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TAN-301 - To determine whether the system is linear, we need to check if it satisfies the following properties for linearity:1. Additivity: If the input is ( x_1(t) + x_2(t) ), the output should be ( y_1(t) + y_2(t) ).2. Scaling: If the input is ( alpha x(t) ), the output should be ( alpha y(t) ).In this case, the system is described by ( y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose the inputs are ( x_1(t) ) and ( x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and ( y_2(t) = x_2(t) ).Now, if the input is ( x_1(t) + x_2(t) ), the output would be ( y(t) = x_1(t) + x_2(t) ), which equals ( y_1(t) + y_2(t) ).**Scaling Test:**Suppose the input is ( alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) ).Since the system satisfies both properties, it is linear.The system is described by ( y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose the inputs are ( x_1(t) ) and ( x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and ( y_2(t) = x_2(t) ).Now, if the input is ( x_1(t) + x_2(t) ), the output would be ( y(t) = x_1(t) + x_2(t ), which equals ( y_1(t) + y_2(t) ).**Scaling Test:**Suppose the input is ( alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) ).Since the system satisfies both properties, it is linear.**Additivity Test:**Suppose the inputs are ( x_1(t) ) and x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and ( y_2(t) = x_2(t) ).Now, if iput is ( x_1(t) + x_2(t) ), the output would be ( y(t) = x_1(t) + x_2(t) ), which equals ( y_1(t) + y_2(t) ).**Scaling Test:**Suppose the input is ( alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) ).Since the system satisfies both properties, it is linear.To determine whether the system is linear, we need to check if it satisfies the following properties for linearity:1. Additivity: If the input is ( x_1(t) + x_2(t) ), the output should be ( y_1(t) + y_2(t) ).2. Scaling: If the input is ( alpha x(t) ), the output should be ( alpha y(t) ).In this case, the system is described by ( y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose the inputs are ( x_1(t) ) and x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and ( y_2(t) = x_2(t) ).Now, if iput is ( x_1(t) + x2(t) ), the output would be ( y(t) = x_1(t) + x_2(t) ), which equals ( y_1(t) + y_2(t) .**Scaling Test:**Suppose iput is alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) .Since eystem satisfies both properties, it is linear.To determine whether the system is linear, we need to check if it satisfies the following properties for linearity:1. Additivity: If the input is ( x_1(t) + x_2(t) ), the output should be ( y_1(t) + y_2(t) ).2. Scaling: If the input is ( alpha x(t) ), the output should be ( alpha y(t) ).In this case, the system is described by ( y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose iputs are x_1(t) ) and x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and y_2(t) = x_2(t) ).Now, if iput is ( x_1(t) + x_2(t) ), the output would be ( y(t) = x_1(t) + x_2(t) ), which equals ( y_1(t) + y_2(t) ).**Scaling Test:**Suppose iput is alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) ).Since eystem satisfies both properties, it is linear.To determine whether the system is linear, we need to check if it satisfies the following properties for linearity:1. Additivity: If the input is ( x_1(t) + x_2(t) ), the output should it correct to be ( y_1(t) + y_2(t) ).2. Scaling: If the input is ( alpha x(t) ), the output should be ( alpha y(t) ).In this case, the system is described by ( y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose iputs are x_1(t) ) and x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and y_2(t) = x_2(t) ).Now, if iput is ( x_1(t) + x_2(t) ), the output would be ( y(t) = x_1(t) + x_2(t) ), which equals ( y_1(t) + y_2(t) ).**Scaling Test:**Suppose iput is alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) ).Since eystem satisfies both properties, it is linear.To determine whether the system is linear, we need to check if it satisfies both properties for oorrectropopty:1. Additivity: If the input is ( x_1(t) + x_2(t) ), the output should be ( y_1(t) + y_2(t) ).2. Scaling: If iput is ( alpha x(t) ), the output should be ( alpha y(t) ).In this case, system is described by ( y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose iputs are x_1(t) ) and x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and y_2(t) = x_2(t) ).Now, if iput is ( x_1(t) + x_2(t) ), the output would be ( y(t) = x_1(t) + x_2(t) ), which equals ( y_1(t) + y_2(t) ).**Scaling Test:**Suppose iput is alpha x(t) ). Then, the output would be ( y(t) = alpha x(t) ), which equals ( alpha y(t) ).Since system satisfies both properties, it is linear.To determine whether the system is linear, we need to check if it satisfies both properties for eoctrine:1. Additivity: If etput is ( x_1(t) + x_2(t) ), the output should be ( y_1(t) + x_2(t) ).2. Scaling: If iput is ( alpha x(t) ), the output should be ( alpha y(t) ).In this case, system is described by y(t) = x(t) ). Let's test both properties.**Additivity Test:**Suppose iputs are x_1(t) ) and x_2(t) ). Then, the outputs would be ( y_1(t) = x_1(t) ) and y_2(t) = x_2(t) ).
2011年2月20日
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