使用 R 从 FFT 频率和强度数据重建时间序列

Question

在对 EEG 测量应用傅立叶变换后，我想以绘图的形式将 FFT 的近似值与原始信号进行比较。我必须将 FFT 中的数据（频率和强度）转换回时间序列。 为了转换原始时间序列，我使用了eegkit package 的eegfft method。我得到一个频率和幅度列表来近似原始信号。

这里将 FFT 的两个结果显示为缩短的示例：

# Frequency in Hz
freq <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)  

# Strength in uV
ampl <- c(4.1135352, 5.1272713, 3.2069741, 1.5336438, 2.4301334, 1.0974758, 1.8238327, 0.9637886, 1.1401306, 0.2224472)

是否有可以用来从 FFT 近似的频率和幅度数据重构时间序列的包或方法？

---

编辑：

对于原始信号的重构，是否还需要eegfft方法在结果中返回的相位信息？

# Phase shift in range -pi to pi
phase <- c(0.0000000, 1.1469542, -2.1930702, 2.7361738,1.1597980, 2.6118647, -0.6609641, -2.1508755,1.6584852, -1.2906986)

Answer 1

我希望这样的事情应该可行。

编辑：我已将phases 设置为默认为零（如果丢失且未传递到data_from_fft）。

freq <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)  
ampl <- c(4.1135352, 5.1272713, 3.2069741, 1.5336438, 2.4301334, 1.0974758, 1.8238327, 0.9637886, 1.1401306, 0.2224472)
phase <- c(0.0000000, 1.1469542, -2.1930702, 2.7361738,1.1597980, 2.6118647, -0.6609641, -2.1508755,1.6584852, -1.2906986)
sampl_freq = 1000

data_from_fft <- function(xmin, xmax, sample_freq, 
                          frequencies, amplitudes, phases = 0) {
  x_vals <- seq(xmin, xmax, length.out = sample_freq * (xmax-xmin))
  y_vals <- x_vals * 0
  for (i in seq_along(x_vals)) {
    # Note, I don't understand why the pi/2 phase adjustment is needed here,
    #   but I couldn't get the right answers out eegfft without it... :-(
    y_vals[i] <- sum(amplitudes * sin(2*pi*frequencies * x_vals[i] + phase + pi/2))
  }
  data.frame(x_vals, y_vals)
}

library(tidyverse)

plot_from_FFT <- data_from_fft(0, 1, sampl_freq, freq, ampl, phase)
ggplot(plot_from_FFT, aes(x_vals, y_vals)) +
  geom_line()

现在，让我们看看我们是否可以使用该输出来重构输入：

eegkit::eegfft(plot_from_FFT$y_vals, lower = 1, upper = 20, Fs = sampl_freq) %>% 
  filter(abs(strength) > 0.1)

   frequency  strength  phase.shift
1          1 4.1158607  0.004451123
2          2 5.1177070  1.154553861
3          3 3.2155744 -2.185185998
4          4 1.5319350  2.739953054
5          5 2.4283426  1.173258629
6          6 1.0813858  2.645126993
7          7 1.8323207 -0.644216053
8          8 0.9598727 -2.138381646
9          9 1.1427380  1.685081744
10        10 0.2312619 -1.265466418

是的！这些非常接近输入。

eegkit::eegfft(plot_from_FFT$y_vals, lower = 1, upper = 20, Fs = sampl_freq) %>% 
      filter(abs(strength) > 0.1) %>%
      left_join(
        tibble(frequency = freq,
               strength_orig = ampl,
               phase_orig   = phase)
      ) %>%
      gather(stat, value, -frequency) %>%
      mutate(category = if_else(stat %>% str_detect("str"), "strength", "phase"),
             version  = if_else(stat %>% str_detect("orig"), "plot inputs", "reconstructed inputs"),) %>%
      ggplot(aes(frequency, value, shape = version, size = version)) + 
      geom_point() +
      scale_x_continuous(breaks = 1:10, minor_breaks = NULL) +
      scale_shape_manual(values = c(16, 21)) +
      scale_size_manual(values = c(1,5)) +
      facet_wrap(~category)