# coding: utf-8

# # Audio Synthesis with Sines

# In[1]:

import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt


# In[2]:

from simple_audio import play


# Let's fix some parameters. The `sample_rate` says how many values (so-called "samples") are played per second.
# 
# The audio hardware takes these 'samples' in blocks of 1024, so we need to do some rounding.

# In[3]:

sample_rate = 44100
count = (sample_rate//1024) * 1024


# In[4]:

t = np.linspace(0, 1, sample_rate)[:count]


# Here are some example tones. What are they?

# In[14]:

#signal = np.sin(440*2*np.pi*t)
#signal = np.sin(880*2*np.pi*t)
#signal = 0.5 * (np.sin(880*2*np.pi*t) + np.sin(440*2*np.pi*t))
signal = 0.5 * (np.sin(1209*2*np.pi*t) + np.sin(697*2*np.pi*t))


# In[15]:

play(signal)


# In[16]:

pt.plot(signal[:500])


# That last tone--wouldn't it be useful to pick that apart into its frequency components?

# In[17]:

n = 2047
assert n % 2 == 1

cos_k = np.arange(0, n//2 + 1, dtype=np.float64)
sin_k = np.arange(1, n//2 + 1, dtype=np.float64)

x = np.linspace(0, 2*np.pi, n, endpoint=False)

DFT = np.zeros((n,n))
DFT[:, ::2] = np.cos(cos_k*x[:, np.newaxis])
DFT[:, 1::2] = np.sin(sin_k*x[:, np.newaxis])

IDFT = la.inv(DFT)


# In[18]:

pt.xlim([-100, 1100])
pt.plot(np.abs(IDFT.dot(signal[:n])))


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