/* 
 * Free FFT and convolution (TypeScript)
 * 
 * Copyright (c) 2022 Project Nayuki. (MIT License)
 * https://www.nayuki.io/page/free-small-fft-in-multiple-languages
 * 
 * Permission is hereby granted, free of charge, to any person obtaining a copy of
 * this software and associated documentation files (the "Software"), to deal in
 * the Software without restriction, including without limitation the rights to
 * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
 * the Software, and to permit persons to whom the Software is furnished to do so,
 * subject to the following conditions:
 * - The above copyright notice and this permission notice shall be included in
 *   all copies or substantial portions of the Software.
 * - The Software is provided "as is", without warranty of any kind, express or
 *   implied, including but not limited to the warranties of merchantability,
 *   fitness for a particular purpose and noninfringement. In no event shall the
 *   authors or copyright holders be liable for any claim, damages or other
 *   liability, whether in an action of contract, tort or otherwise, arising from,
 *   out of or in connection with the Software or the use or other dealings in the
 *   Software.
 */


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This is a wrapper function.
 */
function transform(real: Array<number>|Float64Array, imag: Array<number>|Float64Array): void {
	const n: number = real.length;
	if (n != imag.length)
		throw new RangeError("Mismatched lengths");
	if (n == 0)
		return;
	else if ((n & (n - 1)) == 0)  // Is power of 2
		transformRadix2(real, imag);
	else  // More complicated algorithm for arbitrary sizes
		transformBluestein(real, imag);
}


/* 
 * Computes the inverse discrete Fourier transform (IDFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This is a wrapper function. This transform does not perform scaling, so the inverse is not a true inverse.
 */
function inverseTransform(real: Array<number>|Float64Array, imag: Array<number>|Float64Array): void {
	transform(imag, real);
}


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector's length must be a power of 2. Uses the Cooley-Tukey decimation-in-time radix-2 algorithm.
 */
function transformRadix2(real: Array<number>|Float64Array, imag: Array<number>|Float64Array): void {
	// Length variables
	const n: number = real.length;
	if (n != imag.length)
		throw new RangeError("Mismatched lengths");
	if (n == 1)  // Trivial transform
		return;
	let levels: number = -1;
	for (let i = 0; i < 32; i++) {
		if (1 << i == n)
			levels = i;  // Equal to log2(n)
	}
	if (levels == -1)
		throw new RangeError("Length is not a power of 2");
	
	// Trigonometric tables
	let cosTable = new Array<number>(n / 2);
	let sinTable = new Array<number>(n / 2);
	for (let i = 0; i < n / 2; i++) {
		cosTable[i] = Math.cos(2 * Math.PI * i / n);
		sinTable[i] = Math.sin(2 * Math.PI * i / n);
	}
	
	// Bit-reversed addressing permutation
	for (let i = 0; i < n; i++) {
		const j: number = reverseBits(i, levels);
		if (j > i) {
			let temp: number = real[i];
			real[i] = real[j];
			real[j] = temp;
			temp = imag[i];
			imag[i] = imag[j];
			imag[j] = temp;
		}
	}
	
	// Cooley-Tukey decimation-in-time radix-2 FFT
	for (let size = 2; size <= n; size *= 2) {
		const halfsize: number = size / 2;
		const tablestep: number = n / size;
		for (let i = 0; i < n; i += size) {
			for (let j = i, k = 0; j < i + halfsize; j++, k += tablestep) {
				const l: number = j + halfsize;
				const tpre: number =  real[l] * cosTable[k] + imag[l] * sinTable[k];
				const tpim: number = -real[l] * sinTable[k] + imag[l] * cosTable[k];
				real[l] = real[j] - tpre;
				imag[l] = imag[j] - tpim;
				real[j] += tpre;
				imag[j] += tpim;
			}
		}
	}
	
	// Returns the integer whose value is the reverse of the lowest 'width' bits of the integer 'val'.
	function reverseBits(val: number, width: number): number {
		let result: number = 0;
		for (let i = 0; i < width; i++) {
			result = (result << 1) | (val & 1);
			val >>>= 1;
		}
		return result;
	}
}


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
 * Uses Bluestein's chirp z-transform algorithm.
 */
function transformBluestein(real: Array<number>|Float64Array, imag: Array<number>|Float64Array): void {
	// Find a power-of-2 convolution length m such that m >= n * 2 + 1
	const n: number = real.length;
	if (n != imag.length)
		throw new RangeError("Mismatched lengths");
	let m: number = 1;
	while (m < n * 2 + 1)
		m *= 2;
	
	// Trigonometric tables
	let cosTable = new Array<number>(n);
	let sinTable = new Array<number>(n);
	for (let i = 0; i < n; i++) {
		const j: number = i * i % (n * 2);  // This is more accurate than j = i * i
		cosTable[i] = Math.cos(Math.PI * j / n);
		sinTable[i] = Math.sin(Math.PI * j / n);
	}
	
	// Temporary vectors and preprocessing
	let areal: Array<number> = newArrayOfZeros(m);
	let aimag: Array<number> = newArrayOfZeros(m);
	for (let i = 0; i < n; i++) {
		areal[i] =  real[i] * cosTable[i] + imag[i] * sinTable[i];
		aimag[i] = -real[i] * sinTable[i] + imag[i] * cosTable[i];
	}
	let breal: Array<number> = newArrayOfZeros(m);
	let bimag: Array<number> = newArrayOfZeros(m);
	breal[0] = cosTable[0];
	bimag[0] = sinTable[0];
	for (let i = 1; i < n; i++) {
		breal[i] = breal[m - i] = cosTable[i];
		bimag[i] = bimag[m - i] = sinTable[i];
	}
	
	// Convolution
	let creal = new Array<number>(m);
	let cimag = new Array<number>(m);
	convolveComplex(areal, aimag, breal, bimag, creal, cimag);
	
	// Postprocessing
	for (let i = 0; i < n; i++) {
		real[i] =  creal[i] * cosTable[i] + cimag[i] * sinTable[i];
		imag[i] = -creal[i] * sinTable[i] + cimag[i] * cosTable[i];
	}
}


/* 
 * Computes the circular convolution of the given real vectors. Each vector's length must be the same.
 */
function convolveReal(xvec: Array<number>|Float64Array, yvec: Array<number>|Float64Array, outvec: Array<number>|Float64Array): void {
	const n: number = xvec.length;
	if (n != yvec.length || n != outvec.length)
		throw new RangeError("Mismatched lengths");
	convolveComplex(xvec, newArrayOfZeros(n), yvec, newArrayOfZeros(n), outvec, newArrayOfZeros(n));
}


/* 
 * Computes the circular convolution of the given complex vectors. Each vector's length must be the same.
 */
function convolveComplex(
		xreal: Array<number>|Float64Array, ximag: Array<number>|Float64Array,
		yreal: Array<number>|Float64Array, yimag: Array<number>|Float64Array,
		outreal: Array<number>|Float64Array, outimag: Array<number>|Float64Array): void {
	
	const n: number = xreal.length;
	if (n != ximag.length || n != yreal.length || n != yimag.length
			|| n != outreal.length || n != outimag.length)
		throw new RangeError("Mismatched lengths");
	
	xreal = xreal.slice();
	ximag = ximag.slice();
	yreal = yreal.slice();
	yimag = yimag.slice();
	transform(xreal, ximag);
	transform(yreal, yimag);
	
	for (let i = 0; i < n; i++) {
		const temp: number = xreal[i] * yreal[i] - ximag[i] * yimag[i];
		ximag[i] = ximag[i] * yreal[i] + xreal[i] * yimag[i];
		xreal[i] = temp;
	}
	inverseTransform(xreal, ximag);
	
	for (let i = 0; i < n; i++) {  // Scaling (because this FFT implementation omits it)
		outreal[i] = xreal[i] / n;
		outimag[i] = ximag[i] / n;
	}
}


function newArrayOfZeros(n: number): Array<number> {
	let result: Array<number> = [];
	for (let i = 0; i < n; i++)
		result.push(0);
	return result;
}