Numerical Analysis – Digital Signal Processing Hanyang University Jong-Il Park.

Slides:

Advertisements

Similar presentations

DFT & FFT Computation.

Advertisements

Michael Phipps Vallary S. Bhopatkar

1 Fast Multiplication of Large Numbers Using Fourier Techniques Henry Skiba Advisor: Dr. Marcus Pendergrass.

Digital Kommunikationselektronik TNE027 Lecture 5 1 Fourier Transforms Discrete Fourier Transform (DFT) Algorithms Fast Fourier Transform (FFT) Algorithms.

MATLAB Session 5 ES 156 Signals and Systems 2007 HSEAS Prepared by Frank Tompkins.

Physics 434 Module 4-FFT - T. Burnett 1 Physics 434 Module 4 week 2: the FFT Explore Fourier Analysis and the FFT.

Sampling theorem, I  Suppose function h(t) is sampled at evenly spaced intervals in time; – 1/  : Sampling rate  For any sampling interval , there.

Lecture #17 INTRODUCTION TO THE FAST FOURIER TRANSFORM ALGORITHM Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh,

FFT-based filtering and the Short-Time Fourier Transform (STFT) R.C. Maher ECEN4002/5002 DSP Laboratory Spring 2003.

MSP15 The Fourier Transform (cont’) Lim, MSP16 The Fourier Series Expansion Suppose g(t) is a transient function that is zero outside the interval.

General Functions A non-periodic function can be represented as a sum of sin’s and cos’s of (possibly) all frequencies: F(  ) is the spectrum of the function.

Introduction to Fast Fourier Transform (FFT) Algorithms R.C. Maher ECEN4002/5002 DSP Laboratory Spring 2003.

Chapter 12 Fourier Transforms of Discrete Signals.

Lecture #18 FAST FOURIER TRANSFORM INVERSES AND ALTERNATE IMPLEMENTATIONS Department of Electrical and Computer Engineering Carnegie Mellon University.

Discrete Fourier Transform(2) Prof. Siripong Potisuk.

Fast Fourier Transforms

Discrete-Time and System (A Review)

Numerical Analysis - Simulation -

DISCRETE FOURIER TRANSFORM

CHAPTER 8 DSP Algorithm Implementation Wang Weilian School of Information Science and Technology Yunnan University.

CS 179: GPU Programming Lecture 9 / Homework 3. Recap Some algorithms are “less obviously parallelizable”: – Reduction – Sorts – FFT (and certain recursive.

5.6 Convolution and FFT. 2 Fast Fourier Transform: Applications Applications. n Optics, acoustics, quantum physics, telecommunications, control systems,

09/19/2002 (C) University of Wisconsin 2002, CS 559 Last Time Color Quantization Dithering.

The Discrete Fourier Transform 主講人：虞台文. Content Introduction Representation of Periodic Sequences – DFS (Discrete Fourier Series) Properties of DFS The.

Radix-2 2 Based Low Power Reconfigurable FFT Processor Presented by Cheng-Chien Wu, Master Student of CSIE,CCU 1 Author: Gin-Der Wu and Yi-Ming Liu Department.

Hossein Sameti Department of Computer Engineering Sharif University of Technology.

Real time DSP Professors: Eng. Julian S. Bruno Eng. Jerónimo F. Atencio Sr. Lucio Martinez Garbino.

Digital Signal Processing Chapter 3 Discrete transforms.

Fourier Analysis of Discrete Time Signals

Inverse DFT. Frequency to time domain Sometimes calculations are easier in the frequency domain then later convert the results back to the time domain.

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.

Fourier and Wavelet Transformations Michael J. Watts

Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W.

Professor A G Constantinides 1 Discrete Fourier Transforms Consider finite duration signal Its z-tranform is Evaluate at points on z-plane as We can evaluate.

Fast Fourier Transforms. 2 Discrete Fourier Transform The DFT pair was given as Baseline for computational complexity: –Each DFT coefficient requires.

The Discrete Fourier Transform

CS 179: GPU Programming Lecture 9 / Homework 3. Recap Some algorithms are “less obviously parallelizable”: – Reduction – Sorts – FFT (and certain recursive.

Eeng Chapter 2 Discrete Fourier Transform (DFT) Topics:  Discrete Fourier Transform. Using the DFT to Compute the Continuous Fourier Transform.

EE345S Real-Time Digital Signal Processing Lab Fall 2006 Lecture 17 Fast Fourier Transform Prof. Brian L. Evans Dept. of Electrical and Computer Engineering.

Computer Vision – 2D Signal & System Hanyang University Jong-Il Park.

Dr. Michael Nasief Digital Signal Processing Lec 7 1.

Husheng Li, UTK-EECS, Fall The specification of filter is usually given by the tolerance scheme.  Discrete Fourier Transform (DFT) has both discrete.

بسم الله الرحمن الرحيم Digital Signal Processing Lecture 14 FFT-Radix-2 Decimation in Frequency And Radix -4 Algorithm University of Khartoum Department.

Chapter 9. Computation of Discrete Fourier Transform 9.1 Introduction 9.2 Decimation-in-Time Factorization 9.3 Decimation-in-Frequency Factorization 9.4.

Discrete Fourier Transform

Chapter 4 Discrete-Time Signals and transform

Section II Digital Signal Processing ES & BM.

DIGITAL SIGNAL PROCESSING ELECTRONICS

FFT-based filtering and the

Laplace and Z transforms

Fourier and Wavelet Transformations

Fast Fourier Transform

General Functions A non-periodic function can be represented as a sum of sin’s and cos’s of (possibly) all frequencies: F() is the spectrum of the function.

2D Fourier transform is separable

4.1 DFT In practice the Fourier components of data are obtained by digital computation rather than by analog processing. The analog values have to be.

Chapter 8 The Discrete Fourier Transform

Fast Fourier Transformation (FFT)

APPLICATION of the DFT: Convolution of Finite Sequences.

Lecture 17 DFT: Discrete Fourier Transform

Chapter 9 Computation of the Discrete Fourier Transform

LECTURE 18: FAST FOURIER TRANSFORM

Chapter 2 Discrete Fourier Transform (DFT)

1-D DISCRETE COSINE TRANSFORM DCT

Chapter 8 The Discrete Fourier Transform

 = N  N matrix multiplication N = 3 matrix N = 3 matrix N = 3 matrix

Lecture 4 Image Enhancement in Frequency Domain

Fast Fourier Transform

CE Digital Signal Processing Fall Discrete Fourier Transform (DFT)

LECTURE 18: FAST FOURIER TRANSFORM

Presentation transcript:

Numerical Analysis – Digital Signal Processing Hanyang University Jong-Il Park

Division of Electrical and Computer Engineering, Hanyang University Digital Signal Processing Discrete Fourier Transform Fast Fourier Transform(FFT) Multi-dimensional FFT Convolution

Division of Electrical and Computer Engineering, Hanyang University Sampling and aliasing

Division of Electrical and Computer Engineering, Hanyang University Discrete Fourier Transform Fourier Transform Discrete Fourier Transform DFT: IDFT:

Division of Electrical and Computer Engineering, Hanyang University Fast Fourier Transform(FFT) [Danielson&Lanczos][Cooley&Tukey]

Division of Electrical and Computer Engineering, Hanyang University Decimation-in-time FFT Cooley-Tukey Algorithm

Division of Electrical and Computer Engineering, Hanyang University Sande-Tukey Algorithm

Division of Electrical and Computer Engineering, Hanyang University Decimation-in-frequency FFT(I) Sande-Tukey Algorithm

Division of Electrical and Computer Engineering, Hanyang University Decimation-in-frequency FFT (II)

Division of Electrical and Computer Engineering, Hanyang University Why FFT? Further reading:

Division of Electrical and Computer Engineering, Hanyang University Computation of FFT(I) input and output of four1() in NR in C

Division of Electrical and Computer Engineering, Hanyang University Computation of FFT(II) Eg. FFT

Division of Electrical and Computer Engineering, Hanyang University 2D FFT(I)

Division of Electrical and Computer Engineering, Hanyang University 2D FFT(II) * Generalization to L-dimension

Division of Electrical and Computer Engineering, Hanyang University 2D FFT(III) Eg. 2D FFT

Division of Electrical and Computer Engineering, Hanyang University Convolution(I) Def.

Division of Electrical and Computer Engineering, Hanyang University Convolution(II) Convolution theorem o direct convolution  complex computation o FFT and multiplication  less computation

Division of Electrical and Computer Engineering, Hanyang University Convolution(III) Convolution of discrete sampled function

Division of Electrical and Computer Engineering, Hanyang University Convolution(IV) Trouble in using DFT of finite duration  End effects  Treated by zero padding End effect

Division of Electrical and Computer Engineering, Hanyang University Convolution(V) Zero padding

Division of Electrical and Computer Engineering, Hanyang University Convolution(VI) Convolving very large data sets