def trig_sum(t, h, df, N, f0=0, freq_factor=1,
             oversampling=5, use_fft=True, Mfft=4):
    """Compute (approximate) trigonometric sums for a number of frequencies

    This routine computes weighted sine and cosine sums:

        S_j = sum_i { h_i * sin(2 pi * f_j * t_i) }
        C_j = sum_i { h_i * cos(2 pi * f_j * t_i) }

    Where f_j = freq_factor * (f0 + j * df) for the values j in 1 ... N.
    The sums can be computed either by a brute force O[N^2] method, or
    by an FFT-based O[Nlog(N)] method.

    Parameters
    ----------
    t : array_like
        array of input times
    h : array_like
        array weights for the sum
    df : float
        frequency spacing
    N : int
        number of frequency bins to return
    f0 : float (optional, default=0)
        The low frequency to use
    freq_factor : float (optional, default=1)
        Factor which multiplies the frequency
    use_fft : bool
        if True, use the approximate FFT algorithm to compute the result.
        This uses the FFT with Press & Rybicki's Lagrangian extirpolation.
    oversampling : int (default = 5)
        oversampling freq_factor for the approximation; roughtly the number of
        time samples across the highest-frequency sinusoid. This parameter
        contains the tradeoff between accuracy and speed. Not referenced
        if use_fft is False.
    Mfft : int
        The number of adjacent points to use in the FFT approximation.
        Not referenced if use_fft is False.

    Returns
    -------
    S, C : ndarrays
        summation arrays for frequencies f = df * np.arange(1, N + 1)
    """
    df *= freq_factor
    f0 *= freq_factor

    assert df > 0
    t, h = map(np.ravel, np.broadcast_arrays(t, h))

    if use_fft:
        Mfft = int(Mfft)
        assert(Mfft > 0)

        # required size of fft is the power of 2 above the oversampling rate
        Nfft = bitceil(N * oversampling)
        t0 = t.min()

        if f0 > 0:
            h = h * np.exp(2j * np.pi * f0 * (t - t0))

        tnorm = ((t - t0) * Nfft * df) % Nfft
        grid = extirpolate(tnorm, h, Nfft, Mfft)

        fftgrid = np.fft.ifft(grid)
        if t0 != 0:
            f = f0 + df * np.arange(Nfft)
            fftgrid *= np.exp(2j * np.pi * t0 * f)
        fftgrid = fftgrid[:N]

        C = Nfft * fftgrid.real
        S = Nfft * fftgrid.imag
    else:
        f = f0 + df * np.arange(N)
        C = np.dot(h, np.cos(2 * np.pi * f * t[:, np.newaxis]))
        S = np.dot(h, np.sin(2 * np.pi * f * t[:, np.newaxis]))

    return S, C
Compute (approximate) trigonometric sums for a number of frequencies

    This routine computes weighted sine and cosine sums:

        S_j = sum_i { h_i * sin(2 pi * f_j * t_i) }
        C_j = sum_i { h_i * cos(2 pi * f_j * t_i) }

    Where f_j = freq_factor * (f0 + j * df) for the values j in 1 ... N.
    The sums can be computed either by a brute force O[N^2] method, or
    by an FFT-based O[Nlog(N)] method.

    Parameters
    ----------
    t : array_like
        array of input times
    h : array_like
        array weights for the sum
    df : float
        frequency spacing
    N : int
        number of frequency bins to return
    f0 : float (optional, default=0)
        The low frequency to use
    freq_factor : float (optional, default=1)
        Factor which multiplies the frequency
    use_fft : bool
        if True, use the approximate FFT algorithm to compute the result.
        This uses the FFT with Press & Rybicki's Lagrangian extirpolation.
    oversampling : int (default = 5)
        oversampling freq_factor for the approximation; roughtly the number of
        time samples across the highest-frequency sinusoid. This parameter
        contains the tradeoff between accuracy and speed. Not referenced
        if use_fft is False.
    Mfft : int
        The number of adjacent points to use in the FFT approximation.
        Not referenced if use_fft is False.

    Returns
    -------
    S, C : ndarrays
        summation arrays for frequencies f = df * np.arange(1, N + 1)