![]() ![]() If the output due to input x(t) is y(t), then the output due to input x(t-T) is y(t-T). * "Time invariance" means that whether we apply an input to the system now or "T" seconds from now, the output will be identical, except for a time delay of the "T" seconds. :Then, formally, a linear system is a system that exhibits the following property : If the input of the system is:: x(t) = sum_k c_k x_k(t), :then the output of the system will be:: y(t) = sum_k c_k y_k(t), :for any constants c_k and where each y_k(t) is the output resulting from the sole input x_k(t). ![]() If the output due to input x(t) is y(t), then the output due to input c x(t) is c y(t). :: x(t) = c_1 x_1(t) + c_2 x_2(t), :then the output of the system will be:: y(t) = c_1 y_1(t) + c_2 y_2(t), :where c_1 and c_2 are constants, and y_k(t) is the output resulting from the sole input x_k(t).:It can be shown that, given this superposition property, the scaling property follows for any rational scalar. If the input to the system is the sum of two component signals: * "Linearity" means that the relationship between the input and the output of the system satisfies the superposition property. The defining properties of any linear time-invariant system are, of course, "linearity" and "time invariance": The term "linear shift-invariant" is the corresponding concept for a discrete-time (sampled) system. Thus an alternately used term is "linear translation-invariant". Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate. LTI system theory or linear time-invariant system theory is a theory in the field of electrical engineering, specifically in circuits, signal processing, and control theory, that investigates the response of a linear, time-invariant system to an arbitrary input signal. ![]()
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