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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def time_frequency(gram, frequencies, times, fs, function=np.sin, length=None, n_dec=1):
 """Reverse synthesis of a time-frequency representation of a signal Parameters gram : np.ndarray ``gram[n, m]`` is the magnitude of ``frequencies[n]`` from ``times[m]`` to ``times[m + 1]`` Non-positive magnitudes are interpreted as silence. frequencies : np.ndarray array of size ``gram.shape[0]`` denoting the frequency of each row of gram times : np.ndarray, shape= ``(gram.shape[1],)`` or ``(gram.shape[1], 2)`` Either the start time of each column in the gram, or the time interval corresponding to each column. fs : int desired sampling rate of the output signal function : function function to use to synthesize notes, should be :math:`2\pi`-periodic length : int desired number of samples in the output signal, defaults to ``times[-1]*fs`` n_dec : int the number of decimals used to approximate each sonfied frequency. Defaults to 1 decimal place. Higher precision will be slower. Returns ------- output : np.ndarray synthesized version of the piano roll """

    # Default value for length
    if times.ndim == 1:
        # Convert to intervals
        times = util.boundaries_to_intervals(times)

    if length is None:
        length = int(times[-1, 1] * fs)

    times, _ = util.adjust_intervals(times, t_max=length)

    # Truncate times so that the shape matches gram
    n_times = gram.shape[1]
    times = times[:n_times]

    def _fast_synthesize(frequency):
        """A faster way to synthesize a signal.
            Generate one cycle, and simulate arbitrary repetitions
            using array indexing tricks.
        """
        # hack so that we can ensure an integer number of periods and samples
        # rounds frequency to 1st decimal, s.t. 10 * frequency will be an int
        frequency = np.round(frequency, n_dec)

        # Generate 10*frequency periods at this frequency
        # Equivalent to n_samples = int(n_periods * fs / frequency)
        # n_periods = 10*frequency is the smallest integer that guarantees
        # that n_samples will be an integer, since assuming 10*frequency
        # is an integer
        n_samples = int(10.0**n_dec * fs)

        short_signal = function(2.0 * np.pi * np.arange(n_samples) *
                                frequency / fs)

        # Calculate the number of loops we need to fill the duration
        n_repeats = int(np.ceil(length/float(short_signal.shape[0])))

        # Simulate tiling the short buffer by using stride tricks
        long_signal = as_strided(short_signal,
                                 shape=(n_repeats, len(short_signal)),
                                 strides=(0, short_signal.itemsize))

        # Use a flatiter to simulate a long 1D buffer
        return long_signal.flat

    def _const_interpolator(value):
        """Return a function that returns `value`
            no matter the input.
        """
        def __interpolator(x):
            return value
        return __interpolator

    # Threshold the tfgram to remove non-positive values
    gram = np.maximum(gram, 0)

    # Pre-allocate output signal
    output = np.zeros(length)
    time_centers = np.mean(times, axis=1) * float(fs)

    for n, frequency in enumerate(frequencies):
        # Get a waveform of length samples at this frequency
        wave = _fast_synthesize(frequency)

        # Interpolate the values in gram over the time grid
        if len(time_centers) > 1:
            gram_interpolator = interp1d(
                time_centers, gram[n, :],
                kind='linear', bounds_error=False,
                fill_value=0.0)
        # If only one time point, create constant interpolator
        else:
            gram_interpolator = _const_interpolator(gram[n, 0])

        # Scale each time interval by the piano roll magnitude
        for m, (start, end) in enumerate((times * fs).astype(int)):
            # Clip the timings to make sure the indices are valid
            start, end = max(start, 0), min(end, length)
            # add to waveform
            output[start:end] += (
                wave[start:end] * gram_interpolator(np.arange(start, end)))

    # Normalize, but only if there's non-zero values
    norm = np.abs(output).max()
    if norm >= np.finfo(output.dtype).tiny:
        output /= norm

    return output