{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 序列时域混叠\n", "\n", "同时域采样一样，定义采样序列$p_N(\\\\omega) = \\\\sum_{k=-\\\\infty}^{\\\\infty} \\\\delta (\\\\omega - 2k\\\\pi /N)$。那么，将频域采样前后的关系写为\n", "\n", "\n", "$$\n", "\\begin{split}\n", " X_p(e^{j\\omega}) &= X(e^{j\\omega})\\cdot p_N(\\omega) = \\sum_{k=-\\infty}^\\infty \\delta\\left(\\omega - \\frac{2k\\pi}{N}\\right) \\cdot X(e^{j\\omega}) \\\\\n", " & = \\sum_{k=-\\infty}^\\infty \\delta\\left(\\omega - \\frac{2k\\pi}{N}\\right) \\cdot X(e^{j2k\\pi/N}) \n", "\\end{split}\n", "$$\n", "\n", "\n", "其中$X_p(e^{j\\\\omega})$为$X[k]$的连续形式，表达的信号是等价的。那么对上式进行 IDTFT 变换，就可以得到频域采样后时域的信号，这里记作$\\\\tilde{x}[n]$\n", "\n", "\n", "$$\n", " \\tilde{x}[n] &= \\text{IDTFT}\\{X_p(e^{j\\omega})\\} \\\\\n", " & =\\frac{1}{2 \\pi} \\int_{-\\pi}^{\\pi}\\left(\\sum_{k=-\\infty}^{\\infty} X\\left(e^{j 2 \\pi k / N}\\right) \\delta(\\omega-2 \\pi k / N)\\right) e^{j \\omega n} d \\omega \\\\\n", " & = \\frac{1}{2 \\pi} \\sum_{k=-\\infty}^{\\infty} \\underbrace{X\\left(e^{j 2 \\pi k / N}\\right)}_{=\\sum_{l=-\\infty}^{+\\infty} x[l] e^{-j 2 \\pi k l / N}} \\cdot \\underbrace{\\int_{-\\pi}^{\\pi} \\delta(\\omega-2 \\pi k / N) e^{j \\omega n} d \\omega}_{=e^{j 2 \\pi k n / N, \\ \\text{ for } k\\in [0,N-1]}} \\\\\n", " &= \\frac{1}{2 \\pi} \\sum_{l=-\\infty}^{\\infty} x[l] \\underbrace{\\sum_{k=0}^{N-1} e^{j 2 \\pi k / N(n-l)}}_{=N\\sum_{m=-\\infty}^{\\infty}\\delta[l-n-Nm]} \\\\\n", " &= \\frac{N}{2 \\pi} \\sum_{m=-\\infty}^{\\infty} x[n+mN]\n", "$$\n", "\n", "\n", "上式最后一个等式的求和符号的含义就是，将原信号$x[n]$以$N$为长度，周期延拓。也就是说，如果对 DTFT 变换的频率进行离散化，会导致时域上信号的周期延拓，幅度也会变为原来的$N/2\\\\pi$倍。这个结论与对信号在时域进行采样，频域周期延拓是类似的。" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/mac/anaconda3/envs/Sphinx/lib/python3.6/site-packages/ipykernel_launcher.py:11: UserWarning: In Matplotlib 3.3 individual lines on a stem plot will be added as a LineCollection instead of individual lines. This significantly improves the performance of a stem plot. To remove this warning and switch to the new behaviour, set the \"use_line_collection\" keyword argument to True.\n", " # This is added back by InteractiveShellApp.init_path()\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stderr", "output_type": "stream", "text": [ "/Users/mac/anaconda3/envs/Sphinx/lib/python3.6/site-packages/ipykernel_launcher.py:28: UserWarning: In Matplotlib 3.3 individual lines on a stem plot will be added as a LineCollection instead of individual lines. This significantly improves the performance of a stem plot. To remove this warning and switch to the new behaviour, set the \"use_line_collection\" keyword argument to True.\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stderr", "output_type": "stream", "text": [ "/Users/mac/anaconda3/envs/Sphinx/lib/python3.6/site-packages/ipykernel_launcher.py:43: UserWarning: In Matplotlib 3.3 individual lines on a stem plot will be added as a LineCollection instead of individual lines. This significantly improves the performance of a stem plot. To remove this warning and switch to the new behaviour, set the \"use_line_collection\" keyword argument to True.\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from numpy import *\n", "from matplotlib.pyplot import *\n", "\n", "# 产生长度为L信号\n", "L = 16\n", "n = arange(0,L)\n", "x = ones([L,]) #sin(2*pi*0.5*n/L)\n", "\n", "subplot(311)\n", "stem(n,x)\n", "show()\n", "\n", "\n", "\n", "# 如果L<=N,不会产生混叠\n", "N = 19\n", "xn = zeros([N,])\n", "xn[:L] = x\n", "xt = zeros([N,])\n", "for i in range(0,N):\n", " xt[i] = xn[i]\n", " if i-N >= 0:\n", " xt[i] = xn[i-N] + xt[i]\n", " if i+N N,会产生混叠\n", "N = 10\n", "xn = x\n", "\n", "xt = zeros([N,])\n", "for i in range(0,N):\n", " xt[i] = xn[i]\n", " if i-N >= 0:\n", " xt[i] = xn[i-N] + xt[i]\n", " if i+N < L:\n", " xt[i]=xt[i] + xn[i+N]\n", "subplot(313)\n", "stem(xt)\n", "show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## DFT变换与DTFT变换的关系" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/mac/anaconda3/envs/Sphinx/lib/python3.6/site-packages/ipykernel_launcher.py:12: UserWarning: In Matplotlib 3.3 individual lines on a stem plot will be added as a LineCollection instead of individual lines. This significantly improves the performance of a stem plot. To remove this warning and switch to the new behaviour, set the \"use_line_collection\" keyword argument to True.\n", " if sys.path[0] == '':\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stderr", "output_type": "stream", "text": [ "/Users/mac/anaconda3/envs/Sphinx/lib/python3.6/site-packages/ipykernel_launcher.py:20: UserWarning: In Matplotlib 3.3 individual lines on a stem plot will be added as a LineCollection instead of individual lines. This significantly improves the performance of a stem plot. To remove this warning and switch to the new behaviour, set the \"use_line_collection\" keyword argument to True.\n" ] }, { "data": { "image/png": "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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from numpy import *\n", "from matplotlib.pyplot import *\n", "from scipy import *\n", "\n", "# 产生长度为L信号\n", "L = 16\n", "n = arange(0,L)\n", "x = ones([L,]) #sin(2*pi*0.5*n/L)\n", "\n", "subplot(211)\n", "stem(n,x)\n", "show()\n", "\n", "\n", "# 如果L<=N,不会产生混叠\n", "N = 19\n", "xt = real(ifft(fft(x,N),N))\n", "subplot(212)\n", "stem(xt)\n", "show()\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "hide_input": false, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.8" } }, "nbformat": 4, "nbformat_minor": 2 }