discrete Fourier transform

简明释义

离散型傅里叶变换

英英释义

例句

1.The discrete Fourier transform 离散傅里叶变换 is widely used in digital signal processing.

在数字信号处理中，离散傅里叶变换 discrete Fourier transform 被广泛使用。

2.In image processing, the discrete Fourier transform 离散傅里叶变换 helps in filtering and compression.

在图像处理中，离散傅里叶变换 discrete Fourier transform 有助于过滤和压缩。

3.Engineers often use the discrete Fourier transform 离散傅里叶变换 to analyze vibrations in mechanical systems.

工程师通常使用离散傅里叶变换 discrete Fourier transform 来分析机械系统中的振动。

4.By applying the discrete Fourier transform 离散傅里叶变换, we can analyze the frequency components of a signal.

通过应用离散傅里叶变换 discrete Fourier transform，我们可以分析信号的频率成分。

5.The discrete Fourier transform 离散傅里叶变换 converts time-domain signals into frequency-domain representations.

离散傅里叶变换 discrete Fourier transform 将时域信号转换为频域表示。

作文

The concept of the discrete Fourier transform is fundamental in the field of signal processing and data analysis. It provides a powerful tool for transforming a sequence of complex numbers into another sequence, which reveals the frequency components of the original data. The discrete Fourier transform (DFT) is particularly important because it allows us to analyze signals that are sampled at discrete intervals, making it applicable to digital systems. To understand the discrete Fourier transform, we first need to appreciate the importance of frequency analysis in various applications. For instance, in audio processing, the DFT enables us to decompose sound waves into their constituent frequencies, allowing us to manipulate sounds effectively. This is crucial in music production, where different instruments can be isolated, enhanced, or modified. Similarly, in image processing, the DFT helps in filtering and compressing images by analyzing the frequency content of pixel values. Mathematically, the discrete Fourier transform is defined for a finite sequence of length N as follows: X(k) = Σ[x(n) * e^(-j(2π/N)kn)], where X(k) represents the transformed output, x(n) is the input sequence, j is the imaginary unit, and k ranges from 0 to N-1. This formula illustrates how each output value X(k) corresponds to a specific frequency component of the input sequence. The summation integrates the contributions of all input samples, weighted by complex exponentials that oscillate with frequency k. One of the remarkable aspects of the discrete Fourier transform is its efficiency when implemented using the Fast Fourier Transform (FFT) algorithm. The FFT significantly reduces the computational complexity of calculating the DFT, making it feasible to analyze large datasets in real-time applications. This advancement has revolutionized various fields, including telecommunications, medical imaging, and even finance, where rapid frequency analysis can lead to better decision-making. However, while the discrete Fourier transform is a powerful tool, it is essential to recognize its limitations. For example, the DFT assumes that the input signal is periodic, which may not always be the case in practical scenarios. This assumption can lead to artifacts known as spectral leakage, where energy from one frequency bin spills over into others. To mitigate this issue, techniques such as windowing functions are employed to smooth the input signal before applying the DFT. In conclusion, the discrete Fourier transform is a cornerstone of modern signal processing, enabling us to analyze and manipulate data across various domains. Its ability to convert time-domain signals into frequency-domain representations has opened up numerous possibilities for innovation and efficiency. As technology continues to advance, the significance of the DFT will only grow, making it an essential topic for anyone interested in the fields of engineering, mathematics, and computer science. Understanding the discrete Fourier transform not only enhances our analytical capabilities but also enriches our appreciation of the underlying principles that govern the behavior of signals in our world.

离散傅里叶变换（discrete Fourier transform）的概念在信号处理和数据分析领域是基础性的。它提供了一种强大的工具，可以将一系列复数转换为另一系列，从而揭示原始数据的频率成分。离散傅里叶变换（DFT）尤其重要，因为它允许我们分析以离散间隔采样的信号，使其适用于数字系统。要理解discrete Fourier transform，我们首先需要认识到频率分析在各种应用中的重要性。例如，在音频处理中，DFT使我们能够将声波分解为其组成频率，从而有效地操控声音。这在音乐制作中至关重要，不同乐器可以被隔离、增强或修改。同样，在图像处理中，DFT通过分析像素值的频率内容来帮助过滤和压缩图像。从数学上讲，discrete Fourier transform对长度为N的有限序列的定义如下：X(k) = Σ[x(n) * e^(-j(2π/N)kn)]，其中X(k)表示变换后的输出，x(n)是输入序列，j是虚数单位，k的范围从0到N-1。这个公式展示了每个输出值X(k)如何对应于输入序列的特定频率成分。求和整合了所有输入样本的贡献，权重由以频率k振荡的复指数决定。discrete Fourier transform的一个显著方面是其通过快速傅里叶变换（FFT）算法实现的高效性。FFT显著降低了计算DFT的复杂性，使得在实时应用中分析大型数据集成为可能。这一进展彻底改变了电信、医学成像甚至金融等多个领域，其中快速的频率分析能够带来更好的决策。然而，尽管discrete Fourier transform是一种强大的工具，但必须认识到它的局限性。例如，DFT假设输入信号是周期性的，而这在实际场景中并不总是成立。这一假设可能导致称为谱泄漏的伪影，即一个频率箱中的能量溢出到其他箱中。为了解决这个问题，采用如窗函数等技术在应用DFT之前平滑输入信号。总之，discrete Fourier transform是现代信号处理的基石，使我们能够在各个领域分析和操控数据。它将时域信号转换为频域表示的能力为创新和效率开辟了无数可能性。随着技术的不断进步，DFT的重要性只会增加，使其成为任何对工程、数学和计算机科学感兴趣的人必学的主题。理解discrete Fourier transform不仅增强了我们的分析能力，也丰富了我们对支配我们世界中信号行为的基本原理的理解。