Explained normalization of window function

Author: spors

spectral_estimation_random_signals/welch_method.ipynb

@@ -38,7 +38,7 @@
     "X_l(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\sum_{k = 0}^{N-1} x[k + l \\cdot M] \\, w[k] \\; \\mathrm{e}^{\\,-\\mathrm{j}\\,\\Omega\\,k}\n",
     "\\end{equation}\n",
     "\n",
-    "where the window $w[k]$ defined within $0\\leq k\\leq N-1$ should be normalized as $\\frac{1}{N} \\sum\\limits_{k=0}^{N-1} | w[k] |^2 = 1$. The stepsize $M$ determines the overlap between the segments. In general, $N-M$ number of samples overlap between adjacent segments. For $M = N$ no overlap occurs. The overlap is sometimes given as ratio $\\frac{N-M}{N}\\cdot 100\\%$. \n",
+    "where the window $w[k]$ defined within $0\\leq k\\leq N-1$ should be normalized as $\\frac{1}{N} \\sum\\limits_{k=0}^{N-1} | w[k] |^2 = 1$. The latter condition ensures that the power of the signal is maintained in the estimate. The stepsize $M$ determines the overlap between the segments. In general, $N-M$ number of samples overlap between adjacent segments. For $M = N$ no overlap occurs. The overlap is sometimes given as ratio $\\frac{N-M}{N}\\cdot 100\\%$. \n",
     "\n",
     "Introducing $X_l(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ into the definition of the periodogram yields the periodogram of the $l$-th segment\n",
     "\n",