
from IPython.display import Audio


import matplotlib.pyplot as plt
import numpy as np


fact_cache = [1]
def factorial(x):
    while len(fact_cache) <= x:
        fact_cache.append(len(fact_cache) * fact_cache[-1])
    return fact_cache[x]


def sin_unoptimized(x, num_iterations=100):
    sin_nth_deriv = lambda n_: (n_ & 1) * (1 - (((n_ >> 1) & 1) << 1))
    return sum(sin_nth_deriv(n) * (x ** n)  / factorial(n) for n in range(num_iterations))


def cos_unoptimized(x, num_iterations=100):
    cos_nth_deriv = lambda n_: (1 - (n_ & 1)) * (1 - (((n_ >> 1) & 1) << 1))
    return sum(cos_nth_deriv(n) * (x ** n)  / factorial(n) for n in range(num_iterations))


def newton_raphson(f, df, start=0, num_iterations=100):
    approx = start
    for _ in range(num_iterations):
        approx -= f(approx) / df(approx)
    return approx


pi = newton_raphson(sin_unoptimized, cos_unoptimized, start=3)
f"PI Approximation: {pi}"

'PI Approximation: 3.1415926535897936'


def optimize_trig_parameter(x):
    if x < 0:
        x = -x, -1
    
    approx = int(x / (2 * pi)) * 2 * pi
    return x - approx, 1


def sin_val(x):
    x, invert = optimize_trig_parameter(x)
    return invert * sin_unoptimized(x)

def cos_val(x):
    x, _ = optimize_trig_parameter(x)
    return cos_unoptimized(x)

sin = np.vectorize(sin_val)
cos = np.vectorize(cos_val)


amplitude = 0.5
seconds = 10
sample_rate = 10000
base = np.arange(1, sample_rate * seconds)


frequency = 440


len(base)

99999


def get_frequency(frequency):
    c = (frequency * 2 * pi) / sample_rate
    wavelet_ = amplitude * sin(c * base)
    return wavelet_


a = get_frequency(440)
c_sharp = get_frequency(554.37)
e = get_frequency(659.255)
# g = get_frequency(783.991)


wavelet = a + c_sharp


Audio(wavelet, rate=sample_rate)


plt.plot(test[:1000], test_sin[:1000])

[<matplotlib.lines.Line2D at 0x1671d0550>]

Pitch Reconstruction with Audio Processing¶

Gino Prasad¶

12/25/2022¶