线性时不变系统理论

维基百科，自由的百科全书

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在电子工程中，尤其是在电路、信号处理和控制理论中，线性时不变系统理论研究的是线性、时不变系统对于任意输入信号的响应。尽管标准的独立变量是时间，但是也可以很容易地使用空间（如图像处理和场论中那样）或者其它的坐标系统。在离散系统中对应的术语是线性移位不变系统。

[编辑] 概述

顾名思义，线性时不变系统必须满足同时满足线性和时不变性：

线性，指系统的输入和输出之间同时满足齐次性和叠加性。即，若系统的输入为

$x(t) = Ax_1(t) + Bx_2(t) \,$

那么系统输出为

$y(t) = Ay_1(t) + By_2(t) \,$

对于任何A和B都成立。其中

y i (t)

是输入为

x i (t)

时系统的输出。

时不变性，指如果将系统的输入信号延迟 $τ$ 秒，那么得到的输出除了这 $τ$ 秒延时以外是完全相同的，称这样的系统是“时不变”的。即若系统输入 $x (t)$ ，对应的输出为 $y (t)$ ，则输入为 $x (t + τ)$ 时系统的输出为 $y (t + τ)$ 。

The fundamental result in LTI system theory is that any LTI system can be characterized entirely by a single function called the system's impulse response. The output of the system is simply the convolution of the input to the system with the system's impulse response. This method of analysis is often called the time domain point-of-view. The same result is true of discrete-time linear shift-invariant systems, in which signals are discrete-time samples, and convolution is defined on sequences.

Relationship between the time domain and the frequency domain

Equivalently, any LTI system can be characterized in the frequency domain by the system's transfer function, which is the Laplace transform of the system's impulse response (or Z transform in the case of discrete-time systems). As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the transform of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain.

For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials. This is, if the input to a system is the complex waveform $A exp(s t)$ for some complex amplitude $A$ and complex frequency $s$ , the output will be some complex constant times the input, say $B exp(s t)$ for some new complex amplitude $B$ . The ratio $B / A$ is the transfer function at frequency $s$ .

Because sinusoids are a sum of complex exponentials with complex-conjugate frequencies, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, but always with the same frequency.

LTI system theory is good at describing many important systems. Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or nonlinear case. Any system that can be modeled as a linear homogeneous differential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistors, inductors, and capacitors (RLC circuits). Ideal spring–mass–damper systems are also LTI systems, and are mathematically equivalent to RLC circuits.

Most LTI system concepts are similar between the continuous-time and discrete-time (linear shift-invariant) cases. In image processing, the time variable is replaced with 2 space variables, and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing filter banks and MIMO systems, it is often useful to consider vectors of signals.

[编辑] 连续时间系统

[编辑] 时不变与线性变换

我们从一个脉冲响应是二维函数的时变系统开始看看时不变这个条件是如何将它简化成一维的。例如，假设输入信号是 $x (t)$ ，其中 index set 是实轴，即 $t \in \mathbb{R}$ 。线性算子 $\mathcal{H}$ 表示系统对输入信号的变换。对于这个 index set 来说合适的算子是一个二维函数

$h(t_1, t_2) \mbox{ where } t_1, t_2 \in \mathbb{R}.$

由于 $\mathcal{H}$ 是线性算子，系统对于输入信号 $x (t)$ 所起的变换是下面重叠积分表示的线性变换

$y(t_1) = \int_{-\infty}^{\infty} h(t_1, t_2) \, x(t_2) \, d t_2.$

如果线性算子 $\mathcal{H}$ 也是时不变的，那zh-cn:么;zh-tw:么

$h(t_1, t_2) = h(t_1 + \tau, t_2 + \tau) \qquad \forall \, \tau \in \mathbb{R}.$

取

$\tau = -t_2, \,$

那么得到

$h(t_1, t_2) = h(t_1 - t_2, 0). \,$

出于化简的考虑，我们通常丢掉 $h (t 1, t 2)$ 中的第二个参数 0，这样重叠积分就变成了滤波中常见卷积

$y(t_1) = \int_{-\infty}^{\infty} h(t_1 - t_2) \, x(t_2) \, d t_2 = (h * x) (t_1).$

这样，这个卷积就表示线性、时不变系统对于任意输入函数所起的作用。对于有限维的模拟信号，参见轮换矩阵。

[编辑] 脉冲响应

如果我们给系统输入一个狄拉克δ函数信号，由于狄拉克δ函数是一个理想脉冲，所以线性时不变的结果是脉冲响应，这可以表示为：

$(h * \delta) (t) = \int_{-\infty}^{\infty} h(t - \tau) \, \delta (\tau) \, d \tau = h(t),$

（根据狄拉克δ函数的 sifting 特性）。

注意

$h(t) = h(t, 0) \ (\mbox{with } t = t_1 - t_2)$

这样 $h (t)$ 就是系统的脉冲响应。

根据下面的方法我们可以用脉冲响应计算任意输入信号的响应。再次应用 $δ(t)$ 的 sifting 特性，我们可以将输入写成δ的叠加：

$x(t) = \int_{-\infty}^\infty x(\tau) \delta(t-\tau) \,d\tau$

这个输入经过系统变换，

$\mathcal{H} x(t) = \mathcal{H} \int_{-\infty}^\infty x(\tau) \delta(t-\tau) \,d\tau$

$\quad = \int_{-\infty}^\infty \mathcal{H} x(\tau) \delta(t-\tau) \,d\tau$ (由于 $\mathcal{H}$ 是线性的所以可以在积分间传递)

$\quad = \int_{-\infty}^\infty x(\tau) \mathcal{H} \delta(t-\tau) \,d\tau$ (由于

x (τ)

在 t 中是常量并且 $\mathcal{H}$ 是线性的)

$\quad = \int_{-\infty}^\infty x(\tau) h(t-\tau) \,d\tau$ (根据

h (t)

的定义)

系统的所有信息都包含在脉冲响应 $h (t)$ 中。

[编辑] Exponentials as eigenfunctions

An eigenfunction is a function for which the output of the operator is the same function, just scaled by some amount. In symbols,

$\mathcal{H}f = \lambda f$ ,

where f is the eigenfunction and $λ$ is the eigenvalue, a constant.

The exponential functions $e s t$ , where $s \in \mathbb{C}$ , are eigenfunctions of a linear, time-invariant operator. A simple proof illustrates this concept.

Suppose the input is $x (t) = e s t$ . The output of the system with impulse response $h (t)$ is then

$\int_{-\infty}^{\infty} h(t - \tau) e^{s \tau} d \tau$

which is equivalent to the following by the commutative property of convolution

$\int_{-\infty}^{\infty} h(\tau) \, e^{s (t - \tau)} \, d \tau$

$\quad = e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau$

$\quad = e^{s t} H(s)$ ,

where

$H(s) = \int_{-\infty}^\infty h(t) e^{-s t} d t$

is dependent only on the parameter s.

So, $e s t$ is an eigenfunction of an LTI system because the system response is the same as the input times the constant $H (s)$ .

[编辑] 傅里叶与拉普拉斯变换

The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Laplace transform

$H(s) = \mathcal{L}\{h(t)\} = \int_{-\infty}^\infty h(t) e^{-s t} d t$

is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids, i.e. exponentials of the form $exp(j ω t)$ where $\omega \in \mathbb{R}$ and $j = \sqrt{-1}$ . These are generally called complex exponentials even though the argument is purely imaginary. The Fourier transform $H(j \omega) = \mathcal{F}\{h(t)\}$ gives the eigenvalues for pure complex sinusoids. Both of $H (s)$ and $H (j ω)$ are called the system function, system response, or transfer function.

The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of t less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown with lower limit of integration of negative infinity is formally known as the bilateral Laplace transform).

The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it can not be directly applied to input and output signals that are not square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist.

Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist

$y(t) = (h*x)(t) = \int_{-\infty}^\infty h(t - \tau) x(\tau) d \tau$

$\quad = \mathcal{L}^{-1}\{H(s)X(s)\}$

Not only is it often easier to do the transforms, multiplication, and inverse transform than the original convolution, but one can also gain insight into the behavior of the system from the system response. One can look at the modulus of the system function |H(s)| to see whether the input $exp(s t)$ is passed (let through) the system or rejected or attenuated by the system (not let through).

[编辑] 例子

一个线性时不变算子的简单实例是导数：

$\frac{d}{dt} \left( c_1 x_1(t) + c_2 x_2(t) \right) = c_1 x'_1(t) + c_2 x'_2(t),$

$\frac{d}{dt} x(t-\tau) = x'(t-\tau).$

取导数的拉普拉斯变换，得到一个简单的与拉普拉斯变换变量 s 的乘积。

$\mathcal{L}\left\{\frac{d}{dt}x(t)\right\} = s X(s)$

导数的拉普拉斯变换如此简单一定程度上说明了拉普拉斯变换的用途。

另外一个简单的线性时不变算子是平均算子

$\mathcal{A}\left\{x(t)\right\} = \int_{t-a}^{t+a} x(\lambda) d \lambda$ .

因为积分是线性的所以它也是线性的

$\mathcal{A}\left\{c_1 x_1(t) + c_2 x_2(t) \right\}$

$= \int_{t-a}^{t+a} \left( c_1 x_1(\lambda) + c_2 x_2(\lambda) \right) d \lambda$

$= c_1 \int_{t-a}^{t+a} x_1(\lambda) d \lambda + c_2 \int_{t-a}^{t+a} x_2(\lambda) d \lambda$

$= c_1 \mathcal{A}\left\{x_1(t) \right\} + c_2 \mathcal{A}\left\{x_2(t) \right\}$ .

它也是时不变的

$\mathcal{A}\left\{x(t-\tau)\right\}$

$= \int_{t-a}^{t+a} x(\lambda-\tau) d \lambda$

$= \int_{(t-\tau)-a}^{(t-\tau)+a} x(\xi) d \xi$

$= \mathcal{A}\{x\}(t-\tau)$ .

实际上， $\mathcal{A}$ 可以写成与 box 函数 $Π(t)$ 的卷积。

$\mathcal{A}\left\{x(t)\right\} = \int_{-\infty}^\infty \Pi\left(\frac{\lambda-t}{2a}\right) x(\lambda) d \lambda$ ,

其中 box 函数是

$\Pi(t) = \left\{ \begin{matrix} 1 & |t| < 1/2 \\ 0 & |t| > 1/2 \end{matrix} \right.$ 。

[编辑] 重要的系统特性

因果性和稳定性是描述系统的两个重要性质。因为真实世界是因果的，因此通常要求设计的系统也是因果，以便实现。也可以构建不稳定的系统，并且这种系统在很多场合很有用。甚至也可以构建 non-real系统也可以用在non-real 系统中，并且在很多场合也是非常有用的。

[编辑] 因果性

主条目：因果系统

如果系统输出只与当前以及过去的输入有关，那么该系统就是因果系统。因果性的充分必要条件是

$h(t) = 0 \quad \forall t < 0,$

其中 $h (t)$ 是脉冲响应。由于拉普拉斯变换的逆变换不唯一，所以通常不能根据拉普拉斯变换确定系统的因果性。只有在确定了系统的收敛域之后才能确定该系统的因果性。

[编辑] 稳定性

主条目：有界输入有限输出稳定

如果系统对每个有界输入来说输出都是有界的，那么系统就是有界输入有限输出稳定的（BIBO 稳定），用数学方法表示就是如果

$||x(t)||_\infty < \infty$

和

$||y(t)||_\infty < \infty$

（也就是说 $x (t)$ 和 $y (t)$ 的最大绝对值都是有界的），那么系统就是稳定的。系统稳定的充分必要条是脉冲响应 $h (t)$ 满足

$||h(t)||_1 = \int_{-\infty}^\infty |h(t)| dt < \infty.$

在频域中，收敛域必须包含虚轴 $s = j ω$ 。

[编辑] 离散时间系统

几乎所有的连续时间系统都能找到与之对应的离散时间系统。

[编辑] 连续时间系统中的离散时间系统

在许多情况下，离散时间（DT）系统实际上是较大的连续时间（CT）系统的一部分。例如，数字录音系统记录模拟声音、数字化、或许对数字信号进行处理、然后重放模拟信号。

正式场合下所研究的离散时间信号几乎总是连续时间信号的均匀采样。如果 $x (t)$ 是一个连续时间信号，那么模数转换器将把它转换成离散时间信号 $x [n]$ ，

x [n] = x (n T)

其中 T 是采样周期。为了保证离散信号能够忠实地表示输入信号，非常重要的一点就是需要限制输入信号的频率范围。根据采样定理，离散时间信号所包括的最大频率范围是 $1 / (2 T)$ 。其它频率都成为这个范围的混叠信号。

[编辑] 时不变和线性变换

我们从一个脉冲响应是二维函数的时变系统开始来看看时不变这个条件是如何将系统降到一维的。例如，假设输入信号是 $x [n]$ ，其中 index set 是整数，即 $n \in \mathbb{Z}$ 。线性算子 $\mathcal{H}$ 表示系统在输入信号上的操作，对于这个 index set 来说合适的算子是一个二维函数

$h[n_1, n_2] \mbox{ where } n_1, n_2 \in \mathbb{Z}.$

由于 $\mathcal{H}$ 是一个线性算子，系统在输入信号 $x [n]$ 上的作用就是下面累加和所表示的线性变换

$y[n_1] = \sum_{n_2=-\infty}^{\infty} h[n_1, n_2] \, x[n_2],$

如果线性算子 $\mathcal{H}$ 也是时不变的，那么

$h[n_1, n_2] = h[n_1 + m, n_2 + m] \qquad \forall \, m \in \mathbb{Z}.$

如果取

$m = -n_2, \,$

那么

$h[n_1, n_2] = h[n_1 - n_2, 0]. \,$

为了简化通常我们丢弃 $h [n 1, n 2]$ 的第二个参数零，这样重叠积分现在变成了滤波中常见的卷积和

$y[n_1] = \sum_{n_2=-\infty}^{\infty} h[n_1 - n_2] \, x[n_2] = (h * x) [n_1].$

这样，卷积和表示一个线性时不变系统在任意输入函数上所起的作用，对于类似的有限维参数，参见轮换矩阵（en:circulant matrix）。