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305 lines
79 KiB
305 lines
79 KiB
11 years ago
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{
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"metadata": {
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"name": ""
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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"worksheets": [
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Fast matrix biquad\n",
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"==================\n",
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"\n",
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"This notebook shows the math behind a very fast NEON implementation of biquad IIR filters.\n",
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"One of the main primitives NEON is very highly optimized for is multiplying a matrix by\n",
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"a vector, especially of size 4.\n",
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"\n",
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"A biquad filter has two state variables (this is most apparent in transposed direct form II).\n",
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"For this implementation, the input vector consists of two input values and the two state\n",
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"variables. The output vector is two output values and the new state vector. The relationship\n",
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"is linear, ie the transformation from input to output vector is simply a matrix-vector\n",
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"multiplication. Each iteration computes two samples, using a single 4x4 matrix-vector multiply."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"%pylab inline"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"Populating the interactive namespace from numpy and matplotlib\n"
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]
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}
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],
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"prompt_number": 117
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We start with a simple implementation of biquad for reference; we'll compare the error of the matrix approach against this scalar implementation. This is in direct form I."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def biquad(coefs, inp):\n",
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" b0, b1, b2, a1, a2 = coefs\n",
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" out = zeros(len(inp))\n",
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" x1 = 0\n",
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" x2 = 0\n",
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" y1 = 0\n",
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" y2 = 0\n",
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" for i in range(len(inp)):\n",
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" x = inp[i]\n",
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" y = b0 * x + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2\n",
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" out[i] = y\n",
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" x2, x1 = x1, x\n",
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" y2, y1 = y1, y\n",
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" return out"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 118
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"As a starting step, here's the same biquad in a 2x2 matrix formulation. This version is derived from\n",
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"transposed direct form II, with a little expansion of the effect _this_ input sample has on the _next_\n",
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"state vector (this is the B array below). The A matrix describes the evolution of the IIR state."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def bqmat(coefs, inp):\n",
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" b0, b1, b2, a1, a2 = coefs\n",
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" A = array([[-a1, 1], [-a2, 0]])\n",
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" B = array([b1 - a1 * b0, b2 - a2 * b0])\n",
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" y = array([0, 0])\n",
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" out = np.zeros(len(inp))\n",
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" for i in range(len(inp)):\n",
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" x = inp[i]\n",
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" out[i] = y[0] + x * b0\n",
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" y = dot(A, y) + B * x\n",
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" return out"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 119
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's test it out."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"impulse = zeros(100)\n",
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"impulse[10] = 1\n",
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"coefs = 1.0207, -1.7719, .9376, -1.7719, 0.9583 # peaking\n",
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"plot(biquad(coefs, impulse))\n",
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"plot(bqmat(coefs, impulse))"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
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"output_type": "pyout",
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"prompt_number": 120,
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"text": [
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"[<matplotlib.lines.Line2D at 0x10cecce10>]"
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]
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},
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{
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEACAYAAAC6d6FnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nmw2AiAKf9AHg0g3YhFZ7W7G0
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"text": [
|
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"<matplotlib.figure.Figure at 0x10ced4110>"
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]
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}
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],
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"prompt_number": 120
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"plot(biquad(coefs, impulse) - bqmat(coefs, impulse))"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"metadata": {},
|
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|
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"output_type": "pyout",
|
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|
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"prompt_number": 121,
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"text": [
|
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"[<matplotlib.lines.Line2D at 0x10d3a9e50>]"
|
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|
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]
|
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|
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},
|
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|
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{
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|
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"metadata": {},
|
||
|
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"output_type": "display_data",
|
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|
|
"png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEGCAYAAABsLkJ6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nYOdOWgaDMeZaOADY4cIFWgPI
|
||
|
|
"text": [
|
||
|
|
"<matplotlib.figure.Figure at 0x10d390750>"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"prompt_number": 121
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Note the scale of the error -- it's floating point rounding.\n",
|
||
|
|
"\n",
|
||
|
|
"Now for the 4x4 matrix approach. It's really four 2x2 matrices tiled. The top\n",
|
||
|
|
"left corner is the influence of the two input samples on the two output samples;\n",
|
||
|
|
"it's the first two values of the biquad's impulse response, banded. The top\n",
|
||
|
|
"right corner is the contribution of the state vector to the output. The bottom\n",
|
||
|
|
"left is the effect of the input on the state, and the bottom right is the\n",
|
||
|
|
"evolution of the IIR. Since each iteration is two samples, this matrix is\n",
|
||
|
|
"simply the square of the A matrix above.\n",
|
||
|
|
"\n",
|
||
|
|
"Note that each iteration processes _two_ samples. The matrix-vector product\n",
|
||
|
|
"is 16 multiply-accumulates, so it amortizes to 8 per sample. This compares\n",
|
||
|
|
"well to the 5 multiply-accumulates of the direct form, but organized for extremely\n",
|
||
|
|
"fast evaluation in SIMD, and NEON in particular."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"collapsed": false,
|
||
|
|
"input": [
|
||
|
|
"def biquadm(coefs, inp):\n",
|
||
|
|
" b0, b1, b2, a1, a2 = coefs\n",
|
||
|
|
" c1 = b1 - a1 * b0\n",
|
||
|
|
" c2 = b2 - a2 * b0\n",
|
||
|
|
" A = array([[b0, 0, 1, 0],\n",
|
||
|
|
" [c1, b0, -a1, 1],\n",
|
||
|
|
" [-a1 * c1 + c2, c1, -a2 + a1*a1, -a1],\n",
|
||
|
|
" [-a2 * c1, c2, a1*a2, -a2]])\n",
|
||
|
|
" out = np.zeros(len(inp))\n",
|
||
|
|
" y = zeros(4)\n",
|
||
|
|
" for i in range(0, len(inp), 2):\n",
|
||
|
|
" y[0:2] = inp[i:i+2]\n",
|
||
|
|
" y = dot(A, y)\n",
|
||
|
|
" out[i:i+2] = y[0:2]\n",
|
||
|
|
" return out"
|
||
|
|
],
|
||
|
|
"language": "python",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"prompt_number": 122
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"Again, let's test it out."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"collapsed": false,
|
||
|
|
"input": [
|
||
|
|
"plot(biquad(coefs, impulse))\n",
|
||
|
|
"plot(biquadm(coefs, impulse))"
|
||
|
|
],
|
||
|
|
"language": "python",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "pyout",
|
||
|
|
"prompt_number": 123,
|
||
|
|
"text": [
|
||
|
|
"[<matplotlib.lines.Line2D at 0x10d3b9510>]"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data",
|
||
|
|
"png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEACAYAAAC6d6FnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nmw2AiAKf9AHg0g3YhFZ7W7G0
|
||
|
|
"text": [
|
||
|
|
"<matplotlib.figure.Figure at 0x10d3b94d0>"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"prompt_number": 123
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"The error is similar to that of the 2x2 matrix version, but not quite the same."
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"collapsed": false,
|
||
|
|
"input": [
|
||
|
|
"plot(biquadm(coefs, delta) - biquad(coefs, delta))\n",
|
||
|
|
"plot(biquadm(coefs, delta) - bqmat(coefs, delta))"
|
||
|
|
],
|
||
|
|
"language": "python",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "pyout",
|
||
|
|
"prompt_number": 124,
|
||
|
|
"text": [
|
||
|
|
"[<matplotlib.lines.Line2D at 0x10d70b550>]"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "display_data",
|
||
|
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|
||
|
|
"text": [
|
||
|
|
"<matplotlib.figure.Figure at 0x10d70b5d0>"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"prompt_number": 124
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"collapsed": false,
|
||
|
|
"input": [],
|
||
|
|
"language": "python",
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"prompt_number": 124
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"metadata": {}
|
||
|
|
}
|
||
|
|
]
|
||
|
|
}
|