import numpy import numpy as np import matplotlib.pyplot as plt def h(w): """ 原函数的傅里叶变换 :param w : omega :return: omega 变换后的傅里叶变换 """ return np.sin(np.pi * w) / (w * (1 - w ** 2)) def f(t: np.arange): """ 返回参数为t的原函数 """ return 0.5 * (1 + np.cos(t)) * (np.heaviside(t + np.pi, 0) - np.heaviside(t - np.pi, 0)) def expr1_1_1(): t = np.arange(-np.pi, np.pi, 0.01) f0 = f(t) # f = 0.5 * (1 + np.cos(t)) * (np.heaviside(t + np.pi, 0) - np.heaviside(t - np.pi, 0)) plt.plot(t, f0) plt.grid() plt.xlabel("time[s]") plt.ylabel("f(t)") plt.title("Figure 1.1.1: Origin Function Time Domain Diagram") plt.show() w = np.linspace(-20, 20, 1000) F = h(w) # expr1_1_1() def expr1_1_2(): # 绘制f(t)的频谱 w = np.linspace(-20, 20, 1000) h0 = h(w) # 幅度谱 plt.plot(w, abs(h0)) plt.grid() plt.xlabel("w") plt.ylabel("amplitude") plt.title("Figure 1.1.2: Origin Function Amplitude Spectrum") plt.show() # 相位谱 plt.plot(w, numpy.angle(h0)) plt.grid() plt.xlabel("w") plt.ylabel("phase") plt.title("Figure 1.1.3: Origin Function Phase Spectrum") plt.show() # expr1_1_2() def expr1_2_1(): # 0.5s为间隔 t = np.arange(-np.pi, np.pi, 0.5) # 取样间隔设置为0.5 f0 = f(t) # 时域波形 plt.stem(t, f0) plt.grid() plt.xlabel("time[s]") plt.ylabel("f(t)") plt.title("Figure 1.2.1: Sampling Function(0.5s) Time Domain Diagram") plt.show() w = np.linspace(-20, 20, 1000) F = h(w) for i in range(1, 3): F += h(w + i * 4 * np.pi) F += h(w - i * 4 * np.pi) # 幅度谱 plt.plot(w, abs(F * 2)) plt.grid() plt.xlabel("w") plt.ylabel("Amplitude") plt.title("Figure 1.2.2: Sampling Function(0.5s) Amplitude Spectrum") plt.show() # 相位谱 plt.plot(w, np.angle(F * 2)) plt.grid() plt.xlabel("w") plt.ylabel("phase") plt.title("Figure 1.2.3: Sampling Function(0.5s) Phase Spectrum") plt.show() expr1_2_1() def expr1_2_2(): # 1s为间隔 t = np.arange(-np.pi, np.pi, 1) f0 = f(t) plt.stem(t, f0) plt.grid() plt.xlabel("time[s]") plt.ylabel("f(t)") plt.title("Figure 1.2.4: Sampling Function(1s) Time Domain Diagram") plt.show() w = np.linspace(-20, 20, 1000) F = h(w) for i in range(1, 5): F += h(w + i * 2 * np.pi) F += h(w - i * 2 * np.pi) # 幅度谱 plt.plot(w, abs(F)) plt.grid() plt.xlabel("w") plt.ylabel("Amplitude") plt.title("Figure 1.2.5: Sampling Function(1s) Amplitude Spectrum") plt.show() # 相位谱 plt.plot(w, np.angle(F)) plt.grid() plt.xlabel("w") plt.ylabel("Phase") plt.title("Figure 1.2.6: Sampling Function(1s) Phase Spectrum") plt.show() # expr1_2_2() def expr1_2_3(): # 2s为间隔 t = np.arange(-np.pi, np.pi, 2) f0 = f(t) plt.stem(t, f0) plt.grid() plt.xlabel("time[s]") plt.ylabel("f(t)") plt.title("Figure 1.2.7: Sampling Function(2s) Time Domain Diagram") plt.show() w = np.linspace(-20, 20, 1000) F = h(w) for i in range(1, 10): F += h(w + i * np.pi) F += h(w - i * np.pi) # 幅度谱 plt.plot(w, abs(F / 2)) plt.grid() plt.xlabel("w") plt.ylabel("Amplitude") plt.title("Figure 1.2.8: Sampling Function(2s) Amplitude Spectrum") plt.show() # 相位谱 plt.plot(w, np.angle(F / 2)) plt.grid() plt.xlabel("w") plt.ylabel("Phase") plt.title("Figure 1.2.9: Sampling Function(2s) Phase Spectrum") plt.show() # expr1_2_3() def expr1_3(): # 截取前三个周期 t = np.arange(-np.pi, np.pi, 0.01) f1 = f(t) # h1 = h() w = np.linspace(-20, 20, 1000) # F0 = h(w) # F0为0.5s抽样函数的频谱，初始赋值为原函数的傅里叶变换 F1 = h(w) # F1为1s抽样函数的频谱，初始赋值为原函数的傅里叶变换 # F2 = h(w) # for i in range(1, 3): # F0 += h(w + i * 4 * np.pi) # F0 += h(w - i * 4 * np.pi) for i in range(1, 5): F1 += h(w + i * 2 * np.pi) F1 += h(w - i * 2 * np.pi) # for i in range(1, 10): # omega从1递增到3叠加 # F2 += h(w + i * np.pi) # F2 += h(w - i * np.pi) plt.plot(t, f1) # plt.plot(w, abs(F0 * 2)) plt.plot(w, abs(F1)) # plt.plot(w, abs(F2 / 2)) plt.grid() plt.xlabel("w") plt.ylabel("Amplitude") plt.title("Figure 1.3: Sampling Function(0.5s,1s,2s) Amplitude Spectrum") plt.show() # expr1_3() def expr2_1(): # 取样间隔为0.5s t0 = np.arange(-np.pi, np.pi, 0.5) # 自变量取值范围，间隔0.5 F0 = f(0) * np.sinc(2.4 * t0) # 重建信号初始值 for i in range(1, 10): # K从1递增到10叠加 F0 += f(i * 0.5) * np.sinc(2.4 * (t0 - i * 0.5)) F0 += f(i * 0.5) * np.sinc(2.4 * (t0 + i * 0.5)) plt.stem(t0, f(t0), label='Origin', markerfmt='gD') # 原函数 plt.grid() plt.xlabel("time[s]") plt.ylabel("f") plt.title("Figure 2.1.1: Sampling Function(0.5s) Origin") plt.legend() plt.show() # 取样间隔为1s t1 = np.arange(-np.pi, np.pi, 1) # 自变量取值范围，间隔1 F1 = f(0) * np.sinc(2.4 * t1) # 重建信号初始值 for i in range(1, 10): # K从1递增到10叠加 F1 += f(i * 1) * np.sinc(2.4 * (t1 - i * 1)) F1 += f(i * 1) * np.sinc(2.4 * (t1 + i * 1)) plt.stem(t1, f(t1), label='Origin', markerfmt='gD') # 原函数 plt.grid() plt.xlabel("time[s]") plt.ylabel("f") plt.title("Figure 2.1.2: Sampling Function(1s) Origin") plt.legend() plt.show() # 取样间隔为2s t2 = np.arange(-np.pi, np.pi, 2) # 自变量取值范围，间隔1 F2 = f(0) * np.sinc(2.4 * t2) # 重建信号初始值 for i in range(1, 10): # K从1递增到10叠加 F2 += f(i * 2) * np.sinc(2.4 * (t2 - i * 2)) F2 += f(i * 2) * np.sinc(2.4 * (t2 + i * 2)) plt.stem(t2, f(t2), label='Origin', markerfmt='gD') # 原函数 plt.grid() plt.xlabel("time[s]") plt.ylabel("f") plt.title("Figure 2.1.3: Sampling Function(1s) Origin") plt.legend() plt.show() # expr2_1() def expr2_2(): # 取样间隔为0.5s t0 = np.arange(-np.pi, np.pi, 0.5) # 自变量取值范围，间隔0.5 F0 = f(0) * np.sinc(2.4 * t0) # 重建信号初始值 for i in range(1, 10): # K从1递增到10叠加 F0 += f(i * 0.5) * np.sinc(2.4 * (t0 - i * 0.5)) F0 += f(i * 0.5) * np.sinc(2.4 * (t0 + i * 0.5)) plt.stem(t0, F0 * 2.4 * 0.5 / np.pi, label='Reconstruction', markerfmt='rD') # 重建函数 plt.stem(t0, f(t0), label='Origin', markerfmt='gD') # 原函数 plt.plot(t0, abs(F0 * 2.4 * 0.5 / np.pi - f(t0)), label='Difference', color='orange') # 差值的绝对值 plt.grid() plt.xlabel("time[s]") plt.ylabel("f") plt.title("Figure 2.1.1: Sampling Function(0.5s) Reconstruction") plt.legend() plt.show() # 取样间隔为1s t1 = np.arange(-np.pi, np.pi, 1) # 自变量取值范围，间隔1 F1 = f(0) * np.sinc(2.4 * t1) # 重建信号初始值 for i in range(1, 10): # K从1递增到10叠加 F1 += f(i * 1) * np.sinc(2.4 * (t1 - i * 1)) F1 += f(i * 1) * np.sinc(2.4 * (t1 + i * 1)) plt.stem(t1, F1 * 2.4 * 1 / np.pi, label='Reconstruction', markerfmt='rD') # 重建函数 plt.stem(t1, f(t1), label='Origin', markerfmt='gD') # 原函数 plt.plot(t1, abs(F1 * 2.4 * 1 / np.pi - f(t1)), label='Difference', color='orange') # 差值的绝对值 plt.grid() plt.xlabel("time[s]") plt.ylabel("f") plt.title("Figure 2.1.2: Sampling Function(1s) Reconstruction") plt.legend() plt.show() # 取样间隔为2s t2 = np.arange(-np.pi, np.pi, 2) # 自变量取值范围，间隔1 F2 = f(0) * np.sinc(2.4 * t2) # 重建信号初始值 for i in range(1, 10): # K从1递增到10叠加 F2 += f(i * 2) * np.sinc(2.4 * (t2 - i * 2)) F2 += f(i * 2) * np.sinc(2.4 * (t2 + i * 2)) plt.stem(t2, F2 * 2.4 * 2 / np.pi, label='Reconstruction', markerfmt='rD') # 重建函数 plt.stem(t2, f(t2), label='Origin', markerfmt='gD') # 原函数 plt.plot(t2, abs(F2 * 2.4 * 2 / np.pi - f(t2)), label='Difference', color='orange') # 差值的绝对值 plt.grid() plt.xlabel("time[s]") plt.ylabel("f") plt.title("Figure 2.1.3: Sampling Function(1s) Reconstruction") plt.legend() plt.show() # expr2_2()