The Response of LTI systems to complex Exponential Signals

MaxZeroLight

于 2017-10-21 14:11:30 发布

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本文探讨了线性时不变(LTI)系统对于复指数信号的响应特性，指出这些信号是LTI系统的特征函数，而系统的特征值则由输入信号的幅度决定。文中详细推导了连续时间和离散时间系统下特征值的计算公式。

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1. The eigenfuctions and eigenvalues of LTI systems

The importance of the responses of LTI systems to complex exponential signals lies in the fact that the responses are a constant times the corresponding input values. In other words ,the exponential signals are theeigenfunctions of the LTI system and their amplitudes are the system'seigenvalue.

Specifically,

where the complex amplitude factor H(s) and H(z) are determined by s and z respectively i.e. if s and z is given , we can calculate the values of H((s) and H(z) .

2. The calculation of H(s) and H(z)

let's consider the impulse response of a LTI system is h(t) and the input is x(t)= exp(st) . From our early discussion in the convolution integral representation of LTI systems and using the commutative law of convolution operation , we have

and

Hence , we can conclude that

Later we will know that H(s) and H(z) are actually the Fourier Transforms of unit impulse responses in continuous time and discrete time respectively .