Time Series/Data Processing and Analysis (MATH 587/GEOP 505)

Transcription

1 Time Series/Data Processing and Analysis (MATH 587/GEOP 55) Rick Aster and Brian Borchers October 7, 28 Plotting Spectra Using the FFT Plotting the spectrum of a signal from its FFT is a very common activity. In this set of notes, we discuss how to produce such plots on both absolute and db scales. Suppose that x n represents a voltage. Then the power of the signal at time t = n t, assuming a constant load R is P n = x 2 n/r. (1) For convenience, we ll take R = 1 ohm. The scaling factor of R can easily be reinserted into the equations, but it has no important effects. The average power of the signal is N 1 n= x n 2 t T where the total length of the signal is T = N t. Thus (2) N 1 Recall that by Parseval s theorem for the DFT, N 1 n= x n 2 = 1 N n= x n 2. (3) N N 1 k= X k 2. (4) Thus N 1 k= X k 2 N 2. (5) Now, consider the power spectral density. We want to have r 1 P SD(f)df (6)

2 where r is the sampling frequency. In terms of the discrete Fourier transform, we want N 1 k= where f = r/n. Combining (5) and (7) we see that P SD k f (7) P SD k = X k 2 Nr. (8) The units of the PSD are x 2 /Hz, whatever the units of x are. If we want to plot the spectrum with a db scale, relative to to an amplitude of 1, then we should plot P SDk db X k 2 = 1 log 1 Nr. (9) Note that a factor of 1 is used here because the amplitudes are already squared The frequency associated with point k of the DFT is f k = kr N. We can also shift the range of frequencies to run from r/2 to r/2 if desired. Note that the PSD values will automatically adjust to changes in the length N of the sampled signal and the sampling rate r. We should be able to recover the total power of the signal by integrating the PSD from f = to f = r. A common problem with spectral estimation is that due to short term random variations in the signal (noise), the spectral estimate can be noisy. By computing the spectrum for each of many sections of the input signal and then averaging the spectra, we can average out these variations to get at the long term behavior of the signal. In computing the average, there are several optionswe could average the values of X k, the values of X k 2, or even the db values. This produces subtle changes in the results. In Welch s method, the method most commonly used in practice, values of X k 2 are averaged. In the following example, we consider the spectral analysis of a signal containing what is known as Gaussian white noise. The signal consists of samples that are independent and normally distributed with expected value and standard deviation 1. We ll assume a sampling frequency of r = 1 Hz. For such a signal, the average value of x 2 n is 1, so the average power of the signal should be 1 Watt. It can be shown (later in the course) that the spectrum of such a signal is flat, with equal energy at all frequencies from to r. However, because we have only a random sample of finite length, the actual spectrum that we obtain from our sample will have some sampling variability. We generate a signal of 1,, samples. Both approaches to computing the average power give P = 1.2 Watts. Figure 1 shows the periodogram estimate of the spectrum. One problem with this plot is that 1,, different frequencies are represented and there simply isn t enough resolution on the paper to show all of these frequencies. In 2

3 Figure 2, we ve plotted 1, of the frequencies. Notice that the average value of the PSD is about.1. When this is multiplied by the frequency range of 1 Hz, we get an average power of 1 Watt, as expected. Figure 3 shows the same figure, on a db scale. The average is at -2 db, corresponding to an average x 2 /Hz of.1. Again, when multiplied by 1, this gives a total power of 1 Watt. These spectra are somewhat noisy. We can improve upon them by breaking the signal into sections of length M = 1, computing spectra for each section, and averaging the spectra. Figure 4 shows the averaged spectrum in x 2 /Hz units. Figure 5 shows the same spectrum in db units. These spectra are much smoother than the first three spectra. Note that the vertical axes have changed to cover a much smaller range. Finally, Figure 6 shows the spectrum produced by MATLAB s pwelch command. This closely matches Figure 5. A few small differences can be attributed to the fact that pwelch uses a Hamming window before computing the FFT to help reduce spectral leakage. There are many other varieties of colored noise that have been identified over the years. For example, in red noise, a white noise signal x n is integrated to obtain a new signal y n. Since differentiation effectively mulitplies the Fourier transform of a signal by 2πif, one would expect integration to divide the Fourier transform (and thus the spectrum) by the same factor of 2πif. Since Φ(f) falls off proportionally to 1/f as f increases, we would expect the power spectrum of red noise to fall off as 1/f 2. Figure 7 shows the averaged spectrum of a red noise signal plotted against a logarithmic frequency scale. Notice that the PSD falls off at a rate of 2 db per decade. Similarly, pink noise or 1/f noise, Φ(f) is proportional to 1/ f and the power spectral density is proportional to 1/f as f increases. Here 1/f refers to the power spectral density, not Φ(f). 3

4 PSD (x 2 /Hz) Figure 1: Periodogram of the white noise signal, N = 1,, PSD (x 2 /Hz) Figure 2: Periodogram of the white noise signal, N = 1,,. This graph shows only 1, equally spaced frequency values. 4

5 PSD (db/hz) Figure 3: Periodogram of the white noise signal, N = 1,,. This graph shows only 1, equally spaced frequency values. This version of the plot has db/hz units PSD (x 2 /Hz) Figure 4: Averaged periodogram of the white noise signal, blocks of M = 1,. 5

6 PSD (db/hz) Figure 5: Averaged periodogram of the white noise signal, blocks of M = 1,, db/hz units Power Spectral Density Estimate via Welch Power/frequency (db/hz) Frequency (Hz) Figure 6: PSD estimate produced by pwelch. 6

7 3 2 1 PSD (db/hz) Figure 7: Spectrum of red noise. 7