added roots for filtering

Author: unpingco

Filtering.ipynb

@@ -354,7 +354,7 @@
       "\n",
       "$$ H(z) = \\sum_{n=0}^{M-1} h_n z^{-n} = \\frac{1}{8} \\sum_{n=0}^7 z^{-n} =\\frac{1}{8}  (1+z)(1+z^2)(1+z^4)/z^7 $$\n",
       "\n",
-      "Thus, the first zero occurs when $z=-1$ or when $ \\exp(j\\omega) = -1 \\Rightarrow \\omega=\\pi$. The next pair of zeros occurs when $z= \\pm j$ which corresponds to $ \\omega = \\pm \\pi/2 $. Notice that any filter with this  $ z+1 $ term will  eliminate the $ \\omega=\\pi $ (highest) frequency. Likewise, the term $ z-1 $ means that the filter zeros out $ \\omega=0 $. In general, the roots of the z-transform *do not* lie on the unit circle. One way to understand FIR filter design is as the judicious placement of these zeros in the complex plane so the shape of the resulting transfer function $ H(z) $ evaluated on the unit circle satisfies our design specifications.\n",
+      "Thus, the first zero occurs when $z=-1$ or when $ \\exp(j\\omega) = -1 \\Rightarrow \\omega=\\pi$. The next pair of zeros occurs when $z= \\pm j$ which corresponds to $ \\omega = \\pm \\pi/2 $. Finally, the last four zeros are for $\\omega=\\pm \\pi/4$ and $\\omega=\\pm 3\\pi/4 $.  Notice that any filter with this  $ z+1 $ term will  eliminate the $ \\omega=\\pi $ (highest) frequency. Likewise, the term $ z-1 $ means that the filter zeros out $ \\omega=0 $. In general, the roots of the z-transform *do not* lie on the unit circle. One way to understand FIR filter design is as the judicious placement of these zeros in the complex plane so the shape of the resulting transfer function $ H(z) $ evaluated on the unit circle satisfies our design specifications.\n",
       "\n",
       "We need a special case of the Fourier transform as a tool for our analysis. "
      ]