理解Python中multitaper方法的频率轴

Question

我正在使用 Python 中的频谱包，使用 multitaper 方法来计算一个样本的功率谱密度 (PSD)（参见http://pyspectrum.readthedocs.io/en/latest/_modules/spectrum/mtm.html ） 60 个磁力计本地磁北分量读数，每分钟采样一次，以纳特斯拉测量。

我的目标是仅查看 1.5-6mHz 频段中的 PSD，因为这是我研究中唯一重要的频段。

使用自适应方法运行函数（默认 NFFT=256，NW=1.4，默认 k）我得到以下图

我的理解是 y 轴的单位是 (nT)^2/Hz，这很好，但是如何获得 x 轴中感兴趣的特定 mHz 频带？ 我最初认为 x 轴以 Hz 为单位，但是对于我选择的任何 NFFT 值，图形看起来都相同，但只是从 0 到 x 轴中的 NFFT 值。

有谁知道我如何在 1.5 - 6 mHz 范围内找到功率（或者我是否完全走错了路？

谢谢！

如果它有帮助，这里是函数正在做什么的代码（对于它所指的其他函数，请参阅我上面发布的链接，我刚刚发布了这个特定的函数，因为它可以更深入地了解正在发生的事情（使用 np.FFT .FFT 例如）：

def pmtm(x, NW=None, k=None, NFFT=None, e=None, v=None, method='adapt', show=False):
"""Multitapering spectral estimation

:param array x: the data
:param float NW: The time half bandwidth parameter (typical values are
    2.5,3,3.5,4). Must be provided otherwise the tapering windows and
    eigen values (outputs of dpss) must be provided
:param int k: uses the first k Slepian sequences. If *k* is not provided,
    *k* is set to *NW*2*.
:param NW:
:param e: the window concentrations (eigenvalues)
:param v: the matrix containing the tapering windows
:param str method: set how the eigenvalues are used. Must be
    in ['unity', 'adapt', 'eigen']
:param bool show: plot results
:return: Sk (complex), weights, eigenvalues

Usually in spectral estimation the mean to reduce bias is to use tapering window.
In order to reduce variance we need to average different spectrum. The problem
is that we have only one set of data. Thus we need to decompose a set into several
segments. Such method are well-known: simple daniell's periodogram, Welch's method
and so on. The drawback of such methods is a loss of resolution since the segments
used to compute the spectrum are smaller than the data set.
The interest of multitapering method is to keep a good resolution while reducing
bias and variance.

How does it work? First we compute different simple periodogram with the whole data
set (to keep good resolution) but each periodgram is computed with a different
tapering windows. Then, we average all these spectrum. To avoid redundancy and bias
due to the tapers mtm use special tapers.

.. plot::
    :width: 80%
    :include-source:

    from spectrum import data_cosine, dpss, pmtm

    data = data_cosine(N=2048, A=0.1, sampling=1024, freq=200)
    # If you already have the DPSS windows
    [tapers, eigen] = dpss(2048, 2.5, 4)
    res = pmtm(data, e=eigen, v=tapers, show=False)
    # You do not need to compute the DPSS before end
    res = pmtm(data, NW=2.5, show=False)
    res = pmtm(data, NW=2.5, k=4, show=True)


.. versionchanged:: 0.6.2

    APN modified method to return each Sk as complex values, the eigenvalues
    and the weights

"""
assert method in ['adapt','eigen','unity']

N = len(x)

# if dpss not provided, compute them
if e is None and v is None:
    if NW is not None:
        [tapers, eigenvalues] = dpss(N, NW, k=k)
    else:
        raise ValueError("NW must be provided (e.g. 2.5, 3, 3.5, 4")
elif e is not None and v is not None:
    eigenvalues = e[:]
    tapers = v[:]
else:
    raise ValueError("if e provided, v must be provided as well and viceversa.")
nwin = len(eigenvalues) # length of the eigen values vector to be used later

# set the NFFT
if NFFT==None:
    NFFT = max(256, 2**nextpow2(N))

Sk_complex = np.fft.fft(np.multiply(tapers.transpose(), x), NFFT)
Sk = abs(Sk_complex)**2

# si nfft smaller thqn N, cut otherwise add wero.
# compute
if method in ['eigen', 'unity']:
    if method == 'unity':
        weights = np.ones((nwin, 1))
    elif method == 'eigen':
        # The S_k spectrum can be weighted by the eigenvalues, as in Park et al.
        weights = np.array([_x/float(i+1) for i,_x in enumerate(eigenvalues)])
        weights = weights.reshape(nwin,1)

elif method == 'adapt':
    # This version uses the equations from [2] (P&W pp 368-370).

    # Wrap the data modulo nfft if N > nfft
    sig2 = np.dot(x, x) / float(N)
    Sk = abs(np.fft.fft(np.multiply(tapers.transpose(), x), NFFT))**2
    Sk = Sk.transpose()
    S = (Sk[:,0] + Sk[:,1]) / 2    # Initial spectrum estimate
    S = S.reshape(NFFT, 1)
    Stemp = np.zeros((NFFT,1))
    S1 = np.zeros((NFFT,1))
    # Set tolerance for acceptance of spectral estimate:
    tol = 0.0005 * sig2 / float(NFFT)
    i = 0
    a = sig2 * (1 - eigenvalues)

    # converges very quickly but for safety; set i<100
    while sum(np.abs(S-S1))/NFFT > tol and i<100:
        i = i + 1
        # calculate weights
        b1 = np.multiply(S, np.ones((1,nwin)))
        b2 = np.multiply(S,eigenvalues.transpose()) + np.ones((NFFT,1))*a.transpose()
        b = b1/b2

        # calculate new spectral estimate
        wk=(b**2)*(np.ones((NFFT,1))*eigenvalues.transpose())
        S1 = sum(wk.transpose()*Sk.transpose())/ sum(wk.transpose())
        S1 = S1.reshape(NFFT, 1)
        Stemp = S1
        S1 = S
        S = Stemp   # swap S and S1
    weights=wk

if show is True:
    from pylab import semilogy
    if method == "adapt":
        Sk = np.mean(Sk * weights, axis=1)
    else:
        Sk = np.mean(Sk * weights, axis=0)
    semilogy(Sk)

return Sk_complex, weights, eigenvalues

Answer 1

如果采样正确，以 1/60 Hz 采样的离散时间信号表示从-1/120 Hz到1/120 Hz 的频率： https : //en.wikipedia.org/wiki/Nyquist_frequency

如果样本都是实值，则负频率分量和正频率分量相同。

离散时间信号中的频率是圆形/模块化的，即在您的信号中，频率f和f + 1/60 Hz是相同的。

在您的 PSD 结果中，x 轴在此范围内运行，从0 Hz开始，到1/120 Hz ，这与-1/120Hz相同，并再次继续到0 Hz之前。

bin x表示的频率为f = x / (NFFT*60 Hz) ，您可以忽略所有x >= NFFT/2 。

如果您只对特定频段感兴趣，那么您可以选择您想要的值并丢弃其余的值，选择NFFT来确定分辨率。