


"""simple docstring"""


from __future__ import annotations

from math import pi
from typing import Protocol

import matplotlib.pyplot as plt
import numpy as np


class 			A__	(						__lowercase):


     """simple docstring"""

     def 		a__						(	self:						Any     ,		__a:						float	)->    float:
           return 0.0




def   snake_case		(     UpperCamelCase__       :						np.ndarray			,				UpperCamelCase__       :						int    )   ->      tuple[int | float, int | float]:
      lowerCamelCase					:					str        =	min([-20, np.min(fft_results[1 : samplerate // 2 - 1]    )]    )
      lowerCamelCase					:					str        =	max([20, np.max(fft_results[1 : samplerate // 2 - 1]    )]    )
      return lowest, highest




def   snake_case		(     UpperCamelCase__       :						FilterType			,				UpperCamelCase__       :						int    )   ->      None:
      lowerCamelCase					:					Optional[int]        =	512
      lowerCamelCase					:					List[str]        =	[1] + [0] * (size - 1)
      lowerCamelCase					:					List[Any]        =	[filter_type.process(UpperCamelCase__    ) for item in inputs]

      lowerCamelCase					:					str        =	[0] * (samplerate - size)  # zero-padding
      outputs += filler
      lowerCamelCase					:					List[Any]        =	np.abs(np.fft.fft(UpperCamelCase__    )    )
      lowerCamelCase					:					Tuple        =	20 * np.logaa(UpperCamelCase__    )

      # Frequencies on log scale from 24 to nyquist frequency
      plt.xlim(24			,				samplerate / 2 - 1    )
      plt.xlabel("""Frequency (Hz)"""    )
      plt.xscale("""log"""    )

      # Display within reasonable bounds
      lowerCamelCase					:					Union[str, Any]        =	get_bounds(UpperCamelCase__			,				UpperCamelCase__    )
      plt.ylim(max([-80, bounds[0]]    )			,				min([80, bounds[1]]    )    )
      plt.ylabel("""Gain (dB)"""    )

      plt.plot(UpperCamelCase__    )
      plt.show()






def   snake_case		(     UpperCamelCase__       :						FilterType			,				UpperCamelCase__       :						int    )   ->      None:
      lowerCamelCase					:					str        =	512
      lowerCamelCase					:					str        =	[1] + [0] * (size - 1)
      lowerCamelCase					:					Optional[int]        =	[filter_type.process(UpperCamelCase__    ) for item in inputs]

      lowerCamelCase					:					List[str]        =	[0] * (samplerate - size)  # zero-padding
      outputs += filler
      lowerCamelCase					:					Tuple        =	np.angle(np.fft.fft(UpperCamelCase__    )    )

      # Frequencies on log scale from 24 to nyquist frequency
      plt.xlim(24			,				samplerate / 2 - 1    )
      plt.xlabel("""Frequency (Hz)"""    )
      plt.xscale("""log"""    )

      plt.ylim(-2 * pi			,				2 * pi    )
      plt.ylabel("""Phase shift (Radians)"""    )
      plt.plot(np.unwrap(UpperCamelCase__			,				-2 * pi    )    )
      plt.show()








"""simple docstring"""


from __future__ import annotations

import math




def   snake_case		(     UpperCamelCase__       :						float			,				UpperCamelCase__       :						int    )   ->      float:
      lowerCamelCase					:					Dict        =	u
      for i in range(1			,				UpperCamelCase__    ):
            lowerCamelCase					:					List[str]        =	temp * (u - i)
      return temp




def   snake_case		(     )   ->      None:
      lowerCamelCase					:					List[Any]        =	int(input("""enter the numbers of values: """    )    )
      lowerCamelCase					:					list[list[float]]        =	[]
      for _ in range(UpperCamelCase__    ):
            y.append([]    )
      for i in range(UpperCamelCase__    ):
            for j in range(UpperCamelCase__    ):
                  y[i].append(UpperCamelCase__    )
                  lowerCamelCase					:					Union[str, Any]        =	0

      print("""enter the values of parameters in a list: """    )
      lowerCamelCase					:					Any        =	list(map(UpperCamelCase__			,				input().split()    )    )

      print("""enter the values of corresponding parameters: """    )
      for i in range(UpperCamelCase__    ):
            lowerCamelCase					:					int        =	float(input()    )

      lowerCamelCase					:					Dict        =	int(input("""enter the value to interpolate: """    )    )
      lowerCamelCase					:					List[Any]        =	(value - x[0]) / (x[1] - x[0])

      # for calculating forward difference table

      for i in range(1			,				UpperCamelCase__    ):
            for j in range(n - i    ):
                  lowerCamelCase					:					str        =	y[j + 1][i - 1] - y[j][i - 1]

      lowerCamelCase					:					Any        =	y[0][0]
      for i in range(1			,				UpperCamelCase__    ):
            summ += (ucal(UpperCamelCase__			,				UpperCamelCase__    ) * y[0][i]) / math.factorial(UpperCamelCase__    )

      print(F'the value at {value} is {summ}'    )


if __name__ == "__main__":
    main()




