A few days ago, I learned the principle of automatic control, and suddenly felt that the difference and connection between Fourier transform, Laplace transform and z-transform were not particularly clear, so I put them together and studied them, and recorded them here after sorting out and summarizing them.

1. Fourier transform

The basis of the Fourier transform is the Fourier series. Let me first talk about what a Fourier series is.

1.1 Fourier series

The French mathematician Fourier believes that any periodic function can be represented by an infinite series composed of sine and cosine functions (the sine and cosine functions are chosen as the basis functions because they are orthogonal), later known as Fourier The series is a special kind of trigonometric series.

The Fourier series in triangular form is as follows:

$\text{[math]}$

According to Euler's formula, trigonometric functions can be transformed into exponential form, also called Fourier series as an exponential series. Euler's formula is as follows:

$\text{[math]}$

The Fourier series in exponential form is as follows:

$\text{[math]}$

in,

$\text{[math]}$

When k=0,

$\text{[math]}$

$\text{[math]}$ is the DC component of the function

When k=1,

$\text{[math]}$

This $\text{[math]}$ is usually called the fundamental wave of the function

The trigonometric function when k takes different values (k>1) is called the kth harmonic of the function

It should be noted that in the definition of Fourier series, only periodic functions can be expanded into Fourier series

In fact, any periodic function can be expanded into a Fourier series, but the obtained Fourier series does not necessarily converge

Then, if the periodic function satisfies the Dirichlet condition, its Fourier series is convergent

Dirichlet conditions ( sufficient and unnecessary conditions for the convergence of Fourier series ):
1. Within a period, the function is absolutely integrable ;
2. Within a period, the function is continuous or has only a limited number of discontinuities of the first type ;
3. In a cycle, the number of function maxima and minima is limited .

Take a square wave signal as an example:

$\text{[math]}$ The Fourier series expansion of a square wave signal with a period of T and a pulse width of 2 is:

$\text{[math]}$

The image of the Fourier series is shown in the figure:

At the time $\text{[math]}$ , a sequence of discrete Fourier series would program a continuous curve, the Fourier transform

After studying the Fourier series of periodic functions, people will think about it: can non-periodic functions be expressed in this way, and what changes need to be made to the formula if it is to be expressed? So there is a study of the Fourier transform.

1.2 Fourier transform

For a non-periodic function, it can actually be regarded as a $\text{[math]}$ periodic function with a period of

For example, a periodic function image is as follows:

Order $\text{[math]}$ , get:

Derived from the formula:

Denote $\text{[math]}$ a periodic function by , according to the Fourier series:

$\text{[math]}$

Instead $\text{[math]}$ , as shown in the function image above.

because $\text{[math]}$ , then

$\text{[math]}$

$\text{[math]}$ (from discrete $\text{[math]}$ to continuous $\text{[math]}$ )

so $\text{[math]}$ it becomes

$\text{[math]}$

This is the formula for the Fourier transform

$\text{[math]}$

And now $\text{[math]}$ it becomes

$\text{[math]}$

This is the inverse Fourier transform

Note: If the Fourier transform is taken for the periodic signal, the result obtained is not actually the coefficient value of the Fourier series $\text{[math]}$ , but $\text{[math]}$ the value of

1.3 Limitations of the Fourier Transform

In fact, not all functions can be Fourier transformed, and the Dirichlet condition must be satisfied before the Fourier transform can be performed

Like the sufficient and unnecessary conditions for the convergence of the Fourier series, the Dirichlet condition for the Fourier transform is:

迪利克雷条件（傅里叶变换存在的充分不必要条件）：
1.在整个定义域内，函数是绝对可积的；
2.在整个定义域内，函数连续或者只有有限个第一类间断点；
3.在整个定义域内，函数极大值和极小值的数目是有限个。

其实只需要把傅里叶级数的迪利克雷条件的周期内变成整个定义域内就可以了

1.4 常见函数的傅里叶变换

二、拉普拉斯变换

2.1 为什么需要引入拉普拉斯变换

傅里叶变换在信号频域研究上起到了非常重要的作用，可是并不是所有的信号都可以进行傅里叶变换。那么对于无法进行傅里叶变换的信号，我们想研究它的频域特性，应该怎么办呢？

例如：我们已知 $\text{[math]}$ ，虽然 $\text{[math]}$ 并不满足迪利克雷条件中绝对可积的条件，这也说明了迪利克雷条件其实是充分不必要条件。但是函数 $\text{[math]}$ 显然就无法进行傅里叶变换，这里可以归结为它的增长速度太快，以至于在绝对值积分的时候没法收敛。

那么我们要解决这个问题，针对无法进行傅里叶变换的函数，引入了拉普拉斯变换。其最通俗基本的原理就是给我们的函数乘一个 $\text{[math]}$ ，我们需要取合适的 $\text{[math]}$ 使得它可以快速下降，这样它就可以满足迪利克雷条件的绝对可积这一条件，这样就可以进行傅里叶变换了。

2.2 拉普拉斯变换

对于函数 $\text{[math]}$ ，令 $\text{[math]}$ ，对 $\text{[math]}$ 做傅里叶变换，得：

$\text{[math]}$

此时，变换结果的变量从傅里叶变换的 $\text{[math]}$ 变为了 $\text{[math]}$ 和 $\text{[math]}$ 两个，但其实我们发现 $\text{[math]}$ 总是和虚数单位J在一起，所以我们将 $\text{[math]}$ 和 $\text{[math]}$ 两个变量合成为一个变量 $\text{[math]}$

于是我们得到了拉普拉斯变换的完整公式：

$\text{[math]}$

需要注意的是，我们上面讲了需要取合适的 $\text{[math]}$ 使得 $\text{[math]}$ 可以快速下降，所以对于一些升高比较快的函数，我们需要限定 $\text{[math]}$ 比较大，才可以使得这个函数满足迪利克雷条件。所以，拉普拉斯变换不像傅里叶变换那样没有变量范围的限定，它需要有ROC(Range of Re{s} (or $\text{[math]}$ ) for X(s) to converge)与拉普拉斯变换配套存在。事实上，ROC也是拉普拉斯变换的一部分，对于相同的表达式，不同的ROC，其时域函数有可能完全不同，所以一定需要注意ROC不可以遗漏。

拉普拉斯逆变换公式：

$\text{[math]}$

2.3 常见函数的拉普拉斯变换

三、z变换

我们知道z变换其实是为离散信号而引入的一种变换，其主要原理和拉普拉斯变换很相似，是为了解决一些离散序列无法进行离散时间傅里叶变换而引入的。我们首先介绍离散时间傅里叶变换。

3.1 DTFT离散时间傅里叶变换

我们上面已经介绍了连续时间傅里叶变换，即傅里叶变换，公式如下：

$\text{[math]}$

那么我们将 $\text{[math]}$ 转化为离散的 $\text{[math]}$ ，积分变为求和，即得到了离散时间傅里叶变换：

$\text{[math]}$

至于为什么要用 $\text{[math]}$ 而不是 $\text{[math]}$ ，这个问题其实也让我疑惑，我初步思考的结果是：
为了区分离散和连续。试想，如果你看到这样一个符号 $\text{[math]}$ ，在没有区分的情况下你并不知道这个变换是连续的还是离散的，所以我觉得可能是为了区分。
$\text{[math]}$ 括号中 $\text{[math]}$ 的代表一个复数，即模为1的复数，而不是 $\text{[math]}$ 所代表的纯虚数，这样也是为z变换做准备。
我看到有博客解释说一个是积分一个是求和，没有特别理解。