VecLab

0.1.13

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Swift Real/Complex Vector DSP Library

What's New

2024-04-12T20:48:19Z

Rename DocC folder to VecLab.

VecLab

A real/complex vector library in Swift.

Overview

VecLab is a numeric library for real and complex vector operations and MATLAB-style functions.

Real and complex scalars and vectors.
Overloaded arithmetic operators.
Basic Matlab-style functions.
Vectorized using vDSP.

Example Usage

The libray includes an FFT function using Accelerate, but here is an example creating a complex FFT using a recursive algorithm and its MATLAB equivalent:

Swift

// FFTX Fast Finite Fourier Transform.
public func fftx(_ x: ComplexArray) -> ComplexArray {
    let n = length(x) 
    let omega = exp(-2 * Real.pi * Real.i / Real(n))
    if rem(n, 2) == 0 {
        // Recursive divide and conquer.
        let k = vector(0 ... (n / 2 - 1))
        let w = omega ** k
        let u = fftx(slice(x, 0 ..< n - 1, 2))
        let v = w * fftx(slice(x, 1 ..< n, 2))
        return cat(u + v, u - v)
    } else {
        return x
    }
}

Here's a breakdown of the real and complex vector operations in the function:

x is the complex input array.
omega is a complex exponential number. Real.i is the imaginary unit i.
Tests if the input x length is divisible by 2.
k is a real vector from 0, 1, 2,... n/2-1.
w is the complex vector of omega to power of vector k.
u is the complex result of a recursive call with even x.
v is the complex result of a recursive call with odd x.
The result is the concatenation of the complex vector addition and subtraction of u and v.
The recursion ends when there is one element in the input array. The fft of a single element is itself.

MATLAB

function y = fftx(x)
% FFTX Fast Finite Fourier Transform.
n = length(x);
omega = exp(-2*pi*i/n);
if rem(n,2) == 0
    % Recursive divide and conquer.
    k = (0:n/2-1)';
    w = omega.^k;
    u = fftx(x(1:2:n-1));
    v = w.*fftx(x(2:2:n));
    y = [u+v; u-v];
else
    y = x
end

Library Convention

The library works with existing Swift types, using only arrays and tuples. For convenience, these have been given type aliases for the underlying native types. Only the Real need be defined, the others are all derived from this type.

public typealias Real = Double
public typealias RealArray = [Real]
public typealias Complex = (Real, Real)
public typealias ComplexArray = ([Real], [Real])

Real Numbers

Real numbers are Double types.

Real Arrays

Real arrays are just a normal Swift Array of Double.

Complex Numbers

Complex numbers defined as a tuple of two real numbers, representing the real and imaginary parts of the number.

let c = (10.0, 2.0)

Complex Arrays

A complex array consists of a tuple of two real arrays. This arrangement is sometimes known as split complex.

let realArray = [1.0, 2.0, 3.0, 4.0]
let imagArray = [1.0, 2.0, 3.0, 4.0]
let complexArray = (realArray, imagArray)

The Imaginary Unit

The imaginary unit, i, is defined as an extension to Real, similar to other constants such as pi.

let c = 10 + Real.i

It can be used in any expression. This a a complex exponential:

let phi = 100.0
let c = exp(Real.i * 2 * Real.pi * phi)

Ranges

Ranges can be defined using the Swift Range or ClosedRange types, but with the addition of an optional by value. This has been implemented as an extension to the Array type.

Swift style:

let t = [Double](0...<100)
let s = [Double](1...100, 2)

VecLab style using the vector function:

let t = vector(0..<100)
let s = vector(1...100, 2)

Operators

Overloaded operators for scalar and vectors.

Operator	Description
+	Add
-	Subtract
*	Multiply
/	Divide
**	Power
*~	Right conjugate multiply: a * conj(b)
~*	Left conjugate multiply: conj(a) * b
-	Unary minus

Functions

Group	Functions
Arrays	cat, circshift, flip, length, ones, paddata, repelem, resize, slice, trimdata, zeros.
Basic	abs, all, any, cumsum, disp, iterate, norm, prod, sign, sinc, sum.
Complex	abs, angle, conj, cplxpair, imag, real, unwrap, wrapTo2Pi, wrapToPi.
Conversion	cart2pol, cart2sph, db2mag, db2pow, deg2rad, mag2db, pol2cart, pow2db, rad2deg, sph2cart.
Discrete	factor, factorial, gcd, isprime, lcm, nextprime, nochoosek, perms, prevprime, primes.
Exponents	exp, log, log2, log10, nextpow2, sqrt.
FFT	dft, dftr, fft, fftr, fftshift, fftsymmetric, idft, idftr, ifft, ifftr, ifftshift.
Filter	filter.
Integration	diff, gradient, trapz.
Interpolation	interp1, interpft, sincresample.
Modulo	ceil, fix, floor, mod, rem, round, trunc.
Optimization	fminbnd, fminsearch.
Power	pow.
Random	agwn, rand, randn, rng.
Smoothing	hampel, medfilt1.
Space	freqspace, linspace, logspace.
Special	besseli0.
Statistics	histcounts, max, maxindex, mean, median, min, minindex, mode, rms, stddev, variance.
Timing	tic, toc, timeit.
Trigonometry	acos, asin, atan, atan2, cos, sin, tan.
Window	blackman, blackmanharris, flattopwin, gausswin, hann, hamming, tukeywin, rectwin.

Description

Swift Tools 5.5.0

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Dependencies

None

Last updated: Mon Nov 18 2024 05:06:54 GMT-1000 (Hawaii-Aleutian Standard Time)