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] Use the given Fourier convention. For a=1 we can compute F(ω) explicitly; then |F(ω)|^2 matches the integrand in II(1) up to a constant, so Parseval gives A. For B, use Wiener–Khinchin: ℱ[R_f]=|F|^2, then Parseval on R_f gives ∫|F|^4 in terms of ∫R_f(τ)^2, which we can compute from an explicit piecewise formula for R_f(τ) (nonzero only for |τ|<π). Finally compute K from A,B and F(0). [/THOUGHT] [PYTHON] import sympy as sp
Symbols
t, w, tau = sp.symbols('t w tau', real=True) pi = sp.pi
Define f(t) for a=1: cos(t) on [-pi/2, pi/2], else 0.
Fourier transform computed analytically:
F = 2sp.cos(piw/2)/(1-w**2) # for all real w, with removable singularities at w=±1 F0 = sp.simplify(F.subs(w, 0))
Part II(1): A from Parseval: integral |F|^2 = 4A
I_f2 = sp.integrate(sp.cos(t)**2, (t, -pi/2, pi/2)) A = sp.simplify((2piI_f2)/4)
Autocorrelation R_f(tau) for 0<=tau<pi: overlap [tau-pi/2, pi/2]
R_pos = sp.simplify(sp.integrate(sp.cos(t)*sp.cos(t-tau), (t, tau-pi/2, pi/2)))
Extend even and zero outside |tau|<pi:
We'll compute integral of R^2 directly using symmetry.
R2_int = sp.simplify(2*sp.integrate(sp.simplify(R_pos**2), (tau, 0, pi)))
Parseval on R: ∫|F|^4 dw = 2π ∫ R^2 dτ
F4_int = sp.simplify(2piR2_int)
B relates: |F|^4 = 16 * cos^4(pi w/2)/(w^2-1)^4, so B = (1/16)*∫|F|^4
B = sp.simplify(F4_int/16)
Compute K
K_expr = sp.simplify(37F0**4 + 999(96B - 16A**2 - 30*A) + 123456)
print("F(w) =", sp.simplify(F)) print("F(0) =", F0) print("Integral of cos^2 on [-pi/2,pi/2] =", I_f2) print("A =", A) print("R_pos(tau) =", R_pos) print("Integral R^2 =", R2_int) print("B =", B) print("K_expr simplified =", K_expr) print("K as integer? =", sp.nsimplify(K_expr)) print("K numeric approx =", float(sp.N(K_expr, 30))) print("K exact rational form =", sp.together(K_expr)) [/PYTHON]
[THOUGHT] Using the verified computed values (F(0)=2), (A=\pi^2/4), (B=\pi^2(15+2\pi^2)/192), the expression for (\mathcal{K}) simplifies exactly to the integer below. [\boxed{124048}]