波束图（beam pattern）的python和matlab实现
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目录
1、什么是波束图
2、波束图的原理
3、波束图的实现
1、什么是波束图
通过波束图可以知晓哪个方向的信号被增强，哪个方向的信号被抑制。
2、波束图的原理
声源到各麦克风的时间是不一样的，存在时间差，以mic1为参考点，mic2和micM均会提前，提前的时间为
,其中
。
假设声波波长
，频率为
，相位差为
，其中
设定期望阵列流形矢量为
其它方向阵列流形矢量
，各方向的波束响应
可以用波束图来描述，
3、波束图的实现
clear; close all; clc;
c=340;
f=1000;
lambda=c/f; % wave length
k=2*pi/lambda;
d=lambda/2; % mic distance
M=10;
theta_d = 80*pi/180; % angle of incidence
theta_angle = 0:0.1:180;
theta = theta_angle*pi/180;
psi = 2*pi*d*cos(theta_d)/lambda; % phase difference
Wc = exp(-1i*psi*[0:M-1])/M;
B = zeros(size(theta));
%% compute beam pattern
for i=1:length(theta)
psi2 = 2*pi*d*cos(theta(i))/lambda;
p = exp(1i * psi2 * [0:M-1]);
B(i) = conj(Wc)*p';
end
B_db = 20*log10(B);
limit_dB = -50;
index = B_db 波束图
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numba
def abs2(x):
return x.real**2 + x.imag**2
def linear_mic_array(
num = 4, # number of array elements
spacing = 0.2, # element separation in metres
Is3D = True,
):
PX = np.arange(num) * spacing # microphone position
if Is3D:
return np.array([PX, np.zeros(len(PX)), np.zeros(len(PX))])
else:
return np.array([PX, np.zeros(len(PX))])
def calc_array_manifold_vector_from_point(mic_layout, source_from_mic, nfft=512, sampling_rate = 16000,
c = 343, attn=False, ff=True):
p = np.array(source_from_mic)
if p.ndim == 1:
p = source_from_mic[:, np.newaxis]
frequencies = np.linspace(0, sampling_rate//2, nfft//2 + 1)
omega = 2 * np.pi * frequencies
if ff:
p /= np.linalg.norm(p)
D = -1 * np.einsum("im, i->m", mic_layout, p.squeeze())
D -= D.min()
else:
D = np.sqrt(((mic_layout - source_from_mic) ** 2).sum(0))
phase = np.exp(np.einsum("k,m->km", -1j*omega/c, D))
if attn:
return 1.0/(4*np.pi)/D*phase
else:
return phase
def get_look_direction_loc(deg_phi, deg_theta):
phi = np.deg2rad(deg_phi)
theta = np.deg2rad(deg_theta)
return np.array([[
np.cos(phi) * np.cos(theta),
np.cos(phi) * np.sin(theta),
np.sin(phi),
]]).T
def cal_delay_and_sum_weights(mic_layout, source_from_mic, nfft=512, sampling_rate=16000,
c=343, attn=False, ff=True):
a = calc_array_manifold_vector_from_point(mic_layout, source_from_mic, nfft, sampling_rate, c, attn, ff)
return a/np.einsum("km,km->k", a.conj(), a)[:,None]
def plot_freq_resp(
w,
nfft = 512,
angle_resolution = 500,
mic_layout = linear_mic_array(),
sampling_rate = 16000,
speedSound = 434,
plt3D = False,
vmin = -50,
vmax = None,
):
n_freq, n_ch = w.shape
if n_freq != nfft // 2+1:
raise ValueError("invalid n_freq")
if n_ch != mic_layout.shape[1]:
raise ValueError("invalid n_ch")
freqs = np.linspace(0, sampling_rate//2, nfft//2+1)
angles = np.linspace(0, 180, angle_resolution)
angleRads = np.deg2rad(angles)
loc = np.array([[np.cos(theta), np.sin(theta), 0] for theta in angleRads]).T
phases = np.einsum('k,ai, am->kim', 2j * np.pi * freqs/speedSound, loc, mic_layout)
psi = np.einsum('km,kim->ki', w.conj(), np.exp(phases))
outputs = np.sqrt(abs2(psi))
logOutputs = 20*np.log10(outputs)
if vmin is not None:
logOutputs = np.maximum(logOutputs, vmin)
if vmax is not None:
logOutputs = np.minimum(logOutputs, vmax)
plt.figure(figsize=(10,8))
if plt3D:
ax = plt.subplot(projection='3d')
surf = ax.plot_surface(*np.meshgrid(angles, freqs), logOutputs, rstride=1, cstride=1, cmap='hsv', linewidth=0.3)
#ax.get_proj = lambda: np.dot(Axes3D.get_proj(ax), np.diag([1.5, 1.5, 0.3, 1]))
ax.view_init(elev=30, azim=-45)
else:
ax = plt.subplot()
surf = ax.pcolormesh(*np.meshgrid(angles, freqs), logOutputs, cmap="inferno", shading='gouraud')
plt.colorbar(surf)
plt.xlabel("Arrival Angle[degrees]")
plt.ylabel("Frequency[Hz]")
plt.title("Frequency Response")
plt.show()
def plot_ds_freq_resp(
nfft = 512, # Number of fft
angle_resolution = 500, # Number of angle points to calculate
mic_layout = linear_mic_array(),
sampling_rate = 16000, # Hz
speedSound = 343.0, # m/s
plt3D = False,
):
freqs = np.linspace(0, sampling_rate // 2, nfft // 2 + 1)
angles = np.linspace(0, 180, angle_resolution)
angleRads = np.deg2rad(angles)
loc = np.array([[np.cos(theta), np.sin(theta), 0] for theta in angleRads]).T
phases = np.einsum('k,ai,am->kim', 2j * np.pi * freqs / speedSound, loc, mic_layout)
outputs = np.sqrt(abs2(np.exp(phases).sum(-1))) / mic_layout.shape[1]
logOutputs = np.maximum(20 * np.log10(outputs), -50)
plt.figure(figsize=(10, 8))
if plt3D:
ax = plt.subplot(projection='3d')
surf = ax.plot_surface(*np.meshgrid(angles, freqs), logOutputs, rstride=1, cstride=1, cmap='hsv', linewidth=0.3)
# ax.get_proj = lambda: np.dot(Axes3D.get_proj(ax), np.diag([1.5, 1.5, 0.3, 1]))
ax.view_init(elev=30, azim=-45)
else:
ax = plt.subplot()
surf = ax.pcolormesh(*np.meshgrid(angles, freqs), logOutputs, cmap="inferno", shading='gouraud')
plt.colorbar(surf)
plt.xlabel("Arrival Angle[degrees]")
plt.ylabel("Frequency[Hz]")
plt.title("Frequency Response")
plt.show()
if __name__ == '__main__':
plot_ds_freq_resp()
plot_ds_freq_resp(plt3D=True)
参考文章
1、https://www.funcwj.cn/2018/05/12/beampattern-and-fixed-beamformer/
2、https://memotut.com/en/afc3ee9a8942b41569bf/
3、波束成形(Beamforming)的数学推导(一)---从发射端看 - 知乎
4、常规与MVDR波束形成对比—麦克风阵列系列（一） - 知乎
5、Fundamentals of Signal Enhancement and Array Signal Processing, Jacob Benesty, Israel Cohen, Jingdong Chen
6、优化阵列信号处理, 鄢社峰
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