Formalization of Fourier Transform in HOL. Higher-order Logic (Rough Diamond)

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1 On the Formalization of Fourier Transform in Higher-order Logic (Rough Diamond) Adnan Rashid and Osman Hasan System Analysis and Verification (SAVe) Lab National University of Sciences and Technology (NUST) Islamabad, Pakistan ITP, 2016 Nancy, France August 22, 2016 A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 1 / 19

2 Overview 1 Introduction and Motivation 2 Formalization Details 3 Case Study (Automobile Suspension System) 4 Conclusions A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 2 / 19

3 Fourier Transform: Analogy Smoothie to Recipe Example Banana Filter Orange Filter Milk Filter Water Filter Banana Amount (1 oz) Orange Amount (2 oz) Milk Amount (3 oz) Water Amount (3 oz) Consumer Viewpoint Transformation Producer Viewpoint A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 3 / 19

4 Fourier Transform: Graphical Representation Time Domain s(t) FT Frequency Domain S( A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 4 / 19

5 Fourier Transform An integral-based transform method Mathematically, expressed as: F[f (t)] = F (ω) = + f (t)e jωt dt, ω ɛ R A linear operator Accepts a time varying non-causal function f (t) with real argument t ɛ R Outputs the corresponding ω-domain representation F (ω) with ω ɛ R A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 5 / 19

6 Fourier Transform: Utilization Solve linear Partial Differential Equations (PDEs) using simple algebraic techniques Used to conduct frequency response analysis of the continuous-time engineering and physical systems A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 6 / 19

7 Fourier Transform: Example u t = c2 2 u x 2 Second Order Linear Partial Differential Equation A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 7 / 19

8 Fourier Transform: Example u t = c2 2 u x 2 Second Order Linear Partial Differential Equation u t = c 2 u xx Equivalently written as A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 7 / 19

9 Fourier Transform: Example u t = c2 2 u x 2 Second Order Linear Partial Differential Equation u t = c 2 u xx Equivalently written as F[u t] = F[c 2 u xx ] Taking Fourier Transform on both sides A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 7 / 19

10 Fourier Transform: Example u t = c2 2 u x 2 Second Order Linear Partial Differential Equation u t = c 2 u xx Equivalently written as F[u t] = F[c 2 u xx ] Taking Fourier Transform on both sides U(ω, t) = c 2 (iω) 2 U(ω, t) = c 2 ω 2 U(ω, t) Using the Fourier of derivative of function and the linearity property A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 7 / 19

11 Fourier Transform: Example u t = c2 2 u x 2 Second Order Linear Partial Differential Equation u t = c 2 u xx Equivalently written as F[u t] = F[c 2 u xx ] Taking Fourier Transform on both sides U(ω, t) = c 2 (iω) 2 U(ω, t) = c 2 ω 2 U(ω, t) Using the Fourier of derivative of function and the linearity property t U(ω, t) = c2 ω 2 U(ω, t) Equivalent written as A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 7 / 19

12 Fourier Transform: Example u t = c2 2 u x 2 Second Order Linear Partial Differential Equation u t = c 2 u xx Equivalently written as F[u t] = F[c 2 u xx ] Taking Fourier Transform on both sides U(ω, t) = c 2 (iω) 2 U(ω, t) = c 2 ω 2 U(ω, t) Using the Fourier of derivative of function and the linearity property t U(ω, t) = c2 ω 2 U(ω, t) Equivalent written as U(ω, t) = K(ω)e c2 ω 2 t Solution in ω-domain A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 7 / 19

13 Real-world Applications for Fourier Transform Integral part of analyzing many engineering and physical systems Signal Processing Image Processing Mechanical Networks Optical Systems Antenna Designing Communication Systems A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 8 / 19

14 Transform Methods: Related Work S.H. Taqdees and O. Hasan, Formalization of Laplace Transform Using the Multivariate Calculus Theory of HOL-Light; Logic for Programing Artificial Intelligence and Reasoning (LPAR 2013), Lecture Notes in Computer Science Volume 8312, Springer, pp U. Siddique, M. Y. Mahmoud and S. Tahar, On the Formalization of Z-Transform in HOL; Interactive Theorem Proving (ITP 2014), Lecture Notes in Computer Science Volume 8558, 2014, Springer, pp A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 9 / 19

15 Transform Methods: Comparison Transform Methods Governing Equations Input Function Input Function Type Output Function Laplace Transform F (s) = 0 f (t)e st dt Continuous time domain function Semi-finite / causal function s-domain (s = a+iω) Stable / Convergent function Fourier Transform F (ω) = + f (t)e jωt dt Continuous time domain function ω-domain (s = iω) z-transform F (z) = Discrete + 0 f [n]z n time function Sampled function z-domain (z = e is ) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 10 / 19

16 Formal Definition of Fourier Transform Mathematical Definition: F[f (t)] = F (ω) = + f (t)e jωt dt, ω ɛ R Given f is piecewise smooth and is absolutely integrable on the whole real line, i.e., f is absolutely integrable on both the positive and negative real lines A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 11 / 19

17 Formal Definition of Fourier Transform Mathematical Definition: F[f (t)] = F (ω) = + f (t)e jωt dt, ω ɛ R Given f is piecewise smooth and is absolutely integrable on the whole real line, i.e., f is absolutely integrable on both the positive and negative real lines Definition: Fourier Transform w f. fourier f w = integral UNIV (λt. cexp (--((ii Cx w) Cx (drop t))) f t) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 11 / 19

18 Formal Definition of Fourier Transform Mathematical Definition: F[f (t)] = F (ω) = + f (t)e jωt dt, ω ɛ R Given f is piecewise smooth and is absolutely integrable on the whole real line, i.e., f is absolutely integrable on both the positive and negative real lines Definition: Fourier Transform w f. fourier f w = integral UNIV (λt. cexp (--((ii Cx w) Cx (drop t))) f t) Definition: Fourier Existence f w a b. fourier exists f = ( a b. f piecewise differentiable on interval [lift a, lift b]) f absolutely integrable on {x &0 <= drop x} f absolutely integrable on {x drop x <= &0} A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 11 / 19

19 Formalized Fourier Transform Properties Mathematical Form Formalized Form Existence of the Improper Integral of Fourier Transform f w. fourier exists f f (t)e jωt (λt. cexp (--((ii Cx w) Cx (drop t))) f t) integrable on (, + ) integrable on UNIV Linearity f g w a b. F[αf (t) + βg(t)] = fourier exists f fourier exists g αf (ω) + βg(ω) fourier (λt. a f t + b g t) w = a fourier f w + b fourier g w Frequency Shifting f w w0. fourier exists f F[e iω0t f (t)] = F (ω ω 0) fourier (λt. cexp ((ii Cx (w0)) Cx (drop t)) f t) w = F[cos(ω 0t)f (t)] = F (ω ω 0) + F (ω + ω 0) 2 F[sin(ω 0t)f (t)] = F (ω ω 0) F (ω + ω 0) 2i fourier f (w - w0) Modulation f w w0. fourier exists f fourier (λt. ccos (Cx w0 Cx (drop t)) f t) w = (fourier f (w - w0) + fourier f (w + w0)) / Cx (&2) f w w0. fourier exists f fourier (λt. csin (Cx w0 Cx (drop t)) f t) w = (fourier f (w - w0) - fourier f (w + w0)) / (Cx (&2) ii) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 12 / 19

20 Formalized Fourier Transform Properties Mathematical Form F[f ( t)] = F ( ω) F[ d f (t)] = iωf (ω) dt F[ dn dt n f (t)] = (iω)n F (ω) Formalized Form Time Reversal f w. fourier exists f fourier (λt. f (--t)) w = fourier f (--w) First-order Differentiation f w. fourier exists f fourier exists (λt. vector derivative f (at t)) ( t. f differentiable at t) ((λt. f (lift t)) vec 0) at posinfinity ((λt. f (lift t)) vec 0) at neginfinity fourier (λt. vector derivative f (at t)) w = ii Cx w fourier f w Higher-order Differentiation f w n. fourier exists higher deriv n f ( t. differentiable higher derivative n f t) ( p. p < n ((λt. higher vector derivative p f (lift t)) vec 0) at posinfinity) ( p. p < n ((λt. higher vector derivative p f (lift t)) vec 0) at neginfinity) fourier (λt. higher vector derivative n f t) w = (ii Cx w) pow n fourier f w 3500 lines of HOL-Light code and approximately 300 man-hours A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 13 / 19

21 Case Study: Automobile Suspension System (ASS) A mechanical system Is intended to filter out the rapid variations on the road surface To act as a low-pass filter M d2 y(t) dt 2 Differential Equation: + b dy(t) dt + ky(t) = ku(t) + b du(t) dt Frequency Response: b Y (ω) U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 14 / 19

22 Automobile Suspension System (ASS) Differential Equation: M d 2 y(t) dt 2 + b dy(t) dt + ky(t) = ku(t) + b du(t) dt A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 15 / 19

23 Automobile Suspension System (ASS) Differential Equation: M d 2 y(t) dt 2 + b dy(t) dt + ky(t) = ku(t) + b du(t) dt Definition: Differential Equation of order n n L f t. differential equation n L f t vsum (0..n) (λk. EL k L t higher order derivative k f t ) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 15 / 19

24 Automobile Suspension System (ASS) Differential Equation: M d 2 y(t) dt 2 + b dy(t) dt + ky(t) = ku(t) + b du(t) dt Definition: Differential Equation of order n n L f t. differential equation n L f t vsum (0..n) (λk. EL k L t higher order derivative k f t ) Definition: Differential Equation of ASS y u a b c. diff eq ASS y u M b k ( t. differential equation 2 [Cx k; Cx b ; Cx M] y t = differential equation 1 [Cx k; Cx b] u t) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 15 / 19

25 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

26 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

27 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

28 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

29 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

30 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

31 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

32 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

33 Automobile Suspension System (ASS) Frequency Response: Y (ω) Theorem: Frequency Response of ASS b U(ω) = M (iω) + k M (iω) 2 + b M (iω) + k M y u w a. &0 < M &0 < b &0 < k ( t. differentiable higher derivative 2 y t) ( t. differentiable higher derivative 1 u t) fourier exists higher deriv 2 y fourier exists higher deriv 1 u ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at posinfinity) ( p. p < 2 ((λt. higher vector derivative p y (lift t)) vec 0) at neginfinity) ((λt. u (lift t)) vec 0) at posinfinity ((λt. u (lift t)) vec 0) at neginfinity ( t. diff eq ASS y u a b c) (fourier u w = Cx (&0)) ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M) = Cx (&0)) (fourier y w / fourier u w = (Cx (b / M) ii Cx w + Cx (k / M)) / ((ii Cx w) pow 2 + Cx (b / M) ii Cx w + Cx (k / M)) 500 lines of HOL-Light code and the proof process took just a couple of hours A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 16 / 19

34 Conclusions Formalization of Fourier transform using higher-order logic Foundations Utilized Multivariate Calculus theory of HOL-Light Integration Differential Transcendental Topological Case Study Automobile Suspension System A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 17 / 19

35 Future Work Formalization of inverse Fourier transform Whole framework will provide solutions to differential equations Formalization of 2-dimensional Fourier transform Widely used for the analysis Optical systems Electromagnetics Image processing A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 18 / 19

36 Thanks! For More Information Visit our website Contact A. Rashid and O. Hasan Formalization of Fourier Transform in HOL 19 / 19