Merge pull request #11 from Naereen/patch-1

Fix a typo, "see show these" -> see how these in Frequency_Resolution.ipynb

Author: unpingco

Frequency_Resolution.ipynb

@@ -250,7 +250,7 @@
       "\n",
       "Thus, two frequencies that differ by at least this amount should be resolvable in these plots. \n",
       "\n",
-      "Thus, in our last example, we had $f_s= 64,N_s = 128 \\Rightarrow \\delta f = 1/2$ Hz and we were trying to separate two frequencies 0.5 Hz apart so we were right on the edge in this case.  I invite you to download this IPython notebook and try longer or shorter signal durations to see show these plots change. Incidentally, this where some define the notion of *frequency bin* as the DFT resolution ($ f_s/N $) divided by this minimal resolution, $ f_s/N_s $ which gives $ N_s/N $. In other words, the DFT measures frequency in discrete *bins* of minimal resolution, $ N_s/N $.\n",
+      "Thus, in our last example, we had $f_s= 64,N_s = 128 \\Rightarrow \\delta f = 1/2$ Hz and we were trying to separate two frequencies 0.5 Hz apart so we were right on the edge in this case.  I invite you to download this IPython notebook and try longer or shorter signal durations to see how these plots change. Incidentally, this is where some define the notion of *frequency bin* as the DFT resolution ($ f_s/N $) divided by this minimal resolution, $ f_s/N_s $ which gives $ N_s/N $. In other words, the DFT measures frequency in discrete *bins* of minimal resolution, $ N_s/N $.\n",
       "\n",
       "However, sampling over a longer duration only helps when the signal frequencies are *stable* over the longer duration. If these frequencies drift during the longer sampling interval or otherwise become contaminated with other signals, then advanced techniques become necessary.\n",
       "\n",
@@ -359,11 +359,11 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The figure above shows the <font color=\"Blue\">rectangular window DFT in blue, $R_k$ </font> against the sinusoid <FONT color=\"red\"> input signal in red, $X_k$, </font> for each value of $k$ as the two terms slide past each other from left to right, top to bottom. In other words, the $k^{th}$ term in $Z_k$, the DFT of the product $x_n r_n $, can be thought of as the inner-product of the red and blue lines. This is not exactly true because we are just plotting magnitudes and not the real/imaginary parts, but it's enough to understand the mechanics of the circular convolution.\n",
+      "The figure above shows the <font color=\"blue\">rectangular window DFT in blue, $R_k$</font> against the sinusoid <font color=\"red\">input signal in red, $X_k$</font>, for each value of $k$ as the two terms slide past each other from left to right, top to bottom. In other words, the $k^{th}$ term in $Z_k$, the DFT of the product $x_n r_n $, can be thought of as the inner-product of the red and blue lines. This is not exactly true because we are just plotting magnitudes and not the real/imaginary parts, but it's enough to understand the mechanics of the circular convolution.\n",
       "\n",
       "A good way to think about the rectangular window's `sinc` shape as it slides past the input signal is as a *probe* with a resolution defined by its mainlobe width. For example, in frame $k=12$, we see that the peak of the rectangular window coincides with the peak of the input frequency so we should expect a large value for $Z_{k=12}$ which is shown below. However, if the rectangular window were shorter, corresponding to a wider mainlobe width, then two nearby frequencies could be draped in the same mainlobe and would then be indistinguishable in the resulting DFT because the DFT for that value of $k$ is the inner-product (i.e. a complex number) of the two overlapping graphs.\n",
       "\n",
-      "The figure below shows the the direct computation of the DFT of $Z_k$ matches the circular convolution method using $X_k$ and $R_k$."
+      "The figure below shows the direct computation of the DFT of $Z_k$ matches the circular convolution method using $X_k$ and $R_k$."
      ]
     },
     {
@@ -429,4 +429,4 @@
    "metadata": {}
   }
  ]
-}
+}