| import OAI.Analysis.FiniteFactor.Sampling.Fourier |
| import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds |
| import Mathlib.Algebra.Ring.GeomSum |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Finset |
| open scoped ComplexConjugate |
| noncomputable section |
|
|
| def samplingCharacter (v : ℤ) (t : ℝ) : ℂ := |
| Complex.exp (2 * Real.pi * Complex.I * v * t) |
|
|
| theorem samplingCharacter_norm (v : ℤ) (t : ℝ) : ‖samplingCharacter v t‖ = 1 := by |
| simp [samplingCharacter, Complex.norm_exp, Complex.mul_re, Complex.mul_im] |
|
|
| theorem samplingCharacter_continuous (v : ℤ) : Continuous (samplingCharacter v) := by |
| unfold samplingCharacter |
| fun_prop |
|
|
| theorem samplingCharacter_zero (t : ℝ) : samplingCharacter 0 t = 1 := by simp [samplingCharacter] |
|
|
| theorem samplingCharacter_add (v w : ℤ) (t : ℝ) : |
| samplingCharacter (v+w) t = samplingCharacter v t * samplingCharacter w t := by |
| unfold samplingCharacter |
| rw [← Complex.exp_add] |
| congr 1 |
| push_cast |
| ring |
|
|
| theorem samplingCharacter_neg (v : ℤ) (t : ℝ) : |
| samplingCharacter (-v) t = conj (samplingCharacter v t) := by |
| have htwo : conj (2 : ℂ) = 2 := by |
| change conj ((2 : ℝ) : ℂ) = _ |
| exact Complex.conj_ofReal _ |
| simp [samplingCharacter, ← Complex.exp_conj, mul_assoc, htwo] |
|
|
| theorem samplingCharacter_sub (v w : ℤ) (t : ℝ) : |
| samplingCharacter (v-w) t = samplingCharacter v t * conj (samplingCharacter w t) := by |
| rw [sub_eq_add_neg, samplingCharacter_add, samplingCharacter_neg] |
|
|
| theorem samplingCharacter_integrable (v : ℤ) : |
| Integrable (samplingCharacter v) unitIntervalMeasure := |
| (samplingCharacter_continuous v).integrableOn_Icc.mono_set Set.Ico_subset_Icc_self |
|
|
| theorem samplingCharacter_integral (v : ℤ) : |
| (∫ t, samplingCharacter v t ∂unitIntervalMeasure) = if v = 0 then 1 else 0 := by |
| by_cases hv : v = 0 |
| · subst v; simp [samplingCharacter_zero] |
| rw [ite_eq_right hv] |
| unfold unitIntervalMeasure |
| rw [restrict_Ico_eq_restrict_Ioc, ← intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)] |
| change (∫ t in (0 : ℝ)..1, Complex.exp ((2 * Real.pi * Complex.I * (v : ℂ)) * t)) = 0 |
| have hvC : (v : ℂ) ≠ 0 := by exact_mod_cast hv |
| have hcoef : 2 * (Real.pi : ℂ) * Complex.I * (v : ℂ) ≠ 0 := |
| mul_ne_zero (mul_ne_zero (mul_ne_zero (by norm_num) (Complex.ofReal_ne_zero.mpr Real.pi_ne_zero)) |
| Complex.I_ne_zero) hvC |
| rw [integral_exp_mul_complex hcoef] |
| simp only [Complex.ofReal_one, mul_one, Complex.ofReal_zero, mul_zero, Complex.exp_zero] |
| rw [show 2 * (Real.pi : ℂ) * Complex.I * (v : ℂ) = (v : ℂ)*(2*Real.pi*Complex.I) by ring, |
| Complex.exp_int_mul_two_pi_mul_I] |
| simp |
|
|
| / |
| def samplingKernel (H : ℕ) (t : ℝ) : ℝ := |
| ‖∑ r ∈ Finset.range H, samplingCharacter (r : ℤ) t‖^2 / (H : ℝ) |
|
|
| theorem samplingKernel_nonneg (H : ℕ) (t : ℝ) : 0 ≤ samplingKernel H t := by |
| unfold samplingKernel |
| positivity |
|
|
| theorem samplingKernel_continuous (H : ℕ) : Continuous (samplingKernel H) := by |
| unfold samplingKernel |
| apply Continuous.div_const |
| apply Continuous.pow |
| apply Continuous.norm |
| exact continuous_finsetSum _ (fun r _ => samplingCharacter_continuous _) |
|
|
| theorem samplingKernel_expansion (H : ℕ) (t : ℝ) : |
| (samplingKernel H t : ℂ) = (H : ℂ)⁻¹ * |
| ∑ r ∈ Finset.range H, ∑ q ∈ Finset.range H, samplingCharacter ((r : ℤ)-(q : ℤ)) t := by |
| simp only [samplingKernel, Complex.ofReal_div, Complex.ofReal_pow, Complex.ofReal_natCast, |
| samplingCharacter_sub, ← Finset.mul_sum, ← Finset.sum_mul, ← map_sum] |
| rw [← Complex.ofReal_pow, Complex.sq_norm, Complex.mul_conj] |
| ring |
|
|
| theorem samplingKernel_integral (H : ℕ) (hH : 0 < H) : |
| (∫ t, samplingKernel H t ∂unitIntervalMeasure) = 1 := by |
| have hki : Integrable (samplingKernel H) unitIntervalMeasure := |
| (samplingKernel_continuous H).integrableOn_Icc.mono_set Set.Ico_subset_Icc_self |
| have hc : (∫ t, (samplingKernel H t : ℂ) ∂unitIntervalMeasure) = 1 := by |
| simp_rw [samplingKernel_expansion] |
| rw [integral_const_mul, integral_finsetSum _ (fun r _ => |
| integrable_finsetSum _ (fun q _ => samplingCharacter_integrable _))] |
| simp_rw [integral_finsetSum _ (fun q _ => samplingCharacter_integrable _), samplingCharacter_integral] |
| simp [sub_eq_zero, -Finset.mem_range, hH.ne'] |
| rw [integral_complex_ofReal] at hc |
| exact_mod_cast hc |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
