Add the forward FFT for real data

src/lib.rs

@@ -27,6 +27,63 @@ pub fn forward(data: &mut [c64]) {
     perform(data, n, false);
 }
 
+/// Perform the forward transform of real data.
+///
+/// The number of points should be a power of two. The data are replaced by the
+/// positive frequency half of their complex Fourier transform. The real-valued
+/// first and last components of the complex transform are returned as elements
+/// data[0] and data[1], respectively.
+///
+/// ## References
+///
+/// 1. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P.
+///    Flannery, “Numerical Recipes 3rd Edition: The Art of Scientific
+///    Computing,” Cambridge University Press, 2007.
+pub fn forward_real(data: &mut [f64]) {
+    use std::f64::consts::PI;
+    use std::slice::from_raw_parts_mut;
+
+    let n = data.len();
+    power_of_two!(n);
+
+    forward(unsafe {
+        from_raw_parts_mut(data.as_mut_ptr() as *mut _, n / 2)
+    });
+
+    let (c1, c2) = (0.5, -0.5);
+    let (wpr, wpi) = {
+        let theta = -PI / (n / 2) as f64;
+        let temp = (0.5 * theta).sin();
+        (-2.0 * temp * temp, theta.sin())
+    };
+    let mut wr = 1.0 + wpr;
+    let mut wi = wpi;
+    for i in 1..(n / 2 / 2) {
+        let i1 = i + i;
+        let i2 = 1 + i1;
+        let i3 = n - i1;
+        let i4 = 1 + i3;
+
+        let h1r =  c1 * (data[i1] + data[i3]);
+        let h1i =  c1 * (data[i2] - data[i4]);
+        let h2r = -c2 * (data[i2] + data[i4]);
+        let h2i =  c2 * (data[i1] - data[i3]);
+
+        data[i1] =  h1r + wr * h2r - wi * h2i;
+        data[i2] =  h1i + wr * h2i + wi * h2r;
+        data[i3] =  h1r - wr * h2r + wi * h2i;
+        data[i4] = -h1i + wr * h2i + wi * h2r;
+
+        let temp = wr;
+        wr = wr * wpr - wi * wpi + wr;
+        wi = wi * wpr + temp * wpi + wi;
+    }
+    data[n / 2 + 1] *= -1.0;
+    let temp = data[0];
+    data[0] = temp + data[1];
+    data[1] = temp - data[1];
+}
+
 /// Perform the inverse transform.
 ///
 /// The number of points should be a power of two.
@@ -40,7 +97,26 @@ pub fn inverse(data: &mut [c64], scaling: bool) {
     }
 }
 
-#[inline(always)]
+/// Expand a compressed representation produced by `forward_real`.
+pub fn unpack_real(data: &[f64]) -> Vec<c64> {
+    let n = data.len();
+    power_of_two!(n);
+
+    let mut cdata = Vec::with_capacity(n);
+    unsafe { cdata.set_len(n) };
+
+    cdata[0] = c64(data[0], 0.0);
+    for i in 1..(n / 2) {
+        cdata[i] = c64(data[2 * i], data[2 * i + 1]);
+    }
+    cdata[n / 2] = c64(data[1], 0.0);
+    for i in (n / 2 + 1)..n {
+        cdata[i] = c64(data[2 * (n - i)], -data[2 * (n - i) + 1]);
+    }
+
+    cdata
+}
+
 fn rearrange(data: &mut [c64], n: usize) {
     let mut target = 0;
     for position in 0..n {
@@ -56,7 +132,6 @@ fn rearrange(data: &mut [c64], n: usize) {
     }
 }
 
-#[inline(always)]
 fn perform(data: &mut [c64], n: usize, inverse: bool) {
     use std::f64::consts::PI;
 
@@ -83,10 +158,28 @@ fn perform(data: &mut [c64], n: usize, inverse: bool) {
     }
 }
 
-#[inline(always)]
 fn scale(data: &mut [c64], n: usize) {
     let factor = 1.0 / n as f64;
     for position in 0..n {
         data[position] = data[position] * factor;
     }
 }
+
+#[cfg(test)]
+mod tests {
+    use c64;
+
+    #[test]
+    fn unpack_real() {
+        let data = (0..4).map(|i| (i + 1) as f64).collect::<Vec<_>>();
+        assert_eq!(::unpack_real(&data), vec![
+            c64(1.0, 0.0), c64(3.0, 4.0), c64(2.0, 0.0), c64(3.0, -4.0),
+        ]);
+
+        let data = (0..8).map(|i| (i + 1) as f64).collect::<Vec<_>>();
+        assert_eq!(::unpack_real(&data), vec![
+            c64(1.0, 0.0), c64(3.0, 4.0), c64(5.0, 6.0), c64(7.0, 8.0),
+            c64(2.0, 0.0), c64(7.0, -8.0), c64(5.0, -6.0), c64(3.0, -4.0),
+        ]);
+    }
+}

tests/fixtures.rs

@@ -1,4 +1,4 @@
-pub const TIME_DATA: [f64; 256] = [
+pub const TIME_DATA: [f64; 2 * 128] = [
     0.242578298897752, 0.013469574513598,
     0.383138850044058, 0.414652690484492,
     0.067768972864267, 0.993126929734427,
@@ -129,7 +129,7 @@ pub const TIME_DATA: [f64; 256] = [
     0.452619465744411, 0.175360766321123,
 ];
 
-pub const FREQUENCY_DATA: [f64; 256] = [
+pub const FREQUENCY_DATA_FROM_COMPLEX: [f64; 2 * 128] = [
      6.750656266720340e+01,  6.279874768750683e+01,
      7.714838837855442e-02,  5.819389938527752e+00,
     -5.303978155417409e+00, -2.887549849606836e-01,
@@ -259,3 +259,262 @@ pub const FREQUENCY_DATA: [f64; 256] = [
     -1.213411748796469e+00, -2.824305441697994e+00,
     -2.916435892389174e+00,  2.258137395096740e+00,
 ];
+
+pub const FREQUENCY_DATA_FROM_REAL: [f64; 2 * 256] = [
+     1.303053103547102e+02,  0.000000000000000e+00,
+     2.581170396731584e+00,  1.851687150427199e-01,
+    -4.712992966498973e+00,  3.386970116076343e+00,
+    -3.526081640989057e+00, -6.639837287112855e-01,
+     2.470203860537831e+00,  1.880268489878626e+00,
+    -6.963879517225685e+00, -1.359220918694192e+00,
+    -3.729301277306694e+00,  1.188929523990150e+00,
+     9.470909676459618e-01, -1.133537735465806e+00,
+    -5.515738367157939e-01, -7.532580478755269e-01,
+    -3.549075039135673e+00, -7.723961671240310e+00,
+    -4.554423293554938e+00,  3.928695392899054e+00,
+    -3.376141283756011e-01,  3.873091478779083e-01,
+    -2.239363597649238e+00, -2.622333161216238e+00,
+    -1.209463421383755e+00,  2.081549368638949e+00,
+     2.787306134089560e+00, -4.406059514292826e-01,
+    -5.867907573418165e-01, -1.188914452808486e+00,
+    -1.143465327856217e+00,  3.381662456994260e+00,
+     3.502762058115909e-01,  1.939344791605027e+00,
+    -3.966269981614962e+00,  2.578889617469152e+00,
+     3.152848007718109e+00, -1.882857879463846e+00,
+    -3.804426226346088e+00, -2.360628066063453e+00,
+    -1.645310465960079e+00,  1.955745233096744e-01,
+     8.727435711779785e-02,  5.200789512413666e+00,
+     1.120142690086254e+00, -1.554454960578805e+00,
+    -3.568747367377906e-01, -1.708135182867247e+00,
+    -5.343117610163509e+00,  5.248479799162289e+00,
+     5.372292947253837e-01, -4.845214246729507e+00,
+     6.450571433636654e-02,  1.463660986831282e+00,
+     1.455587976273307e+00, -5.712855301747824e+00,
+    -1.042998871212450e-01, -1.101626615691207e+00,
+    -4.844351629884981e+00,  2.524159012105676e+00,
+    -6.773746016629306e+00, -1.640287552626490e+00,
+    -3.831851193928844e+00,  3.944330086139181e+00,
+    -1.492165597423655e+00, -5.490578426839169e+00,
+     1.689911156425059e+00,  8.268603075350915e-01,
+     1.497677302367308e+00,  4.248256795204092e+00,
+    -3.579498902285684e+00, -1.349669158977704e+00,
+     3.044651859669117e+00,  1.746167472193509e+00,
+     7.601446066545735e-01, -3.263458881514898e+00,
+     5.150135192916220e+00,  2.544519592918017e+00,
+    -7.037646240419698e-01,  2.824304911746514e+00,
+     4.772947230001412e+00,  5.943735485336072e-01,
+     3.480135296267813e+00, -1.372346334547339e+00,
+     4.028770258877906e+00,  7.235631972573928e+00,
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+     7.092481670250197e-01,  2.703433464882523e+00,
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+     2.931287676659427e+00,  4.844159203841096e+00,
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+    -2.126408608816460e+00, -6.496133889843878e-01,
+    -3.574988339289840e+00,  4.774591638105163e-01,
+    -8.057225542680848e-01, -1.502186010252237e+00,
+    -4.468302161079752e+00, -6.560811803588420e-01,
+    -3.901614280067265e+00,  6.215718415805858e+00,
+    -4.681753970646017e+00,  6.445270984734710e+00,
+     2.021391810490292e-02,  1.805385996360099e+00,
+    -8.937461326272578e-01, -7.588584122960702e-01,
+    -2.798464518993120e+00, -2.355688055046297e+00,
+     2.173264828603092e+00, -8.248948121629960e-01,
+     2.639682132748301e+00,  1.701633154204415e+00,
+    -9.959356075165555e-01,  2.054128218014691e+00,
+     1.118615654278817e+00,  1.238455571070932e+00,
+    -2.543476747185558e+00, -1.221476001211498e+00,
+    -3.719950405898309e+00, -3.654794911963335e+00,
+    -1.782620750385735e-01, -6.191786592630609e+00,
+     2.312768769350037e+00, -3.796429682741911e+00,
+     4.028770258877906e+00, -7.235631972573928e+00,
+     3.480135296267813e+00,  1.372346334547339e+00,
+     4.772947230001412e+00, -5.943735485336072e-01,
+    -7.037646240419698e-01, -2.824304911746514e+00,
+     5.150135192916220e+00, -2.544519592918017e+00,
+     7.601446066545735e-01,  3.263458881514898e+00,
+     3.044651859669117e+00, -1.746167472193509e+00,
+    -3.579498902285684e+00,  1.349669158977704e+00,
+     1.497677302367308e+00, -4.248256795204092e+00,
+     1.689911156425059e+00, -8.268603075350915e-01,
+    -1.492165597423655e+00,  5.490578426839169e+00,
+    -3.831851193928844e+00, -3.944330086139181e+00,
+    -6.773746016629306e+00,  1.640287552626490e+00,
+    -4.844351629884981e+00, -2.524159012105676e+00,
+    -1.042998871212450e-01,  1.101626615691207e+00,
+     1.455587976273307e+00,  5.712855301747824e+00,
+     6.450571433636654e-02, -1.463660986831282e+00,
+     5.372292947253837e-01,  4.845214246729507e+00,
+    -5.343117610163509e+00, -5.248479799162289e+00,
+    -3.568747367377906e-01,  1.708135182867247e+00,
+     1.120142690086254e+00,  1.554454960578805e+00,
+     8.727435711779785e-02, -5.200789512413666e+00,
+    -1.645310465960079e+00, -1.955745233096744e-01,
+    -3.804426226346088e+00,  2.360628066063453e+00,
+     3.152848007718109e+00,  1.882857879463846e+00,
+    -3.966269981614962e+00, -2.578889617469152e+00,
+     3.502762058115909e-01, -1.939344791605027e+00,
+    -1.143465327856217e+00, -3.381662456994260e+00,
+    -5.867907573418165e-01,  1.188914452808486e+00,
+     2.787306134089560e+00,  4.406059514292826e-01,
+    -1.209463421383755e+00, -2.081549368638949e+00,
+    -2.239363597649238e+00,  2.622333161216238e+00,
+    -3.376141283756011e-01, -3.873091478779083e-01,
+    -4.554423293554938e+00, -3.928695392899054e+00,
+    -3.549075039135673e+00,  7.723961671240310e+00,
+    -5.515738367157939e-01,  7.532580478755269e-01,
+     9.470909676459618e-01,  1.133537735465806e+00,
+    -3.729301277306694e+00, -1.188929523990150e+00,
+    -6.963879517225685e+00,  1.359220918694192e+00,
+     2.470203860537831e+00, -1.880268489878626e+00,
+    -3.526081640989057e+00,  6.639837287112855e-01,
+    -4.712992966498973e+00, -3.386970116076343e+00,
+     2.581170396731584e+00, -1.851687150427199e-01,
+];

tests/lib.rs

@@ -6,21 +6,44 @@ mod fixtures;
 #[test]
 fn forward() {
     let mut data = fixtures::TIME_DATA.to_vec();
-    fft::forward(reinterpret(&mut data));
-    assert::close(&data, &fixtures::FREQUENCY_DATA[..], 1e-14);
+    fft::forward(as_c64_mut(&mut data));
+    assert::close(&data, &fixtures::FREQUENCY_DATA_FROM_COMPLEX[..], 1e-14);
+}
+
+#[test]
+fn forward_real() {
+    let mut data = fixtures::TIME_DATA.to_vec();
+    {
+        let mut data = to_c64(&data);
+        fft::forward(&mut data);
+        assert::close(as_f64(&data), &fixtures::FREQUENCY_DATA_FROM_REAL[..], 1e-13);
+    }
+    {
+        fft::forward_real(&mut data);
+        let data = fft::unpack_real(&data);
+        assert::close(as_f64(&data), &fixtures::FREQUENCY_DATA_FROM_REAL[..], 1e-13);
+    }
 }
 
 #[test]
 fn inverse() {
-    let mut data = fixtures::FREQUENCY_DATA.to_vec();
-    fft::inverse(reinterpret(&mut data), true);
+    let mut data = fixtures::FREQUENCY_DATA_FROM_COMPLEX.to_vec();
+    fft::inverse(as_c64_mut(&mut data), true);
     assert::close(&data, &fixtures::TIME_DATA[..], 1e-14);
 }
 
-fn reinterpret<'l>(slice: &'l mut [f64]) -> &'l mut [fft::c64] {
+fn as_f64<'l>(slice: &'l [fft::c64]) -> &'l [f64] {
     unsafe {
-        let length = slice.len();
-        assert!(length % 2 == 0, "the number of elements should be even");
-        std::slice::from_raw_parts_mut(slice.as_mut_ptr() as *mut _, length / 2)
+        std::slice::from_raw_parts(slice.as_ptr() as *const _, 2 * slice.len())
     }
 }
+
+fn as_c64_mut<'l>(slice: &'l mut [f64]) -> &'l mut [fft::c64] {
+    unsafe {
+        std::slice::from_raw_parts_mut(slice.as_mut_ptr() as *mut _, slice.len() / 2)
+    }
+}
+
+fn to_c64(slice: &[f64]) -> Vec<fft::c64> {
+    slice.iter().map(|&re| fft::c64(re, 0.0)).collect()
+}