# 数学代写|傅里叶分析代写Fourier analysis代考|AMTH246

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

The Fourier transform is the limiting case of the complex Fourier series where:

• the period tends to infinity, i.e. $T \rightarrow \infty$, and thus
• the finite frequency interval, $\Delta f=1 / T$, tends to the infinitesimally small, i.e. $\Delta f \rightarrow d f$.
This requires that the Fourier series summations for finite $\Delta f$ become Fourier integrals for infinitesimal $d f$. Stated without formal derivation, this yields a pair of infinite integrals: the forward transform, from time to frequency $(-i$ in the complex exponential), and the inverse transform, from frequency to time $(+i$ in the complex exponential), i.e.
\begin{aligned} X(f) &=\int_{-\infty}^{\infty} x(t) e^{-2 \pi i f t} d t \ x(t) &=\int_{-\infty}^{\infty} X(f) e^{2 \pi i f t} d f \end{aligned}
respectively.

The frequency function, $X(f)$, is continuous. It can be considered as a continuous spectrum of frequencies in which the discrete frequency coefficients in a Fourier series have merged into a continuous frequency function as the frequency interval between adjacent frequencies tends to zero (and the number of frequency coefficients in any given finite frequency interval, however small, tends to infinity). This is shown schematically in Figure 4.1, which shows central regions of (a) an infinite real time function (in the upper plot) and (b) the amplitude of the infinite (complex) Fourier transform frequency function (in the lower plot).

Note the symmetry around zero in the frequency function in Figure 4.1: this positive-negative frequency symmetry is characteristic of the Fourier transforms of real functions and we will discuss this in more detail in Sections $4.4$ and 4.7. Also note that although we are considering time-frequency relationships, the Fourier transform and Fourier series are not limited to time and frequency. If we are investigating periodic features in, for example, spatial data, we can transform between distance functions (with distance as a timeanalogue in the formulae in Equation 4.5) and spatial-frequency (cycles per unit distance) functions.

## 数学代写|傅里叶分析代写Fourier analysis代考|Fourier Transforms of Sinusoids

A little consideration shows that the Fourier series for an arbitrary sinusoid of frequency $k$ cycles over period $T$, e.g. $x(t)=A_k \cos \left(2 \pi f_k t-\phi_k\right)$, is the sinusoid itself. If we evaluate the real Fourier series, we obtain a single pair of sine and cosine coefficients $\left(a_k, b_k\right)$ for the sinusoid $a_k \cos \left(2 \pi f_k t\right)+$ $b_k \sin \left(2 \pi f_k t\right)$, which rearranges to the original sinusoid. Similarly, if we evaluate the complex Fourier series, we obtain the single pair of complex coefficients $\left(c_k, c_{-k}\right)$ for the complex exponential $c_k e^{2 \pi i f_k t}+c_{-k} e^{-2 \pi i f_k t}$ which rearranges to the original sinusoid.

Thus, it is reasonable to expect that the Fourier transform of an infinite sinusoid $x(t)=A_k \cos \left(2 \pi f_k t-\phi_k\right)$ will be a single pair of frequency ‘markers’ at $\pm f_k\left(f_{\pm k}\right)$ corresponding to coefficients $c_{\pm k}$ in the complex Fourier series. However, if we try to evaluate the Fourier transform for this sinusoid we obtain
\begin{aligned} X(f)=& \int_{-\infty}^{\infty} A_k \cos \left(2 \pi f_k t-\phi_k\right) e^{-2 \pi i f t} d t \ =& A_k \cos (\phi) \int_{-\infty}^{\infty} \cos \left(2 \pi f_k t\right) e^{-2 \pi i f t} d t \ &+A_k \sin (\phi) \int_{-\infty}^{\infty} \sin \left(2 \pi f_k t\right) e^{-2 \pi i f t} d t . \end{aligned}
As observed in Section 4.2.3, these integrals do not meet the integrability criteria, i.e. neither integral tends to zero (or any limit) as $t \rightarrow \pm \infty$, and so cannot be directly integrated to ‘true’ functions. Various mathematicians conjectured that the Fourier transform of a pure sinusoid, i.e. a function with single frequency $f_k$, must somehow be consistent with complex Fourier series and so contain a pair of markers at $\pm f_k$ and be zero everywhere else. However, it was only when P A M Dirac (1930) formalised the $\delta$-function (‘deltafunction’) that this situation was fully resolved.

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

• 周期趋于无穷大，即 $T \rightarrow \infty$ ， 因此
• 有限的频率间隔， $\Delta f=1 / T$, 趋于无穷小，即 $\Delta f \rightarrow d f$. 这需要有限的傅立叶级数求和 $\Delta f$ 成为无穷小的傅里叶积分 $d f$. 在没有正式推导的情况下，这会产生 一对无限积分：正向变换，从时间到频率( $-i$ 在复指数) 和逆六换，从频率到时间 $(+i$ 在复指数 $)$ ， 即
$$X(f)=\int_{-\infty}^{\infty} x(t) e^{-2 \pi i f t} d t x(t)=\int_{-\infty}^{\infty} X(f) e^{2 \pi i f t} d f$$
分别。
频率函数， $X(f)$ ，是连续的。它可以被认为是一个连续的频率谱，其中傅里叶级数中的禽散频率系数随 着相邻频率之间的频率间隔趋于零（以及任何给定的有限频率间隔中的频率系数的数量) 而合并为一个 连续的频率函数，无论多么小，都趋于无穷大) 。这在图 $4.1$ 中以示意图方式显示，其中显示了 (a) 无 限实时函数 (在上图中) 和 (b) 无限 (筫数) 傅里叶变换频率函数 (在下图中) 的振幅。
注意图 4.1 中频率函数中零附近的对称性: 这种正负频率对称性是实函数傅里叶变换的特征，我们将在 4.4和 4.7。还要注意，虽然我们考虑的是时频关系，但傅里叶变换和傅里叶级数并不局限于时间和频 率。例如，如果我们正在研究空间数据中的周期性特征，我们可以在距离函数 (在公式 $4.5$ 的公式中将 距禽作为时间模拟) 和空间频率 (每单位距离的周期数) 函数之间进行转换。

## 数学代写|傅里叶分析代写Fourier analysis代考|Fourier Transforms of Sinusoids

$x(t)=A_k \cos \left(2 \pi f_k t-\phi_k\right)$, 是正弦曲线本身。如果我们评估真实的傅立叶级数，我们将获得一对正 弦和余弦系数 $\left(a_k, b_k\right)$ 对于正弦波 $a_k \cos \left(2 \pi f_k t\right)+b_k \sin \left(2 \pi f_k t\right)$ ，重新排列为原始正弦曲线。类似 地，如果我们评估㙏数傅立叶级数，我们将获得一对筫数系数 $\left(c_k, c_{-k}\right)$ 对于筫指数 $c_k e^{2 \pi i f_k t}+c_{-k} e^{-2 \pi i f_k t}$ 重新排列为原始正弦曲线。

$$X(f)=\int_{-\infty}^{\infty} A_k \cos \left(2 \pi f_k t-\phi_k\right) e^{-2 \pi i f t} d t=\quad A_k \cos (\phi) \int_{-\infty}^{\infty} \cos \left(2 \pi f_k t\right) e^{-2 \pi i f t} d t+A_k \sin (\phi) \int_{-\infty}^{\infty}$$

$t \rightarrow \pm \infty$ ，因此不能直接集成到”真实”功能中。各种数学家推测纯正弦曲线的傅里叶变换，即单频函数
$f_k$ ，必须以某种方式与复杂的傅里叶级数一致，因此包含一对标记 $\pm f_k$ 在其他地方都为零。然而，直到
PAM Dirac (1930) 将 $\delta$-function (‘deltafunction’) 表明这种情况已完全解决。

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