scilib 1.0.0

A scientific library for the Rust programming language.
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226

//!
//! # Fourier transform algorithms
//! 

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

use std::f64::consts::{     // Using std lib constants
    PI                      // Pi
};

use num_complex::Complex64; // Using complex numbers from the num crate

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

/// # Convolution
/// 
/// Computes the convolution of two vectors, including the edges.
/// 
/// ```
/// # use scilib::signal::convolve;
/// // Creating two vectors to convolve
/// let a1: Vec<f64> = vec![2.8, 2.5, 1.0, 0.5, 3.2, 0.25];
/// let a2: Vec<f64> = vec![0.25, 1.0, 0.5];
/// let res = convolve(&a1, &a2);
/// let expected: Vec<f64> = vec![0.7, 3.425, 4.15, 2.375, 1.8, 3.5125, 1.85, 0.125];
/// 
/// assert_eq!(res, expected);
/// ```
pub fn convolve<T>(a_i: &[T], b_i: &[T]) -> Vec<T>
where T: std::ops::Mul<Output = T> + std::ops::AddAssign + Default + Copy {

    // We check which box is the smallest
    let (a, b): (&[T], &[T]) = match a_i.len() < b_i.len() {
        true => (b_i, a_i),
        false => (a_i, b_i)
    };

    // We initialize our variables
    let mut sum: T;
    let l_a: usize = a.len();
    let l_b: usize = b.len();
    let mut res: Vec<T> = Vec::with_capacity(l_a + 2 * l_b - 2);    // Complete convolution length

    // Box b starts to slide over a
    for n in 0..(l_b - 1) {
        sum = T::default();
        for m in 0..=n {
            sum += a[n-m] * b[m];

        }
        res.push(sum);
    }

    // Box b covers a completely
    for n in (l_b - 1)..=(l_a - 1) {
        sum = T::default();
        for m in 0..l_b {
            sum += a[n-m] * b[m];
        }
        res.push(sum);
    }

    // Box b exits the slide over a
    for n in l_a..(l_a + l_b - 1) {
        sum = T::default();
        for m in (n - l_a + 1)..l_b {
            sum += a[n-m] * b[m];
        }
        res.push(sum);
    }

    res
}

/// # Exact convolution
/// 
/// Computes the convolution of two vectors, only for points where both vectors
/// are completely covered.
/// 
/// ```
/// # use scilib::signal::convolve_full;
/// // Creating two vectors to convolve
/// let a1: Vec<f64> = vec![2.8, 2.5, 1.0, 12.2];
/// let a2: Vec<f64> = vec![0.25, 1.0, 23.2, 0.25, 1.3, 2.1];
/// let res = convolve_full(&a1, &a2);
/// let expected: Vec<f64> = vec![62.75, 39.665, 292.42];
/// 
/// for (e, c) in expected.iter().zip(&res) {
///     assert!((e - c).abs() < 1.0e-10);
/// }
/// ```
pub fn convolve_full<T>(a_i: &[T], b_i: &[T]) -> Vec<T>
where T: std::ops::Mul<Output = T> + std::ops::AddAssign + Default + Copy {

    // We check which box is the smallest
    let (a, b): (&[T], &[T]) = match a_i.len() < b_i.len() {
        true => (b_i, a_i),
        false => (a_i, b_i)
    };

    // We initialize our variables
    let mut sum: T;
    let l_a: usize = a.len();
    let l_b: usize = b.len();
    let mut res: Vec<T> = Vec::with_capacity(l_a - l_b + 1);    // Exact convolution length

    // Box b covers a completely only
    for n in (l_b - 1)..=(l_a - 1) {
        sum = T::default();
        for m in 0..l_b {
            sum += a[n-m] * b[m];
        }
        res.push(sum);
    }

    res
}

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

/// # Fast Fourier transform
/// 
/// Computes the FFT for a one-dimensional array, based on the discrete Fourier transform.
/// This function accepts complex input. The FFT is computed using
/// [Bluestein's algorithm](https://en.wikipedia.org/wiki/Chirp_Z-transform#Bluestein.27s_algorithm).
/// 
/// ```
/// # use num_complex::Complex64;
/// # use scilib::range;
/// # use scilib::signal::fft;
/// // We create a Vec with the value sin(v) + cos(v)i
/// let r = range::linear(0.0, 10.0, 15);
/// let s: Vec<Complex64> = r.iter().map(|val| (Complex64::i() * val.cos()) + val.sin()).collect();
/// let res = fft(&s);
/// 
/// // Checking for some values...
/// assert!((res[0].re - 2.19227873).abs() < 1.0e-8 && (res[0].im - -0.64850436).abs() < 1.0e-8);
/// assert!((res[4].re - 0.73245756).abs() < 1.0e-8 && (res[4].im - 0.44922173).abs() < 1.0e-8);
/// assert!((res[9].re - -0.02709553).abs() < 1.0e-8 && (res[9].im - 1.02037473).abs() < 1.0e-8);
/// assert!((res[14].re - 4.77371673).abs() < 1.0e-8 && (res[14].im - -2.58964065).abs() < 1.0e-8);
/// ```
pub fn fft<T>(data: &[T]) -> Vec<Complex64>
where T: Into<Complex64> + Copy {

    let length: isize = data.len() as isize;

    // The pre-factor in the exponential terms
    let ipi_n: Complex64 = Complex64::i() * PI / length as f64;

    let mut res: Vec<Complex64> = Vec::with_capacity(data.len());

    let mut an: Complex64;
    let mut bkn: Complex64;
    let mut bk_star: Complex64;
    let mut sum_k: Complex64;

    for k in 0..length {
        sum_k = Complex64::default();
        bk_star = (k.pow(2) as f64 * ipi_n).exp().conj();

        for (n, val) in data.iter().enumerate() {
            an = (- (n.pow(2) as f64) * ipi_n).exp() * (*val).into();
            bkn = ((k - n as isize).pow(2) as f64 * ipi_n).exp();
            sum_k += an * bkn;
        }

        res.push(bk_star * sum_k);
    }

    res
}

/// # Inverse fast Fourier transform
/// 
/// Computes the IFFT for a one-dimensional array, based on the discrete Fourier transform.
/// This function accepts complex input. The FFT is computed using
/// [Bluestein's algorithm](https://en.wikipedia.org/wiki/Chirp_Z-transform#Bluestein.27s_algorithm).
/// 
/// This function yields `v = ifft(fft(v))`, within numerical errors.
/// 
/// ```
/// # use num_complex::Complex64;
/// # use scilib::range;
/// # use scilib::signal::{ fft, ifft };
/// // We create a Vec with the value sin(v) + cos(v)i
/// let r = range::linear(0.0, 10.0, 15);
/// let s: Vec<f64> = r.iter().map(|val| val.sin()).collect();
/// let res = ifft(&fft(&s));
/// 
/// for (ori, comp) in s.iter().zip(&res) {
///     assert!((ori - comp.re).abs() < 1.0e-14 && comp.im < 1.0e-14);
/// }
/// ```
pub fn ifft<T>(data: &[T]) -> Vec<Complex64>
where T: Into<Complex64> + Copy {

    let length: isize = data.len() as isize;
    let norm: f64 = length as f64;

    // The pre-factor in the exponential terms
    let ipi_n: Complex64 = -Complex64::i() * PI / length as f64;

    let mut res: Vec<Complex64> = Vec::with_capacity(data.len());

    let mut an: Complex64;
    let mut bkn: Complex64;
    let mut bk_star: Complex64;
    let mut sum_k: Complex64;

    for k in 0..length {
        sum_k = Complex64::default();
        bk_star = (k.pow(2) as f64 * ipi_n).exp().conj();

        for (n, val) in data.iter().enumerate() {
            an = (- (n.pow(2) as f64) * ipi_n).exp() * (*val).into();
            bkn = ((k - n as isize).pow(2) as f64 * ipi_n).exp();
            sum_k += an * bkn;
        }

        res.push(bk_star * sum_k / norm);
    }

    res
}

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////