def bandlimited_interpolate(series, delta_f):
    """Return a new PSD that has been interpolated to the desired delta_f.

    Parameters
    ----------
    series : FrequencySeries
        Frequency series to be interpolated.
    delta_f : float
        The desired delta_f of the output

    Returns
    -------
    interpolated series : FrequencySeries
        A new FrequencySeries that has been interpolated.
    """
    series = FrequencySeries(series, dtype=complex_same_precision_as(series), delta_f=series.delta_f)

    N = (len(series) - 1) * 2
    delta_t = 1.0 / series.delta_f / N

    new_N = int(1.0 / (delta_t * delta_f))
    new_n = new_N / 2 + 1

    series_in_time = TimeSeries(zeros(N), dtype=real_same_precision_as(series), delta_t=delta_t)
    ifft(series, series_in_time)

    padded_series_in_time = TimeSeries(zeros(new_N), dtype=series_in_time.dtype, delta_t=delta_t)
    padded_series_in_time[0:N/2] = series_in_time[0:N/2]
    padded_series_in_time[new_N-N/2:new_N] = series_in_time[N/2:N]

    interpolated_series = FrequencySeries(zeros(new_n), dtype=series.dtype, delta_f=delta_f)
    fft(padded_series_in_time, interpolated_series)

    return interpolated_series

def make_frequency_series(vec):
    """Return a frequency series of the input vector.

    If the input is a frequency series it is returned, else if the input
    vector is a real time series it is fourier transformed and returned as a 
    frequency series. 
    
    Parameters
    ----------
    vector : TimeSeries or FrequencySeries  

    Returns
    -------
    Frequency Series: FrequencySeries
        A frequency domain version of the input vector.
    """
    if isinstance(vec, FrequencySeries):
        return vec
    if isinstance(vec, TimeSeries):
        N = len(vec)
        n = N/2+1    
        delta_f = 1.0 / N / vec.delta_t
        vectilde =  FrequencySeries(zeros(n, dtype=complex_same_precision_as(vec)), 
                                    delta_f=delta_f, copy=False)
        fft(vec, vectilde)   
        return vectilde
    else:
        raise TypeError("Can only convert a TimeSeries to a FrequencySeries")

def make_padded_frequency_series(vec,filter_N=None):
    """Pad a TimeSeries with a length of zeros greater than its length, such
    that the total length is the closest power of 2. This prevents the effects 
    of wraparound.
    """
    if filter_N is None:
        power = ceil(log(len(vec),2))+1
        N = 2 ** power
    else:
        N = filter_N
    n = N/2+1    
    
   
    if isinstance(vec,FrequencySeries):
        vectilde = FrequencySeries(zeros(n, dtype=complex_same_precision_as(vec)),
                                   delta_f=1.0,copy=False)
	if len(vectilde) < len(vec):
	    cplen = len(vectilde)
        else:
            cplen = len(vec)
        vectilde[0:cplen] = vec[0:cplen]  
        delta_f = vec.delta_f
    
        
    if isinstance(vec,TimeSeries):  
        vec_pad = TimeSeries(zeros(N),delta_t=vec.delta_t,
                         dtype=real_same_precision_as(vec))
        vec_pad[0:len(vec)] = vec   
        delta_f = 1.0/(vec.delta_t*N)
        vectilde = FrequencySeries(zeros(n),delta_f=1.0, 
                               dtype=complex_same_precision_as(vec))
        fft(vec_pad,vectilde)
        
    vectilde = FrequencySeries(vectilde * DYN_RANGE_FAC,delta_f=delta_f,dtype=complex64)
    return vectilde

def td_waveform_to_fd_waveform(waveform, out=None, length=None,
                               buffer_length=100):
    """ Convert a time domain into a frequency domain waveform by FFT.
        As a waveform is assumed to "wrap" in the time domain one must be
        careful to ensure the waveform goes to 0 at both "boundaries". To
        ensure this is done correctly the waveform must have the epoch set such
        the merger time is at t=0 and the length of the waveform should be
        shorter than the desired length of the FrequencySeries (times 2 - 1)
        so that zeroes can be suitably pre- and post-pended before FFTing.
        If given, out is a memory array to be used as the output of the FFT.
        If not given memory is allocated internally.
        If present the length of the returned FrequencySeries is determined
        from the length out. If out is not given the length can be provided
        expicitly, or it will be chosen as the nearest power of 2. If choosing
        length explicitly the waveform length + buffer_length is used when
        choosing the nearest binary number so that some zero padding is always
        added.
    """
    # Figure out lengths and set out if needed
    if out is None:
        if length is None:
            N = pnutils.nearest_larger_binary_number(len(waveform) + \
                                                     buffer_length)
            n = int(N//2) + 1
        else:
            n = length
            N = (n-1)*2
        out = zeros(n, dtype=complex_same_precision_as(waveform))
    else:
        n = len(out)
        N = (n-1)*2
    delta_f =  1. / (N * waveform.delta_t)

    # total duration of the waveform
    tmplt_length = len(waveform) * waveform.delta_t
    if len(waveform) > N:
        err_msg = "The time domain template is longer than the intended "
        err_msg += "duration in the frequency domain. This situation is "
        err_msg += "not supported in this function. Please shorten the "
        err_msg += "waveform appropriately before calling this function or "
        err_msg += "increase the allowed waveform length. "
        err_msg += "Waveform length (in samples): {}".format(len(waveform))
        err_msg += ". Intended length: {}.".format(N)
        raise ValueError(err_msg)
    # for IMR templates the zero of time is at max amplitude (merger)
    # thus the start time is minus the duration of the template from
    # lower frequency cutoff to merger, i.e. minus the 'chirp time'
    tChirp = - float( waveform.start_time )  # conversion from LIGOTimeGPS
    waveform.resize(N)
    k_zero = int(waveform.start_time / waveform.delta_t)
    waveform.roll(k_zero)
    htilde = FrequencySeries(out, delta_f=delta_f, copy=False)
    fft(waveform.astype(real_same_precision_as(htilde)), htilde)
    htilde.length_in_time = tmplt_length
    htilde.chirp_length = tChirp
    return htilde

def get_waveform_filter(out, template=None, **kwargs):
    """Return a frequency domain waveform filter for the specified approximant
    """
    n = len(out)

    input_params = props(template, **kwargs)

    if input_params['approximant'] in filter_approximants(_scheme.mgr.state):
        wav_gen = filter_wav[type(_scheme.mgr.state)]
        htilde = wav_gen[input_params['approximant']](out=out, **input_params)
        htilde.resize(n)
        htilde.chirp_length = get_waveform_filter_length_in_time(**input_params)
        htilde.length_in_time = htilde.chirp_length
        return htilde

    if input_params['approximant'] in fd_approximants(_scheme.mgr.state):
        wav_gen = fd_wav[type(_scheme.mgr.state)]
        hp, hc = wav_gen[input_params['approximant']](**input_params)
        hp.resize(n)
        out[0:len(hp)] = hp[:]
        hp = FrequencySeries(out, delta_f=hp.delta_f, copy=False)
        hp.chirp_length = get_waveform_filter_length_in_time(**input_params)
        hp.length_in_time = hp.chirp_length
        return hp

    elif input_params['approximant'] in td_approximants(_scheme.mgr.state):
        # N: number of time samples required
        N = (n-1)*2
        delta_f = 1.0 / (N * input_params['delta_t'])
        wav_gen = td_wav[type(_scheme.mgr.state)]
        hp, hc = wav_gen[input_params['approximant']](**input_params)
        # taper the time series hp if required
        if ('taper' in input_params.keys() and \
            input_params['taper'] is not None):
            hp = wfutils.taper_timeseries(hp, input_params['taper'],
                                          return_lal=False)
        # total duration of the waveform
        tmplt_length = len(hp) * hp.delta_t
        # for IMR templates the zero of time is at max amplitude (merger)
        # thus the start time is minus the duration of the template from
        # lower frequency cutoff to merger, i.e. minus the 'chirp time'
        tChirp = - float( hp.start_time )  # conversion from LIGOTimeGPS
        hp.resize(N)
        k_zero = int(hp.start_time / hp.delta_t)
        hp.roll(k_zero)
        htilde = FrequencySeries(out, delta_f=delta_f, copy=False)
        fft(hp.astype(real_same_precision_as(htilde)), htilde)
        htilde.length_in_time = tmplt_length
        htilde.chirp_length = tChirp
        return htilde

    else:
        raise ValueError("Approximant %s not available" %
                            (input_params['approximant']))

def lfilter(coefficients, timeseries):
    """ Apply filter coefficients to a time series
    
    Parameters
    ----------
    coefficients: numpy.ndarray
        Filter coefficients to apply
    timeseries: numpy.ndarray
        Time series to be filtered.

    Returns
    -------
    tseries: numpy.ndarray
        filtered array
    """
    from pycbc.fft import fft, ifft
    from pycbc.filter import correlate

    # If there aren't many points just use the default scipy method
    if len(timeseries) < 2**7:
        if hasattr(timeseries, 'numpy'):
            timeseries = timeseries.numpy()
        series = scipy.signal.lfilter(coefficients, 1.0, timeseries)
        return series
    else:
        cseries = (Array(coefficients[::-1] * 1)).astype(timeseries.dtype)
        cseries.resize(len(timeseries))
        cseries.roll(len(timeseries) - len(coefficients) + 1)
        timeseries = Array(timeseries, copy=False)

        flen = len(cseries) / 2 + 1
        ftype = complex_same_precision_as(timeseries)

        cfreq = zeros(flen, dtype=ftype)
        tfreq = zeros(flen, dtype=ftype)

        fft(Array(cseries), cfreq)
        fft(Array(timeseries), tfreq)

        cout = zeros(flen, ftype)
        out = zeros(len(timeseries), dtype=timeseries)

        correlate(cfreq, tfreq, cout)   
        ifft(cout, out)

        return out.numpy()  / len(out)

def interpolate_complex_frequency(series, delta_f, zeros_offset=0, side='right'):
    """Interpolate complex frequency series to desired delta_f.

    Return a new complex frequency series that has been interpolated to the
    desired delta_f.

    Parameters
    ----------
    series : FrequencySeries
        Frequency series to be interpolated.
    delta_f : float
        The desired delta_f of the output
    zeros_offset : optional, {0, int}
        Number of sample to delay the start of the zero padding
    side : optional, {'right', str}
        The side of the vector to zero pad
        
    Returns
    -------
    interpolated series : FrequencySeries
        A new FrequencySeries that has been interpolated.
    """
    new_n = int( (len(series)-1) * series.delta_f / delta_f + 1)
    samples = numpy.arange(0, new_n) * delta_f
    old_N = int( (len(series)-1) * 2 )
    new_N = int( (new_n - 1) * 2 )
    time_series = TimeSeries(zeros(old_N), delta_t =1.0/(series.delta_f*old_N),
                             dtype=real_same_precision_as(series))
                             
    ifft(series, time_series)

    time_series.roll(-zeros_offset)
    time_series.resize(new_N)
    
    if side == 'left':
        time_series.roll(zeros_offset + new_N - old_N)
    elif side == 'right':
        time_series.roll(zeros_offset)

    out_series = FrequencySeries(zeros(new_n), epoch=series.epoch,
                           delta_f=delta_f, dtype=series.dtype)
    fft(time_series, out_series)

    return out_series

    def to_frequencyseries(self, delta_f=None):
        """ Return the Fourier transform of this time series

        Parameters
        ----------
        delta_f : {None, float}, optional
            The frequency resolution of the returned frequency series. By
        default the resolution is determined by the duration of the timeseries.

        Returns
        -------
        FrequencySeries:
            The fourier transform of this time series.
        """
        from pycbc.fft import fft
        if not delta_f:
            delta_f = 1.0 / self.duration

        # add 0.5 to round integer
        tlen  = int(1.0 / delta_f / self.delta_t + 0.5)
        flen = int(tlen / 2 + 1)

        if tlen < len(self):
            raise ValueError("The value of delta_f (%s) would be "
                             "undersampled. Maximum delta_f "
                             "is %s." % (delta_f, 1.0 / self.duration))
        if not delta_f:
            tmp = self
        else:
            tmp = TimeSeries(zeros(tlen, dtype=self.dtype),
                             delta_t=self.delta_t, epoch=self.start_time)
            tmp[:len(self)] = self[:]

        f = FrequencySeries(zeros(flen,
                           dtype=complex_same_precision_as(self)),
                           delta_f=delta_f)
        fft(tmp, f)
        return f

def welch(timeseries, seg_len=4096, seg_stride=2048, window='hann',
          avg_method='median', num_segments=None, require_exact_data_fit=False):
    """PSD estimator based on Welch's method.

    Parameters
    ----------
    timeseries : TimeSeries
        Time series for which the PSD is to be estimated.
    seg_len : int
        Segment length in samples.
    seg_stride : int
        Separation between consecutive segments, in samples.
    window : {'hann'}
        Function used to window segments before Fourier transforming.
    avg_method : {'median', 'mean', 'median-mean'}
        Method used for averaging individual segment PSDs.

    Returns
    -------
    psd : FrequencySeries
        Frequency series containing the estimated PSD.

    Raises
    ------
    ValueError
        For invalid choices of `seg_len`, `seg_stride` `window` and
        `avg_method` and for inconsistent combinations of len(`timeseries`),
        `seg_len` and `seg_stride`.

    Notes
    -----
    See arXiv:gr-qc/0509116 for details.
    """
    window_map = {
        'hann': numpy.hanning
    }

    # sanity checks
    if not window in window_map:
        raise ValueError('Invalid window')
    if not avg_method in ('mean', 'median', 'median-mean'):
        raise ValueError('Invalid averaging method')
    if type(seg_len) is not int or type(seg_stride) is not int \
        or seg_len <= 0 or seg_stride <= 0:
        raise ValueError('Segment length and stride must be positive integers')

    if timeseries.precision == 'single':
        fs_dtype = numpy.complex64
    elif timeseries.precision == 'double':
        fs_dtype = numpy.complex128
        
    num_samples = len(timeseries)
    if num_segments is None:
        num_segments = int(num_samples // seg_stride)
        # NOTE: Is this not always true?
        if (num_segments - 1) * seg_stride + seg_len > num_samples:
            num_segments -= 1

    if not require_exact_data_fit:
        data_len = (num_segments - 1) * seg_stride + seg_len

        # Get the correct amount of data
        if data_len < num_samples:
            diff = num_samples - data_len
            start = diff // 2
            end = num_samples - diff // 2
            # Want this to be integers so if diff is odd, catch it here.
            if diff % 2:
                start = start + 1

            timeseries = timeseries[start:end]
            num_samples = len(timeseries)
        if data_len > num_samples:
            err_msg = "I was asked to estimate a PSD on %d " %(data_len)
            err_msg += "data samples. However the data provided only contains "
            err_msg += "%d data samples." %(num_samples)

    if num_samples != (num_segments - 1) * seg_stride + seg_len:
        raise ValueError('Incorrect choice of segmentation parameters')
        
    w = Array(window_map[window](seg_len).astype(timeseries.dtype))

    # calculate psd of each segment
    delta_f = 1. / timeseries.delta_t / seg_len
    segment_tilde = FrequencySeries(numpy.zeros(seg_len / 2 + 1), \
        delta_f=delta_f, dtype=fs_dtype)

    segment_psds = []
    for i in xrange(num_segments):
        segment_start = i * seg_stride
        segment_end = segment_start + seg_len
        segment = timeseries[segment_start:segment_end]
        assert len(segment) == seg_len
        fft(segment * w, segment_tilde)
        seg_psd = abs(segment_tilde * segment_tilde.conj()).numpy()

        #halve the DC and Nyquist components to be consistent with TO10095
        seg_psd[0] /= 2
        seg_psd[-1] /= 2

#.........这里部分代码省略.........

def inverse_spectrum_truncation(psd, max_filter_len, low_frequency_cutoff=None, trunc_method=None):
    """Modify a PSD such that the impulse response associated with its inverse
    square root is no longer than `max_filter_len` time samples. In practice
    this corresponds to a coarse graining or smoothing of the PSD.

    Parameters
    ----------
    psd : FrequencySeries
        PSD whose inverse spectrum is to be truncated.
    max_filter_len : int
        Maximum length of the time-domain filter in samples.
    low_frequency_cutoff : {None, int}
        Frequencies below `low_frequency_cutoff` are zeroed in the output.
    trunc_method : {None, 'hann'}
        Function used for truncating the time-domain filter.
        None produces a hard truncation at `max_filter_len`.

    Returns
    -------
    psd : FrequencySeries
        PSD whose inverse spectrum has been truncated.

    Raises
    ------
    ValueError
        For invalid types or values of `max_filter_len` and `low_frequency_cutoff`.

    Notes
    -----
    See arXiv:gr-qc/0509116 for details.
    """
    # sanity checks
    if type(max_filter_len) is not int or max_filter_len <= 0:
        raise ValueError('max_filter_len must be a positive integer')
    if low_frequency_cutoff is not None and low_frequency_cutoff < 0 \
        or low_frequency_cutoff > psd.sample_frequencies[-1]:
        raise ValueError('low_frequency_cutoff must be within the bandwidth of the PSD')

    N = (len(psd)-1)*2

    inv_asd = FrequencySeries((1. / psd)**0.5, delta_f=psd.delta_f, \
        dtype=complex_same_precision_as(psd))
        
    inv_asd[0] = 0
    inv_asd[N/2] = 0
    q = TimeSeries(numpy.zeros(N), delta_t=(N / psd.delta_f), \
        dtype=real_same_precision_as(psd))

    if low_frequency_cutoff:
        kmin = int(low_frequency_cutoff / psd.delta_f)
        inv_asd[0:kmin] = 0

    ifft(inv_asd, q)
    
    trunc_start = max_filter_len / 2
    trunc_end = N - max_filter_len / 2

    if trunc_method == 'hann':
        trunc_window = Array(numpy.hanning(max_filter_len), dtype=q.dtype)
        q[0:trunc_start] *= trunc_window[max_filter_len/2:max_filter_len]
        q[trunc_end:N] *= trunc_window[0:max_filter_len/2]

    q[trunc_start:trunc_end] = 0
    psd_trunc = FrequencySeries(numpy.zeros(len(psd)), delta_f=psd.delta_f, \
                                dtype=complex_same_precision_as(psd))
    fft(q, psd_trunc)
    psd_trunc *= psd_trunc.conj()
    psd_out = 1. / abs(psd_trunc)

    return psd_out

def get_two_pol_waveform_filter(outplus, outcross, template, **kwargs):
    """Return a frequency domain waveform filter for the specified approximant.
    Unlike get_waveform_filter this function returns both h_plus and h_cross
    components of the waveform, which are needed for searches where h_plus
    and h_cross are not related by a simple phase shift.
    """
    n = len(outplus)

    # If we don't have an inclination column alpha3 might be used
    if not hasattr(template, 'inclination') and 'inclination' not in kwargs:
        if hasattr(template, 'alpha3'):
            kwargs['inclination'] = template.alpha3

    input_params = props(template, **kwargs)

    if input_params['approximant'] in fd_approximants(_scheme.mgr.state):
        wav_gen = fd_wav[type(_scheme.mgr.state)]
        hp, hc = wav_gen[input_params['approximant']](**input_params)
        hp.resize(n)
        hc.resize(n)
        outplus[0:len(hp)] = hp[:]
        hp = FrequencySeries(outplus, delta_f=hp.delta_f, copy=False)
        outcross[0:len(hc)] = hc[:]
        hc = FrequencySeries(outcross, delta_f=hc.delta_f, copy=False)
        hp.chirp_length = get_waveform_filter_length_in_time(**input_params)
        hp.length_in_time = hp.chirp_length
        hc.chirp_length = hp.chirp_length
        hc.length_in_time = hp.length_in_time
        return hp, hc
    elif input_params['approximant'] in td_approximants(_scheme.mgr.state):
        # N: number of time samples required
        N = (n-1)*2
        delta_f = 1.0 / (N * input_params['delta_t'])
        wav_gen = td_wav[type(_scheme.mgr.state)]
        hp, hc = wav_gen[input_params['approximant']](**input_params)
        # taper the time series hp if required
        if 'taper' in input_params.keys() and \
                input_params['taper'] is not None:
            hp = wfutils.taper_timeseries(hp, input_params['taper'],
                                          return_lal=False)
            hc = wfutils.taper_timeseries(hc, input_params['taper'],
                                          return_lal=False)
        # total duration of the waveform
        tmplt_length = len(hp) * hp.delta_t
        # for IMR templates the zero of time is at max amplitude (merger)
        # thus the start time is minus the duration of the template from
        # lower frequency cutoff to merger, i.e. minus the 'chirp time'
        tChirp = - float( hp.start_time )  # conversion from LIGOTimeGPS
        hp.resize(N)
        hc.resize(N)
        k_zero = int(hp.start_time / hp.delta_t)
        hp.roll(k_zero)
        hc.roll(k_zero)
        hp_tilde = FrequencySeries(outplus, delta_f=delta_f, copy=False)
        hc_tilde = FrequencySeries(outcross, delta_f=delta_f, copy=False)
        fft(hp.astype(real_same_precision_as(hp_tilde)), hp_tilde)
        fft(hc.astype(real_same_precision_as(hc_tilde)), hc_tilde)
        hp_tilde.length_in_time = tmplt_length
        hp_tilde.chirp_length = tChirp
        hc_tilde.length_in_time = tmplt_length
        hc_tilde.chirp_length = tChirp
        return hp_tilde, hc_tilde
    else:
        raise ValueError("Approximant %s not available" %
                            (input_params['approximant']))