VecLab

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

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

// 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
    }
}Code language: Swift (swift)

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

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

NumPy

import numpy as np

def fftx(x):
    """
    FFTX Fast Finite Fourier Transform.
    """
    n = len(x)
    omega = np.exp(-2j * np.pi / n)
    
    if n % 2 == 0:
        # Recursive divide and conquer
        k = np.arange(n//2)
        w = omega ** k
        u = fftx(x[::2])
        v = w * fftx(x[1::2])
        y = np.concatenate([u + v, u - v])
    else:
        y = x
    
    return y
```Code language: Python (python)

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
endCode language: Matlab (matlab)

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])Code language: Swift (swift)

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)Code language: JavaScript (javascript)

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)Code language: JavaScript (javascript)

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.iCode language: JavaScript (javascript)

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)Code language: JavaScript (javascript)

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)Code language: JavaScript (javascript)

VecLab style using the vector function:

let t = vector(0..<100) let s = vector(1...100, 2)Code language: JavaScript (javascript)

Github

Swift Package Index

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, wrap.
Conversion	cart2pol, cart2sph, db2mag, db2pow, deg2rad, mar2db, pol2cart, pow2db, rad2deg, sph2cart.
Discrete	factor, factorial, gcd, isprime, lcm, nextprime, nochoosek, perms, prevprime, primes.
Exponents	exp, log, log2, log10, nextpow2, sqrt.
FFT	dft, fft, fftshift, fftsymmetric, idft, ifft, 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, mean, median, min, mode, rms, stddev, variance.
Timing	tic, toc, timeit.
Trigonometry	acos, asin, atan, atan2, cos, sin, tan.
Window	blackman, blackmanharris, flattopwin, gausswin, hann, hamming, rectwin.