import numpy as np
import matplotlib.pyplot as plt

%matplotlib inline

def generate_sinusoid(N, A, f0, fs, phi):
 '''
 N(int) : number of samples
 A(float) : amplitude
 f0(float): frequency in Hz
 fs(float): sample rate
 phi(float): initial phase
 
 return 
 x (numpy array): sinusoid signal which lenght is M
 '''
 
 T = 1/fs
 n = np.arange(N) # [0,1,..., N-1]
 x = A * np.cos( 2*f0*np.pi*n*T + phi )
 
 return x

N = 511
A = 0.8
f0 = 440
fs = 44100
phi = 0

x = generate_sinusoid(N, A, f0, fs, phi)

plt.plot(x)
plt.show()

# 另一种生成正弦信号的方法，生成时长为t的序列
def generate_sinusoid_2(t, A, f0, fs, phi):
 '''
 t (float) : 生成序列的时长
 A (float) : amplitude
 f0 (float) : frequency
 fs (float) : sample rate
 phi(float) : initial phase
 
 returns
 x (numpy array): sinusoid signal sequence
 '''
 
 T = 1.0/fs
 N = t / T
 
 return generate_sinusoid(N, A, f0, fs, phi)

A = 1.0
f0 = 440
fs = 44100
phi = 0
t = 0.02

x = generate_sinusoid_2(t, A, f0, fs, phi)

n = np.arange(0, 0.02, 1/fs)
plt.plot(n, x)

from scipy.fftpack import fft

# generate sinusoid
N = 511
A = 0.8
f0 = 440
fs = 44100
phi = 1.0
x = generate_sinusoid(N, A, f0, fs, phi)

# fft is
X = fft(x)
mX = np.abs(X) # magnitude
pX = np.angle(X) # phase

# plot the magnitude and phase
plt.subplot(2,1,1)
plt.plot(mX)

plt.subplot(2,1,2)
plt.plot(pX)
plt.show()

def generate_complex_sinusoid(k, N):
 '''
 k (int): frequency index
 N (int): length of complex sinusoid in samples
 
 returns
 c_sin (numpy array): the generated complex sinusoid (length N)
 '''
 
 n = np.arange(N)
 
 c_sin = np.exp(1j * 2 * np.pi * k * n / N)
 
 return np.conjugate(c_sin)

def generate_complex_sinusoid_matrix(N):
 '''
 N (int): length of complex sinusoid in samples
 
 returns
 c_sin_matrix (numpy array): the generated complex sinusoid (length N)
 '''
 
 n = np.arange(N)
 n = np.expand_dims(n, axis=1)  # 扩充维度，将1D向量，转为2D矩阵，方便后面的矩阵相乘
 
 k = n
 
 m = n.T * k / N     # [N,1] * [1, N] = [N,N]
 
 S = np.exp(1j * 2 * np.pi * m)  # 计算矩阵 S
 
 return np.conjugate(S)

# 生成信号，用于测试
N = 511
A = 0.8
f0 = 440
fs = 44100
phi = 1.0
x = generate_sinusoid(N, A, f0, fs, phi)

# 第一种方式计算DFT
X_1 = np.array([])
for k in range(N):
 s = generate_complex_sinusoid(k, N)
 X_1 = np.append(X_1, np.sum(x * s))
 
mX = np.abs(X_1)
pX = np.angle(X_1)

# plot the magnitude and phase
plt.subplot(2,1,1)
plt.plot(mX)

plt.subplot(2,1,2)
plt.plot(pX)
plt.show()

# 结果和scipy的结果基本相同

# 第二种方法计算DFT
S = generate_complex_sinusoid_matrix(N)
X_2 = np.dot(S, x)

mX = np.abs(X_2)
pX = np.angle(X_2)

# plot the magnitude and phase
plt.subplot(2,1,1)
plt.plot(mX)

plt.subplot(2,1,2)
plt.plot(pX)
plt.show()