Linearity implies two properties: additivity and homogeneity. We will first check the additivity of the system...
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Linearity implies two properties: additivity and homogeneity. We will first check the additivity of the system y(t) = 2x(t). Create the Simulink diagram shown in Figure 9. You can find the "Gain" and "Sum" blocks from the "Math Operations" library. To copy a block, hold down the control key, left-click on the block and drag the copy to wherever it is needed. Make the amplitudes and frequencies of the two sine waves different. On the scope labeled "Additivity", the first waveform shows y(t) = 2(x, (t) + x₂ (t)) and the second one shows y₁(t) + y₂ (t), where y₁ (t) = 2x₁ (t), y₂ (t) = 2x₂ (t). Use different markers for these two signals to show that they do overlap. The third waveform is the difference of the first two, which shows zero for all time. Therefore, additivity is verified. Save the Simulink diagram for future use. 2 A Sine Wave x1(t) А Sine Wave x2(t) + 2 + 2 x1(t)+x2(t) y1(t)=2x1(t) y2(t)=2x2(t) 2 + 2(x1(t)+x2(t)) y1(t)+y2(t) + Additivity Figure 9 Verification of Additivity of the System y(t)=2x(t) Next, let's verify if it satisfies homogeneity. Create the Simulink diagram shown in Figure 10. (It is easier to modify Figure 9 than to start from scratch. Be sure to save under a new name.) Compare the waveform showing 2(Kx(t)) and the waveform showing Ky(t), for K = 3. The waveforms overlap and the difference is zero for all time. Homogeneity is verified. Again, use different markers to distinguish between signals. Save the Simulink diagram for future use. А Sine Wave x(t) 3 2 Kx(t) y(t)=2x(t) 2 3 2(Kx(t)) Ky(t) 1 Figure 10 Verification of Homogeneity of the System y(t)=2x(t) Homogeneity Now let's check if the system y(t) = 2x(t) is time-invariant. Create the Simulink diagram shown in Figure 11. Compare the two waveforms on Scope "Time Invariance" showing 2(x(t–T)) and y(t-T) respectively, where y(t) = 2x(t). The waveforms overlap and the difference is zero, for the same T in both "Transport Delay" blocks. Time-invariance is verified. Save the diagram for future use. A Sine Wave x(t) a x(t-T) 2 y(t)=2x(t) 2 2x(t-T) y(t-T) - + 1 Time Invariance DAC Figure 11 Verification of Time-invariance of the System y(t)=2x(t) Linearity implies two properties: additivity and homogeneity. We will first check the additivity of the system y(t) = 2x(t). Create the Simulink diagram shown in Figure 9. You can find the "Gain" and "Sum" blocks from the "Math Operations" library. To copy a block, hold down the control key, left-click on the block and drag the copy to wherever it is needed. Make the amplitudes and frequencies of the two sine waves different. On the scope labeled "Additivity", the first waveform shows y(t) = 2(x, (t) + x₂ (t)) and the second one shows y₁(t) + y₂ (t), where y₁ (t) = 2x₁ (t), y₂ (t) = 2x₂ (t). Use different markers for these two signals to show that they do overlap. The third waveform is the difference of the first two, which shows zero for all time. Therefore, additivity is verified. Save the Simulink diagram for future use. 2 A Sine Wave x1(t) А Sine Wave x2(t) + 2 + 2 x1(t)+x2(t) y1(t)=2x1(t) y2(t)=2x2(t) 2 + 2(x1(t)+x2(t)) y1(t)+y2(t) + Additivity Figure 9 Verification of Additivity of the System y(t)=2x(t) Next, let's verify if it satisfies homogeneity. Create the Simulink diagram shown in Figure 10. (It is easier to modify Figure 9 than to start from scratch. Be sure to save under a new name.) Compare the waveform showing 2(Kx(t)) and the waveform showing Ky(t), for K = 3. The waveforms overlap and the difference is zero for all time. Homogeneity is verified. Again, use different markers to distinguish between signals. Save the Simulink diagram for future use. А Sine Wave x(t) 3 2 Kx(t) y(t)=2x(t) 2 3 2(Kx(t)) Ky(t) 1 Figure 10 Verification of Homogeneity of the System y(t)=2x(t) Homogeneity Now let's check if the system y(t) = 2x(t) is time-invariant. Create the Simulink diagram shown in Figure 11. Compare the two waveforms on Scope "Time Invariance" showing 2(x(t–T)) and y(t-T) respectively, where y(t) = 2x(t). The waveforms overlap and the difference is zero, for the same T in both "Transport Delay" blocks. Time-invariance is verified. Save the diagram for future use. A Sine Wave x(t) a x(t-T) 2 y(t)=2x(t) 2 2x(t-T) y(t-T) - + 1 Time Invariance DAC Figure 11 Verification of Time-invariance of the System y(t)=2x(t)
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Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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