• 1 Pages

Lecture 13 as presentation

Allan Hancock College, ELEC 3601

Excerpt: ... ier transform and its inverse The 2D discrete Fourier transform and its inverse Filtering in the frequency domain Some basic lters Correspondence between ltering in the spatial domain and frequency domains Periodicity and the need for padding Correlation Summary of properties of the 2D Fourier transform The convolution theorem ELEC3601/7608 Introduction to Image Formation Lecture 13 Filtering in the frequency domain I Theorem Convolution theorem: Let F (u, v ) and H(u, v ) denote the Fourier transforms of f (x, y ) and h(x, y ) respectively. The convolution f (x, y ) h(x, y ) and the product F (u, v )H(u, v ) constitute a Fourier transform pair written f (x, y ) h(x, y ) F (u, v )H(u, v ). Likewise the product f (x, y )h(x, y ) and the convolution F (u, v ) H(u, v ) constitute a Fourier transform pair written f (x, y )h(x, y ) F (u, v ) H(u, v ). Introduction Introduction and background The Fourier transform and the frequency domain The Fourier transform and its inver ...

• 1 Pages

Note to class 2_19_09

Rensselaer Polytechnic Institute, ECSE 2410

Excerpt: ... February 10, 2009. Enough students in our class have made me aware that e-mailing A#09 so late (Wed evening) created a significant scheduling problem for them in trying to do the homework in the space of a day and a half already packed with other commitments like studying for two exams, athletic competition, etc. Normally, we have to do the things we have to do in the time we have available, but in this case I could not ignore the number of requests. So I will delay A#09 until next Tuesday, Feb 24th. This does not get you off the hook. We will have a quiz tomorrow which will include Wednesdays lecture on Fourier transforms . I will also send out A#10 that will be due on Tuesday, Feb 24th. (Note. Two assignments will be due on Feb 24th.) Make sure you read my Fourier transform notes. We already covered everything thru page 12. You will find pages 6, 11, 12 especially useful for A#09. The videos are also a very good source of information on how to solve Fourier transform problems. ...

• 11 Pages

Lecture 11

UCSC, CMPE 154

Excerpt: ... Lecture 11 Fourier Transforms Modulation Truncation in time functions Fourier Series and Fourier Transform Filter Synthesis Analysis using Fourier Transforms Discrete Equivalents Difference Eqns. Amplitude Modulation ...

• 11 Pages

Lecture 11

UCSC, CMPE 154

Excerpt: ... Lecture 11 Fourier Transforms Modulation Truncation in time functions Fourier Series and Fourier Transform Filter Synthesis Analysis using Fourier Transforms Discrete Equivalents Difference Eqns. Amplitude Modulation ...

• 17 Pages

Lecture 9

UCSC, CMPE 154

Excerpt: ... Lecture 9 Fourier Transforms Differentiation Differential equations / transfer functions Impulse response / transfer function Convolution / Fourier Transform Product ...

• 17 Pages

Lecture 9

UCSC, CMPE 154

Excerpt: ... Lecture 9 Fourier Transforms Differentiation Differential equations / transfer functions Impulse response / transfer function Convolution / Fourier Transform Product ...

• 102 Pages

topic4-ft

Excerpt: ... f the CT Fourier Transform Convergence examples Fourier transform of periodic signals Properties of CT Fourier Transform Summary Appendix Transition: CT Fourier Series to CT Fourier Transform 51 Properties of the CT Fourier Transform The properties are useful in determining the Fourier transform or inverse Fourier transform They help to represent a given signal in term of operations (e.g., convolution, differentiation, shift) on another signal for which the Fourier transform is known Operations on {x(t)} Operations on {X(j)} Help find analytical solutions to Fourier transform problems of complex signals Example: FT { y (t ) = a t u (t 5) } delay and multiplication 52 Properties of the CT Fourier Transform The properties of the CT Fourier transform are very similar to those of the CT Fourier series Consider two signals x(t) and y(t) with Fourier transforms X(j) and Y(j), respectively (or X(f) and Y(f) The following properties can easily been shown using 53 Properties of the CT Fourier ...

• 12 Pages

TheFourierTransformAndItsApplications-Lecture13

Stanford, LSOFTAEE 261

Excerpt: ... is. To define a distribution you need a class, first of all, a class of test functions. So the setup is, you first need you first have to define a class of test functions, or test signals that usually have particularly nice properties for the given problem at hand. And it can vary from problem to problem. For us, for the Fourier transform, it's the class of rapidly decreasing functions. So these typically have particularly nice properties. Sorry for not specifying that terribly carefully. But they come, generally, out of again, sort of, years of bitter experience with working with problems, working with a particular class of applications and trying to decide what the best functions are for the given class. For Fourier transforms , the class of test functions is the rapidly decreasing function, so I won't write down the definition again, but I'll remind you of the main properties in just a minute when we need it. Rapidly decreasing functions these are the functions which are infinitely differentiable an ...

• 5 Pages

Lecture 10

UCSC, CMPE 154

Excerpt: ... Lecture 10 Fourier Transforms Convolution (revisited) Properties of Fourier Transforms Applications Parsevals Theorem Frequency Domain Filtering MATLAB and Fourier Series validation for Full-wave rectified sine wave ...

• 5 Pages

Lecture 10

UCSC, CMPE 154

Excerpt: ... Lecture 10 Fourier Transforms Convolution (revisited) Properties of Fourier Transforms Applications Parseval's Theorem Frequency Domain Filtering MATLAB and Fourier Series validation for Full-wave rectified sine wave ...

• 1 Pages

FTsample

Wright State, EE 321

Excerpt: ... EE321 Fourier Transform Sample Problems Summer 2005 Instructor: Kefu Xue, Ph.D. Instructions: You are permitted to use a self-prepared study-guide limited to two (8 1 11) page (both sides). Show 2 all the intermediate steps for credits. 1. Given a ...

• 5 Pages

lecture8

Utah State, ECE 3640

Excerpt: ... riety of other basis functions for other useful representations. Fourier transforms which can be used to examine frequency response of signals. By means of their properties, we are also lead to consider concepts such as modulation. Fourier transforms do not really address the stability issues that Laplace transforms do, nor can they be used as conveniently for transient analysis. However, by not starting at t = 0, they simplify some other issues. Two more transforms are introduced: The Discrete-time Fourier Transform is to the Z-transform what the Fourier transform is to the Laplace transform. That is, we have an exact frequency component representation of signals that are not periodic by evaluating a (possibly two-sided) Z-transform at z = ej . The DTFT is the study of this set of lecture notes. The Discrete Fourier Transform (DFT) can be used to compute a transform of a finite-length discretely-sampled set of data. The DFT can be used for computational signal analysis, and its implementation in the form of ...

• 6 Pages

EE 303 lab3

SUNY Buffalo, EE 303

Excerpt: ... EE 303 MATLAB Laboratory Experiment: Fourier Transform The Fourier Transform of () is defined by: Spring 2008 = ()exp() Similarly, the inverse Fourier Transform is defined by: = 1 2 ()exp() In MATLAB, the Fourier Transform can be nume ...

• 3 Pages

hw7

Cornell, ECE 2200

Excerpt: ... ECE220 Signals and Information Spring 2008 Homework 7: Due Monday, April 7, at 10:08pm Drop your homework in the collection box marked "ECE220 Spring 2008, homework", located on the second floor of Phillips at the south entrance to 219 Phillips. ...

• 4 Pages

jun1

DePaul, PHY 301

Excerpt: ... PHY 301 (June 1, 2009) Lecture 27 Chapter 10: Fourier Transforms In addition to solving linear second order PDEs with semi infinite and infinite boundaries, the Fourier Transform finds applications in fields from Digital Signals Processing to Image Processing in Astronomy. The Fourier Transform of f (x) is F () = Then, the Inverse Fourier Transform is f (x) = - 1 2 - f (x) eix dx (10.3.6) F () e- ix d (10.3.7) Note:- Some authors use 1 in both in order to make the inverse transform look like the Fourier 2 transform - when using tables, always check the definition. Example Problem: Find the Fourier transform of 1, | x| < 5 f (x) = 0, | x| > 5 The Fourier Transform is F () = 1 2 - 5 2 Function plot f (x) eix dx 1 f(x) 0 1 = 2 1 = 2 (1) eix dx + 0 -5 eix i 5 -1 -8 -6 -4 -2 -5 0 2 x Fourier transform plot 4 6 8 2 = F () = & 2isin(5) & F() 0 -1 -8 ei5 - e- i5 = 2(i) 2() i sin(5) 1 -6 -4 -2 0 2 4 6 8 PHY 301 June 1, 2009 An Important Result: The Fourier Transform o ...

• 1 Pages

mar21

Wisconsin, ECE 330

Excerpt: ... Began chapter 4, getting formula for general (possibly aperiodic) signal x(t) = (1/2 \pi) \int X(jw) exp(jwt) dt where X(jw) is the Fourier transform X(jw) = \int x(t) exp(-jwt) dw. Computed several Fourier transforms . Introduced sinc function, as Fourier transform of square pulse. Noted the duality in computing Fourier/inverse Fourier transforms so that the Fourier transform of a sinc function is a square pulse. Obtained elementary properties (linearity, time-shift, conjugate) - their effect on the Fourier transform. ...

• 3 Pages

Lecture10

UCSD, CASS 246

Excerpt: ... Lecture 10: Fourier Transforms If a function is not periodic and is not defined on a finite interval, we can re-interpret it as a periodic function of infinite period. That would give a fundamental frequency equal to zero. The way to do this right is to compute the limit of the Fourier series for T, or a-, b+ and w0: We have put this in a form that allows us to relate to the Riemann sum that defines an integral: Y- gHwL , w =limDw0 Dw /n=- gHn DwL f HtL = a Y- f HwL - w t , w cn - nD w t f HtL = limDw0 a Dw / n=- aD w We get: where now w is not the fundamental frequency anymore, but rather a real variable that can take any frequency value. And we have defined: which is the equivalent of the old Fourier coefficient, but it is devided by (a Dw) and it is now a function of a real variable w. Given our previous definition for the Fourier coefficients, we get: f Hn DwL cn Ha DwL b where we have used Dw = 2 p T. b 1 f Hn DwL =lima- Ya f Ht ...