傅里叶级数变换恢复原始信号
假设我有三个复杂波形（由许多正弦波组成）：A、B 和 C。每个波形都有以下频率：分别为 550、600、700 Hz。
现在我将三个信号相加，即叠加得到信号D。
我的目标是分别取回三个原始信号A、B和C。
我绘制了傅里叶频谱，其中主峰位于 550、660 和 700 Hz。还有其他较小的峰。我如何知道哪些峰值与哪些波形关联，以便我可以重新创建原始波形 A、B 和 C？谢谢。
Suppose I have three complex waveforms (consisting of many sinewaves): A, B and C. Each one has the following frequency: 550, 600, 700 Hz respectively.
Now I add the three signals i.e. do a superposition to get signal D.
My aim is to get back the three original signals A, B and C separately.
I have plotted the Fourier spectrum where I get the main peaks at 550, 660 and 700 Hz. There are other smaller peaks. How will I know which peaks associate with which waveforms so that I can recreate the original waveforms A, B and C ? Thank you.
如果你对这篇内容有疑问，欢迎到本站社区发帖提问 参与讨论，获取更多帮助，或者扫码二维码加入 Web 技术交流群。
绑定邮箱获取回复消息
由于您还没有绑定你的真实邮箱，如果其他用户或者作者回复了您的评论，将不能在第一时间通知您！
发布评论
评论（4）
如果 A、B 和 C 不是纯正弦波，那么如果没有附加信息，您想要做的事情是不可能的。假设 A 是频率为 100 和 200 Hz 的正弦波之和，
B 为 300 和 400 Hz，C 为 500 和 600 Hz。现在考虑信号 D 的频率
100 和 300 Hz，E 频率为 200 和 500 Hz，F 频率为 400 和 600 Hz，
每个分量与 A、B 和 C 分量具有相同的相位和幅度。
(A+B+C) 与 (D+E+F) 具有相同的 FFT，因此您无法区分它们。
If A, B, and C are not pure sine waves, what you want to do is not possible without additional information. Suppose A is a sum of sine waves with frequencies 100 and 200 Hz,
B with 300 and 400 Hz, and C with 500 and 600 Hz. Now consider signal D with frequencies
100 and 300 Hz, E with frequencies 200 and 500 Hz, and F with frequencies 400 and 600 Hz,
each component with the same phase and amplitude as the components of A, B and C.
(A+B+C) will have the same FFT as (D+E+F), so you can't tell them apart.
重要的是要认识到您需要在 FFT 之前应用窗口函数，否则您将得到频域中的伪影来自您应用于时域数据的隐式矩形窗口的影响。一个好的通用窗口函数是 Hann（又名 Hanning）窗口。
It's important to realise that you need to apply a window function prior to the FFT, otherwise you will get artefacts in the frequency domain from the effect of the implicit rectangular window that you are applying to your time domain data. A good general purpose window function is the Hann (aka Hanning) window.
如果不知道您正在使用哪种技术（DSP 芯片、fftw 等），就不可能为您提供实现细节。但是，是的，应用快速傅里叶变换，然后假设您想要在 550、600 和 700 处重建三个纯正弦波，FFT 将为您提供每个的幅度和相位。然后，
y=a*sin(wt+p)形式的简单正弦表达式将重建信号。 “a”和“p”是来自 FFT 的幅度和相位，w=2*pi*f，其中 f 为 550Hz、600Hz 或 700Hz。Without knowing which technology you're using (a DSP chip, fftw, etc.) it's impossible to give you implementation details. But yes, apply a fast Fourier transform, and then assuming that you want to reconstruct three pure sine waves at 550, 600 and 700, the FFT will give you the amplitude and phase for each. Then a simple sine expression of the form
y=a*sin(wt+p)will reconstruct the signal. "a" and "p" are the amplitude and phase from the FFT, and w=2*pi*f, where f is 550Hz, 600Hz, or 700Hz.您需要添加任何形式的复杂信号吗？他们有模式吗？如果你想检索任何形式的信号，那是不可能的。但在某些情况下，如果您的材料有限，您可以对其进行处理。例如，在 Melodyne 中，他们可以分离音高定义的乐器的一些材料：http://www.melodyne.com youtube.com/watch?v=jFCjv4_jqAY
Do you need to add any form of complex signals? Do they have a pattern? If you want to retrieve any form of signal, it will be impossible. But in some cases, that you have a restrict material, you can work on that. In melodyne,for instance, they can separate some material of pitch-defined instruments: http://www.youtube.com/watch?v=jFCjv4_jqAY