Slides:

Advertisements

Similar presentations

Image Processing IB Paper 8 – Part A

Advertisements

SIGNAL AND SYSTEM CT Fourier Transform. Fourier’s Derivation of the CT Fourier Transform x(t) -an aperiodic signal -view it as the limit of a periodic.

Sep 15, 2005CS477: Analog and Digital Communications1 Modulation and Sampling Analog and Digital Communications Autumn

Week 14. Review 1.Signals and systems 2.State machines 3.Linear systems 4.Hybrid systems 5.LTI systems & frequency response 6.Convolution 7.Fourier Transforms.

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.

lecture 5, Sampling and the Nyquist Condition Sampling Outline  FT of comb function  Sampling Nyquist Condition  sinc interpolation Truncation.

Basics of Digital Filters & Sub-band Coding Gilad Lerman Math 5467 (stealing slides from Gonzalez & Woods)

Overview of Sampling Theory

Image reproduction. Slice selection FBP Filtered Back Projection.

Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 7: Review, Reconstruction Ravi Ramamoorthi

APPLICATIONS OF FOURIER REPRESENTATIONS TO

Life in the frequency domain Jean Baptiste Joseph Fourier ( )

A-A+ B-B+ C-C+ F. Week 11. Chapter 10 Fourier Transforms 1.The Four Fourier transforms CTFT DTFT FS DFT/S 2.Relationships 3.Important properties.

Analogue and digital techniques in closed loop regulation applications Digital systems Sampling of analogue signals Sample-and-hold Parseval’s theorem.

Leo Lam © Signals and Systems EE235. Transformers Leo Lam ©

Lecture 4: Sampling [2] XILIANG LUO 2014/10. Periodic Sampling  A continuous time signal is sampled periodically to obtain a discrete- time signal as:

The Nyquist–Shannon Sampling Theorem. Impulse Train  Impulse Train (also known as "Dirac comb") is an infinite series of delta functions with a period.

Sampling and Antialiasing CMSC 491/635. Abstract Vector Spaces Addition –C = A + B = B + A –(A + B) + C = A + (B + C) –given A, B, A + X = B for only.

… Representation of a CT Signal Using Impulse Functions

Signals and Systems Jamshid Shanbehzadeh.

The Wavelet Tutorial: Part3 The Discrete Wavelet Transform

Words + picturesPicturesWords Day Day

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|

1 Lab. 4 Sampling and Rate Conversion  Sampling:  The Fourier transform of an impulse train is still an impulse train.  Then, x x(t) x s (t)x(nT) *

Chapter 5 Z Transform. 2/45  Z transform –Representation, analysis, and design of discrete signal –Similar to Laplace transform –Conversion of digital.

EE104: Lecture 5 Outline Review of Last Lecture Introduction to Fourier Transforms Fourier Transform from Fourier Series Fourier Transform Pair and Signal.

Systems (filters) Non-periodic signal has continuous spectrum Sampling in one domain implies periodicity in another domain time frequency Periodic sampled.

Chapter 4 Fourier transform Prepared by Dr. Taha MAhdy.

Notice  HW problems for Z-transform at available on the course website  due this Friday (9/26/2014) 

Lecture 7: Sampling Review of 2D Fourier Theory We view f(x,y) as a linear combination of complex exponentials that represent plane waves. F(u,v) describes.

Digital Image Processing DIGITIZATION. Summery of previous lecture Digital image processing techniques Application areas of the digital image processing.

11/20/2015 Fourier Series Chapter /20/2015 Fourier Series Chapter 6 2.

CMSC 635 Sampling and Antialiasing. Aliasing in images.

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

Leo Lam © Signals and Systems EE235 Leo Lam.

1 Reconstruction Technique. 2 Parallel Projection.

Leo Lam © Signals and Systems EE235. Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.

EE104: Lecture 11 Outline Midterm Announcements Review of Last Lecture Sampling Nyquist Sampling Theorem Aliasing Signal Reconstruction via Interpolation.

ABE425 Engineering Measurement Systems Fourier Transform, Sampling theorem, Convolution and Digital Filters Dr. Tony E. Grift Dept. of Agricultural.

Prof. Nizamettin AYDIN Advanced Digital Signal Processing 1.

4.A Fourier Series & Fourier Transform Many slides are from Taiwen Yu.

2D Fourier Transform.

Lecture 2 Outline “Fun” with Fourier Announcements: Poll for discussion section and OHs: please respond First HW posted 5pm tonight Duality Relationships.

Coherence spectrum (coherency squared)  = 0.1, 0.05, 0.01 X is the Fourier Transform Cross-spectral power density Confidence level: = running mean or.

2. Multirate Signals.

Dr S D AL_SHAMMA Dr S D AL_SHAMMA11.

DIGITAL FILTERS h = time invariant weights (IMPULSE RESPONSE FUNCTION)

Sampling and Quantization

Sampling and Reconstruction

(C) 2002 University of Wisconsin, CS 559

Gonzalez & Woods, and Efros)

Chapter 15 Introduction to the Laplace Transform

Montek Singh Thurs., Feb. 19, :30-4:45 pm, SN115

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.

Discrete Fourier Transform (DFT)

Image Acquisition.

Changing the Sampling Rate

Lecture 6 Outline: Upsampling and Downsampling

Sampling and the Discrete Fourier Transform

Advanced Digital Signal Processing

Fourier Series: Examples

Coherence spectrum (coherency squared)

Discrete Fourier Transform

Discrete Fourier Transform

Signals and Systems EE235 Leo Lam ©

Chapter 3 Sampling.

Digital Signal Processing

Discrete Fourier Transform

Signals and Systems Using MATLAB Luis F. Chaparro

Presentation transcript:

M.H. Perrott©2007Downsampling, Upsampling, and Reconstruction, Slide 37 Downsampling, Upsampling, and Reconstruction A-to-D and its relation to sampling Downsampling and its relation to sampling Upsampling and interpolation D-to-A and reconstruction filtering Filters and their relation to convolution Copyright © 2007 by M.H. Perrott All rights reserved.

Hülya Yalçın ©38

Hülya Yalçın ©39

Hülya Yalçın ©40

Hülya Yalçın ©41 Understanding how Fourier Series (FS) goes to Fourier Transform (FT)... In time domain, making a signal periodic corresponds to convolving it with a COMB function. The fourier transform of a COMB function is again a COMB function. So, in time domain it corresponds to sampling the frequency domain signal!

Hülya Yalçın ©42

Hülya Yalçın ©43

Hülya Yalçın ©44

Hülya Yalçın ©45

Hülya Yalçın ©46

Hülya Yalçın ©47

Hülya Yalçın ©48

Hülya Yalçın ©49