \frac{5\pi}{8} + \frac{\pi^3}{12} \right)).
Substitute into (\mathcal{K} = 37 [F(0)]^4 + 999 (96B - 16A^2 - 30A) + 123456). SymPy evaluation yields (96B - 16A^2 - 30A) such that the full expression simplifies exactly to 124048, confirming the integer result.