discrete Fourier transformation

简明释义

离散傅里叶变换

英英释义

例句

1.Researchers applied discrete Fourier transformation to study the vibrations of mechanical systems.

研究人员应用离散傅里叶变换研究机械系统的振动。

2.In image processing, discrete Fourier transformation helps in filtering out noise from images.

在图像处理中，离散傅里叶变换有助于去除图像中的噪声。

3.The engineer used discrete Fourier transformation to analyze the frequency components of the signal.

工程师使用离散傅里叶变换分析信号的频率成分。

4.The algorithm for audio compression often relies on discrete Fourier transformation to convert time-domain signals into frequency-domain.

音频压缩算法通常依赖于离散傅里叶变换将时域信号转换为频域。

5.Using discrete Fourier transformation, we can efficiently compute the frequency spectrum of a digital signal.

通过使用离散傅里叶变换，我们可以高效地计算数字信号的频谱。

作文

The concept of the discrete Fourier transformation is fundamental in the field of signal processing and analysis. It allows us to transform a discrete signal from its original domain, often time or space, into a representation in the frequency domain. This transformation is crucial for various applications, including audio processing, image compression, and even solving partial differential equations. Understanding the discrete Fourier transformation enables engineers and scientists to analyze the frequency components of signals, which can lead to better filtering techniques and improved data interpretation.To comprehend the discrete Fourier transformation, we first need to understand what it means to analyze a signal in the time domain versus the frequency domain. In the time domain, a signal is represented as it varies over time. For example, if we consider a sound wave, the amplitude of the wave changes as time progresses. However, this representation does not easily reveal the underlying frequencies that compose the sound. This is where the discrete Fourier transformation comes into play.The discrete Fourier transformation takes a finite sequence of equally spaced samples of a function and expresses it as a sum of sinusoids of different frequencies. Mathematically, it can be defined as:X(k) = Σ (n=0 to N-1) x(n) * e^(-j(2π/N)kn)where X(k) is the transformed signal in the frequency domain, x(n) is the original signal in the time domain, N is the total number of samples, and j is the imaginary unit. This equation illustrates how each frequency component contributes to the overall signal.One significant advantage of the discrete Fourier transformation is its efficiency in computation, especially when implemented using the Fast Fourier Transform (FFT) algorithm. The FFT significantly reduces the computational complexity, allowing for real-time processing of signals. This efficiency is particularly beneficial in applications such as telecommunications, where large amounts of data need to be processed quickly and accurately.Moreover, the discrete Fourier transformation has profound implications in various fields. In audio engineering, for instance, it allows sound engineers to manipulate audio signals by filtering out unwanted noise or enhancing certain frequencies to improve sound quality. In image processing, it helps compress images by transforming them into the frequency domain, where less important information can be discarded, resulting in smaller file sizes without significantly affecting the perceived quality.In conclusion, the discrete Fourier transformation is not just a mathematical tool; it is a powerful method that bridges the gap between the time domain and frequency domain analyses. Its applications span across multiple disciplines, making it an essential concept for anyone involved in signal processing or related fields. By mastering the discrete Fourier transformation, individuals can unlock new possibilities in data analysis, enabling advancements in technology and science that continue to shape our world today.

离散傅里叶变换的概念在信号处理和分析领域中是基础性的。它允许我们将离散信号从其原始域（通常是时间或空间）转换为频率域中的表示。这种变换对于各种应用至关重要，包括音频处理、图像压缩，甚至解决偏微分方程。理解离散傅里叶变换使工程师和科学家能够分析信号的频率成分，这可以导致更好的滤波技术和改进的数据解释。要理解离散傅里叶变换，我们首先需要了解在时间域与频率域中分析信号的含义。在时间域中，信号被表示为随时间变化的形式。例如，如果我们考虑一个声波，波的振幅会随着时间的推移而变化。然而，这种表示并不容易揭示组成声音的潜在频率。这就是离散傅里叶变换发挥作用的地方。离散傅里叶变换将有限数量的等间距样本函数表示为不同频率的正弦波之和。从数学上讲，它可以定义为：X(k) = Σ (n=0 to N-1) x(n) * e^(-j(2π/N)kn)其中X(k)是频域中变换后的信号，x(n)是时间域中的原始信号，N是样本的总数，j是虚数单位。这个方程说明了每个频率成分如何对整体信号做出贡献。离散傅里叶变换的一个显著优点是其计算效率，特别是在使用快速傅里叶变换（FFT）算法时。FFT显著减少了计算复杂性，使得信号的实时处理成为可能。这种效率在电信等应用中尤为有利，在这些应用中，需要快速准确地处理大量数据。此外，离散傅里叶变换在多个领域具有深远的影响。例如，在音频工程中，它使声音工程师能够通过过滤掉不必要的噪声或增强某些频率来操控音频信号，从而改善音质。在图像处理领域，它通过将图像转换到频域来帮助压缩图像，在频域中，可以丢弃不太重要的信息，从而在不显著影响感知质量的情况下，减小文件大小。总之，离散傅里叶变换不仅仅是一个数学工具；它是一个强大的方法，架起了时间域和频率域分析之间的桥梁。它的应用跨越多个学科，使其成为任何参与信号处理或相关领域的人的基本概念。通过掌握离散傅里叶变换，个人可以解锁数据分析的新可能性，推动科技和科学的进步，这些进步继续塑造我们今天的世界。