Math for CS Fourier Transform

Slides:



Advertisements
Similar presentations
Signals and Fourier Theory
Advertisements

Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering.
Lecture 7: Basis Functions & Fourier Series
Engineering Mathematics Class #15 Fourier Series, Integrals, and Transforms (Part 3) Sheng-Fang Huang.
Aristotle University of Thessaloniki – Department of Geodesy and Surveying A. DermanisSignals and Spectral Methods in Geoinformatics A. Dermanis Signals.
Lecture 3 Laplace transform
Signals and Systems – Chapter 5
Properties of continuous Fourier Transforms
Reminder Fourier Basis: t  [0,1] nZnZ Fourier Series: Fourier Coefficient:
EECS 20 Chapter 8 Part 21 Frequency Response Last time we Revisited formal definitions of linearity and time-invariance Found an eigenfunction for linear.
Ch 6.1: Definition of Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted upon by discontinuous or impulsive.
Autumn Analog and Digital Communications Autumn
PROPERTIES OF FOURIER REPRESENTATIONS
Lecture 8: Fourier Series and Fourier Transform
EE-2027 SaS, L11 1/13 Lecture 11: Discrete Fourier Transform 4 Sampling Discrete-time systems (2 lectures): Sampling theorem, discrete Fourier transform.
中華大學 資訊工程系 Fall 2002 Chap 4 Laplace Transform. Page 2 Outline Basic Concepts Laplace Transform Definition, Theorems, Formula Inverse Laplace Transform.
Meiling chensignals & systems1 Lecture #04 Fourier representation for continuous-time signals.
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell Fourier Transforms.
DEPARTMENT OF MATHEMATI CS [ YEAR OF ESTABLISHMENT – 1997 ] DEPARTMENT OF MATHEMATICS, CVRCE.
Chapter 4 The Fourier Transform EE 207 Dr. Adil Balghonaim.
Chapter 3: The Laplace Transform
Integral Transform Dongsup Kim Department of Biosystems, KAIST Fall, 2004.
Signals and Systems Jamshid Shanbehzadeh.
Laplace Transform BIOE 4200.
ORDINARY DIFFERENTIAL EQUATION (ODE) LAPLACE TRANSFORM.
Example We can also evaluate a definite integral by interpretation of definite integral. Ex. Find by interpretation of definite integral. Sol. By the interpretation.
Fourier Transforms Section Kamen and Heck.
CISE315 SaS, L171/16 Lecture 8: Basis Functions & Fourier Series 3. Basis functions: Concept of basis function. Fourier series representation of time functions.
1 Review of Continuous-Time Fourier Series. 2 Example 3.5 T/2 T1T1 -T/2 -T 1 This periodic signal x(t) repeats every T seconds. x(t)=1, for |t|
Fourier Series. Introduction Decompose a periodic input signal into primitive periodic components. A periodic sequence T2T3T t f(t)f(t)
Chapter 5: Fourier Transform.
Chapter 4 Fourier transform Prepared by Dr. Taha MAhdy.
Course Outline (Tentative) Fundamental Concepts of Signals and Systems Signals Systems Linear Time-Invariant (LTI) Systems Convolution integral and sum.
Astronomical Data Analysis I
7- 1 Chapter 7: Fourier Analysis Fourier analysis = Series + Transform ◎ Fourier Series -- A periodic (T) function f(x) can be written as the sum of sines.
Chapter 7 The Laplace Transform
EE 207 Dr. Adil Balghonaim Chapter 4 The Fourier Transform.
Leo Lam © Signals and Systems EE235 Leo Lam.
1 Roadmap SignalSystem Input Signal Output Signal characteristics Given input and system information, solve for the response Solving differential equation.
Boyce/DiPrima 9 th ed, Ch 6.1: Definition of Laplace Transform Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William.
Frequency domain analysis and Fourier Transform
Eeng360 1 Chapter 2 Fourier Transform and Spectra Topics:  Fourier transform (FT) of a waveform  Properties of Fourier Transforms  Parseval’s Theorem.
Ch # 11 Fourier Series, Integrals, and Transform 1.
بسم الله الرحمن الرحيم University of Khartoum Department of Electrical and Electronic Engineering Third Year – 2015 Dr. Iman AbuelMaaly Abdelrahman
Fourier Transform and Spectra
The Fourier Transform.
Math for CS Fourier Transforms
EE104: Lecture 6 Outline Announcements: HW 1 due today, HW 2 posted Review of Last Lecture Additional comments on Fourier transforms Review of time window.
case study on Laplace transform
Convergence of Fourier series It is known that a periodic signal x(t) has a Fourier series representation if it satisfies the following Dirichlet conditions:
الفريق الأكاديمي لجنة الهندسة الكهربائية 1 Discrete Fourier Series Given a periodic sequence with period N so that The Fourier series representation can.
Subject : Advance engineering mathematics Topic : Fourier series & Fourier integral.
Then,  Fourier Series: Suppose that f(x) can be expressed as the following series sum because Fourier Series cf. Orthogonal Functions Note: At this point,
CHAPTER III LAPLACE TRANSFORM
Review of DSP.
Laplace Transforms.
Advanced Engineering Mathematics 6th Edition, Concise Edition
Derivative and properties of functions
Chapter 15 Introduction to the Laplace Transform
Chapter 2. Fourier Representation of Signals and Systems
UNIT II Analysis of Continuous Time signal
B.Sc. II Year Mr. Shrimangale G.W.
Fourier Analysis Lecture-8 Additional chapters of mathematics
Fourier Analysis.
Fourier Transform and Spectra
Signals & Systems (CNET - 221) Chapter-5 Fourier Transform
Discrete-Time Signal processing Chapter 3 the Z-transform
4. The Continuous time Fourier Transform
Chapter 9: An Introduction to Laplace Transforms
Review of DSP.
Presentation transcript:

Math for CS Fourier Transform Lecture 11 Fourier Transform Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Fourier Series in exponential form Math for CS Fourier Series in exponential form Consider the Fourier series of the 2T periodic function: Due to the Euler formula It can be rewritten as With the decomposition coefficients calculated as: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (1) (2) Math for CS Lecture 11

Fourier transform (3) (4) The frequencies are and Math for CS Fourier transform The frequencies are and Therefore (1) and (2) are represented as Since, on one hand the function with period T has also the periods kT for any integer k, and on the other hand any non-periodic function can be considered as a function with infinite period, we can run the T to infinity, and obtain the Riemann sum with ∆w→∞, converging to the integral: (3) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (4) Math for CS Lecture 11

Fourier transform definition Math for CS Fourier transform definition The integral (4) suggests the formal definition: The funciotn F(w) is called a Fourier Transform of function f(x) if: The function Is called an inverse Fourier transform of F(w). (5) Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. (6) Math for CS Lecture 11

Example 1 The Fourier transform of is The inverse Fourier transform is Math for CS Example 1 The Fourier transform of is The inverse Fourier transform is Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Math for CS Fourier Integral If f(x) and f’(x) are piecewise continuous in every finite interval, and f(x) is absolutely integrable on R, i.e. converges, then Remark: the above conditions are sufficient, but not necessary. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Properties of Fourier transform Math for CS Properties of Fourier transform 1 Linearity: For any constants a, b the following equality holds: Proof is by substitution into (5). Scaling: For any constant c, the following equality holds: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Properties of Fourier transform 2 Math for CS Properties of Fourier transform 2 Time shifting: Proof: Frequency shifting: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Properties of Fourier transform 3 Math for CS Properties of Fourier transform 3 Symmetry: Proof: The inverse Fourier transform is therefore Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Properties of Fourier transform 4 Math for CS Properties of Fourier transform 4 Modulation: Proof: Using Euler formula, properties 1 (linearity) and 4 (frequency shifting): Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Differentiation in time Math for CS Differentiation in time Transform of derivatives Suppose that f(n) is piecewise continuous, and absolutely integrable on R. Then In particular and Proof: From the definition of F{f(n)(t)} via integrating by parts. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Math for CS Example 2 The property of Fourier transform of derivatives can be used for solution of differential equations: Setting F{y(t)}=Y(w), we have Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Example 2 Then Therefore Math for CS Lecture 11 Math for CS Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Frequency Differentiation Math for CS Frequency Differentiation In particular and Which can be proved from the definition of F{f(t)}. Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Math for CS Convolution The convolution of two functions f(t) and g(t) is defined as: Theorem: Proof: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11

Dirac Delta Function Consider the function: Define Math for CS Dirac Delta Function Consider the function: Define It is called the Dirac delta function. It has the following properties: Outline: Central Scientific Problem – Artificial Intelligence Machine Learning: Definition Specifics Requirements Existing Solutions and their limitations Multiresolution Approximation: Limitation Our Approach. Results. Binarization. Plans. Math for CS Lecture 11