Digital Signal Octave Codes (0A)
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1 Digital Signal Periodic Conditions
2 Copyright (c) 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 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.
3 Based on M.J. Roberts, Fundamentals of Signals and Systems S.K. Mitra, Digital Signal Processing : a computer-based approach 2nd ed S.D. Stearns, Digital Signal Processing with Examples in MATLAB 3
4 Sampling and Normalized Frequency ω0 t = 2π f 0t t = nt s ω0 n T s = 2 π f 0 n T s f0= = 2π nts T0 Ts : sampling period T0 T0 : signal period Ts = 2πn T0 Ts f = 0 T0 fs normalization = 2 π n F0 f0 F0 = f 0T s = fs normalization 4
5 Analog and Digital Frequencies Analog Signal Frequency ω0 t = 2 π f 0 t t = nt s t = nt s n ω0 T s = 2 π n f 0 T s ω 0 T s = Ω0 f 0 T s = F0 Ω0 n = 2 π n F 0 Digital Signal Frequency 5
6 Multiplying by Ts Normalization ω0 t = 2 π f 0 t T s T s Ω0 n = 2 π n F 0 6 Analog Signal Normalization Digital Signal
7 Normalization F 0 = f 0 T s f 0 T s Multiplied by Ts = f0 / fs f0 / fs Divided by fs = Ts / T0 Ω0 = 2 π F 0 7
8 Normalized Cyclic and Radian Frequencies Normalized Cyclic Frequency f 0 cycles/ second F 0 cycles/ sample = f s samples/ second Normalized Radian Frequency ω 0 cycles / second Ω0 cycles / sample = f s samples/ second 8
9 Periodic Relation : N0 and F0 e j (2 π (n+ N 0 ) F 0 ) =e j (2 π n F 0 ) e j2πm = Digital Signal Period N0 : the smallest integer e j 2 π N 0 F0 e j2πm 2 π N 0 F0 = 2 π m 9 = Periodic Condition : integer m N 0 F0 = m
10 Periodic Condition : N0 and F0 2 π N 0 F0 = 2 π m T0 m N0 = = m F0 Ts Digital Signal Period N0 : the smallest integer Periodic Condition : the smallest integer m N 0 F0 = m Integer * Rational : must be an Integer m Ts m= p reduced form 0
11 Periodic Condition : N0 and F0 in a reduced form Integer reduced form T0 m q N0 = = m = m F0 Ts p Integer * Rational : must be an Integer Rational numbers
12 N0 and F0 in a reduced form : Examples reduced form p F0 = q Rational m q N0 = = m F0 p integer N0 q m p integers the smallest integer m = = = F m=3 N0 = 2 m 4.07 N = = = F m=3 N0 = 2 m 5 N 0 0 2
13 Periodic Relations Analog and Digital Cases e j (2 π(n+ N 0 ) F 0 ) =e j (2 π n F 0 ) e Digital Signal Period N0 : the smallest integer =e j(2 π f 0 )t Analog Signal Period T0 : the smallest real number T0 m N0 = = m F0 Ts T0 = f0 k T 0 f 0 = k k N 0 F 0 = k m integer multiple of m : some integers m j (2 π f 0 )(t +T 0 ) all integers 3
14 Periodic Conditions Analog and Digital Cases N F 0 = k m T f 0 = k k m N= F0 k T= f0 Integer N0 Rational F0 Minimum Integer N0 N0 = q m= p Real T0 Real f0 Minimum Real T0 T0 = f0 p F0 = q m= reduced form m N0 = p/ q 4
15 Periodic Conditions Examples N F 0 = k m T f 0 = k m N0 = F0 T0 = f0 Integer N given F0 = Real T given f0= km: multiples of k: all integers N 0 = m = T 0 = = 2 N 0 = 72 2 m = 72 2T 0 = 2 2 = 2 3 N 0 = 08 3 m = 08 3T 0 = 3 3 = 3 5
16 Periodic Condition of a Sampled Signal g(n T s ) = A cos(2 π f 0 T s n+θ) F 0 = f 0 T s = f 0/ f s g [n] = A cos(2 π F 0 n + θ) 2 π F0n = 2 π m F0 = m n F0n = m n, m integers Rational Number integers F0 = m n n, m The Smallest Integer n N 0 = min(n) F0 = m N0 M.J. Roberts, Fundamentals of Signals and Systems 6
17 F0 and N0 of a Sampled Signal Integer N0 Rational Number F0 F0 = m p = n q f0 Ts F0 = = fs T0 integer n, m, p, q real N 0 F0 = m f 0, f s,t s, T 0 N0 = 2 π F0 n T f m q = m 0 = m s = m F0 Ts f0 p 2π f 0T s n M.J. Roberts, Fundamentals of Signals and Systems 7
18 A cosine waveform example n= [0:]; x= cos(2*pi**(n/0)); n T s = n 0 2 π F0n = 2 π f 0Ts n f0 Ts F0 = f 0T s = = fs T0 n= [0:]; x= cos(2*pi*(/0)*n); n T s = n 2π f 0 nt s 2 π f 0 nt s = 2 π n 0 = 2 π n 0 T s = 0. Ts = f 0 = (T 0 = ) f 0 = 0. (T 0 =0) F 0 = f 0 T s = 0. F 0 = f 0 T s = 0. U of Rhode Island, ELE 4, FFT Tutorial 8
19 Two cases of the same F0 = f0 Ts cos (0.2 π n) cos(0.2 π n) cos(2 π f 0 n T s ) cos(2 π f 0 n T s ) cos(2 π 0. n) cos(2 π 0. n) T s = 0. Ts = f0= f 0 = 0. F 0 = 0. F 0 = 0. U of Rhode Island, ELE 4, FFT Tutorial
20 The same sampled waveform examples cos(0.2 π n) cos(0.2 π n) 2 π f 0 nt s 2 π n/0 n= [0:]; x= cos(2*pi**0.*n); 0. = f 0 T s 0. = F 0 2 π F0n = 2 π f 0Ts n n= [0:]; x= cos(2*pi*0.**n); T s = 0. T s = 0.2 T s = 0.5 Ts = f0 = f 0 = 0.5 f 0 = 0.2 f 0 = 0. F 0 = 0. F 0 = 0. F 0 = 0. F 0 = 0. U of Rhode Island, ELE 4, FFT Tutorial 20
21 Many waveforms share the same sampled data The same sampled data T s = 0. T0 = 2 π n/ 0 2π nf 0T s ` 0. = f 0 T s 0. = F 0 3 T s = 0. T0 = Ts = T 0 = 0 Ts = T 0 = 0 30 U of Rhode Island, ELE 4, FFT Tutorial 2
22 Different number of data points x= cos(2*pi*n/0); [0,, ] 20 data points [0:]; t = [0:]/0; y = cos(2*pi*t); stem(t, y) hold on t2 = [0:9]/00; y2 = cos(2*pi*t2); plot(t2, y2) size([0:], 2) = 20 [0,,9] 200 data points [0:9]; size([0:9], 2) = 200 U of Rhode Island, ELE 4, FFT Tutorial 22
23 Normalized data points x= cos(2*pi*n/0); t = [0:]/0; [0.0,,.90 ] 20 data points coarse resolution t = [0:]/0; y = cos(2*pi*t); stem(t, y) hold on t2 = [0:9]/00; y2 = cos(2*pi*t2); plot(t2, y2) [0.0,,.90 ] 2 cycles t2 = [0:9]/00; [0,, 4 π ] [0.0,,.99 ] 200 data points fine resolution [0.0,,.99 ] 2 cycles [0,, 4 π ] U of Rhode Island, ELE 4, FFT Tutorial 23
24 Different number of data points [0,, ] 20 data points [0:]; size([0:], 2) = 20 x= cos(0.2*pi*n); t = [0:]; y = cos(0.2*pi*t); stem(t, y) hold on t2 = [0:9]/0; y2 = cos(0.2*pi*t2); plot(t2, y2) [0,,9] 200 data points [0:9]; size([0:9], 2) = 200 U of Rhode Island, ELE 4, FFT Tutorial 24
25 Normalized data points t = [0:]; [0.0,,.0 ] 20 data points coarse resolution [0.0,,.0 ] 2 cycles [0,, 4 π ] x= cos(0.2*pi*n); t = [0:]; y = cos(0.2*pi*t); stem(t, y) hold on t2 = [0:0]/0; y2 = cos(0.2*pi*t2); plot(t2, y2) t2 = [0:9]/0; [0.0,,.9 ] 200 data points fine resolution [0.0,,.9 ] 2 cycles [0,, 4 π ] U of Rhode Island, ELE 4, FFT Tutorial 25
26 Plotting sampled cosine waves x= cos(2*pi*n/0); t = [0:]/0; y = cos(2*pi*t); stem(t, y) hold on t2 = [0:9]/00; y2 = cos(2*pi*t2); plot(t2, y2) t = [0:]/0; [0.0,,.9 ] 20 data points y = cos(2*pi*t); stem(t, y) t2 = [0:9]/00; [0.0,,.99 ] 200 data points y = cos(2*pi*t2); plot(t2, y) coarse resolution fine resolution x= cos(0.2*pi*n); t = [0:]; y = cos(0.2*pi*t); stem(t, y) hold on t2 = [0:0]/0; y2 = cos(0.2*pi*t2); plot(t2, y2) t = [0:]; [0.0,,.9 ] 20 data points y = cos(0.2*pi*t); stem(t, y) t2 = [0:9]/00; y = cos(0.2*pi*t2); coarse resolution [0.0,,.99 ] 200 data points plot(t2, y) fine resolution U of Rhode Island, ELE 4, FFT Tutorial 26
27 Two waveforms with the same normalized frequency x= cos(2*pi*n/0); t = [0:]/0; y = cos(2*pi*t); stem(t, y) hold on t2 = [0:9]/00; y2 = cos(2*pi*t2); plot(t2, y2) x= cos(0.2*pi*n); t = [0:]; y = cos(0.2*pi*t); stem(t, y) hold on t2 = [0:0]/0; y2 = cos(0.2*pi*t2); plot(t2, y2) cos(2 π t) [0.0,,.9 ] 2 cycles cos(2 π t) cos(0.2 π t) F 0 = 0. f0 = T0 = f s = 0 T s = 0. F 0 = 0. [0.,,.] 2 cycles cos(2 π 0. t) f 0 = 0. T 0 = 0 fs= Ts = U of Rhode Island, ELE 4, FFT Tutorial 27
28 Cosine Wave x= cos(2*pi*n/0); cos(2 π t ) T0 = cos(2 π t) f0 = T0 = f s = 0 T s = 0. t = [0:29]/0; y = cos(2*pi*t); stem(t, y) hold on t2 = [0:299]/00; y2 = cos(2*pi*t2); plot(t2, y2) f0 = T s = 0. F 0 = f 0 T s = 0. U of Rhode Island, ELE 4, FFT Tutorial 28
29 Cosine Wave 2 x= cos(0.2*pi*n); cos(0.2 π t ) T 0 = 0 cos(2 π 0. t) f 0 = 0. T 0 = 0 fs= Ts = t = [0:29]; y = cos(0.2*pi*t); stem(t, y) hold on t2 = [0:299]/0; y2 = cos(0.2*pi*t2); plot(t2, y2) f 0 = 0. Ts = F 0 = f 0 T s = 0. U of Rhode Island, ELE 4, FFT Tutorial 29
30 Sampled Sinusoids g [n] = A cos(2 π F 0 n + θ) F0 2 π F0 g [n] = A cos(2 π n m/ N 0 + θ) m/ N0 2 πm/ N0 g[n] = A cos(ω0 n + θ) Ω0 / 2 π Ω0 N0 = g[n] = A e βn g[n] = A z n m F0 N0 F0 z = eβ M.J. Roberts, Fundamentals of Signals and Systems 30
31 Sampling Period Ts and Frequency fs g (t ) = A cos(2 π f 0 t +θ) F 0 f 0 T s f 0 F 0 f s Ts g [n] = A cos(2 π F 0 n + θ) Ts = fs =fs Ts sampling period sampling frequency sampling rate M.J. Roberts, Fundamentals of Signals and Systems 3
32 Periodic Condition Examples g (t ) = A cos(2 π f 0 t +θ) g[n] = A cos(2 π F 0 n + θ) 72 π t ( ) = 4 cos ( 2 π t ) g[t ] = 4 cos f0= 72 π n ( ) = 4 cos ( 2 π n ) g[n] = 4 cos F0 = F 0 = f 0 T s = f 0/ f s F0 = f 0T s = f 0 Ts = M.J. Roberts, Fundamentals of Signals and Systems 32
33 Periodic Condition Examples 72 π t ( ) = 4 cos ( 2 π t ) g (t ) = 4 cos ( 2 π (t + T )) g[t ] = 4 cos 0 T0 = 72 π n ( ) = 4 cos ( 2 π n ) g [n] = 4 cos ( 2 π (n + N ) ) g[n] = 4 cos Ts = 0 N 0 = Fundamental Period of g [n] Fundamental Period of g(t ) N0 F0 N0 = q F0 q = F0 N0 M.J. Roberts, Fundamentals of Signals and Systems 33
34 Periodic Condition Examples g (t ) = 4 cos 2 π (t + T 0 ) ( ) g[n] = 4 cos 2 π (n + N 0 ) ( q F0 = = N0 Fundamental Period of g(t ) N 0 = Fundamental Period of g [n] T0 = ) the number of cycles in N0 samples the smallest integer : fundamental period N0 F0 N0 = q F0 q = F0 N0 M.J. Roberts, Fundamentals of Signals and Systems 34
35 Periodic Condition Examples q F0 = = N0 the number of cycles in N0 samples the smallest integer : fundamental period When F0 is not the reciprocal of an integer (q=), a discrete-time sinusoid may not be immediately recognizable from its graph as a sinusoid. M.J. Roberts, Fundamentals of Signals and Systems 35
36 Periodic Condition Examples g[n] = 4 cos 2 π (n + N 0 ) ( ) (n + N 0 ) integer N 0 = g (t ) = 4 cos 2 π (t + T 0 ) ( ) N = k 0 integer T 0 = k integer integer integer Fundamental period of g [n] (t + T 0 ) T0 = N0 T0 integer Fundamental period of g(t ) M.J. Roberts, Fundamentals of Signals and Systems
37 Periodic Condition Examples g [n] = 4 cos 2 π n ) 2 g [n] = 4 cos 2 π n ) 3 g [n] = 4 cos 2 π n ) ( ( ( g [n] = 4 cos 2 π n ( ) clf n = [0:]; t = [0:00]/00; y = 4*cos(2*pi*(/)*n); y2 = 4*cos(2*pi*(2/)*n); y3 = 4*cos(2*pi*(3/)*n); y4 = 4*cos(2*pi*(/)*n); yt = 4*cos(2*pi*(/)*t); yt2 = 4*cos(2*pi*(2/)*t); yt3 = 4*cos(2*pi*(3/)*t); yt4 = 4*cos(2*pi*(/)*t); subplot(4,,); stem(n, y); hold on; plot(t, yt); subplot(4,,2); stem(n, y2); hold on; plot(t, yt2); subplot(4,,3); stem(n, y3); hold on; plot(t, yt3); subplot(4,,4); stem(n, y4); hold on; plot(t, yt4); M.J. Roberts, Fundamentals of Signals and Systems 37
38 Periodic Condition Examples g [n] = 4 cos 2 π n ) 2 g [n] = 4 cos 2 π n ) 3 g [n] = 4 cos 2 π n ) g [n] = 4 cos 2 π n ) ( ( ( ( N 0 = Ts = cycle 2 cycles 3 cycles cycles 38
39 Periodic Condition Examples g (t ) = A cos(2 π f 0 t +θ) g (t ) = 4 cos ( 2 π t ) g2 (t ) = 4 cos ( 2 π 2 t ) g3 (t ) = 4 cos ( 2 π 3 t ) t n T t nt2 t nt3 T = T2 = T3 = g [n] = A cos(2 π F 0 n + θ) t n T t nt2 t nt3 g [n] = 4 cos ( 2 π n T s ) g2 [n] = 4 cos ( 2 π n T s 2 ) g3 [n] = 4 cos ( 2 π n T s 3 ) n = 0,, 2, 3, n = 0,, 2, 3, n = 0,, 2, 3, t = 0, 0., 0.2, 0.3, 2 t = 0, 0., 0.2, 0.3, 3 t = 0, 0., 0.2, 0.3, { g [n] } { g2 [n ] } { g2 [n] } M.J. Roberts, Fundamentals of Signals and Systems 39
40 Periodic Condition Examples g (t ) = A cos(2 π f 0 t +θ) 2 π F0n = 2 π m g [n] = A cos(2 π F 0 n + θ) n=m g [n] = 4 cos ( 72 π n m = n ) ( ( )) ( ( = 4 cos 2 π = 4 cos 2 π f0 m = = n fs n (n + N 0 ) smallest smallest n = / N 0 )) F0 = N 0 = q N0 = F0 = Ω0 /2 π M.J. Roberts, Fundamentals of Signals and Systems 40
41 Periodic Condition Examples 4 cos ( ω t ) 4 cos ( ω n ) t = 2t 2 n Ts = n = 2 n2 4 cos ( 2ω t 2 ) 4 cos ( ω 2n 2 ) n 2 2 Ts = 2 M.J. Roberts, Fundamentals of Signals and Systems 4
42 Periodic Condition Examples Ts = Ts = Ts = 0 ω = 2 π 20 ω = 2 π 2 30 clf n = [0:0]; t = [0:000]/000; y = 4*cos(2*pi**n/0); y2 = 4*cos(2*pi*2*n/20); y3 = 4*cos(2*pi*3*n/30); yt = 4*cos(2*pi*t); yt2 = 4*cos(2*pi*2*t); yt3 = 4*cos(2*pi*3*t); subplot(3,,); stem(n, y); hold on; plot(t, yt); subplot(3,,2); stem(n/20, y2); hold on; plot(t, yt2); subplot(3,,3); stem(n/30, y3); hold on; plot(t, yt3); ω = 2 π 3 M.J. Roberts, Fundamentals of Signals and Systems 42
43 References [] [2] [3] [4] [5] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 M.J. Roberts, Fundamentals of Signals and Systems S.J. Orfanidis, Introduction to Signal Processing K. Shin, et al., Fundamentals of Signal Processing for Sound and Vibration Engineerings [6] A graphical interpretation of the DFT and FFT, by Steve Mann 43
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Generation Copyright (c) 2016-2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
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Relations (3A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
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General (2A Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
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Background ODEs (2A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
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Definitions of the Laplace Transform (A) Copyright (c) 24 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
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Matrix (A Copyright (c 01 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published
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Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015 Copyright (c) 011-015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
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Higher Order ODE's, (3A) Initial Value Problems, and Boundary Value Problems Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms
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Hilbert nner Product Space (2B) Copyright (c) 2009-2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
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Complex Series (3A) 8/7/3 Copyright (c) 202, 203 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 or
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Introduction to ODE's (0A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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Separable Equations (1A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
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Surface Integrals (6A) Surface Integral Stokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation
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Linear Equations with Constant Coefficients (2A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
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Hamiltonian Cycle (3A) Copyright (c) 015 018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any
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Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
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Complex Functions (1A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
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Second Order ODE's (2A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
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/6/5 Copyright (c) 2 25 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 or any later version published
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Introuction to ODE's (0P) Copyright (c) 2011-2014 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
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(6A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
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