LaTeXiTが動かなくなってしまった...

M1チップ搭載Macでは，MacTexを入れてもLaTexiTのパス設定が上手くいっていないのか，動きません．M1チップ以前のMacでは，勝手に設定されていたように思うのですが，今回で導入した際には設定が出来ていませんでした．

設定を行った（パスを通した）際のメモです．

もしかして，上記のように設定しても動かない場合はパスが間違っていると思われますので，以下のようにしてターミナルから調べてみてください．

% where gs

/opt/homebrew/bin/gs

Python による実開口レーダーの実装

 

まずは，ライブラリをインポートします．

import numpy as np

import matplotlib.pyplot as plt

from tqdm import tqdm

続いて，周波数，時間，波長，サンプル数などのパラメータを設定します．

# Examples Parameters

freqency = 2 * (1E3 ** 2)

time_sec = 1E-8

wavelength = 0.025

num_sample = 1024 * 3

print(f'- Frequency: {freqency} Hz \n- Time: {time_sec} s \n- Wavelength: {wavelength} m')

times = np.linspace(0, time_sec, num=num_sample)

描画してみます．

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Time')

plt.plot(times, label='Time')

plt.legend(loc='upper left',)

plt.xlabel('Sample')

plt.ylabel('Time (s)')

plt.savefig(f'section_03_001_Time.png', bbox_inches='tight')

plt.show()

図-1 経過時間

図-1の横軸はサンプリング，縦軸はサンプリングした時の経過時間です．続いて，送信する矩形の送信信号を生成して描画してみます．

pulse_sample = 128

pulse_times = np.linspace(0, pulse_sample, pulse_sample)

pulse = np.zeros_like(pulse_times, dtype=np.complex64)

pulse[16:128-16] = 1

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Pulse Pattarn')

plt.plot(pulse.real, label='Real', alpha=0.75)

plt.plot(pulse.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_002_pulse_pattarn.png', bbox_inches='tight')

plt.show()

図-2  送信する信号

図-2の横軸はサンプリング，縦軸は信号の大きさです．この送信信号から受信信号を作成してみます．

points_pulse = [

256,

1024, 1024+256,

1024+1024, 1024+1024+128, 1024+1024+256,

]

respons_signal = np.zeros_like(times, dtype=np.complex64)

for point in points_pulse:

respons_signal[point:point + pulse_sample] += pulse

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Recieved Signal')

plt.plot(respons_signal.real, label='Real', alpha=0.75)

plt.plot(respons_signal.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_003_recieved_signal.png', bbox_inches='tight')

plt.show()

図-3 受信信号

def impulse_response(respons_signal, pulse):

length_pulse, length_signal = len(pulse), len(respons_signal)

# padding

_respons_signal = np.zeros((length_pulse+length_signal), dtype=np.complex64)

_respons_signal[:length_signal] = respons_signal

# process the signal corration

corr = np.zeros_like(_respons_signal)

for i in tqdm(range(len(respons_signal))):

corr[i] = np.sum(np.dot(_respons_signal[i:i+length_pulse] ,np.conj(pulse)))

corr = np.abs(corr)

return corr[:length_signal]

impulse_response_signal = impulse_response(respons_signal, pulse)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title('impulse Response')

plt.plot(impulse_response_signal ,)

plt.axhline(80, color='r', linestyle='--', label='Threshold')

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_004_impulse_response.png', bbox_inches='tight')

plt.show()

ノイズ

noises = np.random.normal(0, 1, len(impulse_response_signal))

noises = np.abs(noises)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title('Random Noise')

plt.plot(noises,)

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_005_noise.png', bbox_inches='tight')

plt.show()

impulse_response_signal_noise = impulse_response(respons_signal+ noises, pulse)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title('impulse Response with Noise')

plt.plot(impulse_response_signal_noise,)

plt.axhline(150, color='r', linestyle='--', label='Threshold')

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_006_impulse_response_with_noise.png', bbox_inches='tight')

plt.show()

noises = np.random.normal(0, 1, len(impulse_response_signal))

noises = np.abs(noises)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title('Random Noise')

plt.plot(noises,)

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_005_noise.png', bbox_inches='tight')

plt.show()

impulse_response_signal_noise = impulse_response(respons_signal+ noises, pulse)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title('impulse Response with Noise')

plt.plot(impulse_response_signal_noise,)

plt.axhline(150, color='r', linestyle='--', label='Threshold')

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_006_impulse_response_with_noise.png', bbox_inches='tight')

plt.show()

noises = np.random.normal(0, 1, len(impulse_response_signal))

ratio = 0.9 # SNR Ratio

impulse_response_signal_noise = impulse_response(respons_signal *ratio + (1 - ratio) * noises, pulse)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title(f'impulse Response with Noise {1 - ratio:.2f}')

plt.plot(impulse_response_signal_noise,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_007_impulse_response_with_noise_ratio{ratio:.2f}.png', bbox_inches='tight')

plt.show()

noises = np.random.normal(0, 1, len(impulse_response_signal))

ratio = 0.5 # SNR Ratio

impulse_response_signal_noise = impulse_response(respons_signal *ratio + (1 - ratio) * noises, pulse)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title(f'impulse Response with Noise {1 - ratio:.2f}')

plt.plot(impulse_response_signal_noise,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_007_impulse_response_with_noise_ratio{ratio:.2f}.png', bbox_inches='tight')

plt.show()

noises = np.random.normal(0, 1, len(impulse_response_signal))

ratio = 0.1 # SNR Ratio

impulse_response_signal_noise = impulse_response(respons_signal *ratio + (1 - ratio) * noises, pulse)

plt.figure(figsize=(16, 4), dpi=200, facecolor='w', edgecolor='k')

plt.title(f'impulse Response with Noise {1 - ratio:.2f}')

plt.plot(impulse_response_signal_noise,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_007_impulse_response_with_noise_ratio{ratio:.2f}.png', bbox_inches='tight')

plt.show()

Chirp Pulse

ratio = 0.25 # SNR Ratio

pulse_sample = 128

pulse_times = np.linspace(0, pulse_sample, pulse_sample)

pulse = np.exp(2j * np.pi * pulse_times)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Sin Pulse')

plt.plot(pulse.real, label='Real', alpha=0.75)

plt.plot(pulse.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_008_sin_pulse.png', bbox_inches='tight')

plt.show()

respons_signal = np.zeros_like(times, dtype=np.complex64)

for point in points_pulse:

respons_signal[point:point + pulse_sample] += pulse

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Recieved Signal Sin')

plt.plot(respons_signal.real, label='Real', alpha=0.75)

plt.plot(respons_signal.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_009_recieved_signal_sin.png', bbox_inches='tight')

plt.show()

impulse_response_signal = impulse_response(respons_signal, pulse)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('impulse Response')

plt.plot(impulse_response_signal,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_010_impulse_response_sin.png', bbox_inches='tight')

plt.show()

noises = np.random.normal(0, 1, len(impulse_response_signal))

impulse_response_signal_noise = impulse_response(respons_signal *ratio + (1 - ratio) * noises, pulse)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title(f'impulse Response with Noise {1 - ratio:.2f}')

plt.plot(impulse_response_signal_noise,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_011_impulse_response_sin_with_noise_ratio{ratio:.2f}.png', bbox_inches='tight')

plt.show()

pulse_sample = 128

pulse_times = np.linspace(0, pulse_sample, pulse_sample)

pulse = np.exp(1j * np.pi * pulse_times)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Chirp Pulse')

plt.plot(pulse.real, label='Real', alpha=0.75)

plt.plot(pulse.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_012_chirp_pulse.png', bbox_inches='tight')

plt.show()

respons_signal = np.zeros_like(times, dtype=np.complex64)

for point in points_pulse:

respons_signal[point:point + pulse_sample] += pulse

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Recieved Signal Chirp')

plt.plot(respons_signal.real, label='Real', alpha=0.75)

plt.plot(respons_signal.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_013_recieved_signal_chirp.png', bbox_inches='tight')

plt.show()

impulse_response_signal = impulse_response(respons_signal, pulse)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('impulse Response')

plt.plot(impulse_response_signal,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_014_impulse_response_chirp.png', bbox_inches='tight')

plt.show()

impulse_response_signal_noise = impulse_response(respons_signal *ratio + (1 - ratio) * noises, pulse)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title(f'impulse Response with Noise {1 - ratio:.2f}')

plt.plot(impulse_response_signal_noise,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_015_impulse_response_chirp_with_noise_ratio{ratio:.2f}.png', bbox_inches='tight')

plt.show()

respons_signal = np.zeros_like(times, dtype=np.complex64)

for point in points_pulse:

respons_signal[point:point + pulse_sample] += pulse

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('Recieved Signal Chirp')

plt.plot(respons_signal.real, label='Real', alpha=0.75)

plt.plot(respons_signal.imag, label='Imaginary', alpha=0.75)

plt.legend()

plt.xlabel('Sample')

plt.ylabel('Amplitude')

plt.savefig(f'section_03_017_recieved_signal_chirp.png', bbox_inches='tight')

plt.show()

impulse_response_signal = impulse_response(respons_signal, pulse)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title('impulse Response')

plt.plot(impulse_response_signal,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_018_impulse_response_chirp.png', bbox_inches='tight')

plt.show()

impulse_response_signal_noise = impulse_response(respons_signal *ratio + (1 - ratio) * noises, pulse)

plt.figure(figsize=(16, 3), dpi=200, facecolor='w', edgecolor='k')

plt.title(f'impulse Response with Noise {1 - ratio:.2f}')

plt.plot(impulse_response_signal_noise,)

plt.xlabel('Sample')

plt.ylabel('Correlation')

plt.savefig(f'section_03_019_impulse_response_chirp_with_noise_ratio{ratio:.2f}.png', bbox_inches='tight')

plt.show()

Appendix

Numpy implimentation `impulse Response`

def impulse_response(respons_signal, pulse):

return np.convolve(respons_signal, pulse)

Movie

import os

import warnings, gc

from scipy.signal import chirp, spectrogram, correlate

warnings.simplefilter('ignore')

skip_step = 10

t_sec = 60

t_sec_chirp_start = 30

tau_p_sec = 10

fs_sampling = 1E2

K_r = 1.01

noise_top = 1.

SNR = 0.25

assert 0 < SNR < 1

tau = np.arange(-tau_p_sec/2, tau_p_sec/2, 1/fs_sampling)

phase = 1j*np.pi*K_r*tau**2

ra_chirp_temp = np.exp(phase)

slice_chirp = slice(int(t_sec_chirp_start*fs_sampling), int((t_sec_chirp_start+tau_p_sec)*fs_sampling))

T = np.arange(0, t_sec, 1/fs_sampling)

noise = np.random.uniform(low=-noise_top, high=noise_top, size=T.shape)

reflect_noise = noise.copy()

noise_tau_sec = np.random.uniform(low=-noise_top, high=noise_top, size=tau.shape)

corr_T = []

for i, t1 in tqdm(enumerate(T), total=len(T)):

# shortcut plot

if i % skip_step != 0:

continue

# pulse width time

if t1 > T.max() - tau_p_sec:

break

_slice_chirp = slice(round(t1*fs_sampling), round((t1+tau_p_sec)*fs_sampling))

# reference pulse

kernel = np.zeros(T.shape)

kernel[_slice_chirp] = ra_chirp_temp

# correlation

corr = correlate(kernel.real[_slice_chirp], reflect_noise.real[_slice_chirp], mode='same')

corr_center = corr[int(len(corr)/2)]

if i == 0:

corr_T.append(corr_center)

else:

corr_T.extend([corr_center]*skip_step)

# visualization

plt.figure(figsize=(26, 18), dpi=200, facecolor='w', edgecolor='k')

plt.subplot(6, 1, 1)

plt.grid(True,)

plt.plot(tau, corr, alpha=0.8, color='green', label='Correlation Value')

plt.scatter([0], corr_center, label='Target Center', color='orange')

plt.scatter([tau[np.argmax(corr)]], corr[np.argmax(corr)], label='Maximum Peak Correlation', color='black')

plt.legend()

plt.ylim(-50, 400)

plt.title(f'Local Correlation Time:{t1: .2f}')

plt.ylabel('Correlation')

plt.subplot(6, 1, 2)

plt.plot(T, kernel, color='orange', label='Kernel')

plt.grid(True,)

plt.title('Reference Pulse Chirp Signal')

plt.xlabel('Time axis [sec]')

plt.ylabel('Amplitude')

reflect_noise = noise.copy()

reflect_noise[slice_chirp] = ((1 - SNR)*ra_chirp_temp + SNR*noise_tau_sec)

plt.subplot(6, 1, 3)

plt.grid(True,)

plt.plot(T, reflect_noise.real, 'c')

plt.xlabel('Time axis [sec]')

plt.ylabel('Real')

plt.subplot(6, 1, 4)

plt.grid(True,)

plt.plot(T, kernel, 'navy', alpha=0.7, label='Kernel')

plt.plot(T, reflect_noise.real, 'magenta', alpha=0.4, label='Noise Reflect')

plt.legend()

plt.ylabel('Amplitude')

plt.subplot(6, 1, 5)

plt.grid(True,)

plt.plot(range(i+1), np.array(corr_T), 'b', alpha=0.8, label='Peak')

plt.scatter([i], corr_center, label='Center Correlation', color='red')

plt.legend()

plt.xlim(0, int(len(T) - len(ra_chirp_temp)))

plt.ylim(-50, 400)

plt.title('Correlation Value')

plt.xlabel('Sample [n]')

plt.ylabel('Correlation')

f, t, Sxx = spectrogram(reflect_noise, fs_sampling)

plt.subplot(6, 1, 6)

plt.grid(True,)

plt.pcolormesh(t, f, Sxx, shading='gouraud', label='spectrum', vmax=1e-1, cmap='inferno')

plt.vlines(t1 + round(tau_p_sec/2), ymin=0, ymax=50, color='red', label='position')

plt.legend()

plt.ylabel('Spectrum')

plt.xlabel('Time [sec]')

plt.tight_layout()

os.makedirs('frames', exist_ok=True)

plt.savefig(os.path.join('frames', f'impulse_respose_frame_{str(i).zfill(5)}.png'), format='png')

plt.clf()

plt.close()