Topic 6. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)
|
|
- Lionel Pope
- 5 years ago
- Views:
Transcription
1 Topic 6 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)
2 Why bother? The ear processes sound by decomposing it into sine waves at different frequencies. A machine that does the same would be a step towards one that hears things as humans do. So how do we do this by machine? Peripheral Auditory system 1100 Hz Cochlea, Auditory nerve 660 Hz 220 Hz
3 Jean Baptiste Joseph Fourier A French mathematician and physicist who lived from presented a paper in 1807 to the Institut de France claiming any continuous periodic signal could be represented as the sum of properly chosen sinusoidal waves. Among the reviewers were two famous mathematicians Joseph Louis Lagrange ( ) Pierre Simon de Laplace ( ) Lagrange said sine waves could not perfectly represent signals with discontinuous slopes, like square waves. (He was, technically, right) Thus, the Institut de France did not publish Fourier s work until 15 years later, after Lagrange died.
4 The Fourier Series Given a periodic function x(t). x( t mt) x( t) m Integers the period A series of sine and cosine functions reproduces x(t) 0 n n n n n1 x t A A t B t ( ) cos( ) sin( ) Where the frequency n of each sine and cosine is an integer multiple (n) of a fundamental frequency o n 2 n / T n 0
5 Fundamental Frequency The lowest frequency that a sine or cosine can have and still fit exactly into one period of the function. f / 2 1/ 0 0 T
6 Fourier Series Quite a few! The frequency of this component x() t A cos( t) n n A 0 B sin( t) n1 n n The original signal Amplitude: the contribution of this component
7 Kinds of Fourier Transforms Fourier Transform Signals are continuous and aperiodic Fourier Series Signals are continuous and periodic Discrete Time Fourier Transform Signals are discrete and aperiodic What we use! Discrete Fourier Transform Signals are discrete and periodic
8 AMPLITUDE Digital Sampling An analog signal is sampled into sequence of discrete sample points, x[n] sample interval quantization increment TIME
9 Window function A function that is zero-valued outside of some chosen interval. When a signal (data) is multiplied by a window function, the product is zero-valued outside the interval: all that is left is the "view" through the window.
10 amplitude amplitude amplitude Some famous windows Rectangular wn 1 sample Note: we assume w[n] = 0 outside some range [0,N] Hann (Julius von Hann) wn Bartlett w[ n] 2 n 0.51cos N 1 N 2 1 N n N 2 sample sample
11 Windowing x[n] is windowed by function w[n] (multiply the ith value of x by the ith value of w) x[n] w[n] z[n] x = Example: windowing x[n] with a rectangular window
12 Why window shape matters Don t forget that a DFT assumes the signal in the window is periodic The boundary conditions mess things up unless you manage to have a window whose length is exactly 1 period of your signal Making the edges of the window less prominent helps suppress undesirable artifacts
13 Only what s in the window Do the DFT only on the values in the window z[n] x[n] w[n] Ignore Ignore x =
14 Discrete Fourier Transform Represents a finite sequence of complex values as a finite number of discrete real and imaginary sinusoids Time domain xn [ ] Real portion 0 N-1 Imaginary portion 0 N-1 DFT IDFT Frequency domain Xn [ ] Real portion 0 N/2 N-1 Imaginary portion 0 N/2 N-1
15 Discrete Fourier Transform A series of complex amplitudes in the time domain become a series of complex amplitudes in the frequency domain Time domain xn [ ] Real portion 0 N-1 Imaginary portion 0 N-1 DFT IDFT Frequency domain Xk [ ] Real portion 0 N/2 N-1 Imaginary portion 0 N/2 N-1
16 Discrete Fourier Transform If the time-domain signal has no imaginary part (like an audio signal) then the frequency-domain signal is symmetric around N/2..so we can (mostly) ignore frequencies over N/2 Time domain xn [ ] Real portion 0 N-1 Imaginary portion 0 N-1 DFT IDFT Frequency domain Xk [ ] Real portion 0 N/2 N-1 Imaginary portion 0 N/2 N-1
17 Some numbers The highest frequency represented depends on the sample rate of the signal Number of points in the sample window More sample points = finer frequency resolution Sample rate n s 1 frequencies spaced evenly from 0 to 2 2
18 2 Fundamental Frequencies Fundamental frequency of analysis : based number of points in the window & the sample rate Fundamental frequency of the signal : based on the period of the signal. n s 1 frequencies spaced evenly from 0 to 2 2
19 Questions If n = 16 and s = 8000 Hz, what is the fundamental frequency of analysis? what frequencies are represented in the DFT? What if n = 32 and s = 8000 Hz? What if n = 32 and s = Hz? n s 1 frequencies spaced evenly from 0 to 2 2
20 Discrete Fourier Transform A 16 point real-valued sequence is represented by (yes we re ignoring the ones above N/2) (16/2)+1 = 9 Cosine Waves and (16/2)+1 = 9 Sine Waves
21 Complex Numbers z x iy A y A cos isin x x Acos y Asin 2 2 A x y
22 Euler s Formula Useful for relating polar coordinates to rectangular coordinates e i cos i sin Thus... z Ae i PHASE AMPLITUDE
23 Multiplying Complex Numbers POLAR notation EASIER for this i z Ae z A e i z A A e i 1 2
24 Multiplying Complex Numbers Cartesian works as follows z x iy z x iy z x x y y i x y x y
25 Complex Conjugate the complex conjugate of a complex number is given by changing the sign of the imaginary part. A complex number Its complex conjugate z a ib z a ib
26 Back to the DFT Discrete Fourier Transform Inverse Discrete Fourier Transform What s different between them N 1 X[ k] x[ n] e n0 N 1 k 0 2i kn N 1 x[ n] X[ k] e N 2i N kn
27 Put in Cartesian Coordinates N 1 N 1 X[ k] x[ n] e n0 2i kn N 2kn 2kn x[ n] cos isin N N n0 REMEMBER EULER S FORMULA REMEMBER COMPLEX MULTIPLICATION
28 You can code this up! N 1 2kn 2kn X[ k] x[ n] cos isin n0 N N Time domain xn [ ] Real portion 0 N-1 Imaginary portion 0 N-1 DFT IDFT Frequency domain Xk [ ] Real portion 0 N/2 N-1 Imaginary portion 0 N/2 N-1
29 The Inverse DFT N 1 1 x[ n] X[ k] e N complex number k 0 2i kn N N 1 1 2kn 2kn X[ k] cos isin N N N k 0 Seriously, remember complex multiplication here. You ll need it.
30 You can code this up! N 1 1 2kn 2kn x[ n] X[ k] cos isin N k0 N N Time domain xn [ ] Real portion 0 N-1 Imaginary portion 0 N-1 DFT IDFT Frequency domain Xk [ ] Real portion 0 N/2 N-1 Imaginary portion 0 N/2 N-1
31 What about N/2? I said that you can (mostly) ignore frequencies over N/2. What is up with that? Let s have a look. When the input signal in the time domain x[n] is all real values, the signal in the frequency domain X[n] is symmetric around N/2
32 Computational complexity How many operations does this take for each frequency? How many operations total? N 1 2kn 2kn X[ k] x[ n] cos isin N N n0
33 The FFT Fast Fourier Transform A much, much faster way to do the DFT Introduced by Carl F. Gauss Rediscovered by J.W. Cooley and John Turkey in 1965 The Cooley-Turkey algorithm is the one we use today (mostly) Big O notation for this is O(N log N) Matlab functions fft and ifft are standard.
34 Short time Fourier Transform Break signal into windows Calculate DFT of each window
35 The Spectrogram spectrogram(y,256,128,256,fs,'yaxis'); A series of short term DFTs Typically just displays the magnitudes in X of the frequencies up to ½ sampe rate There is a spectrogram function in matlab
36 The Spectrogram spectrogram(y,1024,512,1024,fs,'yaxis'); A series of short term DFTs Typically just displays the magnitudes in X of the frequencies up to ½ sampe rate There is a spectrogram function in matlab
Topic 2. Signal Processing Review. (Some slides are adapted from Bryan Pardo s course slides on Machine Perception of Music)
Topic 2 Signal Processing Review (Some slides are adapted from Bryan Pardo s course slides on Machine Perception of Music) Recording Sound Mechanical Vibration Pressure Waves Motion->Voltage Transducer
More informationThe Discrete Fourier Transform. Claudia Feregrino-Uribe, Alicia Morales-Reyes Original material: Dr. René Cumplido
The Discrete Fourier Transform Claudia Feregrino-Uribe, Alicia Morales-Reyes Original material: Dr. René Cumplido CCC-INAOE Autumn 2015 The Discrete Fourier Transform Fourier analysis is a family of mathematical
More informationDiscrete Fourier Transform (DFT)
Amplitude Amplitude Discrete Fourier Transform (DFT) DFT transforms the time domain signal samples to the frequency domain components. DFT Signal Spectrum Time Frequency DFT is often used to do frequency
More informationLecture 3 Complex Exponential Signals
Lecture 3 Complex Exponential Signals Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/3/1 1 Review of Complex Numbers Using Euler s famous formula for the complex exponential The
More informationThe Discrete Fourier Transform
CHAPTER The Discrete Fourier Transform Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. The discrete Fourier transform (DFT) is the family member
More informationFFT analysis in practice
FFT analysis in practice Perception & Multimedia Computing Lecture 13 Rebecca Fiebrink Lecturer, Department of Computing Goldsmiths, University of London 1 Last Week Review of complex numbers: rectangular
More informationDISCRETE FOURIER TRANSFORM AND FILTER DESIGN
DISCRETE FOURIER TRANSFORM AND FILTER DESIGN N. C. State University CSC557 Multimedia Computing and Networking Fall 2001 Lecture # 03 Spectrum of a Square Wave 2 Results of Some Filters 3 Notation 4 x[n]
More informationG(f ) = g(t) dt. e i2πft. = cos(2πf t) + i sin(2πf t)
Fourier Transforms Fourier s idea that periodic functions can be represented by an infinite series of sines and cosines with discrete frequencies which are integer multiples of a fundamental frequency
More informationDiscrete Fourier Transform, DFT Input: N time samples
EE445M/EE38L.6 Lecture. Lecture objectives are to: The Discrete Fourier Transform Windowing Use DFT to design a FIR digital filter Discrete Fourier Transform, DFT Input: time samples {a n = {a,a,a 2,,a
More informationFrequency Division Multiplexing Spring 2011 Lecture #14. Sinusoids and LTI Systems. Periodic Sequences. x[n] = x[n + N]
Frequency Division Multiplexing 6.02 Spring 20 Lecture #4 complex exponentials discrete-time Fourier series spectral coefficients band-limited signals To engineer the sharing of a channel through frequency
More informationLABORATORY - FREQUENCY ANALYSIS OF DISCRETE-TIME SIGNALS
LABORATORY - FREQUENCY ANALYSIS OF DISCRETE-TIME SIGNALS INTRODUCTION The objective of this lab is to explore many issues involved in sampling and reconstructing signals, including analysis of the frequency
More informationNotes on Fourier transforms
Fourier Transforms 1 Notes on Fourier transforms The Fourier transform is something we all toss around like we understand it, but it is often discussed in an offhand way that leads to confusion for those
More informationCMPT 318: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals
CMPT 318: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 16, 2006 1 Continuous vs. Discrete
More informationContinuous vs. Discrete signals. Sampling. Analog to Digital Conversion. CMPT 368: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals
Continuous vs. Discrete signals CMPT 368: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 22,
More information(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters
FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according
More informationLaboratory Assignment 4. Fourier Sound Synthesis
Laboratory Assignment 4 Fourier Sound Synthesis PURPOSE This lab investigates how to use a computer to evaluate the Fourier series for periodic signals and to synthesize audio signals from Fourier series
More informationDigital Signal Processing Fourier Analysis of Continuous-Time Signals with the Discrete Fourier Transform
Digital Signal Processing Fourier Analysis of Continuous-Time Signals with the Discrete Fourier Transform D. Richard Brown III D. Richard Brown III 1 / 11 Fourier Analysis of CT Signals with the DFT Scenario:
More informationURBANA-CHAMPAIGN. CS 498PS Audio Computing Lab. Audio DSP basics. Paris Smaragdis. paris.cs.illinois.
UNIVERSITY ILLINOIS @ URBANA-CHAMPAIGN OF CS 498PS Audio Computing Lab Audio DSP basics Paris Smaragdis paris@illinois.edu paris.cs.illinois.edu Overview Basics of digital audio Signal representations
More informationProblem Set 1 (Solutions are due Mon )
ECEN 242 Wireless Electronics for Communication Spring 212 1-23-12 P. Mathys Problem Set 1 (Solutions are due Mon. 1-3-12) 1 Introduction The goals of this problem set are to use Matlab to generate and
More information(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters
FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according
More informationChapter Three. The Discrete Fourier Transform
Chapter Three. The Discrete Fourier Transform The discrete Fourier transform (DFT) is one of the two most common, and powerful, procedures encountered in the field of digital signal processing. (Digital
More informationThe Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D.
The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. Home The Book by Chapters About the Book Steven W. Smith Blog Contact Book Search Download this chapter in PDF
More informationFrequency Domain Representation of Signals
Frequency Domain Representation of Signals The Discrete Fourier Transform (DFT) of a sampled time domain waveform x n x 0, x 1,..., x 1 is a set of Fourier Coefficients whose samples are 1 n0 X k X0, X
More informationSAMPLING THEORY. Representing continuous signals with discrete numbers
SAMPLING THEORY Representing continuous signals with discrete numbers Roger B. Dannenberg Professor of Computer Science, Art, and Music Carnegie Mellon University ICM Week 3 Copyright 2002-2013 by Roger
More informationShort-Time Fourier Transform and Its Inverse
Short-Time Fourier Transform and Its Inverse Ivan W. Selesnick April 4, 9 Introduction The short-time Fourier transform (STFT) of a signal consists of the Fourier transform of overlapping windowed blocks
More informationBiomedical Signals. Signals and Images in Medicine Dr Nabeel Anwar
Biomedical Signals Signals and Images in Medicine Dr Nabeel Anwar Noise Removal: Time Domain Techniques 1. Synchronized Averaging (covered in lecture 1) 2. Moving Average Filters (today s topic) 3. Derivative
More informationLab 8. Signal Analysis Using Matlab Simulink
E E 2 7 5 Lab June 30, 2006 Lab 8. Signal Analysis Using Matlab Simulink Introduction The Matlab Simulink software allows you to model digital signals, examine power spectra of digital signals, represent
More informationSignals and Systems EE235. Leo Lam
Signals and Systems EE235 Leo Lam Today s menu Lab detailed arrangements Homework vacation week From yesterday (Intro: Signals) Intro: Systems More: Describing Common Signals Taking a signal apart Offset
More informationINTEGRATING AND DECIPHERING SIGNAL BY FOURIER TRANSFORM
INTEGRATING AND DECIPHERING SIGNAL BY FOURIER TRANSFORM Yoyok Heru Prasetyo Isnomo 1, M. Nanak Zakaria 2, Lis Diana Mustafa 3 Electrical Engineering Department, Malang State Polytechnic, INDONESIA. 1 urehkoyoy@yahoo.co.id,
More informationDigital Image Processing COSC 6380/4393
Digital Image Processing COSC 638/4393 Lecture 9 Sept 26 th, 217 Pranav Mantini Slides from Dr. Shishir K Shah and Frank (Qingzhong) Liu, S. Narasimhan HISTOGRAM SHAPING We now describe methods for histogram
More informationThe Fast Fourier Transform
The Fast Fourier Transform Basic FFT Stuff That s s Good to Know Dave Typinski, Radio Jove Meeting, July 2, 2014, NRAO Green Bank Ever wonder how an SDR-14 or Dongle produces the spectra that it does?
More informationSinusoids. Lecture #2 Chapter 2. BME 310 Biomedical Computing - J.Schesser
Sinusoids Lecture # Chapter BME 30 Biomedical Computing - 8 What Is this Course All About? To Gain an Appreciation of the Various Types of Signals and Systems To Analyze The Various Types of Systems To
More information6 Sampling. Sampling. The principles of sampling, especially the benefits of coherent sampling
Note: Printed Manuals 6 are not in Color Objectives This chapter explains the following: The principles of sampling, especially the benefits of coherent sampling How to apply sampling principles in a test
More informationFourier Transform. Any signal can be expressed as a linear combination of a bunch of sine gratings of different frequency Amplitude Phase
Fourier Transform Fourier Transform Any signal can be expressed as a linear combination of a bunch of sine gratings of different frequency Amplitude Phase 2 1 3 3 3 1 sin 3 3 1 3 sin 3 1 sin 5 5 1 3 sin
More informationDesign of FIR Filters
Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 1 FIR as a
More informationModulation. Digital Data Transmission. COMP476 Networked Computer Systems. Analog and Digital Signals. Analog and Digital Examples.
Digital Data Transmission Modulation Digital data is usually considered a series of binary digits. RS-232-C transmits data as square waves. COMP476 Networked Computer Systems Analog and Digital Signals
More informationFIR Filter Design by Frequency Sampling or Interpolation *
OpenStax-CX module: m689 FIR Filter Design by Frequency Sampling or Interpolation * C. Sidney Burrus This work is produced by OpenStax-CX and licensed under the Creative Commons Attribution License 2.
More informationFourier Transform Pairs
CHAPTER Fourier Transform Pairs For every time domain waveform there is a corresponding frequency domain waveform, and vice versa. For example, a rectangular pulse in the time domain coincides with a sinc
More informationEE 215 Semester Project SPECTRAL ANALYSIS USING FOURIER TRANSFORM
EE 215 Semester Project SPECTRAL ANALYSIS USING FOURIER TRANSFORM Department of Electrical and Computer Engineering Missouri University of Science and Technology Page 1 Table of Contents Introduction...Page
More informationLecture 7 Frequency Modulation
Lecture 7 Frequency Modulation Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/3/15 1 Time-Frequency Spectrum We have seen that a wide range of interesting waveforms can be synthesized
More informationTransforms and Frequency Filtering
Transforms and Frequency Filtering Khalid Niazi Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University 2 Reading Instructions Chapter 4: Image Enhancement in the Frequency
More informationy(n)= Aa n u(n)+bu(n) b m sin(2πmt)= b 1 sin(2πt)+b 2 sin(4πt)+b 3 sin(6πt)+ m=1 x(t)= x = 2 ( b b b b
Exam 1 February 3, 006 Each subquestion is worth 10 points. 1. Consider a periodic sawtooth waveform x(t) with period T 0 = 1 sec shown below: (c) x(n)= u(n). In this case, show that the output has the
More informationFourier transforms and series
Fourier transforms and series A Fourier transform converts a function of time into a function of frequency f is frequency in hertz t is time in seconds t = 1 and f = 1 f t ω = πf i is ( 1) e ia = cos(a)
More informationECE 2713 Homework 7 DUE: 05/1/2018, 11:59 PM
Spring 2018 What to Turn In: ECE 2713 Homework 7 DUE: 05/1/2018, 11:59 PM Dr. Havlicek Submit your solution for this assignment electronically on Canvas by uploading a file to ECE-2713-001 > Assignments
More informationDFT: Discrete Fourier Transform & Linear Signal Processing
DFT: Discrete Fourier Transform & Linear Signal Processing 2 nd Year Electronics Lab IMPERIAL COLLEGE LONDON Table of Contents Equipment... 2 Aims... 2 Objectives... 2 Recommended Textbooks... 3 Recommended
More informationIt is the speed and discrete nature of the FFT that allows us to analyze a signal's spectrum with MATLAB.
MATLAB Addendum on Fourier Stuff 1. Getting to know the FFT What is the FFT? FFT = Fast Fourier Transform. The FFT is a faster version of the Discrete Fourier Transform(DFT). The FFT utilizes some clever
More informationThe 29 th Annual ARRL and TAPR Digital Communications Conference. DSP Short Course Session 1: DSP Intro and Basics. Rick Muething, KN6KB/AAA9WK
The 29 th Annual ARRL and TAPR Digital Communications Conference DSP Short Course Session 1: DSP Intro and Basics Rick Muething, KN6KB/AAA9WK Session 1 Overview What is DSP? Why is DSP better/different
More informationME scope Application Note 01 The FFT, Leakage, and Windowing
INTRODUCTION ME scope Application Note 01 The FFT, Leakage, and Windowing NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-3600 Advanced Signal Processing
More informationELT COMMUNICATION THEORY
ELT 41307 COMMUNICATION THEORY Matlab Exercise #1 Sampling, Fourier transform, Spectral illustrations, and Linear filtering 1 SAMPLING The modeled signals and systems in this course are mostly analog (continuous
More informationELECTRONOTES APPLICATION NOTE NO Hanshaw Road Ithaca, NY Nov 7, 2014 MORE CONCERNING NON-FLAT RANDOM FFT
ELECTRONOTES APPLICATION NOTE NO. 416 1016 Hanshaw Road Ithaca, NY 14850 Nov 7, 2014 MORE CONCERNING NON-FLAT RANDOM FFT INTRODUCTION A curiosity that has probably long been peripherally noted but which
More informationTopic. Spectrogram Chromagram Cesptrogram. Bryan Pardo, 2008, Northwestern University EECS 352: Machine Perception of Music and Audio
Topic Spectrogram Chromagram Cesptrogram Short time Fourier Transform Break signal into windows Calculate DFT of each window The Spectrogram spectrogram(y,1024,512,1024,fs,'yaxis'); A series of short term
More informationECEn 487 Digital Signal Processing Laboratory. Lab 3 FFT-based Spectrum Analyzer
ECEn 487 Digital Signal Processing Laboratory Lab 3 FFT-based Spectrum Analyzer Due Dates This is a three week lab. All TA check off must be completed by Friday, March 14, at 3 PM or the lab will be marked
More informationTHE CITADEL THE MILITARY COLLEGE OF SOUTH CAROLINA. Department of Electrical and Computer Engineering. ELEC 423 Digital Signal Processing
THE CITADEL THE MILITARY COLLEGE OF SOUTH CAROLINA Department of Electrical and Computer Engineering ELEC 423 Digital Signal Processing Project 2 Due date: November 12 th, 2013 I) Introduction In ELEC
More informationLecture 2: SIGNALS. 1 st semester By: Elham Sunbu
Lecture 2: SIGNALS 1 st semester 1439-2017 1 By: Elham Sunbu OUTLINE Signals and the classification of signals Sine wave Time and frequency domains Composite signals Signal bandwidth Digital signal Signal
More informationDigital Image Processing. Image Enhancement: Filtering in the Frequency Domain
Digital Image Processing Image Enhancement: Filtering in the Frequency Domain 2 Contents In this lecture we will look at image enhancement in the frequency domain Jean Baptiste Joseph Fourier The Fourier
More informationReading: Johnson Ch , Ch.5.5 (today); Liljencrants & Lindblom; Stevens (Tues) reminder: no class on Thursday.
L105/205 Phonetics Scarborough Handout 7 10/18/05 Reading: Johnson Ch.2.3.3-2.3.6, Ch.5.5 (today); Liljencrants & Lindblom; Stevens (Tues) reminder: no class on Thursday Spectral Analysis 1. There are
More informationB.Tech III Year II Semester (R13) Regular & Supplementary Examinations May/June 2017 DIGITAL SIGNAL PROCESSING (Common to ECE and EIE)
Code: 13A04602 R13 B.Tech III Year II Semester (R13) Regular & Supplementary Examinations May/June 2017 (Common to ECE and EIE) PART A (Compulsory Question) 1 Answer the following: (10 X 02 = 20 Marks)
More informationMeasurement of RMS values of non-coherently sampled signals. Martin Novotny 1, Milos Sedlacek 2
Measurement of values of non-coherently sampled signals Martin ovotny, Milos Sedlacek, Czech Technical University in Prague, Faculty of Electrical Engineering, Dept. of Measurement Technicka, CZ-667 Prague,
More informationDIGITAL SIGNAL PROCESSING CCC-INAOE AUTUMN 2015
DIGITAL SIGNAL PROCESSING CCC-INAOE AUTUMN 2015 Fourier Transform Properties Claudia Feregrino-Uribe & Alicia Morales Reyes Original material: Rene Cumplido "The Scientist and Engineer's Guide to Digital
More informationSignals, systems, acoustics and the ear. Week 3. Frequency characterisations of systems & signals
Signals, systems, acoustics and the ear Week 3 Frequency characterisations of systems & signals The big idea As long as we know what the system does to sinusoids...... we can predict any output to any
More informationRepresenting Images and Sounds
11755 Machine earning for Signal Processing Representing Images and Sounds Class 4 3 Sep 2009 Instructor: Bhiksha Raj Representing an Elephant n It was six men of Indostan, To learning much inclined, ho
More informationChapter 5 Window Functions. periodic with a period of N (number of samples). This is observed in table (3.1).
Chapter 5 Window Functions 5.1 Introduction As discussed in section (3.7.5), the DTFS assumes that the input waveform is periodic with a period of N (number of samples). This is observed in table (3.1).
More informationFourier Signal Analysis
Part 1B Experimental Engineering Integrated Coursework Location: Baker Building South Wing Mechanics Lab Experiment A4 Signal Processing Fourier Signal Analysis Please bring the lab sheet from 1A experiment
More information1. In the command window, type "help conv" and press [enter]. Read the information displayed.
ECE 317 Experiment 0 The purpose of this experiment is to understand how to represent signals in MATLAB, perform the convolution of signals, and study some simple LTI systems. Please answer all questions
More informationAcoustics, signals & systems for audiology. Week 3. Frequency characterisations of systems & signals
Acoustics, signals & systems for audiology Week 3 Frequency characterisations of systems & signals The BIG idea: Illustrated 2 Representing systems in terms of what they do to sinusoids: Frequency responses
More informationSection 7.1 Graphs of Sine and Cosine
Section 7.1 Graphs of Sine and Cosine OBJECTIVE 1: Understanding the Graph of the Sine Function and its Properties In Chapter 7, we will use a rectangular coordinate system for a different purpose. We
More informationMachine Learning for Signal Processing. Sounds. Class Sep Instructor: Bhiksha Raj. 13 Sep /
-755 Machine earning for Signal Processing Representing Images and Sounds Class 5 3 Sep 20 Instructor: Bhiksha Raj Administrivia Basics of probability: ill not be covered Very nice lecture by Aarthi Singh
More informationDigital Filters IIR (& Their Corresponding Analog Filters) Week Date Lecture Title
http://elec3004.com Digital Filters IIR (& Their Corresponding Analog Filters) 2017 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date
More informationSection 5.2 Graphs of the Sine and Cosine Functions
A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in
More informationESE 150 Lab 04: The Discrete Fourier Transform (DFT)
LAB 04 In this lab we will do the following: 1. Use Matlab to perform the Fourier Transform on sampled data in the time domain, converting it to the frequency domain 2. Add two sinewaves together of differing
More informationSpectrum Analysis - Elektronikpraktikum
Spectrum Analysis Introduction Why measure a spectra? In electrical engineering we are most often interested how a signal develops over time. For this time-domain measurement we use the Oscilloscope. Like
More informationAnalyzing A/D and D/A converters
Analyzing A/D and D/A converters 2013. 10. 21. Pálfi Vilmos 1 Contents 1 Signals 3 1.1 Periodic signals 3 1.2 Sampling 4 1.2.1 Discrete Fourier transform... 4 1.2.2 Spectrum of sampled signals... 5 1.2.3
More informationIntroduction to Wavelet Transform. Chapter 7 Instructor: Hossein Pourghassem
Introduction to Wavelet Transform Chapter 7 Instructor: Hossein Pourghassem Introduction Most of the signals in practice, are TIME-DOMAIN signals in their raw format. It means that measured signal is a
More informationPART I: The questions in Part I refer to the aliasing portion of the procedure as outlined in the lab manual.
Lab. #1 Signal Processing & Spectral Analysis Name: Date: Section / Group: NOTE: To help you correctly answer many of the following questions, it may be useful to actually run the cases outlined in the
More informationData Acquisition Systems. Signal DAQ System The Answer?
Outline Analysis of Waveforms and Transforms How many Samples to Take Aliasing Negative Spectrum Frequency Resolution Synchronizing Sampling Non-repetitive Waveforms Picket Fencing A Sampled Data System
More informationDesign of FIR Filter for Efficient Utilization of Speech Signal Akanksha. Raj 1 Arshiyanaz. Khateeb 2 Fakrunnisa.Balaganur 3
IJSRD - International Journal for Scientific Research & Development Vol. 3, Issue 03, 2015 ISSN (online): 2321-0613 Design of FIR Filter for Efficient Utilization of Speech Signal Akanksha. Raj 1 Arshiyanaz.
More information10. Phase Cycling and Pulsed Field Gradients Introduction to Phase Cycling - Quadrature images
10. Phase Cycling and Pulsed Field Gradients 10.1 Introduction to Phase Cycling - Quadrature images The selection of coherence transfer pathways (CTP) by phase cycling or PFGs is the tool that allows the
More informationChapter 4. Digital Audio Representation CS 3570
Chapter 4. Digital Audio Representation CS 3570 1 Objectives Be able to apply the Nyquist theorem to understand digital audio aliasing. Understand how dithering and noise shaping are done. Understand the
More informationMusic 171: Amplitude Modulation
Music 7: Amplitude Modulation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) February 7, 9 Adding Sinusoids Recall that adding sinusoids of the same frequency
More informationIntroduction to Digital Signal Processing (Discrete-time Signal Processing)
Introduction to Digital Signal Processing (Discrete-time Signal Processing) Prof. Chu-Song Chen Research Center for Info. Tech. Innovation, Academia Sinica, Taiwan Dept. CSIE & GINM National Taiwan University
More informationPrinciples of Communications ECS 332
Principles of Communications ECS 332 Asst. Prof. Dr. Prapun Suksompong prapun@siit.tu.ac.th 5. Angle Modulation Office Hours: BKD, 6th floor of Sirindhralai building Wednesday 4:3-5:3 Friday 4:3-5:3 Example
More informationESE 150 Lab 04: The Discrete Fourier Transform (DFT)
LAB 04 In this lab we will do the following: 1. Use Matlab to perform the Fourier Transform on sampled data in the time domain, converting it to the frequency domain 2. Add two sinewaves together of differing
More informationTRANSFORMS / WAVELETS
RANSFORMS / WAVELES ransform Analysis Signal processing using a transform analysis for calculations is a technique used to simplify or accelerate problem solution. For example, instead of dividing two
More informationSignal Characteristics
Data Transmission The successful transmission of data depends upon two factors:» The quality of the transmission signal» The characteristics of the transmission medium Some type of transmission medium
More informationQäf) Newnes f-s^j^s. Digital Signal Processing. A Practical Guide for Engineers and Scientists. by Steven W. Smith
Digital Signal Processing A Practical Guide for Engineers and Scientists by Steven W. Smith Qäf) Newnes f-s^j^s / *" ^"P"'" of Elsevier Amsterdam Boston Heidelberg London New York Oxford Paris San Diego
More informationSampling and Reconstruction of Analog Signals
Sampling and Reconstruction of Analog Signals Chapter Intended Learning Outcomes: (i) Ability to convert an analog signal to a discrete-time sequence via sampling (ii) Ability to construct an analog signal
More informationLab 3 FFT based Spectrum Analyzer
ECEn 487 Digital Signal Processing Laboratory Lab 3 FFT based Spectrum Analyzer Due Dates This is a three week lab. All TA check off must be completed prior to the beginning of class on the lab book submission
More informationThinking in Frequency
Thinking in Frequency Computer Vision Brown James Hays Slides: Hoiem, Efros, and others Recap of Wednesday linear filtering convolution differential filters filter types boundary conditions. Review: questions
More informationTerminology (1) Chapter 3. Terminology (3) Terminology (2) Transmitter Receiver Medium. Data Transmission. Direct link. Point-to-point.
Terminology (1) Chapter 3 Data Transmission Transmitter Receiver Medium Guided medium e.g. twisted pair, optical fiber Unguided medium e.g. air, water, vacuum Spring 2012 03-1 Spring 2012 03-2 Terminology
More informationRepresenting Images and Sounds
11-755 Machine Learning for Signal Processing Representing Images and Sounds Class 4. 2 Sep 2010 Instructor: Bhiksha Raj 2 Sep 2010 1 Administrivia Homework up Basics of probability: Will not be covered
More informationSignals A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2
Signals A Preliminary Discussion EE442 Analog & Digital Communication Systems Lecture 2 The Fourier transform of single pulse is the sinc function. EE 442 Signal Preliminaries 1 Communication Systems and
More informationDigital Signal Processing. VO Embedded Systems Engineering Armin Wasicek WS 2009/10
Digital Signal Processing VO Embedded Systems Engineering Armin Wasicek WS 2009/10 Overview Signals and Systems Processing of Signals Display of Signals Digital Signal Processors Common Signal Processing
More informationCSC475 Music Information Retrieval
CSC475 Music Information Retrieval Sinusoids and DSP notation George Tzanetakis University of Victoria 2014 G. Tzanetakis 1 / 38 Table of Contents I 1 Time and Frequency 2 Sinusoids and Phasors G. Tzanetakis
More informationTABLE OF CONTENTS TOPIC NUMBER NAME OF THE TOPIC 1. OVERVIEW OF SIGNALS & SYSTEMS 2. ANALYSIS OF LTI SYSTEMS- Z TRANSFORM 3. ANALYSIS OF FT, DFT AND FFT SIGNALS 4. DIGITAL FILTERS CONCEPTS & DESIGN 5.
More informationSignal processing preliminaries
Signal processing preliminaries ISMIR Graduate School, October 4th-9th, 2004 Contents: Digital audio signals Fourier transform Spectrum estimation Filters Signal Proc. 2 1 Digital signals Advantages of
More informationFinal Exam Solutions June 14, 2006
Name or 6-Digit Code: PSU Student ID Number: Final Exam Solutions June 14, 2006 ECE 223: Signals & Systems II Dr. McNames Keep your exam flat during the entire exam. If you have to leave the exam temporarily,
More informationApplication of Fourier Transform in Signal Processing
1 Application of Fourier Transform in Signal Processing Lina Sun,Derong You,Daoyun Qi Information Engineering College, Yantai University of Technology, Shandong, China Abstract: Fourier transform is a
More informationSECTION 7: FREQUENCY DOMAIN ANALYSIS. MAE 3401 Modeling and Simulation
SECTION 7: FREQUENCY DOMAIN ANALYSIS MAE 3401 Modeling and Simulation 2 Response to Sinusoidal Inputs Frequency Domain Analysis Introduction 3 We ve looked at system impulse and step responses Also interested
More informationLecture #2. EE 313 Linear Systems and Signals
Lecture #2 EE 313 Linear Systems and Signals Preview of today s lecture What is a signal and what is a system? o Define the concepts of a signal and a system o Why? This is essential for a course on Signals
More informationDigital Signal Processing
Digital Signal Processing System Analysis and Design Paulo S. R. Diniz Eduardo A. B. da Silva and Sergio L. Netto Federal University of Rio de Janeiro CAMBRIDGE UNIVERSITY PRESS Preface page xv Introduction
More information