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* @license Apache-2.0
*
* Copyright (c) 2019 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var setReadOnly = require( '@stdlib/utils/define-nonenumerable-read-only-property' );
var isPositiveInteger = require( '@stdlib/assert/is-positive-integer' ).isPrimitive;
var isEven = require( '@stdlib/math/base/assert/is-even' );
var iteratorSymbol = require( '@stdlib/symbol/iterator' );
var format = require( '@stdlib/string/format' );
var sinpi = require( '@stdlib/math/base/special/sinpi' );
var floor = require( '@stdlib/math/base/special/floor' );
var validate = require( './validate.js' );
// MAIN //
/**
* Returns an iterator which generates a periodic sinc waveform.
*
* ## Method
*
* - The periodic sinc function, or Dirichlet function, is defined as
*
* ```tex
* D_N(x; A) = \begin{cases}
* A \cdot \frac{\sin(Nx/2)}{N\sin(x/2)} & x \neq 2\pi k\ \textrm{for} k = 0, \pm 1, \pm 2, \pm 3, \ldots \\
* A \cdot (-1)^{k(N-1)} & x = 2\pi k\ \textrm{for} k = 0, \pm 1, \pm 2, \pm 3, \ldots
* \end{cases}
* ```
*
* where, for odd \\( N \\), the waveform period is \\( 2\pi \\), and, for even \\( N \\), the waveform period is \\( 4\pi \\).
*
* - In order to evaluate the periodic sinc function in terms of an iteration number \\( t \\) and a specified period (period \\( \tau \\)), rather than radians, for odd \\( N \\), we define
*
* ```tex
* x = \frac{2\pi(t-\varphi)}{\tau}
* ```
*
* and, for even \\( N \\), we define
*
* ```tex
* x = \frac{4\pi(t-\varphi)}{\tau}
* ```
*
* and substitute accordingly.
*
* @param {PositiveInteger} n - order
* @param {Options} [options] - function options
* @param {PositiveInteger} [options.period=100] - number of iterations before a waveform repeats
* @param {NonNegativeNumber} [options.amplitude=1.0] - peak amplitude
* @param {integer} [options.offset=0] - phase offset (in units of iterations; zero-based)
* @param {NonNegativeInteger} [options.iter=1e308] - number of iterations
* @throws {TypeError} first argument must be a positive integer
* @throws {TypeError} options argument must be an object
* @throws {TypeError} must provide valid options
* @returns {Iterator} iterator
*
* @example
* var iter = iterPeriodicSinc( 7 );
*
* var v = iter.next().value;
* // returns <number>
*
* v = iter.next().value;
* // returns <number>
*
* v = iter.next().value;
* // returns <number>
*
* // ...
*/
function iterPeriodicSinc( n, options ) {
var opts;
var iter;
var FLG;
var err;
var sgn;
var hwf;
var hw;
var t;
var s;
var i;
if ( !isPositiveInteger( n ) ) {
throw new TypeError( format( 'invalid argument. First argument must be a positive integer. Value: `%s`.', n ) );
}
opts = {
'period': 100,
'amplitude': 1.0,
'offset': 0,
'iter': 1e308
};
if ( arguments.length > 1 ) {
err = validate( opts, options );
if ( err ) {
throw err;
}
}
t = ( opts.period-opts.offset ) % opts.period;
if ( t < 0 ) {
t += opts.period; // normalize such that t ϵ [0,τ]
}
if ( isEven( n ) ) {
if ( opts.offset >= 0 ) {
sgn = -1.0;
} else {
sgn = 1.0;
}
// Note: when `n` is even, the waveform period (in radians) is 4π, so we need to adjust the "period" parameter to ensure that the waveform repeats within the specified number of iterations:
hw = opts.period / 2;
hwf = floor( hw ); // note: τ could be an odd number; in which case, negative peaks happen "between" two iterations (t%τ and t%(τ+1))
s = n / hw;
} else {
s = n / opts.period;
}
t -= 1;
i = 0;
// Create an iterator protocol-compliant object:
iter = {};
setReadOnly( iter, 'next', ( sgn === void 0 ) ? next2 : next1 );
setReadOnly( iter, 'return', end );
// If an environment supports `Symbol.iterator`, make the iterator iterable:
if ( iteratorSymbol ) {
setReadOnly( iter, iteratorSymbol, factory );
}
return iter;
/**
* Returns an iterator protocol-compliant object containing the next iterated value.
*
* @private
* @returns {Object} iterator protocol-compliant object
*/
function next1() {
var v;
i += 1;
if ( FLG || i > opts.iter ) {
return {
'done': true
};
}
t += 1;
t %= opts.period;
if ( t === 0 || t === hw ) {
sgn *= -1.0;
v = sgn * opts.amplitude;
} else {
if ( t === hwf ) {
// We just passed a negative peak, so we need flip the sign in order to ensure that the waveform at `t%τ=0` is the correct sign:
sgn *= -1.0;
}
v = opts.amplitude * sinpi(s*t) / ( n*sinpi(t/hw) );
}
return {
'value': v,
'done': false
};
}
/**
* Returns an iterator protocol-compliant object containing the next iterated value.
*
* @private
* @returns {Object} iterator protocol-compliant object
*/
function next2() {
var v;
i += 1;
if ( FLG || i > opts.iter ) {
return {
'done': true
};
}
t += 1;
t %= opts.period;
if ( t === 0 ) {
v = opts.amplitude;
} else {
v = opts.amplitude * sinpi(s*t) / ( n*sinpi(t/opts.period) );
}
return {
'value': v,
'done': false
};
}
/**
* Finishes an iterator.
*
* @private
* @param {*} [value] - value to return
* @returns {Object} iterator protocol-compliant object
*/
function end( value ) {
FLG = true;
if ( arguments.length ) {
return {
'value': value,
'done': true
};
}
return {
'done': true
};
}
/**
* Returns a new iterator.
*
* @private
* @returns {Iterator} iterator
*/
function factory() {
return iterPeriodicSinc( n, opts );
}
}
// EXPORTS //
module.exports = iterPeriodicSinc;
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