# 数学代写|傅里叶分析代写Fourier analysis代考|MAT3105

## 数学代写|傅里叶分析代写Fourier analysis代考|Symmetry

Preservation of symmetry is an important property of Fourier series: the symmetry of a Fourier series corresponds exactly to the symmetry of the decomposed function. A function $x(t)$ is:

• odd-symmetric if $x(t)=-x(-t)$, sometimes referred to as anti-symmetric;
• even-symmetric if $x(t)=x(-t)$.
In general, functions have mixtures of odd- and even-symmetry and any function can be written in terms of odd- and even-symmetric constituent functions. Consider, for the general function $x(t)$ :
• an odd-symmetric constituent function, $x_o(t)=\frac{1}{2}(x(t)-x(-t))$;
• an even-symmetric constituent function, $x_e(t)=\frac{1}{2}(x(t)+x(-t))$;
• and, clearly, $x(t)=x_o(t)+x_e(t)$.
Sines and cosines are odd- and even-symmetric functions respectively, and so:
• an odd-symmetric function decomposes to a Fourier series comprised entirely of sines;
• an even-symmetric function decomposes to a Fourier series comprised entirely of cosines;
• a general function decomposes to a Fourier series comprised of sines and cosines, the sine and cosine terms corresponding to the odd- and even-symmetric parts of the function respectively.
These observations might be a little surprising at first glance. However, if we have an odd-symmetric function on the left-hand side (e.g. of Equation 3.4) then the series on the right-hand side must also be odd-symmetric and any cosine terms in the right-hand side would add some even-symmetry which would be inconsistent with the left-hand side. Similarly, an even function on the left-hand side cannot have odd-symmetric terms in the series on the righthand side.
• These basic symmetry properties carry forward into the Fourier transform and DFT.

## 数学代写|傅里叶分析代写Fourier analysis代考|Pure Odd- and Even-Symmetry

In order to illustrate Fourier series and preservation of symmetry, let us consider an illustrative basic function such as the square wave. In its simplest forms, a square wave can be either odd-symmetric, $s_o(t)$, or even-symmetric, $s_e(t)$, as shown in Figures $3.1$ and $3.2$ respectively. Note that a square wave over a finite time interval is piecewise continuous, i.e. has a finite number of finite discontinuities (two discontinuities per cycle), and so we can integrate to obtain the Fourier series coefficients.

If we work through the calculations to extract the coefficients, we obtain the following Fourier series for the odd- and even-symmetric unit-amplitude square waves $s_o(t)$ and $s_e(t)$ :
\begin{aligned} s_o(t)=& \frac{4}{\pi}\left(\sin \left(2 \pi f_1 t\right)+\frac{1}{3} \sin \left(2 \pi f_3 t\right)+\frac{1}{5} \sin \left(2 \pi f_5 t\right)\right.\ &\left.+\frac{1}{7} \sin \left(2 \pi f_7 t\right)+\cdots\right) \ s_e(t)=& \frac{4}{\pi}\left(\cos \left(2 \pi f_1 t\right)-\frac{1}{3} \cos \left(2 \pi f_3 t\right)+\frac{1}{5} \cos \left(2 \pi f_5 t\right)\right.\ &\left.-\frac{1}{7} \cos \left(2 \pi f_7 t\right)+\cdots\right) . \ \text { where } f_1=1 / T, f_3=3 f_1, f_5=5 f_1, f_7=7 f_1, \ldots \end{aligned}
Note that the even harmonic coefficients are all zero. Figures $3.1$ and $3.2$ show the Fourier series up to the fourth non-zero term, i.e. fourth partial-sum (sum of the first four terms, $\left.f_1+f_3+f_5+f_7\right)$, for $s_o(t)$ and $s_e(t)$ respectively.

## 数学代写|傅里叶分析代写Fourier analysis代考|Symmetry

• 奇对称如果 $x(t)=-x(-t)$ ，有时称为反对称;
• 偶对称如果 $x(t)=x(-t)$.
通常，函数具有奇对称和偶对称的混合，并且任何函数都可以根据奇对称和偶对称组成函数来编写。
考虑一下，对于一般功能 $x(t)$ :
• 奇对称构成函数， $x_o(t)=\frac{1}{2}(x(t)-x(-t))$;
• 偶对称构成函数， $x_e(t)=\frac{1}{2}(x(t)+x(-t))$;
• 并且, 很明显, $x(t)=x_o(t)+x_e(t)$.
正弦和余弦分别是奇对称和偶对称函数，因此:
• 奇对称函数分解为完全由正弦组成的傅立叶级数；
• 偶对称函数分解为完全由余弦组成的傅立叶级数；
• 一般函数分解为由正弦和余弦组成的傅立叶级数，正弦项和余弦项分别对应于函数的奇对称和偶对称 部分。
乍一看，这些观察结果可能有点令人惊讶。但是，如果我们在左侧有一个奇对称函数（例如方程
3.4），那么右侧的级数也必须是奇对称的，并且右侧的任何余弦项都会增加一些与左侧不一致的偶 对称。类似地，左侧的偶函数在右侧的级数中不能有奇对称顶。
• 这些基本的对称性特性被应用到傅里叶变换和 DFT 中。

## 数学代写|傅里叶分析代写Fourier analysis代考|Pure Odd- and Even-Symmetry

$$s_o(t)=\frac{4}{\pi}\left(\sin \left(2 \pi f_1 t\right)+\frac{1}{3} \sin \left(2 \pi f_3 t\right)+\frac{1}{5} \sin \left(2 \pi f_5 t\right) \quad+\frac{1}{7} \sin \left(2 \pi f_7 t\right)+\cdots\right) s_e(t)=\frac{4}{\pi}\left(\cos \left(2 \pi f_1 t\right)\right.$$

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