Fixed equations for zero-padding

Author: spors

nonrecursive_filters/fast_convolution.ipynb

@@ -68,8 +68,15 @@
     "To compute the linear convolution by the periodic convolution one has to take care that the result of the linear convolution fits into one period of the periodic convolution. Hence, the periodicity has to be chosen as $M \\geq N+L-1$. This can be achieved by zero-padding of $x_L[k]$ and $h_N[k]$ to a total length of $M$\n",
     "\n",
     "\\begin{align}\n",
-    "x_M[k] &= \\left[ x_L[0], x_L[1], \\dots, x_L[L-1], 0, \\dots, 0 \\right]^\\text{T} \\\\\n",
-    "h_M[k] &= \\left[ h[0], h[1], \\dots, h[N-1], 0, \\dots, 0 \\right]^\\text{T}\n",
+    "x_M[k] &= \\begin{cases}\n",
+    "x_L[k] & \\mathrm{for} \\; k=0, 1, \\dots, L-1 \\\\\n",
+    "0 & \\mathrm{for} \\; k=L, L+1, \\dots, M-1\n",
+    "\\end{cases}\n",
+    "\\\\\n",
+    "h_M[k] &= \\begin{cases}\n",
+    "h_N[k] & \\mathrm{for} \\; k=0, 1, \\dots, N-1 \\\\\n",
+    "0 & \\mathrm{for} \\; k=N, N+1, \\dots, M-1\n",
+    "\\end{cases}\n",
     "\\end{align}\n",
     "\n",
     "This results in the desired equality of linear and periodic convolution\n",
@@ -87,7 +94,7 @@
    "source": [
     "#### Example\n",
     "\n",
-    "The following example computes the linear, periodic and linear by periodic convolution of a rectangular signal $x[k] = \\text{rect}_L[k]$ of length $N$ with a triangular signal $h[k] = \\Lambda_N[k]$ of length $N$."
+    "The following example computes the linear, periodic and linear by periodic convolution of a rectangular signal $x[k] = \\text{rect}_L[k]$ of length $L$ with a triangular signal $h[k] = \\Lambda_N[k]$ of length $N$."
    ]
   },
   {

spectral_analysis_deterministic_signals/zero_padding.ipynb

@@ -398,7 +398,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.1"
+   "version": "3.5.2"
   }
  },
  "nbformat": 4,