ANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8
|
|
- Frank Ford
- 5 years ago
- Views:
Transcription
1 ANALOG AND DIGITAL SIGNAL PROCESSING ADSP - Chapter 8 Fm n N fnt ( ) e j2mn N X() X() 2 X() X() 3 W Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 W W 3 W W x () x () x () 2 x () 3
2 2 Chapter 8 Discrete Fourier Transform (DFT), FFT Introduction to discrete Fourier transform Time-limited signal made periodic Negative aspect and solution of the periodicity Mathematical approach to the DFT Basic concept DFT application example Fast Fourier transform (FFT) Matrix formulation - Intuitive development Applications Choosing the sampling frequency Windowing - Blind FFT output interpretation Problems Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
3 INTRODUCTION TO DISCRETE FOURIER TRANSFORM: Basic concept: In previous chapters, we dealt with the Fourier Series and the Fourier Transforms of continuous waveforms. When it comes to computing one of them from a real signal (continuous or discretized), we face a practical problem: 3 In most situations we exactly know our signal within a time-window starting at t and ending at t 2 ; however, we totally ignore what happened before t and what will happen after t 2. SOLUTION: To make PERIODIC the Time limited signal Example : Signal made Time PERIODIC limited signal with period T (T = t 2 t ) t t 2 Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
4 Example cont Decaying sine-wave approximate frequency: Hz ( samples/cycle) 256 samples for period of the periodic signal T = µs 256 = 25.6ms ( 4Hz) Fourier Series coefficient magnitudes: Fseries n 5 Number of samples g Hz khz 2kHz 3kHz 25 4 = Hz (harmonic 24 or 25 multiplied by fundamental frequency) n Harmonic number (ts = µs) Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 4kHz
5 Magnitude Spectrum after Sampling Frequency [Hz] fs/2 Potential problems due to the time-limited signal made periodic: Discontinuities Generate unwanted high frequencies components Example 2: Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
6 Example 3: Pure sine-wave a) Integer number of periods 8 khz sine-wave Acquisition window time [s] 6 Sampling frequency: fs =.24 MHz, number of samples: N = 24 Sampled 8kHz sinewave samples Sampled 8kHz sinewave samples Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
7 Example 3: Pure sine-wave b) Non-integer number of periods 7.5 khz sine-wave Tw Tw: acquisition time window, ts: sampling interval Tw = Fundamental frequency: f = Magnitude of Fourier Coefficients 8 khz Hn Harmonic number / Tw = fs / N = khz 5 5 N ts = N / fs 7.5 khz Hn Harmonic number Discontinuities in the time-domain many spectral components Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
8 SOLUTION: WINDOWING Sampled 7.5kHz sinewave xw n A sin 2 28 n 75.5 cos 2 n samples samples Cosine type window Windowed sampled 7.5kHz sinewave samples windowed sampled 7.5 khz sine + period Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, Magnitude of Fourier Coefficients g Hn Harmonic number
9 5 8 g khz without windowing khz with windowing Hn Harmonic number Hn Harmonic number 9 w(t) window x(t) X x(t) w(t) = xw(t) X(jω) * WINDOWING DOWNSIDE: xw(t) W(jω) = XW(jω) Frequency domain convolution of the window spectrum with the signal spectrum! Broadening of the signal spectrum Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
10 MATHEMATICAL APPROACH OF THE DFT Basic concept: If f(t) is our continuous waveform, fd(t) is its discretized version: f(t) fd() t n f( n ts) ( t n ts) Where ts is the time between consecutive samples. By definition, the Fourier Transform of fd(t) is: Ffdt ( ()) Fd( ) n fn ( ts) ( n n ts) e j t dt Fd( ) Previously, we introduced a time-limitation. The simplest way to do this is to set the lower bound of the summation at and the upper bound at N-, thus limiting the total number of samples considered to N. Rewriting Fd() gives: Fdw( ) N n Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 n fn ( ts) e j nts fn ( ts) e j nts WDFT
11 The letter W used to characterize this new definition refers to window since our discretized waveform f(n ts) is effectively windowed, that is, multiplied by a time-window function. In order to illustrate the implication of the WDFT, consider a pulse represented by 8 equally spaced samples (N=8): Fdw ( ) 7 n fnts ( ) e j n ts e j ts e j ts e j 2 ts e j 3 ts From this set of samples, we get the following spectrum (with ts = ) : Fdw n ( ) The periodicity in comes from the sampling theorem (period: 2/ts = s ). Since the theoretical Power Spectrum (from its Fouriertransform)ofapulseexpendfrom- to +, that produces a noticeable aliasing (blue)! Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
12 If the initial waveform is artificially made periodic with a period of 8ts, then we can compute the Fourier coefficients by using the definition of the Fourier series. The next figures compare the spectrum obtained by the two approaches: Fourier Series Coefficients and Discrete Fourier Transform. m Fourier Series It can be easily shown that the Fourier Series Coefficients can be obtained by simply sampling the Discrete Fourier Transform. What is commonly referred to as the Discrete Fourier Transform (DFT) is, in reality, a Discrete- Time Fourier Series (DTFS) derived after transforming a limited number of samples (8 in our example) into a discrete periodic waveform. The fundamental frequency of this new signal is /(N ts). Thus, the spectral lines will appear at m/(n ts) Hz where m is an integer. 4 2 Discrete Fourier Transform Fdw ( ) Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
13 So finally, we can write the DTFS or DFT in the most common form as: F 2 m N ts Fm ( ) n N fn ( ts) e j2 m n N DFT 3 The inverse DFT is: f( n ts) N m N Fm ( ) e j2 m n InvDFT As already mentioned in the beginning of this chapter, due to the limited number of samples used in the computation of the WDFT, numerous problems can appear that one has to be aware of. N Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
14 SIMPLE DFT APPLICATIONS # (integer number of cycle) f(t) = cos(2t) ; a cosine function with a period of s, sampled every ts second, Tw = ts (N-) N = 6, ts = /6 T = s 4 cos 2 t N F m n t j2mn cos2nts N e Tw t n ts F m cos 2nts Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 m n
15 SIMPLE DFT APPLICATIONS #2 (non-integer number of cycle) f(t) = cos(2t) ; N = 6, ts = /2 Tw =.25s cos 2 t cos 2 nts 5 N F m n.5 j2mn N cos2ntse t Tw F m Since the number of periods used to compute the WDFT is not an integer number, a discontinuity in the time domain is created, thus generating artifacts (unwanted high frequencies spectral components) in the power spectrum. In order to solve this problem, various window types have been proposed, each one of them with particular benefits and draw-backs!! The optimum choice will depend upon the features one wishes to emphasize. Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 n
16 FAST FOURIER TRANSFORM (FFT) 6 The FFT is simply an algorithm that can compute the discrete Fourier transform much more rapidly than other available algorithms. In the case of the DFT, the approximate number of multiplication grows with the square of N. If Ncanbewrittenas2 K (where K is an integer), a substantial saving can be achieved on the number of multiplication which becomes in the order of : 2Nlog 2 N. Many algorithms have been designed and the interested reader is encouraged to refer to one of the classical text books covering this topic. A simple matrix factoring example is used to intuitively justify the FFT algorithm. Matrix formulation Consider the discrete Fourier transform: 8- Fm ( ) N n j2m n N f ()e n (8-) describes the computation of N equations. Let N = 4 and 8-2 W e 8-3 F ( ) f ( ) W f ( ) W f ( 2) W f ( 3) W F ( ) f ( ) W f ( ) W f ( 2) f ( 3) W 3 F2 ( ) f ( ) W f ( ) f ( 2) W 4 f ( 3) W 6 F3 ( ) f ( ) W f ( ) W 3 f ( 2) W 6 f ( 3) W j2 mn m N (8-4) reveals that since W and f o (n) are complex matrices, thus N 2 complex multiplication and N (N-) complex additions are necessary to perform the required matrix computation. Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 F ( ) F ( ) F2 ( ) F3 ( ) W W W W W W W 3 W W 4 W 6 N W W 3 W 6 W 9 f ( ) f ( ) f ( 2) f ( 3)
17 INTUITIVE DEVELOPMENT To illustrate the FFT algorithm, it is convenient to choose the number of sample points of f o (n) accordingtotherelation N=2,where is an integer. The first step in developing the FFT algorithm for this example is to rewrite (8-4) as: F ( ) F ( ) F2 ( ) F3 ( ) W W 3 The next step is to factor the square matrix in (8-5) as follows: W W 3 W f ( ) f ( ) f ( 2) f ( 3) 8-7 Matrix Eq. (8-5) was derived from (8-4) by using the relationship W m n = W m n mod(n). Than: 8-6 W 6 = The method of factorization is based on the theory of the FFT algorithm. It is easily shown that multiplication of the two matrices of (8-7) yields the square matrix of (8-5) with the exception that rows and 2 have been interchanged (the rows are numbered,, 2 and 3). F ( ) F2 ( ) F ( ) F3 ( ) Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2 W W W 3 W W f ( ) f ( ) f ( 2) f ( 3)
18 Having accepted the fact that (8-7) is correct, although the result is scrambled, one should then examine the number of multiplication required to compute the equation. First let: 8-8 f ( ) f ( ) f ( 2) f ( 3) W W f ( ) f ( ) f ( 2) f ( 3) Element f () is computed by one complex multiplication and one complex addition. 8-9 f ( ) f ( ) W f ( 2) 8 Element f () is also determined by one complex multiplication and one complex addition. Only one complex addition is required to compute f (2). This from the fact that W = - ; hence: where the complex multiplication W f (2) has already been computed in the determination of f (). By the same reasoning, f (3) is computed by only one complex addition and no multiplications. The intermediate vector f (n) is then determined by four complex additions and two complex multiplications. Let us continue by completing the computation of (8-7): 8- F ( ) F2 ( ) F ( ) F3 ( ) f 2 ( ) f 2 ( ) f 2 ( 2) f 2 ( 3) 8- W f ( 2) f ( ) f ( 2) f ( ) W f ( 2) W W 3 f ( ) f ( ) f ( 2) f ( 3) Term f 2 () is computed by one complex multiplication and addition! 8-2 f 2 ( ) f ( ) W f ( ) Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
19 Element f 2 () is computed by one addition because W =-. By similar reasoning, f 2 (2) is determined by one complex multiplication and addition, and f 2 (3) by only one addition. Thus: - Computation of F(m) by means of Eq. (8-7) requires a total of: - Computation of F(m) by (8-4) requires: four complex multiplication and eight complex additions sixteen complex multiplication and twelve complex additions 9 However, the matrix factoring procedure does introduce one discrepancy. That is: 8-3 Fscrab( m) F ( ) F2 ( ) F ( ) F3 ( ) instead of Fm ( ) This rearrangement is inherent in the matrix factoring process and is a minor problem because it is straightforward to generalize a technique for unscrambling Fscrab(m) and obtain F(m). F ( ) F ( ) F2 ( ) F3 ( ) Thus, if N can be written as 2 K (where K is an integer), a substantial saving can be achieved on the number of multiplication which becomes in the order of : 2Nlog 2 N. Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
20 APPLICATIONS # CHOOSING THE SAMPLING FREQUENCY 3dB 2 fs = 5 khz 2 Hz - Hz-2 3dB kHz fs = 7.6 khz 2kHz Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
21 APPLICATIONS fs = 5 khz, 248 samples (). khz, 5V (sinus) (3) Hz, V (square-wave) #2 WINDOWING No-window after windowing (Hanning) 2 Hz Power Spectrum 4dB kHz Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
22 APPLICATIONS #3 ACQUISITION WINDOW (Tw) LENGTH fs = 5 khz, X samples (). khz, 5V (sinus) (3) Hz, V (square-wave) 892 samples samples samples Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
23 APPLICATIONS #4 FFT FFT 2 Power Spectrum 3 Hz square-wave, Vpp - sampling frequency : 5 khz 44 e-3 FFT -27dB 2Log FFT 23.9 e-3 FFT 2 7dBm Power Spectrum Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
24 APPLICATIONS Consider the following Power Spectrum (red and green): #5 BLIND FFT INTERPRETATION 3Hz 2.5kHz.5kHz What signals produced these spectrums? Chirp (5 Hz 2 khz) + Band-passed filtered (5Hz 2 khz) Gaussian noise 4ms Band-passed filtered (5Hz 2 khz) Gaussian noise 4ms 24 Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
25 PROBLEMS Problem 8. In p.8-4, prove that F = 8 Problem 8.2 With f(nt) = (t 5T) and N = 6, determine F m for m < 8 25 Problem 8.3 With N = 8, compute the module, the real and imaginary part of F(m) for m < 4 f() = f(2) = f(4) = f(6) = and f() = f(3) = f(5) = f(7) = - Problem 8.4 (SystemView) The input signal x(t) of an acquisition system has the following form : x(t) = A sin( t) + B Sqw(t), Sqw(t) : symmetrical square-wave of ms period The sampling rate (fs) can be chosen between 5 khz and khz and f is smaller then khz. Draw several possible FFT plots (magnitude) you may obtain when varying the following parameters: fs, N, with and without windowing, f, A/B Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
26 PROBLEMS Problem 8.5 (SystemView) The input signal x(t) of an acquisition system has the following form : x(t) = A sin( t) + B sin( t) + C sin( 2 t) + D sin( 3 t) with f = khz, f = f + f, f 2 = f + f, f 3 = f 2 + f, f = 5 Hz A, B C and D can take the following values: + or (constant over the sampling window) Assume that fs = /T = 2 khz, how do you choose N? Problem 8.6 Problem Chap. 8 Discrete Fourier Transform (DFT), FFT Prof. J.-P. Sandoz, 2-2
The Discrete Fourier Transform
In [ ]: cd matlab pwd The Discrete Fourier Transform Scope and Background Reading This session introduces the z-transform which is used in the analysis of discrete time systems. As for the Fourier and
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationExperimental Fourier Transforms
Chapter 5 Experimental Fourier Transforms 5.1 Sampling and Aliasing Given x(t), we observe only sampled data x s (t) = x(t)s(t; T s ) (Fig. 5.1), where s is called sampling or comb function and can be
More informationTutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling
2.39 utorial Sheet #2 discrete vs. continuous functions, periodicity, sampling We will encounter two classes of signals in this class, continuous-signals and discrete-signals. he distinct mathematical
More informationLecture 27 Frequency Response 2
Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period
More informationL6: Short-time Fourier analysis and synthesis
L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude
More informationEE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, Cover Sheet
NAME: NAME EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, 2017 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet
More information3.2 Complex Sinusoids and Frequency Response of LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationChirp Transform for FFT
Chirp Transform for FFT Since the FFT is an implementation of the DFT, it provides a frequency resolution of 2π/N, where N is the length of the input sequence. If this resolution is not sufficient in a
More informationChapter 8 The Discrete Fourier Transform
Chapter 8 The Discrete Fourier Transform Introduction Representation of periodic sequences: the discrete Fourier series Properties of the DFS The Fourier transform of periodic signals Sampling the Fourier
More informationANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)
3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic
More informationChapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum
Chapter 4 Discrete Fourier Transform (DFT) And Signal Spectrum CEN352, DR. Nassim Ammour, King Saud University 1 Fourier Transform History Born 21 March 1768 ( Auxerre ). Died 16 May 1830 ( Paris ) French
More informationDiscrete Fourier Transform
Discrete Fourier Transform Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz Diskrete Fourier transform (DFT) We have just one problem with DFS that needs to be solved.
More informationVibration Testing. an excitation source a device to measure the response a digital signal processor to analyze the system response
Vibration Testing For vibration testing, you need an excitation source a device to measure the response a digital signal processor to analyze the system response i) Excitation sources Typically either
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 2011) Final Examination December 19, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 1 pm 4 2 pm Grades will be determined by the correctness of your answers
More informationE : Lecture 1 Introduction
E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation
More informationDistortion Analysis T
EE 435 Lecture 32 Spectral Performance Windowing Spectral Performance of Data Converters - Time Quantization - Amplitude Quantization Quantization Noise . Review from last lecture. Distortion Analysis
More informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationCorrelation, discrete Fourier transforms and the power spectral density
Correlation, discrete Fourier transforms and the power spectral density visuals to accompany lectures, notes and m-files by Tak Igusa tigusa@jhu.edu Department of Civil Engineering Johns Hopkins University
More informationLecture 4 - Spectral Estimation
Lecture 4 - Spectral Estimation The Discrete Fourier Transform The Discrete Fourier Transform (DFT) is the equivalent of the continuous Fourier Transform for signals known only at N instants separated
More informationV(t) = Total Power = Calculating the Power Spectral Density (PSD) in IDL. Thomas Ferree, Ph.D. August 23, 1999
Calculating the Power Spectral Density (PSD) in IDL Thomas Ferree, Ph.D. August 23, 1999 This note outlines the calculation of power spectra via the fast Fourier transform (FFT) algorithm. There are several
More informationFILTERING IN THE FREQUENCY DOMAIN
1 FILTERING IN THE FREQUENCY DOMAIN Lecture 4 Spatial Vs Frequency domain 2 Spatial Domain (I) Normal image space Changes in pixel positions correspond to changes in the scene Distances in I correspond
More informationI. Signals & Sinusoids
I. Signals & Sinusoids [p. 3] Signal definition Sinusoidal signal Plotting a sinusoid [p. 12] Signal operations Time shifting Time scaling Time reversal Combining time shifting & scaling [p. 17] Trigonometric
More informationEE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, Cover Sheet
EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, 2012 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators
More informationSNR Calculation and Spectral Estimation [S&T Appendix A]
SR Calculation and Spectral Estimation [S&T Appendix A] or, How not to make a mess of an FFT Make sure the input is located in an FFT bin 1 Window the data! A Hann window works well. Compute the FFT 3
More informationPS403 - Digital Signal processing
PS403 - Digital Signal processing III. DSP - Digital Fourier Series and Transforms Key Text: Digital Signal Processing with Computer Applications (2 nd Ed.) Paul A Lynn and Wolfgang Fuerst, (Publisher:
More informationGrades will be determined by the correctness of your answers (explanations are not required).
6.00 (Fall 20) Final Examination December 9, 20 Name: Kerberos Username: Please circle your section number: Section Time 2 am pm 4 2 pm Grades will be determined by the correctness of your answers (explanations
More informationRepresenting a Signal
The Fourier Series Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity and timeinvariance of the system and represents the
More informationFig. 1: Fourier series Examples
FOURIER SERIES AND ITS SOME APPLICATIONS P. Sathyabama Assistant Professor, Department of Mathematics, Bharath collage of Science and Management, Thanjavur, Tamilnadu Abstract: The Fourier series, the
More informationEE123 Digital Signal Processing
EE123 Digital Signal Processing Lecture 1 Time-Dependent FT Announcements! Midterm: 2/22/216 Open everything... but cheat sheet recommended instead 1am-12pm How s the lab going? Frequency Analysis with
More informationThe Fourier transform allows an arbitrary function to be represented in terms of simple sinusoids. The Fourier transform (FT) of a function f(t) is
1 Introduction Here is something I wrote many years ago while working on the design of anemometers for measuring shear stresses. Part of this work required modelling and compensating for the transfer function
More informationVibration Testing. Typically either instrumented hammers or shakers are used.
Vibration Testing Vibration Testing Equipment For vibration testing, you need an excitation source a device to measure the response a digital signal processor to analyze the system response Excitation
More informationSignal Processing COS 323
Signal Processing COS 323 Digital Signals D: functions of space or time e.g., sound 2D: often functions of 2 spatial dimensions e.g. images 3D: functions of 3 spatial dimensions CAT, MRI scans or 2 space,
More informationEE 435. Lecture 32. Spectral Performance Windowing
EE 435 Lecture 32 Spectral Performance Windowing . Review from last lecture. Distortion Analysis T 0 T S THEOREM?: If N P is an integer and x(t) is band limited to f MAX, then 2 Am Χ mnp 1 0 m h N and
More informationSound & Vibration Magazine March, Fundamentals of the Discrete Fourier Transform
Fundamentals of the Discrete Fourier Transform Mark H. Richardson Hewlett Packard Corporation Santa Clara, California The Fourier transform is a mathematical procedure that was discovered by a French mathematician
More informationImage Enhancement in the frequency domain. GZ Chapter 4
Image Enhancement in the frequency domain GZ Chapter 4 Contents In this lecture we will look at image enhancement in the frequency domain The Fourier series & the Fourier transform Image Processing in
More informationIB Paper 6: Signal and Data Analysis
IB Paper 6: Signal and Data Analysis Handout 5: Sampling Theory S Godsill Signal Processing and Communications Group, Engineering Department, Cambridge, UK Lent 2015 1 / 85 Sampling and Aliasing All of
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
More informationSampling. Alejandro Ribeiro. February 8, 2018
Sampling Alejandro Ribeiro February 8, 2018 Signals exist in continuous time but it is not unusual for us to process them in discrete time. When we work in discrete time we say that we are doing discrete
More informationESS Finite Impulse Response Filters and the Z-transform
9. Finite Impulse Response Filters and the Z-transform We are going to have two lectures on filters you can find much more material in Bob Crosson s notes. In the first lecture we will focus on some of
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationFrequency-domain representation of discrete-time signals
4 Frequency-domain representation of discrete-time signals So far we have been looing at signals as a function of time or an index in time. Just lie continuous-time signals, we can view a time signal as
More informationSummary of lecture 1. E x = E x =T. X T (e i!t ) which motivates us to define the energy spectrum Φ xx (!) = jx (i!)j 2 Z 1 Z =T. 2 d!
Summary of lecture I Continuous time: FS X FS [n] for periodic signals, FT X (i!) for non-periodic signals. II Discrete time: DTFT X T (e i!t ) III Poisson s summation formula: describes the relationship
More informationFourier analysis of discrete-time signals. (Lathi Chapt. 10 and these slides)
Fourier analysis of discrete-time signals (Lathi Chapt. 10 and these slides) Towards the discrete-time Fourier transform How we will get there? Periodic discrete-time signal representation by Discrete-time
More informationFourier Transform in Image Processing. CS/BIOEN 6640 U of Utah Guido Gerig (slides modified from Marcel Prastawa 2012)
Fourier Transform in Image Processing CS/BIOEN 6640 U of Utah Guido Gerig (slides modified from Marcel Prastawa 2012) Basis Decomposition Write a function as a weighted sum of basis functions f ( x) wibi(
More informationPART 1. Review of DSP. f (t)e iωt dt. F(ω) = f (t) = 1 2π. F(ω)e iωt dω. f (t) F (ω) The Fourier Transform. Fourier Transform.
PART 1 Review of DSP Mauricio Sacchi University of Alberta, Edmonton, AB, Canada The Fourier Transform F() = f (t) = 1 2π f (t)e it dt F()e it d Fourier Transform Inverse Transform f (t) F () Part 1 Review
More informationAccurate Fourier Analysis for Circuit Simulators
Accurate Fourier Analysis for Circuit Simulators Kenneth S. Kundert Cadence Design Systems (Based on Presentation to CICC 94) Abstract A new approach to Fourier analysis within the context of circuit simulation
More informationDESIGN OF CMOS ANALOG INTEGRATED CIRCUITS
DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete
More information06EC44-Signals and System Chapter Fourier Representation for four Signal Classes
Chapter 5.1 Fourier Representation for four Signal Classes 5.1.1Mathematical Development of Fourier Transform If the period is stretched without limit, the periodic signal no longer remains periodic but
More informationEE 435. Lecture 30. Data Converters. Spectral Performance
EE 435 Lecture 30 Data Converters Spectral Performance . Review from last lecture. INL Often Not a Good Measure of Linearity Four identical INL with dramatically different linearity X OUT X OUT X REF X
More informationEvaluating Fourier Transforms with MATLAB
ECE 460 Introduction to Communication Systems MATLAB Tutorial #2 Evaluating Fourier Transforms with MATLAB In class we study the analytic approach for determining the Fourier transform of a continuous
More informationNeed for transformation?
Z-TRANSFORM In today s class Z-transform Unilateral Z-transform Bilateral Z-transform Region of Convergence Inverse Z-transform Power Series method Partial Fraction method Solution of difference equations
More informationPolyphase filter bank quantization error analysis
Polyphase filter bank quantization error analysis J. Stemerdink Verified: Name Signature Date Rev.nr. A. Gunst Accepted: Team Manager System Engineering Manager Program Manager M. van Veelen C.M. de Vos
More informationCentre for Mathematical Sciences HT 2017 Mathematical Statistics
Lund University Stationary stochastic processes Centre for Mathematical Sciences HT 2017 Mathematical Statistics Computer exercise 3 in Stationary stochastic processes, HT 17. The purpose of this exercise
More informationComputational Methods CMSC/AMSC/MAPL 460. Fourier transform
Computational Methods CMSC/AMSC/MAPL 460 Fourier transform Ramani Duraiswami, Dept. of Computer Science Several slides from Prof. Healy s course at UMD Fourier Methods Fourier analysis ( harmonic analysis
More informationTHE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3
THE FOURIER TRANSFORM (Fourier series for a function whose period is very, very long) Reading: Main 11.3 Any periodic function f(t) can be written as a Fourier Series a 0 2 + a n cos( nωt) + b n sin n
More informationKey Intuition: invertibility
Introduction to Fourier Analysis CS 510 Lecture #6 January 30, 2017 In the extreme, a square wave Graphic from http://www.mechatronics.colostate.edu/figures/4-4.jpg 2 Fourier Transform Formally, the Fourier
More informationTime and Spatial Series and Transforms
Time and Spatial Series and Transforms Z- and Fourier transforms Gibbs' phenomenon Transforms and linear algebra Wavelet transforms Reading: Sheriff and Geldart, Chapter 15 Z-Transform Consider a digitized
More informationCast of Characters. Some Symbols, Functions, and Variables Used in the Book
Page 1 of 6 Cast of Characters Some s, Functions, and Variables Used in the Book Digital Signal Processing and the Microcontroller by Dale Grover and John R. Deller ISBN 0-13-081348-6 Prentice Hall, 1998
More informationIntroduction to Digital Signal Processing
Introduction to Digital Signal Processing 1.1 What is DSP? DSP is a technique of performing the mathematical operations on the signals in digital domain. As real time signals are analog in nature we need
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 4 Digital Signal Processing Pro. Mark Fowler ote Set # Using the DFT or Spectral Analysis o Signals Reading Assignment: Sect. 7.4 o Proakis & Manolakis Ch. 6 o Porat s Book /9 Goal o Practical Spectral
More informationSistemas de Aquisição de Dados. Mestrado Integrado em Eng. Física Tecnológica 2016/17 Aula 3, 3rd September
Sistemas de Aquisição de Dados Mestrado Integrado em Eng. Física Tecnológica 2016/17 Aula 3, 3rd September The Data Converter Interface Analog Media and Transducers Signal Conditioning Signal Conditioning
More informationCorrelator I. Basics. Chapter Introduction. 8.2 Digitization Sampling. D. Anish Roshi
Chapter 8 Correlator I. Basics D. Anish Roshi 8.1 Introduction A radio interferometer measures the mutual coherence function of the electric field due to a given source brightness distribution in the sky.
More information3. Lecture. Fourier Transformation Sampling
3. Lecture Fourier Transformation Sampling Some slides taken from Digital Image Processing: An Algorithmic Introduction using Java, Wilhelm Burger and Mark James Burge Separability ² The 2D DFT can be
More informationAspects of Continuous- and Discrete-Time Signals and Systems
Aspects of Continuous- and Discrete-Time Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling the
More informationDiscrete Fourier Transform
Discrete Fourier Transform Virtually all practical signals have finite length (e.g., sensor data, audio records, digital images, stock values, etc). Rather than considering such signals to be zero-padded
More informationFourier Series and Integrals
Fourier Series and Integrals Fourier Series et f(x) beapiece-wiselinearfunctionon[, ] (Thismeansthatf(x) maypossessa finite number of finite discontinuities on the interval). Then f(x) canbeexpandedina
More information/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by
Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,
More informationThe Fourier Transform (and more )
The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Processing Prof. Mark Fowler Note Set #21 Using the DFT to Implement FIR Filters Reading Assignment: Sect. 7.3 of Proakis & Manolakis Motivation: DTFT View of Filtering There are
More informationECSE 512 Digital Signal Processing I Fall 2010 FINAL EXAMINATION
FINAL EXAMINATION 9:00 am 12:00 pm, December 20, 2010 Duration: 180 minutes Examiner: Prof. M. Vu Assoc. Examiner: Prof. B. Champagne There are 6 questions for a total of 120 points. This is a closed book
More informationNew Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Final Exam
New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Name: Solve problems 1 3 and two from problems 4 7. Circle below which two of problems 4 7 you
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationLab Fourier Analysis Do prelab before lab starts. PHSX 262 Spring 2011 Lecture 5 Page 1. Based with permission on lectures by John Getty
Today /5/ Lecture 5 Fourier Series Time-Frequency Decomposition/Superposition Fourier Components (Ex. Square wave) Filtering Spectrum Analysis Windowing Fast Fourier Transform Sweep Frequency Analyzer
More information1 Understanding Sampling
1 Understanding Sampling Summary. In Part I, we consider the analysis of discrete-time signals. In Chapter 1, we consider how discretizing a signal affects the signal s Fourier transform. We derive the
More informationFrequency Response and Continuous-time Fourier Series
Frequency Response and Continuous-time Fourier Series Recall course objectives Main Course Objective: Fundamentals of systems/signals interaction (we d like to understand how systems transform or affect
More informationElec4621 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis
Elec461 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis Dr. D. S. Taubman May 3, 011 In this last chapter of your notes, we are interested in the problem of nding the instantaneous
More informationMultirate Digital Signal Processing
Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-sampler - Used to increase the sampling rate by an integer factor Down-sampler - Used to decrease the sampling rate by an integer
More informationSignals and Systems Lecture (S2) Orthogonal Functions and Fourier Series March 17, 2008
Signals and Systems Lecture (S) Orthogonal Functions and Fourier Series March 17, 008 Today s Topics 1. Analogy between functions of time and vectors. Fourier series Take Away Periodic complex exponentials
More informationFourier Transform Chapter 10 Sampling and Series
Fourier Transform Chapter 0 Sampling and Series Sampling Theorem Sampling Theorem states that, under a certain condition, it is in fact possible to recover with full accuracy the values intervening between
More informationECG782: Multidimensional Digital Signal Processing
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu ECG782: Multidimensional Digital Signal Processing Filtering in the Frequency Domain http://www.ee.unlv.edu/~b1morris/ecg782/ 2 Outline Background
More informationGeneration of bandlimited sync transitions for sine waveforms
Generation of bandlimited sync transitions for sine waveforms Vadim Zavalishin May 4, 9 Abstract A common way to generate bandlimited sync transitions for the classical analog waveforms is the BLEP method.
More informationLecture 4 Filtering in the Frequency Domain. Lin ZHANG, PhD School of Software Engineering Tongji University Spring 2016
Lecture 4 Filtering in the Frequency Domain Lin ZHANG, PhD School of Software Engineering Tongji University Spring 2016 Outline Background From Fourier series to Fourier transform Properties of the Fourier
More informationAPPENDIX A. The Fourier integral theorem
APPENDIX A The Fourier integral theorem In equation (1.7) of Section 1.3 we gave a description of a signal defined on an infinite range in the form of a double integral, with no explanation as to how that
More informationIntroduction to Fourier Analysis Part 2. CS 510 Lecture #7 January 31, 2018
Introduction to Fourier Analysis Part 2 CS 510 Lecture #7 January 31, 2018 OpenCV on CS Dept. Machines 2/4/18 CSU CS 510, Ross Beveridge & Bruce Draper 2 In the extreme, a square wave Graphic from http://www.mechatronics.colostate.edu/figures/4-4.jpg
More informationIntroduction to Biomedical Engineering
Introduction to Biomedical Engineering Biosignal processing Kung-Bin Sung 6/11/2007 1 Outline Chapter 10: Biosignal processing Characteristics of biosignals Frequency domain representation and analysis
More informationFundamentals of the DFT (fft) Algorithms
Fundamentals of the DFT (fft) Algorithms D. Sundararajan November 6, 9 Contents 1 The PM DIF DFT Algorithm 1.1 Half-wave symmetry of periodic waveforms.............. 1. The DFT definition and the half-wave
More informationClosed book (equation sheet handed out with this midterm) Calculators OK You must show your work to get full credit
UNIVERSITY OF CALIFORNIA College of Engineering Electrical Engineering and Computer Sciences Department EECS 145M: Microcomputer Interfacing Laboratory Spring Midterm #2 Monday, April 19, 1999 Closed book
More informationG52IVG, School of Computer Science, University of Nottingham
Image Transforms Fourier Transform Basic idea 1 Image Transforms Fourier transform theory Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is F ( u) f ( x)exp[ j2πux]
More informationOSE801 Engineering System Identification. Lecture 09: Computing Impulse and Frequency Response Functions
OSE801 Engineering System Identification Lecture 09: Computing Impulse and Frequency Response Functions 1 Extracting Impulse and Frequency Response Functions In the preceding sections, signal processing
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationComplex symmetry Signals and Systems Fall 2015
18-90 Signals and Systems Fall 015 Complex symmetry 1. Complex symmetry This section deals with the complex symmetry property. As an example I will use the DTFT for a aperiodic discrete-time signal. The
More informationFOURIER TRANSFORM AND
Version: 11 THE DISCRETE TEX d: Oct. 23, 2013 FOURIER TRANSFORM AND THE FFT PREVIEW Classical numerical analysis techniques depend largely on polynomial approximation of functions for differentiation,
More informationDiscrete Fourier Transform
Last lecture I introduced the idea that any function defined on x 0,..., N 1 could be written a sum of sines and cosines. There are two different reasons why this is useful. The first is a general one,
More informationDigital Speech Processing Lecture 10. Short-Time Fourier Analysis Methods - Filter Bank Design
Digital Speech Processing Lecture Short-Time Fourier Analysis Methods - Filter Bank Design Review of STFT j j ˆ m ˆ. X e x[ mw ] [ nˆ m] e nˆ function of nˆ looks like a time sequence function of ˆ looks
More informationEE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet
EE538 Final Exam Fall 005 3:0 pm -5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five
More information1 1.27z z 2. 1 z H 2
E481 Digital Signal Processing Exam Date: Thursday -1-1 16:15 18:45 Final Exam - Solutions Dan Ellis 1. (a) In this direct-form II second-order-section filter, the first stage has
More informationNew Algorithms for Removal of DC Offset and Subsynchronous. Resonance terms in the Current and Voltage Signals under Fault.
ew Algorithms for Removal of DC Offset and Subsynchronous Resonance terms in the Current and Voltage Signals under Fault Conditions KAALESH KUAR SHARA Department of Electronics & Communication Engineering
More informationFourier transform. Alejandro Ribeiro. February 1, 2018
Fourier transform Alejandro Ribeiro February 1, 2018 The discrete Fourier transform (DFT) is a computational tool to work with signals that are defined on a discrete time support and contain a finite number
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More information