做饲料推广哪个网站好,网站空间流量查询,百度关键词挖掘,比较好的网站建设论坛目录取样和取样函数的傅里叶变换取样取样后的函数的傅里叶变换取样定理混叠由取样后的数据重建#xff08;复原#xff09;函数取样和取样函数的傅里叶变换
取样 fˉ(t)f(t)sΔT(t)∑n−∞∞f(t)δ(t−nΔT)(4.27)\bar f(t) f(t)s_{\Delta T}(t) \sum_{n-\infty}^{\infty}…
目录取样和取样函数的傅里叶变换取样取样后的函数的傅里叶变换取样定理混叠由取样后的数据重建复原函数取样和取样函数的傅里叶变换
取样
fˉ(t)f(t)sΔT(t)∑n−∞∞f(t)δ(t−nΔT)(4.27)\bar f(t) f(t)s_{\Delta T}(t) \sum_{n-\infty}^{\infty}f(t) \delta(t - n\Delta T) \tag{4.27}fˉ(t)f(t)sΔT(t)n−∞∑∞f(t)δ(t−nΔT)(4.27) fk∫−∞∞f(t)δ(t−kΔT)dtf(kΔT)(4.28)f_k \int_{-\infty}^{\infty} f(t)\delta(t - k\Delta T) dt f(k\Delta T) \tag{4.28}fk∫−∞∞f(t)δ(t−kΔT)dtf(kΔT)(4.28)
# 取样
x np.arange(-10, 10, 0.01)
y_1 1.5 - np.sin(x)fig plt.figure(figsize(9, 6))
ax_1 setup_axes(fig, 211)
ax_1.plot(x, y_1), ax_1.set_title(f(t), loccenter, y1.05), ax_1.set_ylim([0, 4]), ax_1.set_yticks([]), ax_1.set_xticks([])x_2 x[::50]
y_2 y_1[::50]
ax_2 setup_axes(fig, 212)
ax_2.scatter(x_2, y_2), ax_2.set_title(f_k, loccenter, y1.05), ax_2.set_ylim([0, 4]), ax_2.set_yticks([]), ax_2.set_xticks([])plt.tight_layout()
plt.show()取样后的函数的傅里叶变换
取样后的函数fˉ(t)\bar f(t)fˉ(t)的傅里叶变换Fˉ(μ)\bar F(\mu)Fˉ(μ)是 Fˉ(μ)J{fˉ(t)}J{f(t)sΔT(t)}(F⋆S)(μ)(4.29)\bar F(\mu) \mathfrak{J} \{ \bar f(t) \} \mathfrak{J} \{ f(t) s_{\Delta T}(t)\} (F \star S)(\mu) \tag{4.29}Fˉ(μ)J{fˉ(t)}J{f(t)sΔT(t)}(F⋆S)(μ)(4.29)
S(μ)1ΔT∑n−∞∞δ(μ−nΔT)(4.30)S(\mu) \frac{1}{\Delta T} \sum_{n-\infty}^{\infty} \delta(\mu - \frac{n}{\Delta T}) \tag{4.30}S(μ)ΔT1n−∞∑∞δ(μ−ΔTn)(4.30)
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# 过取样、临界取样和欠取样
x np.arange(0, 6, 0.01)
y 2 - x
y np.where(x 2, y, 0)
x_1 np.concatenate((-x[::-1], x), axis0)
y_1 np.pad(y, (y.shape[0], 0), modereflect)fig plt.figure(figsize(10, 8))
# 带限函数的傅里叶变换
ax_1 setup_axes(fig, 411)
ax_1.plot(x_1, y_1), ax_1.set_title(F(u), loccenter, y1.05), ax_1.set_ylim([0, 3]), ax_1.set_yticks([]), ax_1.set_xticks([])# 过取样
y_2 y_1[600-250:600250]
y_2 np.tile(y_2, 5)
x_2 np.linspace(-6, 6, y_2.shape[0])ax_2 setup_axes(fig, 412)
ax_2.plot(x_2, y_2), ax_2.set_title(F(u), loccenter, y1.05), ax_2.set_ylim([0, 3]), ax_2.set_yticks([]), ax_2.set_xticks([])# 临界取样
y_3 y_1[600-200:600200]
y_3 np.tile(y_3, 5)
x_3 np.linspace(-6, 6, y_3.shape[0])ax_3 setup_axes(fig, 413)
ax_3.plot(x_3, y_3), ax_3.set_title(F(u), loccenter, y1.05), ax_3.set_ylim([0, 3]), ax_3.set_yticks([]), ax_3.set_xticks([])# 欠取样
y_4 y_1[600-200:600200]
y_4 np.where(y_4 1, y_4, 1)
y_4 np.where(y_4 -1, y_4, 1)
y_4 np.tile(y_4, 5)
x_4 np.linspace(-6, 6, y_4.shape[0])ax_4 setup_axes(fig, 414)
ax_4.plot(x_4, y_4), ax_4.set_title(F(u), loccenter, y1.05), ax_4.set_ylim([0, 3]), ax_4.set_yticks([]), ax_4.set_xticks([])plt.tight_layout()
plt.show()取样定理 带限函数 对于以原点为中心的有限区间(带宽)[−μmax,μmax][-\mu_{max}, \mu_{max}][−μmax,μmax]外的频率值傅里叶变换为零的函数 奈奎斯特定理 如果以超过函数最高频率2倍的取样率来得到样本那么连续带限函数就能够完全由其样本集复原
1ΔT2μmax(4.32)\frac{1}{\Delta T} 2 \mu_{max} \tag{4.32}ΔT12μmax(4.32) 低通滤波器 Hμ{ΔT,−μmax≤μ≤μmax0,others(4.33)H{\mu} \begin{cases} \Delta T, -\mu_{max} \leq \mu \leq \mu_{max} \\0, \text{others} \end{cases} \tag{4.33}Hμ{ΔT,0,−μmax≤μ≤μmaxothers(4.33) 滤波器乘以傅里叶变换后的函数 F(μ)H(μ)F~(μ)(4.34)F(\mu) H(\mu)\tilde F(\mu) \tag{4.34}F(μ)H(μ)F~(μ)(4.34) 傅里叶反变换复原f(t)f(t)f(t): f(t)∫−∞∞F(μ)ej2πμtdμ(4.35)f(t) \int_{-\infty}^{\infty} F(\mu) e^{j2\pi\mu t} d\mu \tag{4.35}f(t)∫−∞∞F(μ)ej2πμtdμ(4.35)
# 采样、滤波
x np.arange(0, 6, 0.01)
y 2 - x
y np.where(x 2, y, 0)
x_1 np.concatenate((-x[::-1], x), axis0)
y_1 np.pad(y, (y.shape[0], 0), modereflect)fig plt.figure(figsize(10, 8))# 取样
y_2 y_1[600-250:600250]
y_2 np.tile(y_2, 5)
x_2 np.linspace(-6, 6, y_2.shape[0])# 带限函数的傅里叶变换
ax_1 setup_axes(fig, 311)
ax_1.plot(x_2, y_2), ax_1.set_title(r\tilde F(u), loccenter, y1.05), ax_1.set_ylim([0, 3]), ax_1.set_yticks([]), ax_1.set_xticks([])x_1 np.linspace(-6, 6, y_2.shape[0])
y_3 np.where(x_1, x_1 -1, 0)
y_3 np.where(x_1 1, y_3, 0)ax_3 setup_axes(fig, 312)
ax_3.plot(x_1, y_3), ax_3.set_title(H(u), loccenter, y1.05), ax_3.set_ylim([0, 3]), ax_3.set_yticks([]), ax_3.set_xticks([])# 滤波
y_4 y_2 * y_3ax_4 setup_axes(fig, 313)
ax_4.plot(x_1, y_4), ax_4.set_title(rF(u) H(u)\tilde F(u), loccenter, y1.05), ax_4.set_ylim([0, 3]), ax_4.set_yticks([]), ax_4.set_xticks([])plt.tight_layout()
plt.show()混叠
在信号处理领域混叠是指取样后不同信号变得彼此无法区分的取样现象或者一个信号“伪装”成另一个信号的现象。
这种称为混叠对 这种混叠对在取样后是无法区分的。出现这种函数混叠的原因是我们所用的取样率太粗也就是说欠取样
带限函数的取样率小于奈奎斯特率的话不管使用何种滤波器器都不可能分离出来一个周期。也就不可能完美的复原函数。
# 混叠
x np.arange(0, 6*np.pi, 0.1)y_1 2 - np.sin(4 * x)
y_2 2 - np.sin(x)sample_1_f 21
sample_2_f 21
sample_1 y_1[::sample_1_f]
sample_2 y_2[::sample_2_f]
fig plt.figure(figsize(14, 8))
plt.subplot(2, 2, 1), plt.plot(x, y_1), plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 3), plt.plot(x, y_2), plt.xticks([]), plt.yticks([])plt.subplot(2, 2, 2), plt.stem(x[::sample_1_f], sample_1, linefmt--, markerfmto, basefmtC0-)
plt.xticks([]), plt.yticks([])
plt.subplot(2, 2, 4), plt.stem(x[::sample_2_f], sample_2, linefmt--, markerfmto, basefmtC0-)
plt.xticks([]), plt.yticks([])
plt.tight_layout()
plt.show()由取样后的数据重建复原函数
f(t)J{F(μ)}J{H(μ)F~(μ)}h(t)⋆f~(t)(4.37)f(t) \mathfrak{J} \{ F(\mu) \} \mathfrak{J} \{ H(\mu) \tilde F(\mu) \} h(t) \star\tilde f(t)\tag{4.37}f(t)J{F(μ)}J{H(μ)F~(μ)}h(t)⋆f~(t)(4.37)
得到空间域表达式 f(t)∑−∞∞f(nΔT)sinc[(t−nΔT)/ΔT](4.38)f(t) \sum_{-\infty}^{\infty} f(n\Delta T) sinc[(t - n \Delta T) / \Delta T]\tag{4.38}f(t)−∞∑∞f(nΔT)sinc[(t−nΔT)/ΔT](4.38)
