基于傅里叶变换的频域滤波
____________________IFFT_____________________
| |
v |
高能信号\ FFT 频域滤波 |
>含噪信号----->含噪频谱---------->高能频谱
低能噪声/
代码：
# -*- coding: utf-8 -*-
from __future__ import unicode_literals
import numpy as np
import numpy.fft as nf
import scipy.io.wavfile as wf
import matplotlib.pyplot as mp
sample_rate, noised_sigs = wf.read(
'../../data/noised.wav')
noised_sigs = noised_sigs / 2 ** 15
times = np.arange(len(noised_sigs)) / sample_rate
freqs = nf.fftfreq(times.size, 1 / sample_rate)
noised_ffts = nf.fft(noised_sigs)
noised_pows = np.abs(noised_ffts)
fund_freq = np.abs(freqs[noised_pows.argmax()])
print(fund_freq)
noised_indices = np.where(np.abs(freqs) != fund_freq)
filter_ffts = noised_ffts.copy()
filter_ffts[noised_indices] = 0
filter_pows = np.abs(filter_ffts)
filter_sigs = nf.ifft(filter_ffts).real
wf.write('../../data/filter.wav', sample_rate,
(filter_sigs * 2 ** 15).astype(np.int16))
mp.figure('Filter', facecolor='lightgray')
mp.subplot(221)
mp.title('Time Domain', fontsize=16)
mp.ylabel('Signal', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(times[:178], noised_sigs[:178],
c='orangered', label='Noised')
mp.legend()
mp.subplot(222)
mp.title('Frequency Domain', fontsize=16)
mp.ylabel('Power', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.semilogy(freqs[freqs >= 0],
noised_pows[freqs >= 0], c='limegreen',
label='Noised')
mp.legend()
mp.subplot(223)
mp.xlabel('Time', fontsize=12)
mp.ylabel('Signal', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(times[:178], filter_sigs[:178],
c='hotpink', label='Filter')
mp.legend()
mp.subplot(224)
mp.xlabel('Frequency', fontsize=12)
mp.ylabel('Power', fontsize=12)
mp.tick_params(labelsize=10)
mp.grid(linestyle=':')
mp.plot(freqs[freqs >= 0], filter_pows[freqs >= 0],
c='dodgerblue', label='Filter')
mp.legend()
mp.tight_layout()
mp.show()3.随机数模块(random)
- 生成服从特定统计规律的随机数序列
- 二项分布：np.random.binomial(n, p, size)
产生size个随机数，每个随机数来自n次尝试中的成功次数，其中每次尝试成功的概率为p。
猜硬币游戏：初始筹码1000，每轮猜9次，猜对5次及5次以上为赢，筹码加1，否则为输，筹码减1，求10000轮的过程中手中筹码的变化。
代码：bi.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np import matplotlib.pyplot as mp outcomes = np.random.binomial(9, 0.5, 10000) chips = [1000] for outcome in outcomes: if outcome >= 5: chips.append(chips[-1] + 1) else: chips.append(chips[-1] - 1) chips = np.array(chips) mp.figure('Binomial Distribution', facecolor='lightgray') mp.title('Binomial Distribution', fontsize=20) mp.xlabel('Round', fontsize=14) mp.ylabel('Chip', fontsize=14) mp.tick_params(labelsize=12) mp.grid(linestyle=':') o, h, l, c = 0, chips.argmax(), chips.argmin(), \ chips.size - 1 if chips[o] < chips[c]: color = 'orangered' elif chips[c] < chips[o]: color = 'limegreen' else: color = 'dodgerblue' mp.plot(chips, c=color, label='Chip') mp.axhline(y=chips[o], linestyle='--', color='deepskyblue', linewidth=1) mp.axhline(y=chips[h], linestyle='--', color='crimson', linewidth=1) mp.axhline(y=chips[l], linestyle='--', color='seagreen', linewidth=1) mp.axhline(y=chips[c], linestyle='--', color='orange', linewidth=1) mp.legend() mp.show()
1 2 3 4 5 6
5 1 2 0 0 2
0 2 0 6 1 1
... - 超几何分布：np.random.hypergeometric(ngood,
nbad, nsample, size)
产生size个随机数，每个随机数来自随机抽取nsample个样本中
好样本的个数，总样本由ngood个好样本和nbad个坏样本组成
模球游戏：将25个好球和1个坏球放在一起，每次模3个球，全为好球加1分，只要摸到了坏球减6分，求100轮的过程中分值的变化。
代码：hyper.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np import matplotlib.pyplot as mp outcomes = np.random.hypergeometric(25, 1, 3, 100) scores = [0] for outcome in outcomes: if outcome == 3: scores.append(scores[-1] + 1) else: scores.append(scores[-1] - 6) scores = np.array(scores) mp.figure('Hypergeometric Distribution', facecolor='lightgray') mp.title('Hypergeometric Distribution', fontsize=20) mp.xlabel('Round', fontsize=14) mp.ylabel('Score', fontsize=14) mp.tick_params(labelsize=12) mp.grid(linestyle=':') o, h, l, c = 0, scores.argmax(), scores.argmin(), \ scores.size - 1 if scores[o] < scores[c]: color = 'orangered' elif scores[c] < scores[o]: color = 'limegreen' else: color = 'dodgerblue' mp.plot(scores, c=color, label='Score') mp.axhline(y=scores[o], linestyle='--', color='deepskyblue', linewidth=1) mp.axhline(y=scores[h], linestyle='--', color='crimson', linewidth=1) mp.axhline(y=scores[l], linestyle='--', color='seagreen', linewidth=1) mp.axhline(y=scores[c], linestyle='--', color='orange', linewidth=1) mp.legend() mp.show() - 标准正态分布：np.random.norm(size)
产生size个随机数，服从标准正态(平均值=0, 标准差=1)分布。
2
x
- ---
2
e
标准正态分布概率密度 = --------
_____
\/ 2 pi
代码：norm.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np import matplotlib.pyplot as mp samples = np.random.normal(size=10000) mp.figure('Normal Distribution', facecolor='lightgray') mp.title('Normal Distribution', fontsize=20) mp.xlabel('Sample', fontsize=14) mp.ylabel('Occurrence', fontsize=14) mp.tick_params(labelsize=12) mp.grid(axis='y', linestyle=':') bins = mp.hist(samples, 100, normed=True, edgecolor='steelblue', facecolor='deepskyblue', label='Normal')[1] probs = np.exp(-bins ** 2 / 2) / np.sqrt(2 * np.pi) mp.plot(bins, probs, 'o-', c='orangered', label='Probability') mp.legend() mp.show()
八.杂项功能
1.排序
- 联合间接排序：numpy.lexsort((参考序列, 待排序列))
->有序索引
[张三 李四 王五 赵六]
[70 60 80 70]<-[30 20 30 20]
numpy.sort_complex(复数数组)
按照实部的升序排列，对于实部相同的元素，参考虚部的升序，直接返回排序后的结果数组。
numpy.searchsorted(有序序列, 待插序列)
->位置数组，表示将待插序列中的元素插入到有序序列中的哪些位置处，结果依然有序
numpy.insert(被插序列, 位置序列, 待插序列)
->将待插序列中的元素，按照位置序列中的位置，插入到被插序列中，返回插入后的结果
代码：sort.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np ages = np.array([30, 20, 30, 20]) scores = np.array([70, 60, 80, 70]) names = np.array(['zhangsan', 'lisi', 'wangwu', 'zhaoliu']) # 按照成绩的升序打印姓名，成 # 绩相同的按照年龄的升序排列 print(np.take(names, np.lexsort((ages, scores)))) compleies = scores + ages * 1j print(compleies) sorted_compleies = np.sort_complex(compleies) print(sorted_compleies) # 0 1 2 3 4 5 6 a = np.array([1, 2, 4, 5, 6, 8, 9]) b = np.array([7, 3]) c = np.searchsorted(a, b) print(c) d = np.insert(a, c, b) print(d)
2.插值
- import scipy.interpolate as si
si.interp1d(离散水平坐标, 离散垂直坐标,
kind=插值算法(缺省为线性插值))->插值器
插值器(水平坐标)->垂直坐标
代码：inter.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np import scipy.interpolate as si import matplotlib.pyplot as mp min_x, max_x = -2.5, 2.5 con_x = np.linspace(min_x, max_x, 1001) con_y = np.sinc(con_x) dis_x = np.linspace(min_x, max_x, 11) dis_y = np.sinc(dis_x) # 线性插值 linear = si.interp1d(dis_x, dis_y) lin_x = np.linspace(min_x, max_x, 51) lin_y = linear(lin_x) # 三次样条插值 cubic = si.interp1d(dis_x, dis_y, kind='cubic') cub_x = np.linspace(min_x, max_x, 51) cub_y = cubic(cub_x) mp.figure('Interpolation', facecolor='lightgray') mp.subplot(221) mp.title('Continuous', fontsize=16) mp.ylabel('y', fontsize=12) mp.tick_params(labelsize=10) mp.grid(linestyle=':') mp.plot(con_x, con_y, c='hotpink', label='Continuous') mp.legend() mp.subplot(222) mp.title('Discrete', fontsize=16) mp.tick_params(labelsize=10) mp.grid(linestyle=':') mp.scatter(dis_x, dis_y, c='orangered', s=80, label='Discrete') mp.legend() mp.subplot(223) mp.title('Linear', fontsize=16) mp.xlabel('x', fontsize=12) mp.ylabel('y', fontsize=12) mp.tick_params(labelsize=10) mp.grid(linestyle=':') mp.plot(lin_x, lin_y, 'o-', c='limegreen', label='Linear') mp.scatter(dis_x, dis_y, c='orangered', s=80, zorder=3) mp.legend() mp.subplot(224) mp.title('Cubic', fontsize=16) mp.xlabel('x', fontsize=12) mp.tick_params(labelsize=10) mp.grid(linestyle=':') mp.plot(cub_x, cub_y, 'o-', c='dodgerblue', label='Cubic') mp.scatter(dis_x, dis_y, c='orangered', s=80, zorder=3) mp.legend() mp.tight_layout() mp.show()
3.积分
- import scipy.integrate as si
si.quad(积分函数, 积分下限, 积分上限)->积分值, 最大误差
代码：integ.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np import scipy.integrate as si import matplotlib.pyplot as mp import matplotlib.patches as mc def f(x): return 2 * x ** 2 + 3 * x + 4 a, b = -5, 5 x1 = np.linspace(a, b, 1001) y1 = f(x1) area = si.quad(f, a, b)[0] print(area) n = 50 x2 = np.linspace(a, b, n + 1) y2 = f(x2) area = 0 for i in range(n): area += (y2[i] + y2[i + 1]) * (x2[i + 1] - x2[i]) / 2 print(area) mp.figure('Integral', facecolor='lightgray') mp.title('Integral', fontsize=20) mp.xlabel('x', fontsize=14) mp.ylabel('y', fontsize=14) mp.tick_params(labelsize=10) mp.grid(linestyle=':') mp.plot(x1, y1, c='orangered', linewidth=6, label=r'$y=2x^2+3x+4$', zorder=0) for i in range(n): mp.gca().add_patch(mc.Polygon([ [x2[i], 0], [x2[i], y2[i]], [x2[i + 1], y2[i + 1]], [x2[i + 1], 0]], fc='deepskyblue', ec='dodgerblue', alpha=0.5)) mp.legend() mp.show()
4.图像
- scipy.ndimage中提供了一些简单的图像处理，如高斯模糊、任意角度旋转、边缘识别等功能。
代码：image.py# -*- coding: utf-8 -*- from __future__ import unicode_literals import numpy as np import scipy.misc as sm import scipy.ndimage as sn import matplotlib.pyplot as mp original = sm.imread('../../data/lily.jpg', True) median = sn.median_filter(original, (31, 31)) rotate = sn.rotate(original, 45) prewitt = sn.prewitt(original) mp.figure('Image', facecolor='lightgray') mp.subplot(221) mp.title('Original', fontsize=16) mp.axis('off') mp.imshow(original, cmap='gray') mp.subplot(222) mp.title('Median', fontsize=16) mp.axis('off') mp.imshow(median, cmap='gray') mp.subplot(223) mp.title('Rotate', fontsize=16) mp.axis('off') mp.imshow(rotate, cmap='gray') mp.subplot(224) mp.title('Prewitt', fontsize=16) mp.axis('off') mp.imshow(prewitt, cmap='gray') mp.tight_layout() mp.show()
5.金融
代码：fin.py
# -*- coding: utf-8 -*-
from __future__ import unicode_literals
import numpy as np
# 终值 = fv(利率, 期数, 每期支付, 现值)
# 将1000元以1%的年利率存入银行5年，每年加存100元，
# 到期后本息合计多少钱？
fv = np.fv(0.01, 5, -100, -1000)
print(round(fv, 2))
# 现值 = pv(利率, 期数, 每期支付, 终值)
# 将多少钱以1%的年利率存入银行5年，每年加存100元，
# 到期后本息合计fv元？
pv = np.pv(0.01, 5, -100, fv)
print(pv)
# 净现值 = npv(利率, 现金流)
# 将1000元以1%的年利率存入银行5年，每年加存100元，
# 相当于一次性存入多少钱？
npv = np.npv(0.01, [
-1000, -100, -100, -100, -100, -100])
print(round(npv, 2))
fv = np.fv(0.01, 5, 0, npv)
print(round(fv, 2))
# 内部收益率 = irr(现金流)
# 将1000元存入银行5年，以后逐年提现100元、200元、
# 300元、400元、500元，银行利率达到多少，可在最后
# 一次提现后偿清全部本息，即净现值为0元？
irr = np.irr([-1000, 100, 200, 300, 400, 500])
print(round(irr, 2))
npv = np.npv(irr, [-1000, 100, 200, 300, 400, 500])
print(npv)
# 每期支付 = pmt(利率, 期数, 现值)
# 以1%的年利率从银行贷款1000元，分5年还清，
# 平均每年还多少钱？
pmt = np.pmt(0.01, 5, 1000)
print(round(pmt, 2))
# 期数 = nper(利率, 每期支付, 现值)
# 以1%的年利率从银行贷款1000元，平均每年还pmt元，
# 多少年还清？
nper = np.nper(0.01, pmt, 1000)
print(int(nper))
# 利率 = rate(期数, 每期支付, 现值, 终值)
# 从银行贷款1000元，平均每年还pmt元，nper年还清，
# 年利率多少？
rate = np.rate(nper, pmt, 1000, 0)
print(round(rate, 2))