![]() ![]() Let to+T>a, then input is u(t-T), t>=to+T. u (t ) for t 0 ≤ t ≤ a y (t ) = 0 for other t Shift the initial time to to+T. Define the initial time of input to, system input is u(t), t>=to. The system is called a truncation operator, which chops off the input after time a. ![]() Is the ideal lowpass filter causal? Is is possible to built the filter in the real world? Translation: 理想低通滤波器的冲激响应如式所示。对于所有的 t,w 和 to,都是常数。理 想低通滤波器是因果的吗?现实世界中有可能构造这种滤波器吗? Answer: Consider two different time: ts and tr, ts a where a is a fixed constant. 2.2 The impulse response of an ideal lowpass filter is given byįor all t, where w and to are constants. ( y1 + y 2 ) = a1 * u1 + a 2 * u 2 So it does not has the property of additivity, therefore, is not a linear system. Choose two different input, get the outputs: Y = a (u ) * u a(u) is a function of input u. The input-output relation in Fig 2.1(c) can be described as: Easy to testify that it is a linear system. Z = y −b z = a *u z is the new output introduced. But we can introduce a new output so that it is linear. ( y1 + y 2 ) = a * (u1 + u 2 ) + 2 * b So it does not has the property of additivity, therefore, is not a linear system. Testify whether it has the property of additivity. Y = a *u + b Here a and b are all constants. The input-output relation in Fig 2.1(b) can be described as: ![]() 2.1 Consider the memoryless system with characteristics shown in Fig 2.19, in which u denotes the input and y the output. ![]()
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February 2023
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