Extended discussion of zero-phase filters

Author: spors

filter_design/window_method.ipynb

@@ -26,7 +26,7 @@
    "source": [
     "### Causal Filters\n",
     "\n",
-    "Let's assume that the desired frequency characteristics of the filter are given by the frequency response (i.e. the DTFT spectrum) $H_\\text{d}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$. The corresponding impulse response is computed by its inverse discrete-time Fourier transform (IDTFT)\n",
+    "Let's assume that the desired frequency characteristics of the filter are given by its frequency response (i.e. discrete-time Fourier transform) $H_\\text{d}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$. The corresponding impulse response is computed by its inverse discrete-time Fourier transform (IDTFT)\n",
     "\n",
     "\\begin{equation}\n",
     "h_\\text{d}[k] = \\frac{1}{2 \\pi} \\int\\limits_{- \\pi}^{\\pi} H_\\text{d}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) \\, \\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega\\,k} \\; \\mathrm{d}\\Omega\n",
@@ -38,7 +38,7 @@
     "h[k] = h_\\text{d}[k] \\cdot w[k]\n",
     "\\end{equation}\n",
     "\n",
-    "where $h[k]$ denotes the impulse response of the designed filter. Its frequency response $H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ is given by the multiplication theorem of the discrete-time Fourier transform (DTFT)\n",
+    "where $h[k]$ denotes the impulse response of the designed filter and $w[k] = 0$ for $k < 0$. Its frequency response $H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ is given by the multiplication theorem of the discrete-time Fourier transform (DTFT)\n",
     "\n",
     "\\begin{equation}\n",
     "H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{1}{2 \\pi} \\; H_\\text{d}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) \\circledast W(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})\n",
@@ -182,15 +182,15 @@
    "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, as most of the idealized filters.\n",
+    "Above results show that an ideal-low pass cannot be realized very well by 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",
     "\\begin{equation}\n",
     "h_d[k] = h_d^*[-k]\n",
     "\\end{equation}\n",
     "\n",
-    "due to the symmetry relations of the DTFT. Hence, a transfer function with zero-phase cannot be realized by a causal non-recursive filter. This observation motivates to replace the zero-phase by a linear-phase in such situations. This is illustrated in the following."
+    "due to the symmetry relations of the DTFT. If the desired impulse response has large values around $k=0$, removal of the anti-causal part by windowing will result in large deviations between the desired transfer function and the designed filter. Such filters cannot be realized very well by simple windowing of the desired impulse response. This observation motivates to replace the zero-phase by a linear-phase in such situations in order to shift the large values of the desired impulse response to the center of the window. This is illustrated in the following."
    ]
   },
   {
@@ -221,7 +221,7 @@
     "\n",
     "Introducing the symmetry relations of the impulse response $h[k]$ into the DTFT and comparing the result with above definition of a generalized linear phase system reveals four different types of linear-phase systems. These can be discriminated with respect to their phase and magnitude characteristics\n",
     "\n",
-    "| Type | Length $N$ | Impulse Response $h[k]\\;$ | Constant Group Delay $\\alpha$ in Samples| Constant Phase $\\beta$ | Transfer Function $A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ |\n",
+    "| Type | Length $N$ | Impulse Response $h[k]\\;$ | Group Delay $\\alpha$ in Samples| Constant Phase $\\beta$ | Transfer Function $A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ |\n",
     "| :---:  | :---: | :---: | :---:| :---: | :---: |\n",
     "| 1 | odd | $h[k] = h[N-1-k]$ | $\\alpha = \\frac{N-1}{2} \\in \\mathbb{N}$ | $\\beta = \\{0, \\pi\\}$ | $A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = A(\\mathrm{e}^{-\\,\\mathrm{j}\\,\\Omega})$, all filter characteristics|\n",
     "| 2 | even| $h[k] = h[N-1-k]$ | $\\alpha = \\frac{N-1}{2} \\notin \\mathbb{N}$ | $\\beta = \\{0, \\pi\\}$ | $A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = A(\\mathrm{e}^{-\\,\\mathrm{j}\\,\\Omega})$, $A(\\mathrm{e}^{\\,\\mathrm{j}\\,\\pi}) = 0$, only lowpass or bandpass|\n",