Comments for the design of ideal low-pass filters

Author: spors

filter_design/frequency_sampling_method.ipynb

@@ -165,7 +165,9 @@
     "**Exercises**\n",
     "\n",
     "* What phase behavior does the designed filter have?\n",
-    "* Increase the length `N` of the filter. Does the attenuation in the stop-band improve?"
+    "* Increase the length `N` of the filter. Does the attenuation in the stop-band improve?\n",
+    "\n",
+    "The reason for the poor perfomance of the designed filter is the zero-phase of the desired transfer function which cannot be realized by a causal non-recursive system. This was [already discussed for the window method](../filter_design/window_method.ipynb#Zero-Phase-Filters). In comparison to the window method, the freqeuncy sampling method suffers from addtional time-domain aliasing due to the periodicy of the DFT. Again a linear-phase design is better in such situations."
    ]
   },
   {

filter_design/window_method.ipynb

@@ -180,7 +180,7 @@
    "source": [
     "### Zero-Phase Filters\n",
     "\n",
-    "Above results show that an ideal-low pass cannot be realized very well with the window method. The reason is that an ideal-low pass has zero-phase. \n",
+    "Above results show that an ideal-low pass cannot be realized very well with the window method. The reason is that an ideal-low pass has zero-phase, as most of the idealized filters.\n",
     "\n",
     "Lets assume a general zero-phase filter with transfer function $H_d(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ with amplitude $A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})  \\in \\mathbb{R}$. Its impulse response $h_d[k] = \\mathcal{F}_*^{-1} \\{ H_d(e^{j \\Omega} \\}$ is conjugate complex symmetric\n",
     "\n",