Bb Lab Notes & Assignments - Digit x Bb ENGR 203 C A ENGR...

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Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C A ENGR 203 3.9 Signal Energy and Power X blackboard.vsu.edu/bbcswebdav/pid-1694284-dt-content-rid-20149868_1/courses/CPEG-423-01-SPR22/Lab3-signal%20energy%20and%20power%281%29.... 1 2 Type here to search O X Graw 2/2 | sum 3. Use MATLAB to Calculate (refer to the book, Signals and Systems, Chapter 3.9) signal energy for x1[n] (0<n<10) and signal power for x2[n]. 150% + Understand the following Matlab functions and operators: ^ Array power 2 11 Messages - Solutioninn.com X + L ww 6 S W 96% 29°F 4) 1:35 AM 2/9/2022 x ⠀ Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C A 115 SIGNAL ENERGY X prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Signal energy is defined by 3.9 SIGNAL ENERGY AND POWER 3.9 Signal Energy and Power X Type here to search and its units are simply the square of the units of the signal itself. 00 Ex = Σ \x [n] 11=-00 1 EXAMPLE 3.5 Signal energy of a signal Find the signal energy of x[n] = (1/2)"u[n]. From the definition of signal energy, Messages - Solutioninn.com L w S W X + 96% 29°F Aa Read aloud (3.8) G 1:35 AM 2/9/2022 X : Bb Lab Notes & Assignments - Digit x Bb ENGR 203 + > C 84 i prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d 13 X tw 3.9 Signal Energy and Power X - Messages - Solutioninn.com EXAMPLE 3.5 Signal energy of a signal Find the signal energy of x[n] = (1/2)mu[n]. From the definition of signal energy, :00 201 Ex = Σ \x[n] = Σ Σ (3) - Σ (3)| - Σ (3) Jum =Σ(;) un ==00 This infinite series can be rewritten as Type here to search We can use the formula for the summation of an infinite geometric series Ξ " Ex = 1 + Ex=1+++ 4 4² ' 00 Σ = - p= n=0 Ι L \r\ < 1 w = 1 + S W 1 2² X + + 24 96% 29°F Ε Aa Ε 1:35 AM 2/9/2022 X Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 SIGNAL POWER X 3.9 Signal Energy and Power X prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d For many signals encountered in signal and system analysis, the summation Type here to search 00 Ex = Σ kx [n]³² 11=-00 1 N→ 2N Px = lim does not converge because the signal energy is infinite, and this usually occurs because the signal is not time limited. The unit sequence is an example of a signal with infinite energy. For signals of this type, it is more convenient to deal with the average signal power of the signal instead of the signal energy. The definition of average signal power is Messages - Solutioninn.com 9 N-1 Σ 1x [n]12 n=-N X + and this is the average signal power over all time. (Why is the upper summation limit N - 1 instead of N?) 81 L S W Page 103 96% 29°F (3.9) Aa ← 1:35 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 > prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d X and this is the average signal power over all time. (Why is the upper summation limit N - 1 instead of N?) (a) x[n] (0.9) sin(2/4) For periodic signals, the average signal power calculation may be simpler. The average value of any periodic function is the average over any period and Px = = Type here to search 3.9 Signal Energy and Power X 1 N O no+N-1 Σkx[n]|² n=no = EXAMPLE 3.6 Finding signal energy and power of signals using MATLAB Using MATLAB find the signal energy or power of the signals, where the notation notation En=(x) means the summation over any range of consecutive n's that is N in length, where N can be any period of Ix[n]1². Then compare the results with analytical calculations. 1 N n=(N) 9 Messages - Solutioninn.com X x [n]², no any integer and (b) x[n] = 485[n]- 787[n]. L ww + S W 96% 29°F (3.10) @ Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 EXAMPLE 3.6 prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Finding signal energy and power of signals using MATLAB Using MATLAB find the signal energy or power of the signals, X (a) x[n] = (0.9) sin(2/4) and (b) x[n] = 485[n]- 787[n]. Then compare the results with analytical calculations. % 3.9 Signal Energy and Power X n = -100:100; % Program to compute the signal energy or power of some example signals (a) % Type here to search Messages - Solutioninn.com Compute the value of the function and its square x = (0.9). ^abs (n). *sin(2*pi*n/4) ; xsq = x.^2; Ex = sum(xsq); 1 % Set up a vector of discrete times at % which to compute the value of the % function % Use the sum function in MATLAB to % find the total energy and display L S W X + 96% 29°F @ Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 > CX Sul X4 disp(['(b) Ex = ', num2str (Ex)]); % (b) prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d NO = 35; n = 0: NO-1; % X disp(['(d) Px = ', num2str (Px)]); The output of this program is. Type here to search 3.9 Signal Energy and Power X x = 4* impND(5,n) - 7*impND (7,n); xsq = x.^2; X Px sum(xsq)/NO; 21 11 Compute the value of the function and its square 70 use the Sulll Tunction MATLAD LU Messages - Solutioninn.com % find the total energy and display % the result. % The fundamental period is 35 % Set up a vector of discrete times % over one period at which to compute % the value of the function % Use the sum function in MATLAB to % find the average power and display % the result. L ww S W X + 96% Page 104 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 < (a) Ex = 4.7107 (b) Px = 8.6 prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Analytical computations: Type here to search X 00 (a) Ex = \x Σ 1x In] = Σ |(0.9)" sin (2nn/4) 11=-00 11=-00 00 3.9 Signal Energy and Power X Ex = (0.9)" sin (2n/4)|² + n=0 00 n=0 21 0 11=-00 Ex = (0.9)²¹ sin² (2n/4) + (0.9)-2" sin² (2лn/4) 3' (0.9)" sin (2n/4)|² = |x [0]|² =0 00 Ex=(0.9)²" (1 - cos (n)) + 2 2 n=() L 11=-00 Messages - Solutioninn.com X + 0 1 Σ(0.9)-2 (1-сos (n)) n=-00 w S W 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 13 X prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Type here to search 00 1 Ex=(0.9)²¹ (1 - cos (an)) + n=0 3.9 Signal Energy and Power X Ex = ((0.9)2-(0.9) 2n Σ. n=0) O Using the even symmetry of the cosine function, and letting n→-n in the second summation, 21 11 ein + e-jan 2 Using the formula for the sum of an infinite geometric series, 3' 00 n=0 Ex = (0.9)2" (1-cos (an)) 00 Σ n=0 n=8 12--00 00 | = Σ (0.81)" . - n=0 ph= Messages - Solutioninn.com X L (0.9)-2 (1-сos (n)) ww 1 2 , Irl < 1 + 00 00 Σ (0.81€*)" + Σ (0.81e)" S W 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 13 (b) prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Px P₁ = Type here to search 1 No Ex Σ Ix [n]²² n=(No) Px = X = 1 35 1 No Ex E1-081-21+0.81 +1+081] 3.9 Signal Energy and Power X 1 1 1 -0.81 2 No-1 Σ 1x [n]] 7=( 00 Σr = \rl < 1 n=0) 2 11 - 1 35 34 1 1-r n=0 1 1 -0.81ej = Messages - Solutioninn.com + L 1 1 -0.81e-j 1 10.81 1485 [n] - 787 [n]1² The impulses in the two impulse train functions only coincide at integer multiples of 35. Therefore in this summation range they coincide only at n = 0. The net impulse strength at n = 0 is therefore -3. All the other impulses occur alone and the sum of the squares is the same as the square of the sum. Therefore Page 105 i 1 1+0.81 (-3)² + 4²2+(-7)² +42 + (−7)² +4² +4² + (−7)² + 4² + (−7)² + 4² n=0 n=5 n=7 n=10 n=14 n=15 n=20 n=21 n=25 n=28 n=30, X S + W = 4.7107 Check. 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C A 13 (b) prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Px = Type here to search 1 No Ex = Σ kx[n]P² n=(No) 1 1 -0.81 Px = = 1 35 X 1 No 1 - 3.9 Signal Energy and Power X 1 21+0.81 No-1 Σx [n]1² 71=0 Px = = + 2 11 1 1 +0.81 1 35 34 35 n=() The impulses in the two impulse train functions only coincide at integer multiples of 35. Therefore in this summation range they coincide only at n = 0. The net impulse strength at n = 0 is therefore -3. All the other impulses occur alone and the sum of the squares is the same as the square of the sum. Therefore Page 105 = 9+6x4² +4x(-7)² 1485 [n] - 787 [n]1² (3-* (-3)² +42 + (−7)² +42 + (−7)² +4² +4² + (−7)² + 4² + (−7)² + 4² n=21 n=25 n=28 n=30, n=() n=5 n=7 n=10 n=14 n=15 n=20 Messages - Solutioninn.com X + 1 10.81 = L 1 1+0.81 9 + 96 + 196 35 ww = 4.7107 Check. = 8.6 Check. S W 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C A ENGR 203 3.9 Signal Energy and Power X blackboard.vsu.edu/bbcswebdav/pid-1694284-dt-content-rid-20149868_1/courses/CPEG-423-01-SPR22/Lab3-signal%20energy%20and%20power%281%29.... 1 2 Type here to search O X Graw 2/2 | sum 3. Use MATLAB to Calculate (refer to the book, Signals and Systems, Chapter 3.9) signal energy for x1[n] (0<n<10) and signal power for x2[n]. 150% + Understand the following Matlab functions and operators: ^ Array power 2 11 Messages - Solutioninn.com X + L ww 6 S W 96% 29°F 4) 1:35 AM 2/9/2022 x ⠀ Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C A 115 SIGNAL ENERGY X prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Signal energy is defined by 3.9 SIGNAL ENERGY AND POWER 3.9 Signal Energy and Power X Type here to search and its units are simply the square of the units of the signal itself. 00 Ex = Σ \x [n] 11=-00 1 EXAMPLE 3.5 Signal energy of a signal Find the signal energy of x[n] = (1/2)"u[n]. From the definition of signal energy, Messages - Solutioninn.com L w S W X + 96% 29°F Aa Read aloud (3.8) G 1:35 AM 2/9/2022 X : Bb Lab Notes & Assignments - Digit x Bb ENGR 203 + > C 84 i prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d 13 X tw 3.9 Signal Energy and Power X - Messages - Solutioninn.com EXAMPLE 3.5 Signal energy of a signal Find the signal energy of x[n] = (1/2)mu[n]. From the definition of signal energy, :00 201 Ex = Σ \x[n] = Σ Σ (3) - Σ (3)| - Σ (3) Jum =Σ(;) un ==00 This infinite series can be rewritten as Type here to search We can use the formula for the summation of an infinite geometric series Ξ " Ex = 1 + Ex=1+++ 4 4² ' 00 Σ = - p= n=0 Ι L \r\ < 1 w = 1 + S W 1 2² X + + 24 96% 29°F Ε Aa Ε 1:35 AM 2/9/2022 X Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 SIGNAL POWER X 3.9 Signal Energy and Power X prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d For many signals encountered in signal and system analysis, the summation Type here to search 00 Ex = Σ kx [n]³² 11=-00 1 N→ 2N Px = lim does not converge because the signal energy is infinite, and this usually occurs because the signal is not time limited. The unit sequence is an example of a signal with infinite energy. For signals of this type, it is more convenient to deal with the average signal power of the signal instead of the signal energy. The definition of average signal power is Messages - Solutioninn.com 9 N-1 Σ 1x [n]12 n=-N X + and this is the average signal power over all time. (Why is the upper summation limit N - 1 instead of N?) 81 L S W Page 103 96% 29°F (3.9) Aa ← 1:35 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 > prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d X and this is the average signal power over all time. (Why is the upper summation limit N - 1 instead of N?) (a) x[n] (0.9) sin(2/4) For periodic signals, the average signal power calculation may be simpler. The average value of any periodic function is the average over any period and Px = = Type here to search 3.9 Signal Energy and Power X 1 N O no+N-1 Σkx[n]|² n=no = EXAMPLE 3.6 Finding signal energy and power of signals using MATLAB Using MATLAB find the signal energy or power of the signals, where the notation notation En=(x) means the summation over any range of consecutive n's that is N in length, where N can be any period of Ix[n]1². Then compare the results with analytical calculations. 1 N n=(N) 9 Messages - Solutioninn.com X x [n]², no any integer and (b) x[n] = 485[n]- 787[n]. L ww + S W 96% 29°F (3.10) @ Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 EXAMPLE 3.6 prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Finding signal energy and power of signals using MATLAB Using MATLAB find the signal energy or power of the signals, X (a) x[n] = (0.9) sin(2/4) and (b) x[n] = 485[n]- 787[n]. Then compare the results with analytical calculations. % 3.9 Signal Energy and Power X n = -100:100; % Program to compute the signal energy or power of some example signals (a) % Type here to search Messages - Solutioninn.com Compute the value of the function and its square x = (0.9). ^abs (n). *sin(2*pi*n/4) ; xsq = x.^2; Ex = sum(xsq); 1 % Set up a vector of discrete times at % which to compute the value of the % function % Use the sum function in MATLAB to % find the total energy and display L S W X + 96% 29°F @ Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 > CX Sul X4 disp(['(b) Ex = ', num2str (Ex)]); % (b) prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d NO = 35; n = 0: NO-1; % X disp(['(d) Px = ', num2str (Px)]); The output of this program is. Type here to search 3.9 Signal Energy and Power X x = 4* impND(5,n) - 7*impND (7,n); xsq = x.^2; X Px sum(xsq)/NO; 21 11 Compute the value of the function and its square 70 use the Sulll Tunction MATLAD LU Messages - Solutioninn.com % find the total energy and display % the result. % The fundamental period is 35 % Set up a vector of discrete times % over one period at which to compute % the value of the function % Use the sum function in MATLAB to % find the average power and display % the result. L ww S W X + 96% Page 104 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 115 < (a) Ex = 4.7107 (b) Px = 8.6 prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Analytical computations: Type here to search X 00 (a) Ex = \x Σ 1x In] = Σ |(0.9)" sin (2nn/4) 11=-00 11=-00 00 3.9 Signal Energy and Power X Ex = (0.9)" sin (2n/4)|² + n=0 00 n=0 21 0 11=-00 Ex = (0.9)²¹ sin² (2n/4) + (0.9)-2" sin² (2лn/4) 3' (0.9)" sin (2n/4)|² = |x [0]|² =0 00 Ex=(0.9)²" (1 - cos (n)) + 2 2 n=() L 11=-00 Messages - Solutioninn.com X + 0 1 Σ(0.9)-2 (1-сos (n)) n=-00 w S W 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 13 X prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Type here to search 00 1 Ex=(0.9)²¹ (1 - cos (an)) + n=0 3.9 Signal Energy and Power X Ex = ((0.9)2-(0.9) 2n Σ. n=0) O Using the even symmetry of the cosine function, and letting n→-n in the second summation, 21 11 ein + e-jan 2 Using the formula for the sum of an infinite geometric series, 3' 00 n=0 Ex = (0.9)2" (1-cos (an)) 00 Σ n=0 n=8 12--00 00 | = Σ (0.81)" . - n=0 ph= Messages - Solutioninn.com X L (0.9)-2 (1-сos (n)) ww 1 2 , Irl < 1 + 00 00 Σ (0.81€*)" + Σ (0.81e)" S W 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C 84 13 (b) prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Px P₁ = Type here to search 1 No Ex Σ Ix [n]²² n=(No) Px = X = 1 35 1 No Ex E1-081-21+0.81 +1+081] 3.9 Signal Energy and Power X 1 1 1 -0.81 2 No-1 Σ 1x [n]] 7=( 00 Σr = \rl < 1 n=0) 2 11 - 1 35 34 1 1-r n=0 1 1 -0.81ej = Messages - Solutioninn.com + L 1 1 -0.81e-j 1 10.81 1485 [n] - 787 [n]1² The impulses in the two impulse train functions only coincide at integer multiples of 35. Therefore in this summation range they coincide only at n = 0. The net impulse strength at n = 0 is therefore -3. All the other impulses occur alone and the sum of the squares is the same as the square of the sum. Therefore Page 105 i 1 1+0.81 (-3)² + 4²2+(-7)² +42 + (−7)² +4² +4² + (−7)² + 4² + (−7)² + 4² n=0 n=5 n=7 n=10 n=14 n=15 n=20 n=21 n=25 n=28 n=30, X S + W = 4.7107 Check. 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1 Bb Lab Notes & Assignments - Digit x Bb ENGR 203 ← → C A 13 (b) prod.reader-ui.prod.mheducation.com/epub/sn_0d00/data-uuid-40c3907268e04d20acf96e3d1b29c28d Px = Type here to search 1 No Ex = Σ kx[n]P² n=(No) 1 1 -0.81 Px = = 1 35 X 1 No 1 - 3.9 Signal Energy and Power X 1 21+0.81 No-1 Σx [n]1² 71=0 Px = = + 2 11 1 1 +0.81 1 35 34 35 n=() The impulses in the two impulse train functions only coincide at integer multiples of 35. Therefore in this summation range they coincide only at n = 0. The net impulse strength at n = 0 is therefore -3. All the other impulses occur alone and the sum of the squares is the same as the square of the sum. Therefore Page 105 = 9+6x4² +4x(-7)² 1485 [n] - 787 [n]1² (3-* (-3)² +42 + (−7)² +42 + (−7)² +4² +4² + (−7)² + 4² + (−7)² + 4² n=21 n=25 n=28 n=30, n=() n=5 n=7 n=10 n=14 n=15 n=20 Messages - Solutioninn.com X + 1 10.81 = L 1 1+0.81 9 + 96 + 196 35 ww = 4.7107 Check. = 8.6 Check. S W 96% 29°F Aa ← 1:36 AM 2/9/2022 X 1

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