From 1e5d6501baafe5ca6061c730c4db103c2d0571fa Mon Sep 17 00:00:00 2001
From: Michel Crucifix <michel.crucifix@uclouvain.be>
Date: Sun, 6 Oct 2024 22:49:27 +0200
Subject: [PATCH] debugged
---
R/mfft_real_quatro.R | 116 +-
R/mfft_real_ter.R | 2 +-
notebook/Untitled.ipynb | 2425 +++++++++++++++++++--------------------
3 files changed, 1243 insertions(+), 1300 deletions(-)
diff --git a/R/mfft_real_quatro.R b/R/mfft_real_quatro.R
index b06b501..bee407e 100644
--- a/R/mfft_real_quatro.R
+++ b/R/mfft_real_quatro.R
@@ -6,6 +6,7 @@ Qsecond0 <- 2/pi^2 - 1./3.
analyse_quatro <- function(xdata, nfreq, fast = TRUE, nu = NULL){
+ # nu = c(0.0429, 0.0186)
# nu can be provided, in which case the frequencies
# are considered to be given
if (!is.null(nu))
@@ -20,10 +21,6 @@ analyse_quatro <- function(xdata, nfreq, fast = TRUE, nu = NULL){
nu[2*nfreq] <- NA
phase[2*nfreq] <- NA
}
- print ('nu 2')
- print(nu)
- print ('phase 2')
- print(phase)
} else
{
nu <- rep(NA,2*nfreq)
@@ -43,6 +40,7 @@ analyse_quatro <- function(xdata, nfreq, fast = TRUE, nu = NULL){
# x_m what remains of the signal at time step m
S <- rep(0,2*nfreq)
+ Prod <- rep(0,2*nfreq)
amp <- rep(NA,2*nfreq)
A <- matrix(0,2*nfreq,2*nfreq)
Q <- matrix(0,2*nfreq,2*nfreq)
@@ -185,14 +183,28 @@ analyse_quatro <- function(xdata, nfreq, fast = TRUE, nu = NULL){
A[m,] = A[m,] / sqrt(norm)
- # S[m] = x_orig %*% B[[m]] = sum hprod(x[[1]],f[[j]])*A[m,j]
+ Prod[m] = hprod(x[[1]],f[[m]])
+ # les prods sont corrects
+
S[m] = 0.
- for (j in seq(m)) S[m] = S[m] + hprod(x[[1]],f[[j]])*A[m,j]
+ for (j in seq(m)) S[m] = S[m] + Prod[j]*A[m,j]
+ # for (j in seq(m)) S[m] = S[m] + A[m,j] * hprod(x[[1]], f[[j]])
+
+ # les Sm sont les projections dans la nouvelle base
# not necessary, for verification only
- B[[m]]=0
- for (j in seq(m)) B[[m]] = B[[m]] + A[m,j] * f[[j]]
-
+ # computes the B and verify they are orthonormal
+ B[[m]]=0
+ for (j in seq(m)) B[[m]] = B[[m]] + A[m,j] * f[[j]]
+
+
+ #print ('S[m] calcule de deux facons')
+ #print (m)
+ #print (S[m])
+ #print (hprod(x[[1]], B[[m]]))
+ #print ('end test')
+
+ # for (j in seq(m)) print (sprintf("B%i%i : %3g", i, j, hprod(B[[j]], B[[m]])))
# if you are curious, the amplitude of the signal will be
# amp[j] = contribution of the sum( S[m]*B[m]) to the f[[m]]
# amp[j] = sum(m in seq(j)) S(m) * A[m,j]
@@ -200,38 +212,85 @@ analyse_quatro <- function(xdata, nfreq, fast = TRUE, nu = NULL){
# for (i in seq(m)) amp[j] = amp[j]+S[m]*A[m,j]
# and the residual signal
- # f[[m+1]] = f[[m]] - S[m] * B[[m]]
+ # x[[m+1]] = x[[m]] - S[m] * B[[m]]
+
x[[m+1]] = x[[m]]
- for (j in seq(m)) x[[m+1]] = x[[m+1]] - S[m] * A[m,j]*f[[j]]
+ # for (j in seq(m)) x[[m+1]] = x[[m+1]] - S[m] * A[m,j]*f[[j]]
+ x[[m+1]] = x[[m+1]] - S[m] * B[[m]]
+# print('residu final')
+# print(m)
+# print (x[[m+1]][seq(20)])
}
}
+ #print ('TESTESTEST')
+ #xtest <- 0
+ #xtest2 <- 0
+ #coeftests <- rep(0,m)
+ #phi1 <- 0.4 ; phi2 <- 0.1234
+ # coefreals <- c(cos(phi1), -sin(phi1), cos(phi2), -sin(phi2))
+ # for (j in seq(m)) xtest <- xtest + coefreals[j] * f[[j]]
+ # for (j in seq(m)) for (i in seq(j)) xtest2 <- xtest2 + S[j] * A[j,i] * f[[i]]
+ # for (j in seq(m)) for (i in seq(j)) coeftests[i] <- coeftests[i] + S[j] * A[j,i]
+ # print ((xtest - x[[1]])[seq(20)])
+ # print ('ENDTESTESTET1')
+ # print ((xtest2 - x[[1]])[seq(20)])
+ # print ('ENDTESTESTET2')
+ # print ("cofreals")
+ # print (coefreals)
+ # print ("coftests")
+ # print (coeftests)
+
+ # donc: j'ai demontre que les deux xtests sont les memes
+ # coefreals est la bonne solution
+ # et je sais ussi que B = sum A_mj f_j
+ # donc je devroais simplement avour les Sj A_mj
+
+
+ # at htis point I would like to make a littel check
+ # make A full
+ # is A Q t(A) = BtB orthonormal ?
+ #print ('TEST')
+ #for (m in seq(2,2*nfreq)) for (j in seq(m-1)) A[j,m]=A[m,j]
+ #for (m in seq(2,2*nfreq)) for (j in seq(m-1)) Q[j,m]=Q[m,j]
+ #print (A*t(Q)*t(A))
+ print ('END TEST')
+
+ print (A)
+ print (A %*% S)
# amplitudes
- for (m in seq(2*nfreq)){
- if (!is.na(nu[m])) {
- amp[m]=0;
- for (j in seq(m)) amp[m] = amp[m] + A[m,j]*S[m]
+ mmax <- 2*nfreq
+ if (is.na(nu[mmax])) mmax <- mmax - 1
+
+ amp[1:mmax] = 0;
+ for (m in seq(mmax)) for (j in seq(m)) amp[j] <- amp[j] + S[m] * A[m,j]
+# print ('ampp...')
+# print (amp)
+
+ amp2m <- amp;
+
+ for (m in seq(mmax)){
# if the perivous "m" was already of the same frequency, we can merge amplitudes and set phase
if ((m > 1) && (nu[m-1] == -nu[m])){
- print("phase computation")
- print (c(m-1, m))
- print (c(nu[m-1], nu[m]))
- print (c(amp[m-1], amp[m]))
- print("phase computation (2)")
phase[m] <- Arg (-1i * amp[m] + amp[m-1])
amp[m] <- sqrt(amp[m-1]^2 + amp[m]^2)
print ("phase")
- print (c(phase[m], amp[m]))
+ print (c(m, phase[m], amp[m], amp[m-1]))
amp[m-1] <- NA
phase[m-1] <- NA
}
- }
}
+# print ('amp2m')
+# print (amp2m)
+# print ('S')
+# print (S)
+# print ('PRod')
+# print (Prod)
# at this point we should have exactly nfreq non na values
# we print a message if this is not right, and in that case I suspect some debugging will be needed.
@@ -240,7 +299,12 @@ analyse_quatro <- function(xdata, nfreq, fast = TRUE, nu = NULL){
nu <- -nu[valid_frequencies]
amp <- amp[valid_frequencies]
phase <- phase[valid_frequencies]
- if (length(valid_frequencies) != nfreq) message (sprintf("something goes wrong : %i valid frequencies, and nfreq = %i"), valid_frequencies, nfreq)
+ if (length(valid_frequencies) != nfreq) message (sprintf("something goes wrong : %i valid frequencies, and nfreq = %i", valid_frequencies, nfreq))
+# print ("nu")
+# print(nu)
+# print ("amp")
+# print (amp)
+
OUT = data.frame(nu=nu, amp=amp, phase=phase)
return(OUT)
}
@@ -298,15 +362,15 @@ mfft_real_quatro <- function(xdata, nfreq=5, correction=1, fast=TRUE){
for (j in seq(nfreq)){
epsilon = OUT$amp[j] * Qprime(-2 * OUT$nu[j] * N2)*cos(2 * OUT$nu[j] * N2 + 2 * OUT$phase[j])
- print ('epsilon')
- print (c(epsilon,j, OUT$amp[j], OUT$nu[j], OUT$phase[j]))
+# print ('epsilon')
+# print (c(epsilon,j, OUT$amp[j], OUT$nu[j], OUT$phase[j]))
if ((j+1) <= nfreq) { for (s in seq(j+1, nfreq)) {
epsilon = epsilon + OUT$amp[s] *
(
Qprime( (OUT$nu[s] - OUT$nu[j])*N2)*cos((OUT$nu[j] - OUT$nu[s])*N2 + OUT$phase[j] - OUT$phase[s] ) -
Qprime(( OUT$nu[s] + OUT$nu[j])*N2)*cos((OUT$nu[j] + OUT$nu[s])*N2 + OUT$phase[j] + OUT$phase[s] ) )
}}
- print(sprintf("j= %i, OUT$nu[j]= %.4f, OUT$amp[j] = %.4f", j, OUT$nu[j], OUT$amp[j]))
+# print(sprintf("j= %i, OUT$nu[j]= %.4f, OUT$amp[j] = %.4f", j, OUT$nu[j], OUT$amp[j]))
epsilon = epsilon / Qsecond0 / N2 / OUT$amp[j]
OUT$nu[j] = OUT$nu[j] - epsilon
diff --git a/R/mfft_real_ter.R b/R/mfft_real_ter.R
index 171d76d..b9be57b 100644
--- a/R/mfft_real_ter.R
+++ b/R/mfft_real_ter.R
@@ -30,8 +30,8 @@ analyse <- function(xdata, nfreq, fast = TRUE, nu = NULL){
sinp <- sin(fp)*Qp
M <- 0.5 * matrix(c(
cosm + cosp ,
- -sinm + sinp ,
sinm + sinp ,
+ - sinm + sinp ,
cosm - cosp ), 2 , 2 )
return(M)
}} else
diff --git a/notebook/Untitled.ipynb b/notebook/Untitled.ipynb
index c00fdc0..483ac8a 100644
--- a/notebook/Untitled.ipynb
+++ b/notebook/Untitled.ipynb
@@ -10,7 +10,7 @@
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 66,
"id": "cf4e6e41",
"metadata": {},
"outputs": [],
@@ -30,7 +30,7 @@
},
{
"cell_type": "code",
- "execution_count": 2,
+ "execution_count": 3,
"id": "20d32dda",
"metadata": {},
"outputs": [],
@@ -44,7 +44,7 @@
},
{
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+ "execution_count": 4,
"id": "285d9179",
"metadata": {},
"outputs": [
@@ -86,28 +86,10 @@
},
{
"cell_type": "code",
- "execution_count": 4,
+ "execution_count": 110,
"id": "15010567",
"metadata": {},
"outputs": [
- {
- "data": {
- "text/html": [
- "50.05"
- ],
- "text/latex": [
- "50.05"
- ],
- "text/markdown": [
- "50.05"
- ],
- "text/plain": [
- "[1] 50.05"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
{
"data": {
"text/html": [
@@ -147,7 +129,7 @@
],
"source": [
"T <- (diff(times)[1] * length(times))/2 \n",
- "T\n",
+ "N <- T*2\n",
"length(times)/2. \n",
"analytical_estimate <- exp(1i*(w1-w2)*T) * Q((w1-w2)*T)\n",
"analytical_estimate"
@@ -163,7 +145,7 @@
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": 6,
"id": "0045eb04",
"metadata": {},
"outputs": [
@@ -208,7 +190,7 @@
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{
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+ "execution_count": 7,
"id": "3d958631",
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@@ -364,7 +346,7 @@
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+ "execution_count": 8,
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@@ -398,7 +380,7 @@
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+ "execution_count": 9,
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@@ -509,7 +491,7 @@
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+ "execution_count": 10,
"id": "b677cd5a",
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@@ -619,7 +601,7 @@
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@@ -729,7 +711,7 @@
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@@ -839,7 +821,7 @@
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- "execution_count": 12,
+ "execution_count": 13,
"id": "76775be8",
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@@ -963,7 +945,7 @@
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{
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+ "execution_count": 14,
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@@ -1085,7 +1067,7 @@
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@@ -1214,7 +1196,7 @@
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@@ -1294,7 +1276,7 @@
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@@ -1372,7 +1354,7 @@
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@@ -1521,7 +1503,7 @@
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@@ -1626,7 +1608,7 @@
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@@ -1636,7 +1618,7 @@
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@@ -1672,7 +1654,7 @@
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@@ -1758,7 +1740,7 @@
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@@ -1871,7 +1853,7 @@
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@@ -1917,7 +1899,7 @@
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+ "execution_count": 172,
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@@ -1946,7 +1928,7 @@
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+ "execution_count": 173,
"id": "fa60bdf6-7211-432f-91fd-cff8b1d8253a",
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@@ -1975,7 +1957,7 @@
},
{
"cell_type": "code",
- "execution_count": 30,
+ "execution_count": 174,
"id": "a2022719-e821-4124-bc81-3d1df326120b",
"metadata": {},
"outputs": [
@@ -2004,7 +1986,7 @@
},
{
"cell_type": "code",
- "execution_count": 31,
+ "execution_count": 175,
"id": "c50e05f5-919f-4fce-b03f-1333956b3460",
"metadata": {},
"outputs": [
@@ -2033,7 +2015,7 @@
},
{
"cell_type": "code",
- "execution_count": 34,
+ "execution_count": 176,
"id": "8b438f78-5a24-476e-b929-2db5360e64bc",
"metadata": {},
"outputs": [
@@ -2077,1034 +2059,139 @@
"M*4995265/50002740"
]
},
+ {
+ "cell_type": "markdown",
+ "id": "1f6adeec-9cbb-4c87-a81c-04c1e94e936c",
+ "metadata": {},
+ "source": [
+ "We will now attempt to understand better what is going one with the wrong phase estimation whene there are several signals"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 198,
+ "id": "c68a8c75-ad15-496c-a9fb-d3b6d47119d4",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1] 2.527826e-27\n",
+ "[1] 0.4\n",
+ "[1] 0.1234\n",
+ "[1] 1\n",
+ "[1] 1\n"
+ ]
+ }
+ ],
+ "source": [
+ "signal <- cos(omega1*times + phi1) + 1*cos(omega2*times + phi2)\n",
+ "v1 <- cos(omega1*times)\n",
+ "v2 <- sin(omega1*times)\n",
+ "v3 <- cos(omega2*times)\n",
+ "v4 <- sin(omega2*times)\n",
+ "\n",
+ "a1 <- cos(phi1)\n",
+ "a2 <- -sin(phi1)\n",
+ "a3 <- cos(phi2)\n",
+ "a4 <- -sin(phi2)\n",
+ "\n",
+ "check_reconstruction <- a1*v1 + a2*v2 + a3*v3 + a4*v4\n",
+ "\n",
+ "# a few sanity checks\n",
+ "\n",
+ "print(sum(signal - check_reconstruction)^2)\n",
+ "\n",
+ "print(atan(a2/(-a1)))\n",
+ "print(atan(a4/(-a3)))\n",
+ "print(a1^2 + a2^2)\n",
+ "print(a3^2 + a4^2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 178,
+ "id": "cc4e4508-9934-487e-8629-1c1fdc50c1e4",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "signal_single <- cos(omega1*times + phi1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 302,
+ "id": "21e04034-0030-4762-9938-c31a752af43d",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ " quarto <- function(f1,f2){\n",
+ " fm <- (f1-f2)*T\n",
+ " fp <- (f1+f2)*T\n",
+ " Qm <- ifelse(fm == 0, 1 , Q(fm))\n",
+ " Qp <- ifelse(fp == 0, 1 , Q(fp))\n",
+ " cosm <- cos(fm)*Qm\n",
+ " cosp <- cos(fp)*Qp\n",
+ " sinm <- sin(fm)*Qm\n",
+ " sinp <- sin(fp)*Qp\n",
+ " M <- 0.5 * matrix(c( \n",
+ " cosm + cosp , \n",
+ " sinm + sinp , \n",
+ " - sinm + sinp , \n",
+ " cosm - cosp ), 2 , 2 )\n",
+ " return(M)\n",
+ " }"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 303,
+ "id": "1baeafc8-f5de-4d8c-ac69-39fb7d0900e5",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "sinbit <- hprod(signal_single, sin(omega1*times), times)\n",
+ "cosbit <- hprod(signal_single, cos(omega1*times), times)\n",
+ "Qmat <- quarto(omega1,omega1)"
+ ]
+ },
{
"cell_type": "code",
- "execution_count": 35,
- "id": "cd62aed5-f81f-46b1-9925-a520ecd46017",
+ "execution_count": 304,
+ "id": "323d28ae-efa6-4ead-a7fe-57cff1a333b7",
"metadata": {},
"outputs": [
{
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class=list-inline><li>1</li><li>0.999079936120883</li><li>0.996321437518614</li><li>0.991729580183044</li><li>0.985312813718316</li><li>0.977082945794521</li><li>0.967055120420072</li><li>0.955247790074796</li><li>0.94168268175501</li><li>0.926384756993079</li><li>0.909382165924999</li><li>0.890706195490557</li><li>0.870391211861362</li><li>0.848474597202697</li><li>0.824996680885562</li><li>0.800000665275479</li><li>0.773532546234616</li><li>0.745641028483529</li><li>0.716377435978251</li><li>0.685795617467657</li><li>0.653951847404885</li><li>0.620904722395154</li><li>0.586715053370525</li><li>0.551445753690015</li><li>0.515161723370979</li><li>0.477929729664789</li><li>0.439818284196557</li><li>0.400897516894995</li><li>0.361239046944388</li><li>0.32091585099614</li><li>0.280002128882418</li><li>0.238573167078973</li><li>0.196705200168417</li><li>0.154475270558841</li><li>0.11196108671595</li><li>0.0692408801695507</li><li>0.0263932615575467</li><li>-0.01650292402768</li><li>-0.059368742124311</li><li>-0.102125314150588</li><li>-0.144693962551477</li><li>-0.186996355575427</li><li>-0.228954651414794</li><li>-0.270491641444715</li><li>-0.311530892296844</li><li>-0.35199688650651</li><li>-0.391815161474503</li><li>-0.43091244648777</li><li>-0.469216797546886</li><li>-0.506657729752205</li><li>-0.543166347005084</li><li>-0.5786754687855</li><li>-0.613119753772794</li><li>-0.646435820082049</li><li>-0.678562361894854</li><li>-0.709440262269843</li><li>-0.73901270192542</li><li>-0.767225263794497</li><li>-0.794026033158846</li><li>-0.819365693178819</li><li>-0.843197615642628</li><li>-0.865477946768217</li><li>-0.886165687899817</li><li>-0.905222770950718</li><li>-0.922614128453406</li><li>-0.938307758088188</li><li>-0.952274781571545</li><li>-0.964489497795866</li><li>-0.974929430122767</li><li>-0.983575367742979</li><li>-0.990411401026692</li><li>-0.995424950799303</li><li>-0.99860679148871</li><li>-0.999951068101537</li><li>-0.999455306997073</li><li>-0.99712042043909</li><li>-0.992950704917154</li><li>-0.986953833240542</li><li>-0.979140840419288</li><li>-0.969526103358357</li><li>-0.958127314402303</li><li>-0.944965448779096</li><li>-0.930064726003017</li><li>-0.913452565307665</li><li>-0.89515953519106</li><li>-0.875219297165702</li><li>-0.853668543817087</li><li>-0.830546931284662</li><li>-0.805897006289464</li><li>-0.779764127842714</li><li>-0.752196383779446</li><li>-0.723244502270742</li><li>-0.692961758477418</li><li>-0.661403876516926</li><li>-0.62862892692385</li><li>-0.594697219792713</li><li>-0.55967119379969</li><li>-0.523615301307472</li><li>-0.486595889764683</li><li>-0.448681079618095</li><li>-0.409940638962306</li><li>-0.370445855157534</li><li>-0.330269403651764</li><li>-0.289485214248639</li><li>-0.248168335067177</li><li>-0.206394794443643</li><li>-0.164241461029697</li><li>-0.121785902344259</li><li>-0.0791062420393542</li><li>-0.0362810161426243</li><li>0.00661097145900751</li><li>0.0494907940285487</li><li>0.09227954721422</li><li>0.134898494243546</li><li>0.177269210809069</li><li>0.219313729379103</li><li>0.260954682667946</li><li>0.302115446001568</li><li>0.342720278316812</li><li>0.382694461534615</li><li>0.421964438050826</li><li>0.460457946091593</li><li>0.498104152684256</li><li>0.534833783999072</li><li>0.570579252821911</li><li>0.605274782923359</li><li>0.638856530095391</li><li>0.671262699632865</li><li>0.702433660043676</li><li>0.732312052778324</li><li>0.760842897776964</li><li>0.787973694639749</li><li>0.813654519234269</li><li>0.837838115562333</li><li>0.860479982717044</li><li>0.881538456770152</li><li>0.900974787439006</li><li>0.918753209392024</li><li>0.934841008061474</li><li>0.949208579842452</li><li>0.961829486567309</li><li>0.972680504155245</li><li>0.981741665347591</li><li>0.988996296450116</li><li>0.994431048014751</li><li>0.998035919404285</li><li>0.999804277194807</li><li>0.999732867382062</li><li>0.997821821369228</li><li>0.99407465572512</li><li>0.988498265713255</li><li>0.981102912603684</li><li>0.971902204790947</li><li>0.960913072752884</li><li>0.948155737896398</li><li>0.93365367534748</li><li>0.917433570753978</li><li>0.899525271180594</li><li>0.879961730186478</li><li>0.858778947186462</li><li>0.836015901207541</li><li>0.811714479162481</li><li>0.785919398772555</li><li>0.758678126281212</li><li>0.730040789110133</li><li>0.70006008361837</li><li>0.668791178134308</li><li>0.6362916114389</li><li>0.60262118688695</li><li>0.56784186236131</li><li>0.532017636262451</li><li>0.495214429743235</li><li>0.457499965405572</li><li>0.418943642682172</li><li>0.379616410132738</li><li>0.339590634889538</li><li>0.29893996949264</li><li>0.257739216359835</li><li>0.216064190140618</li><li>0.173991578207564</li><li>0.131598799541752</li><li>0.0889638622719511</li><li>0.0461652201297063</li><li>0.00328162808443396</li><li>-0.0396080025757696</li><li>-0.0824247494509826</li><li>-0.125089824256767</li><li>-0.167524717804663</li><li>-0.209651344469138</li><li>-0.251392185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+ "<table class=\"dataframe\">\n",
+ "<caption>A matrix: 2 × 2 of type dbl</caption>\n",
+ "<tbody>\n",
+ "\t<tr><td> 5.000274e-01</td><td>-4.656294e-05</td></tr>\n",
+ "\t<tr><td>-4.656294e-05</td><td> 4.999726e-01</td></tr>\n",
+ "</tbody>\n",
+ "</table>\n"
],
"text/latex": [
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- "\\item 0.811714479162481\n",
- "\\item 0.785919398772555\n",
- "\\item 0.758678126281212\n",
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- "\\item -0.562427197806194\n",
- "\\item -0.597370577585482\n",
- "\\item -0.631214719183006\n",
- "\\item ⋯\n",
- "\\item -0.981121386747115\n",
- "\\item -0.988512704384468\n",
- "\\item -0.994085032355116\n",
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- "\\item -0.53475308532285\n",
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- "\\item 0.998601747401947\n",
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- "\\item 0.990398202781889\n",
- "\\item 0.983558125105396\n",
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- "\\item 0.964464268942509\n",
- "\\item 0.952245625590845\n",
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- "\\item 0.886121430621864\n",
- "\\item 0.865430100710767\n",
- "\\item 0.843146268848539\n",
- "\\item 0.819310940132752\n",
- "\\item 0.793967974613405\n",
- "\\item 0.767164006584824\n",
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- "\\item 0.709372951573506\n",
- "\\item 0.678492207515269\n",
- "\\item 0.646362951112247\n",
- "\\item 0.61304430430099\n",
- "\\item 0.578597577648355\n",
- "\\item 0.543086157532248\n",
- "\\item 0.506575389502547\n",
- "\\item 0.469132458036984\n",
- "\\item 0.430826262913104\n",
- "\\item 0.391727292423855\n",
- "\\item 0.351907493670166\n",
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- "\\item -0.280093810745427\n",
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- "\\item -0.967079428013591\n",
- "\\item -0.977103270074187\n",
- "\\item -0.985329117284859\n",
- "\\item -0.991741833035817\n",
- "\\item -0.996329617110805\n",
- "\\item -0.999084027400996\n",
- "\\item -0.999999995439557\n",
- "\\item -0.999075835728275\n",
- "\\item -0.996313248839089\n",
- "\\item -0.991717318284818\n",
- "\\item -0.985296501164848\n",
- "\\item -0.977062612602993\n",
- "\\item -0.967030804006154\n",
- "\\item -0.955219535183797\n",
- "\\item -0.941650540379541\n",
- "\\item -0.926348788277374\n",
- "\\item -0.90934243605609\n",
- "\\item -0.890662777576479\n",
- "\\item -0.870344185796626\n",
- "\\item -0.848424049521271\n",
- "\\item -0.824942704601635\n",
- "\\item -0.799943359712312\n",
- "\\item -0.773472016841758\n",
- "\\item -0.745577386642805\n",
- "\\item -0.716310798798779\n",
- "\\item -0.685726107570356\n",
- "\\item -0.653879592696851\n",
- "\\item -0.620829855834276\n",
- "\\item -0.58663771272084\n",
- "\\item -0.551366081268205\n",
- "\\item -0.515079865784473\n",
- "\\item -0.477845837542009\n",
- "\\item -0.439732511909722\n",
- "\\item -0.400810022276078\n",
- "\\item -0.361149990994658\n",
- "\\item -0.320825397589923\n",
- "\\item -0.279910444465543\n",
- "\\item -0.238480420362475\n",
- "\\item -0.19661156181811\n",
- "\\item -0.154380912881246\n",
- "\\item -0.111866183341248\n",
- "\\item -0.0691456057320828\n",
- "\\item -0.0262977913744432\n",
- "\\item 0.0165984142790776\n",
- "\\item 0.0594640767297409\n",
- "\\item 0.102220317682203\n",
- "\\item 0.144788460190643\n",
- "\\item 0.187090173434413\n",
- "\\item 0.229047616856748\n",
- "\\item 0.270583583401346\n",
- "\\item 0.311621641583212\n",
- "\\item 0.352086276132335\n",
- "\\item 0.391903026951452\n",
- "\\item 0.430998626132134\n",
- "\\item 0.469301132777109\n",
- "\\end{enumerate*}\n"
+ "A matrix: 2 × 2 of type dbl\n",
+ "\\begin{tabular}{ll}\n",
+ "\t 5.000274e-01 & -4.656294e-05\\\\\n",
+ "\t -4.656294e-05 & 4.999726e-01\\\\\n",
+ "\\end{tabular}\n"
],
"text/markdown": [
- "1. 1\n",
- "2. 0.999079936120883\n",
- "3. 0.996321437518614\n",
- "4. 0.991729580183044\n",
- "5. 0.985312813718316\n",
- "6. 0.977082945794521\n",
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- "8. 0.955247790074796\n",
- "9. 0.94168268175501\n",
- "10. 0.926384756993079\n",
- "11. 0.909382165924999\n",
- "12. 0.890706195490557\n",
- "13. 0.870391211861362\n",
- "14. 0.848474597202697\n",
- "15. 0.824996680885562\n",
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- "25. 0.515161723370979\n",
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- "32. 0.238573167078973\n",
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- "198. -0.562427197806194\n",
- "199. -0.597370577585482\n",
- "200. -0.631214719183006\n",
- "201. ⋯\n",
- "202. -0.981121386747115\n",
- "203. -0.988512704384468\n",
- "204. -0.994085032355116\n",
- "205. -0.997828116863683\n",
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- "359. -0.967030804006154\n",
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- "361. -0.941650540379541\n",
- "362. -0.926348788277374\n",
- "363. -0.90934243605609\n",
- "364. -0.890662777576479\n",
- "365. -0.870344185796626\n",
- "366. -0.848424049521271\n",
- "367. -0.824942704601635\n",
- "368. -0.799943359712312\n",
- "369. -0.773472016841758\n",
- "370. -0.745577386642805\n",
- "371. -0.716310798798779\n",
- "372. -0.685726107570356\n",
- "373. -0.653879592696851\n",
- "374. -0.620829855834276\n",
- "375. -0.58663771272084\n",
- "376. -0.551366081268205\n",
- "377. -0.515079865784473\n",
- "378. -0.477845837542009\n",
- "379. -0.439732511909722\n",
- "380. -0.400810022276078\n",
- "381. -0.361149990994658\n",
- "382. -0.320825397589923\n",
- "383. -0.279910444465543\n",
- "384. -0.238480420362475\n",
- "385. -0.19661156181811\n",
- "386. -0.154380912881246\n",
- "387. -0.111866183341248\n",
- "388. -0.0691456057320828\n",
- "389. -0.0262977913744432\n",
- "390. 0.0165984142790776\n",
- "391. 0.0594640767297409\n",
- "392. 0.102220317682203\n",
- "393. 0.144788460190643\n",
- "394. 0.187090173434413\n",
- "395. 0.229047616856748\n",
- "396. 0.270583583401346\n",
- "397. 0.311621641583212\n",
- "398. 0.352086276132335\n",
- "399. 0.391903026951452\n",
- "400. 0.430998626132134\n",
- "401. 0.469301132777109\n",
"\n",
+ "A matrix: 2 × 2 of type dbl\n",
+ "\n",
+ "| 5.000274e-01 | -4.656294e-05 |\n",
+ "| -4.656294e-05 | 4.999726e-01 |\n",
"\n"
],
"text/plain": [
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- " [11] 9.093822e-01 8.907062e-01 8.703912e-01 8.484746e-01 8.249967e-01\n",
- " [16] 8.000007e-01 7.735325e-01 7.456410e-01 7.163774e-01 6.857956e-01\n",
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- " [261] 1.577638e-01 1.999684e-01 2.418051e-01 2.831968e-01 3.240674e-01\n",
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- " [401] -1.184806e-01 -7.578686e-02 -3.295363e-02 9.940242e-03 5.281582e-02\n",
- " [406] 9.559421e-02 1.381967e-01 1.805449e-01 2.225608e-01 2.641673e-01\n",
- " [411] 3.052876e-01 3.458461e-01 3.857683e-01 4.249806e-01 4.634108e-01\n",
- " [416] 5.009884e-01 5.376440e-01 5.733103e-01 6.079217e-01 6.414144e-01\n",
- " [421] 6.737268e-01 7.047995e-01 7.345752e-01 7.629992e-01 7.900193e-01\n",
- " [426] 8.155855e-01 8.396510e-01 8.621715e-01 8.831054e-01 9.024143e-01\n",
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- " [436] 9.823695e-01 9.894834e-01 9.947764e-01 9.982390e-01 9.998646e-01\n",
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- " [451] 8.980658e-01 8.783752e-01 8.570684e-01 8.341844e-01 8.097654e-01\n",
- " [456] 7.838564e-01 7.565049e-01 7.277614e-01 6.976787e-01 6.663122e-01\n",
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- " [911] 2.288617e-01 1.869025e-01 1.445995e-01 1.020303e-01 5.927341e-02\n",
- " [916] 1.640743e-02 -2.648873e-02 -6.933615e-02 -1.120560e-01 -1.545696e-01\n",
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- " [936] -7.457047e-01 -7.735931e-01 -8.000580e-01 -8.250506e-01 -8.485251e-01\n",
- " [941] -8.704382e-01 -8.907496e-01 -9.094219e-01 -9.264207e-01 -9.417148e-01\n",
- " [946] -9.552760e-01 -9.670794e-01 -9.771033e-01 -9.853291e-01 -9.917418e-01\n",
- " [951] -9.963296e-01 -9.990840e-01 -1.000000e+00 -9.990758e-01 -9.963132e-01\n",
- " [956] -9.917173e-01 -9.852965e-01 -9.770626e-01 -9.670308e-01 -9.552195e-01\n",
- " [961] -9.416505e-01 -9.263488e-01 -9.093424e-01 -8.906628e-01 -8.703442e-01\n",
- " [966] -8.484240e-01 -8.249427e-01 -7.999434e-01 -7.734720e-01 -7.455774e-01\n",
- " [971] -7.163108e-01 -6.857261e-01 -6.538796e-01 -6.208299e-01 -5.866377e-01\n",
- " [976] -5.513661e-01 -5.150799e-01 -4.778458e-01 -4.397325e-01 -4.008100e-01\n",
- " [981] -3.611500e-01 -3.208254e-01 -2.799104e-01 -2.384804e-01 -1.966116e-01\n",
- " [986] -1.543809e-01 -1.118662e-01 -6.914561e-02 -2.629779e-02 1.659841e-02\n",
- " [991] 5.946408e-02 1.022203e-01 1.447885e-01 1.870902e-01 2.290476e-01\n",
- " [996] 2.705836e-01 3.116216e-01 3.520863e-01 3.919030e-01 4.309986e-01\n",
- "[1001] 4.693011e-01"
+ " [,1] [,2] \n",
+ "[1,] 5.000274e-01 -4.656294e-05\n",
+ "[2,] -4.656294e-05 4.999726e-01"
]
},
"metadata": {},
@@ -3112,234 +2199,1026 @@
}
],
"source": [
- "cos(omega1*times)"
+ "Qmat"
]
},
{
"cell_type": "code",
- "execution_count": 38,
- "id": "579c47bd-3334-4836-bd1d-f232c572b20a",
+ "execution_count": 305,
+ "id": "d35c9e03-8c82-4304-90cf-0648900e2f0c",
"metadata": {},
"outputs": [
{
- "name": "stdout",
- "output_type": "stream",
- "text": [
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- " [16] 4.429179e-03 5.038909e-03 5.687839e-03 6.375945e-03 7.103199e-03\n",
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- " [241] 9.357061e-01 9.419712e-01 9.482386e-01 9.545081e-01 9.607794e-01\n",
- " [246] 9.670522e-01 9.733263e-01 9.796015e-01 9.858774e-01 9.921539e-01\n",
- " [251] 9.984308e-01 1.004708e+00 1.010984e+00 1.017261e+00 1.023536e+00\n",
- " [256] 1.029811e+00 1.036084e+00 1.042356e+00 1.048627e+00 1.054895e+00\n",
- " [261] 1.061162e+00 1.067426e+00 1.073687e+00 1.079945e+00 1.086200e+00\n",
- " [266] 1.092452e+00 1.098700e+00 1.104945e+00 1.111185e+00 1.117421e+00\n",
- " [271] 1.123652e+00 1.129878e+00 1.136099e+00 1.142315e+00 1.148525e+00\n",
- " [276] 1.154729e+00 1.160928e+00 1.167119e+00 1.173305e+00 1.179483e+00\n",
- " [281] 1.185655e+00 1.191819e+00 1.197975e+00 1.204124e+00 1.210265e+00\n",
- " [286] 1.216397e+00 1.222521e+00 1.228636e+00 1.234742e+00 1.240839e+00\n",
- " [291] 1.246926e+00 1.253004e+00 1.259072e+00 1.265129e+00 1.271176e+00\n",
- " [296] 1.277212e+00 1.283238e+00 1.289252e+00 1.295255e+00 1.301246e+00\n",
- " [301] 1.307226e+00 1.313193e+00 1.319148e+00 1.325090e+00 1.331020e+00\n",
- " [306] 1.336936e+00 1.342839e+00 1.348729e+00 1.354605e+00 1.360467e+00\n",
- " [311] 1.366315e+00 1.372148e+00 1.377967e+00 1.383770e+00 1.389559e+00\n",
- " [316] 1.395333e+00 1.401090e+00 1.406832e+00 1.412558e+00 1.418268e+00\n",
- " [321] 1.423961e+00 1.429637e+00 1.435297e+00 1.440939e+00 1.446564e+00\n",
- " [326] 1.452172e+00 1.457762e+00 1.463333e+00 1.468886e+00 1.474421e+00\n",
- " [331] 1.479937e+00 1.485435e+00 1.490913e+00 1.496372e+00 1.501811e+00\n",
- " [336] 1.507230e+00 1.512630e+00 1.518009e+00 1.523368e+00 1.528706e+00\n",
- " [341] 1.534024e+00 1.539320e+00 1.544595e+00 1.549849e+00 1.555081e+00\n",
- " [346] 1.560291e+00 1.565479e+00 1.570645e+00 1.575788e+00 1.580909e+00\n",
- " [351] 1.586006e+00 1.591081e+00 1.596132e+00 1.601160e+00 1.606165e+00\n",
- " [356] 1.611145e+00 1.616101e+00 1.621033e+00 1.625940e+00 1.630823e+00\n",
- " [361] 1.635681e+00 1.640514e+00 1.645322e+00 1.650104e+00 1.654861e+00\n",
- " [366] 1.659592e+00 1.664296e+00 1.668975e+00 1.673627e+00 1.678253e+00\n",
- " [371] 1.682852e+00 1.687424e+00 1.691969e+00 1.696487e+00 1.700978e+00\n",
- " [376] 1.705440e+00 1.709875e+00 1.714282e+00 1.718661e+00 1.723012e+00\n",
- " [381] 1.727334e+00 1.731627e+00 1.735892e+00 1.740127e+00 1.744334e+00\n",
- " [386] 1.748511e+00 1.752658e+00 1.756776e+00 1.760864e+00 1.764923e+00\n",
- " [391] 1.768951e+00 1.772948e+00 1.776915e+00 1.780852e+00 1.784758e+00\n",
- " [396] 1.788633e+00 1.792477e+00 1.796290e+00 1.800071e+00 1.803821e+00\n",
- " [401] 1.807539e+00 1.811225e+00 1.814879e+00 1.818501e+00 1.822091e+00\n",
- " [406] 1.825649e+00 1.829174e+00 1.832666e+00 1.836126e+00 1.839553e+00\n",
- " [411] 1.842946e+00 1.846306e+00 1.849633e+00 1.852927e+00 1.856187e+00\n",
- " [416] 1.859413e+00 1.862605e+00 1.865764e+00 1.868888e+00 1.871978e+00\n",
- " [421] 1.875034e+00 1.878055e+00 1.881041e+00 1.883993e+00 1.886910e+00\n",
- " [426] 1.889792e+00 1.892639e+00 1.895451e+00 1.898228e+00 1.900969e+00\n",
- " [431] 1.903675e+00 1.906345e+00 1.908979e+00 1.911578e+00 1.914140e+00\n",
- " [436] 1.916667e+00 1.919157e+00 1.921612e+00 1.924030e+00 1.926411e+00\n",
- " [441] 1.928756e+00 1.931065e+00 1.933337e+00 1.935572e+00 1.937770e+00\n",
- " [446] 1.939931e+00 1.942055e+00 1.944142e+00 1.946192e+00 1.948205e+00\n",
- " [451] 1.950180e+00 1.952118e+00 1.954018e+00 1.955881e+00 1.957706e+00\n",
- " [456] 1.959493e+00 1.961242e+00 1.962954e+00 1.964628e+00 1.966263e+00\n",
- " [461] 1.967861e+00 1.969421e+00 1.970942e+00 1.972425e+00 1.973870e+00\n",
- " [466] 1.975276e+00 1.976644e+00 1.977973e+00 1.979264e+00 1.980516e+00\n",
- " [471] 1.981730e+00 1.982905e+00 1.984041e+00 1.985139e+00 1.986198e+00\n",
- " [476] 1.987218e+00 1.988198e+00 1.989141e+00 1.990044e+00 1.990908e+00\n",
- " [481] 1.991733e+00 1.992519e+00 1.993265e+00 1.993973e+00 1.994642e+00\n",
- " [486] 1.995271e+00 1.995861e+00 1.996412e+00 1.996923e+00 1.997396e+00\n",
- " [491] 1.997829e+00 1.998223e+00 1.998577e+00 1.998892e+00 1.999168e+00\n",
- " [496] 1.999404e+00 1.999601e+00 1.999759e+00 1.999877e+00 1.999956e+00\n",
- " [501] 1.999995e+00 1.999995e+00 1.999956e+00 1.999877e+00 1.999759e+00\n",
- " [506] 1.999601e+00 1.999404e+00 1.999168e+00 1.998892e+00 1.998577e+00\n",
- " [511] 1.998223e+00 1.997829e+00 1.997396e+00 1.996923e+00 1.996412e+00\n",
- " [516] 1.995861e+00 1.995271e+00 1.994642e+00 1.993973e+00 1.993265e+00\n",
- " [521] 1.992519e+00 1.991733e+00 1.990908e+00 1.990044e+00 1.989141e+00\n",
- " [526] 1.988198e+00 1.987218e+00 1.986198e+00 1.985139e+00 1.984041e+00\n",
- " [531] 1.982905e+00 1.981730e+00 1.980516e+00 1.979264e+00 1.977973e+00\n",
- " [536] 1.976644e+00 1.975276e+00 1.973870e+00 1.972425e+00 1.970942e+00\n",
- " [541] 1.969421e+00 1.967861e+00 1.966263e+00 1.964628e+00 1.962954e+00\n",
- " [546] 1.961242e+00 1.959493e+00 1.957706e+00 1.955881e+00 1.954018e+00\n",
- " [551] 1.952118e+00 1.950180e+00 1.948205e+00 1.946192e+00 1.944142e+00\n",
- " [556] 1.942055e+00 1.939931e+00 1.937770e+00 1.935572e+00 1.933337e+00\n",
- " [561] 1.931065e+00 1.928756e+00 1.926411e+00 1.924030e+00 1.921612e+00\n",
- " [566] 1.919157e+00 1.916667e+00 1.914140e+00 1.911578e+00 1.908979e+00\n",
- " [571] 1.906345e+00 1.903675e+00 1.900969e+00 1.898228e+00 1.895451e+00\n",
- " [576] 1.892639e+00 1.889792e+00 1.886910e+00 1.883993e+00 1.881041e+00\n",
- " [581] 1.878055e+00 1.875034e+00 1.871978e+00 1.868888e+00 1.865764e+00\n",
- " [586] 1.862605e+00 1.859413e+00 1.856187e+00 1.852927e+00 1.849633e+00\n",
- " [591] 1.846306e+00 1.842946e+00 1.839553e+00 1.836126e+00 1.832666e+00\n",
- " [596] 1.829174e+00 1.825649e+00 1.822091e+00 1.818501e+00 1.814879e+00\n",
- " [601] 1.811225e+00 1.807539e+00 1.803821e+00 1.800071e+00 1.796290e+00\n",
- " [606] 1.792477e+00 1.788633e+00 1.784758e+00 1.780852e+00 1.776915e+00\n",
- " [611] 1.772948e+00 1.768951e+00 1.764923e+00 1.760864e+00 1.756776e+00\n",
- " [616] 1.752658e+00 1.748511e+00 1.744334e+00 1.740127e+00 1.735892e+00\n",
- " [621] 1.731627e+00 1.727334e+00 1.723012e+00 1.718661e+00 1.714282e+00\n",
- " [626] 1.709875e+00 1.705440e+00 1.700978e+00 1.696487e+00 1.691969e+00\n",
- " [631] 1.687424e+00 1.682852e+00 1.678253e+00 1.673627e+00 1.668975e+00\n",
- " [636] 1.664296e+00 1.659592e+00 1.654861e+00 1.650104e+00 1.645322e+00\n",
- " [641] 1.640514e+00 1.635681e+00 1.630823e+00 1.625940e+00 1.621033e+00\n",
- " [646] 1.616101e+00 1.611145e+00 1.606165e+00 1.601160e+00 1.596132e+00\n",
- " [651] 1.591081e+00 1.586006e+00 1.580909e+00 1.575788e+00 1.570645e+00\n",
- " [656] 1.565479e+00 1.560291e+00 1.555081e+00 1.549849e+00 1.544595e+00\n",
- " [661] 1.539320e+00 1.534024e+00 1.528706e+00 1.523368e+00 1.518009e+00\n",
- " [666] 1.512630e+00 1.507230e+00 1.501811e+00 1.496372e+00 1.490913e+00\n",
- " [671] 1.485435e+00 1.479937e+00 1.474421e+00 1.468886e+00 1.463333e+00\n",
- " [676] 1.457762e+00 1.452172e+00 1.446564e+00 1.440939e+00 1.435297e+00\n",
- " [681] 1.429637e+00 1.423961e+00 1.418268e+00 1.412558e+00 1.406832e+00\n",
- " [686] 1.401090e+00 1.395333e+00 1.389559e+00 1.383770e+00 1.377967e+00\n",
- " [691] 1.372148e+00 1.366315e+00 1.360467e+00 1.354605e+00 1.348729e+00\n",
- " [696] 1.342839e+00 1.336936e+00 1.331020e+00 1.325090e+00 1.319148e+00\n",
- " [701] 1.313193e+00 1.307226e+00 1.301246e+00 1.295255e+00 1.289252e+00\n",
- " [706] 1.283238e+00 1.277212e+00 1.271176e+00 1.265129e+00 1.259072e+00\n",
- " [711] 1.253004e+00 1.246926e+00 1.240839e+00 1.234742e+00 1.228636e+00\n",
- " [716] 1.222521e+00 1.216397e+00 1.210265e+00 1.204124e+00 1.197975e+00\n",
- " [721] 1.191819e+00 1.185655e+00 1.179483e+00 1.173305e+00 1.167119e+00\n",
- " [726] 1.160928e+00 1.154729e+00 1.148525e+00 1.142315e+00 1.136099e+00\n",
- " [731] 1.129878e+00 1.123652e+00 1.117421e+00 1.111185e+00 1.104945e+00\n",
- " [736] 1.098700e+00 1.092452e+00 1.086200e+00 1.079945e+00 1.073687e+00\n",
- " [741] 1.067426e+00 1.061162e+00 1.054895e+00 1.048627e+00 1.042356e+00\n",
- " [746] 1.036084e+00 1.029811e+00 1.023536e+00 1.017261e+00 1.010984e+00\n",
- " [751] 1.004708e+00 9.984308e-01 9.921539e-01 9.858774e-01 9.796015e-01\n",
- " [756] 9.733263e-01 9.670522e-01 9.607794e-01 9.545081e-01 9.482386e-01\n",
- " [761] 9.419712e-01 9.357061e-01 9.294435e-01 9.231836e-01 9.169268e-01\n",
- " [766] 9.106733e-01 9.044233e-01 8.981770e-01 8.919348e-01 8.856968e-01\n",
- " [771] 8.794633e-01 8.732346e-01 8.670109e-01 8.607924e-01 8.545794e-01\n",
- " [776] 8.483721e-01 8.421708e-01 8.359757e-01 8.297871e-01 8.236051e-01\n",
- " [781] 8.174302e-01 8.112624e-01 8.051021e-01 7.989494e-01 7.928047e-01\n",
- " [786] 7.866681e-01 7.805399e-01 7.744204e-01 7.683098e-01 7.622083e-01\n",
- " [791] 7.561161e-01 7.500336e-01 7.439609e-01 7.378983e-01 7.318461e-01\n",
- " [796] 7.258043e-01 7.197735e-01 7.137536e-01 7.077450e-01 7.017480e-01\n",
- " [801] 6.957626e-01 6.897893e-01 6.838282e-01 6.778796e-01 6.719436e-01\n",
- " [806] 6.660206e-01 6.601107e-01 6.542142e-01 6.483314e-01 6.424624e-01\n",
- " [811] 6.366074e-01 6.307668e-01 6.249408e-01 6.191295e-01 6.133333e-01\n",
- " [816] 6.075522e-01 6.017867e-01 5.960368e-01 5.903028e-01 5.845850e-01\n",
- " [821] 5.788835e-01 5.731987e-01 5.675306e-01 5.618796e-01 5.562459e-01\n",
- " [826] 5.506296e-01 5.450310e-01 5.394504e-01 5.338879e-01 5.283438e-01\n",
- " [831] 5.228183e-01 5.173115e-01 5.118238e-01 5.063553e-01 5.009063e-01\n",
- " [836] 4.954769e-01 4.900674e-01 4.846780e-01 4.793089e-01 4.739603e-01\n",
- " [841] 4.686324e-01 4.633255e-01 4.580397e-01 4.527753e-01 4.475324e-01\n",
- " [846] 4.423113e-01 4.371122e-01 4.319353e-01 4.267807e-01 4.216487e-01\n",
- " [851] 4.165395e-01 4.114533e-01 4.063903e-01 4.013506e-01 3.963346e-01\n",
- " [856] 3.913423e-01 3.863740e-01 3.814299e-01 3.765102e-01 3.716150e-01\n",
- " [861] 3.667446e-01 3.618992e-01 3.570788e-01 3.522838e-01 3.475144e-01\n",
- " [866] 3.427706e-01 3.380527e-01 3.333610e-01 3.286954e-01 3.240564e-01\n",
- " [871] 3.194439e-01 3.148583e-01 3.102997e-01 3.057682e-01 3.012641e-01\n",
- " [876] 2.967875e-01 2.923386e-01 2.879176e-01 2.835247e-01 2.791600e-01\n",
- " [881] 2.748237e-01 2.705160e-01 2.662370e-01 2.619869e-01 2.577659e-01\n",
- " [886] 2.535741e-01 2.494118e-01 2.452790e-01 2.411759e-01 2.371028e-01\n",
- " [891] 2.330597e-01 2.290468e-01 2.250643e-01 2.211124e-01 2.171911e-01\n",
- " [896] 2.133006e-01 2.094412e-01 2.056129e-01 2.018159e-01 1.980504e-01\n",
- " [901] 1.943164e-01 1.906142e-01 1.869439e-01 1.833056e-01 1.796995e-01\n",
- " [906] 1.761257e-01 1.725844e-01 1.690757e-01 1.655997e-01 1.621566e-01\n",
- " [911] 1.587465e-01 1.553695e-01 1.520258e-01 1.487156e-01 1.454388e-01\n",
- " [916] 1.421958e-01 1.389865e-01 1.358112e-01 1.326699e-01 1.295628e-01\n",
- " [921] 1.264899e-01 1.234515e-01 1.204476e-01 1.174784e-01 1.145440e-01\n",
- " [926] 1.116444e-01 1.087799e-01 1.059504e-01 1.031562e-01 1.003973e-01\n",
- " [931] 9.767385e-02 9.498595e-02 9.233372e-02 8.971724e-02 8.713663e-02\n",
- " [936] 8.459199e-02 8.208341e-02 7.961100e-02 7.717485e-02 7.477507e-02\n",
- " [941] 7.241173e-02 7.008494e-02 6.779479e-02 6.554136e-02 6.332476e-02\n",
- " [946] 6.114506e-02 5.900234e-02 5.689671e-02 5.482823e-02 5.279699e-02\n",
- " [951] 5.080307e-02 4.884655e-02 4.692750e-02 4.504600e-02 4.320213e-02\n",
- " [956] 4.139596e-02 3.962755e-02 3.789698e-02 3.620432e-02 3.454963e-02\n",
- " [961] 3.293298e-02 3.135443e-02 2.981404e-02 2.831188e-02 2.684801e-02\n",
- " [966] 2.542247e-02 2.403534e-02 2.268665e-02 2.137647e-02 2.010485e-02\n",
- " [971] 1.887184e-02 1.767748e-02 1.652183e-02 1.540492e-02 1.432681e-02\n",
- " [976] 1.328753e-02 1.228712e-02 1.132564e-02 1.040310e-02 9.519556e-03\n",
- " [981] 8.675035e-03 7.869572e-03 7.103199e-03 6.375945e-03 5.687839e-03\n",
- " [986] 5.038909e-03 4.429179e-03 3.858675e-03 3.327418e-03 2.835429e-03\n",
- " [991] 2.382728e-03 1.969332e-03 1.595259e-03 1.260522e-03 9.651344e-04\n",
- " [996] 7.091086e-04 4.924543e-04 3.151801e-04 1.772929e-04 7.879812e-05\n",
- "[1001] 1.969972e-05\n"
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],
"source": [
- "print(haan(times, T))"
+ "Qmat[1,1] - hprod(cos(omega1*times), cos(omega1*times), times)\n",
+ "Qmat[1,2] - hprod(cos(omega1*times), sin(omega1*times), times)\n",
+ "Qmat[2,1] - hprod(sin(omega1*times), cos(omega1*times), times)\n",
+ "Qmat[2,2] - hprod(sin(omega1*times), sin(omega1*times), times)\n"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "id": "563ee16f-8066-41b2-b678-52d366b56d51",
+ "execution_count": 306,
+ "id": "b9d5a7b0-66d5-4103-975d-62258b17ae10",
"metadata": {},
- "outputs": [],
- "source": []
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<table class=\"dataframe\">\n",
+ "<caption>A matrix: 2 × 2 of type dbl</caption>\n",
+ "<tbody>\n",
+ "\t<tr><td> 0.0011444968</td><td>0.0003944598</td></tr>\n",
+ "\t<tr><td>-0.0005148489</td><td>0.0009811628</td></tr>\n",
+ "</tbody>\n",
+ "</table>\n"
+ ],
+ "text/latex": [
+ "A matrix: 2 × 2 of type dbl\n",
+ "\\begin{tabular}{ll}\n",
+ "\t 0.0011444968 & 0.0003944598\\\\\n",
+ "\t -0.0005148489 & 0.0009811628\\\\\n",
+ "\\end{tabular}\n"
+ ],
+ "text/markdown": [
+ "\n",
+ "A matrix: 2 × 2 of type dbl\n",
+ "\n",
+ "| 0.0011444968 | 0.0003944598 |\n",
+ "| -0.0005148489 | 0.0009811628 |\n",
+ "\n"
+ ],
+ "text/plain": [
+ " [,1] [,2] \n",
+ "[1,] 0.0011444968 0.0003944598\n",
+ "[2,] -0.0005148489 0.0009811628"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "6.25971973908235e-12"
+ ],
+ "text/latex": [
+ "6.25971973908235e-12"
+ ],
+ "text/markdown": [
+ "6.25971973908235e-12"
+ ],
+ "text/plain": [
+ "[1] 6.25972e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-2.93520443410758e-12"
+ ],
+ "text/latex": [
+ "-2.93520443410758e-12"
+ ],
+ "text/markdown": [
+ "-2.93520443410758e-12"
+ ],
+ "text/plain": [
+ "[1] -2.935204e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-4.16815030841039e-12"
+ ],
+ "text/latex": [
+ "-4.16815030841039e-12"
+ ],
+ "text/markdown": [
+ "-4.16815030841039e-12"
+ ],
+ "text/plain": [
+ "[1] -4.16815e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-3.37742876201086e-12"
+ ],
+ "text/latex": [
+ "-3.37742876201086e-12"
+ ],
+ "text/markdown": [
+ "-3.37742876201086e-12"
+ ],
+ "text/plain": [
+ "[1] -3.377429e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "Qmat_test <- quarto(omega1,omega2)\n",
+ "Qmat_test\n",
+ "Qmat_test[1,1] - hprod(cos(omega1*times), cos(omega2*times), times)\n",
+ "Qmat_test[1,2] - hprod(cos(omega1*times), sin(omega2*times), times)\n",
+ "Qmat_test[2,1] - hprod(sin(omega1*times), cos(omega2*times), times)\n",
+ "Qmat_test[2,2] - hprod(sin(omega1*times), sin(omega2*times), times)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 307,
+ "id": "3217984c-e212-4837-8111-84c61ebd1a70",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "sol <- solve(Qmat, c(cosbit, sinbit))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 308,
+ "id": "d506983e-5e41-48d0-a3a9-f87ca1031959",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1] 0.4000256\n",
+ "[1] 0.4\n"
+ ]
+ },
+ {
+ "data": {
+ "text/html": [
+ "0.999999999976419"
+ ],
+ "text/latex": [
+ "0.999999999976419"
+ ],
+ "text/markdown": [
+ "0.999999999976419"
+ ],
+ "text/plain": [
+ "[1] 1"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "phase_rough <- atan(sinbit / -cosbit)\n",
+ "phase_corrected <- atan(sol[2] / -sol[1])\n",
+ "print(phase_rough)\n",
+ "print(phase_corrected)\n",
+ "sqrt ( (sol[1]^2 + sol[2]^2) ) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "db79227d-8b1e-4565-ac59-6b7d81401ff7",
+ "metadata": {},
+ "source": [
+ "So as we see it we recover almost perfectly the signal phase if we account for the cross product between sin and cos. This was expected but there is a tiny bias on the amplitude. Maybe that's the thing to worry about."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 309,
+ "id": "77c32f36-c186-4cd3-9795-1aa9c22da24a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "sinbit <- hprod(signal, sin(omega1*times), times)\n",
+ "cosbit <- hprod(signal, cos(omega1*times), times)\n",
+ "Qmat <- quarto(omega1,omega1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 310,
+ "id": "47635f3f-8d00-4ede-a7f1-1137834e53e6",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1] 0.4003161\n"
+ ]
+ }
+ ],
+ "source": [
+ "sol <- solve(Qmat, c(cosbit, sinbit))\n",
+ "print ( atan(sol[2] / -sol[1])) "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2826b791-760e-4d84-91c5-e78b9d151602",
+ "metadata": {},
+ "source": [
+ "So this time I have a slight phase shift becaeuse I have not orthoganalised with respect to the omega2 part of the signal. But we could to this:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 340,
+ "id": "1203347d-221a-4b7b-a6f3-603617fb5e56",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<style>\n",
+ ".list-inline {list-style: none; margin:0; padding: 0}\n",
+ ".list-inline>li {display: inline-block}\n",
+ ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
+ "</style>\n",
+ "<ol class=list-inline><li>0.4616610795258</li><li>-0.195373102615154</li><li>0.497631690310978</li><li>-0.0615816292261174</li></ol>\n"
+ ],
+ "text/latex": [
+ "\\begin{enumerate*}\n",
+ "\\item 0.4616610795258\n",
+ "\\item -0.195373102615154\n",
+ "\\item 0.497631690310978\n",
+ "\\item -0.0615816292261174\n",
+ "\\end{enumerate*}\n"
+ ],
+ "text/markdown": [
+ "1. 0.4616610795258\n",
+ "2. -0.195373102615154\n",
+ "3. 0.497631690310978\n",
+ "4. -0.0615816292261174\n",
+ "\n",
+ "\n"
+ ],
+ "text/plain": [
+ "[1] 0.46166108 -0.19537310 0.49763169 -0.06158163"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "<style>\n",
+ ".list-inline {list-style: none; margin:0; padding: 0}\n",
+ ".list-inline>li {display: inline-block}\n",
+ ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
+ "</style>\n",
+ "<ol class=list-inline><li>0.9210609939703</li><li>-0.389418342286748</li><li>0.99239587668744</li><li>-0.123087058208318</li></ol>\n"
+ ],
+ "text/latex": [
+ "\\begin{enumerate*}\n",
+ "\\item 0.9210609939703\n",
+ "\\item -0.389418342286748\n",
+ "\\item 0.99239587668744\n",
+ "\\item -0.123087058208318\n",
+ "\\end{enumerate*}\n"
+ ],
+ "text/markdown": [
+ "1. 0.9210609939703\n",
+ "2. -0.389418342286748\n",
+ "3. 0.99239587668744\n",
+ "4. -0.123087058208318\n",
+ "\n",
+ "\n"
+ ],
+ "text/plain": [
+ "[1] 0.9210610 -0.3894183 0.9923959 -0.1230871"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "sinbit2 <- hprod(signal, sin(omega2*times), times)\n",
+ "cosbit2 <- hprod(signal, cos(omega2*times), times)\n",
+ "Qmat <- matrix (NA, 4,4)\n",
+ "Qmat[1:2,1:2] <- quarto(omega1, omega1)\n",
+ "Qmat[3:4,1:2] <- quarto(omega2, omega1)\n",
+ "Qmat[1:2,3:4] <- quarto(omega1, omega2)\n",
+ "Qmat[3:4,3:4] <- quarto(omega2, omega2)\n",
+ "S <- c(cosbit, sinbit, cosbit2, sinbit2)\n",
+ "S\n",
+ "sol <- solve(Qmat, S)\n",
+ "sol"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 312,
+ "id": "eaa53773-27f8-4acd-90ac-1dfb8683f641",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1] 0.4\n"
+ ]
+ }
+ ],
+ "source": [
+ "print ( atan(sol[2] / -sol[1]))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 313,
+ "id": "b9721036-bc47-4ec5-bafa-72adeaf4a198",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1] 0.1234\n"
+ ]
+ }
+ ],
+ "source": [
+ "print ( atan(sol[4] / -sol[3]))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 314,
+ "id": "ed292840-4353-4dd1-a592-b6b98da792aa",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "0.1234"
+ ],
+ "text/latex": [
+ "0.1234"
+ ],
+ "text/markdown": [
+ "0.1234"
+ ],
+ "text/plain": [
+ "[1] 0.1234"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "phi2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 315,
+ "id": "9aa9a028-ed42-4989-8c32-13416f3d4822",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<table class=\"dataframe\">\n",
+ "<caption>A matrix: 4 × 4 of type dbl</caption>\n",
+ "<tbody>\n",
+ "\t<tr><td> 5.000274e-01</td><td>-4.656294e-05</td><td> 1.144497e-03</td><td> 3.944598e-04</td></tr>\n",
+ "\t<tr><td>-4.656294e-05</td><td> 4.999726e-01</td><td>-5.148489e-04</td><td> 9.811628e-04</td></tr>\n",
+ "\t<tr><td> 1.144497e-03</td><td>-5.148489e-04</td><td> 5.001754e-01</td><td>-4.123781e-05</td></tr>\n",
+ "\t<tr><td> 3.944598e-04</td><td> 9.811628e-04</td><td>-4.123781e-05</td><td> 4.998246e-01</td></tr>\n",
+ "</tbody>\n",
+ "</table>\n"
+ ],
+ "text/latex": [
+ "A matrix: 4 × 4 of type dbl\n",
+ "\\begin{tabular}{llll}\n",
+ "\t 5.000274e-01 & -4.656294e-05 & 1.144497e-03 & 3.944598e-04\\\\\n",
+ "\t -4.656294e-05 & 4.999726e-01 & -5.148489e-04 & 9.811628e-04\\\\\n",
+ "\t 1.144497e-03 & -5.148489e-04 & 5.001754e-01 & -4.123781e-05\\\\\n",
+ "\t 3.944598e-04 & 9.811628e-04 & -4.123781e-05 & 4.998246e-01\\\\\n",
+ "\\end{tabular}\n"
+ ],
+ "text/markdown": [
+ "\n",
+ "A matrix: 4 × 4 of type dbl\n",
+ "\n",
+ "| 5.000274e-01 | -4.656294e-05 | 1.144497e-03 | 3.944598e-04 |\n",
+ "| -4.656294e-05 | 4.999726e-01 | -5.148489e-04 | 9.811628e-04 |\n",
+ "| 1.144497e-03 | -5.148489e-04 | 5.001754e-01 | -4.123781e-05 |\n",
+ "| 3.944598e-04 | 9.811628e-04 | -4.123781e-05 | 4.998246e-01 |\n",
+ "\n"
+ ],
+ "text/plain": [
+ " [,1] [,2] [,3] [,4] \n",
+ "[1,] 5.000274e-01 -4.656294e-05 1.144497e-03 3.944598e-04\n",
+ "[2,] -4.656294e-05 4.999726e-01 -5.148489e-04 9.811628e-04\n",
+ "[3,] 1.144497e-03 -5.148489e-04 5.001754e-01 -4.123781e-05\n",
+ "[4,] 3.944598e-04 9.811628e-04 -4.123781e-05 4.998246e-01"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "Qmat"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 316,
+ "id": "2c0af5bc-abf1-45c0-8e7a-cfb8c179cadb",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "3.37742876201086e-12"
+ ],
+ "text/latex": [
+ "3.37742876201086e-12"
+ ],
+ "text/markdown": [
+ "3.37742876201086e-12"
+ ],
+ "text/plain": [
+ "[1] 3.377429e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "[1] 0.9210610 -0.3894183 0.9923959 -0.1230871\n",
+ "[1] 0.9210610 -0.3894183 0.9923959 -0.1230871\n"
+ ]
+ }
+ ],
+ "source": [
+ "hprod(sin(omega1*times), sin(omega2*times), times) - Qmat[2,4]\n",
+ "\n",
+ "print(sol)\n",
+ "print(c(a1,a2,a3,a4))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 317,
+ "id": "4ecbfdc6-3a58-4b58-83f7-a4c74878f706",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "100.1"
+ ],
+ "text/latex": [
+ "100.1"
+ ],
+ "text/markdown": [
+ "100.1"
+ ],
+ "text/plain": [
+ "[1] 100.1"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "N"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 324,
+ "id": "78f3e607-8967-4841-a67e-62cb9403d5cf",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "0.4616610795258"
+ ],
+ "text/latex": [
+ "0.4616610795258"
+ ],
+ "text/markdown": [
+ "0.4616610795258"
+ ],
+ "text/plain": [
+ "[1] 0.4616611"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "0.460573783396488"
+ ],
+ "text/latex": [
+ "0.460573783396488"
+ ],
+ "text/markdown": [
+ "0.460573783396488"
+ ],
+ "text/plain": [
+ "[1] 0.4605738"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "0.4616610795258"
+ ],
+ "text/latex": [
+ "0.4616610795258"
+ ],
+ "text/markdown": [
+ "0.4616610795258"
+ ],
+ "text/plain": [
+ "[1] 0.4616611"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "6.15074657872583e-12"
+ ],
+ "text/latex": [
+ "6.15074657872583e-12"
+ ],
+ "text/markdown": [
+ "6.15074657872583e-12"
+ ],
+ "text/plain": [
+ "[1] 6.150747e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-1.04640939129256e-11"
+ ],
+ "text/latex": [
+ "-1.04640939129256e-11"
+ ],
+ "text/markdown": [
+ "-1.04640939129256e-11"
+ ],
+ "text/plain": [
+ "[1] -1.046409e-11"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "6.25971973908235e-12"
+ ],
+ "text/latex": [
+ "6.25971973908235e-12"
+ ],
+ "text/markdown": [
+ "6.25971973908235e-12"
+ ],
+ "text/plain": [
+ "[1] 6.25972e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-4.16815030841039e-12"
+ ],
+ "text/latex": [
+ "-4.16815030841039e-12"
+ ],
+ "text/markdown": [
+ "-4.16815030841039e-12"
+ ],
+ "text/plain": [
+ "[1] -4.16815e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-3.19687642514905e-13"
+ ],
+ "text/latex": [
+ "-3.19687642514905e-13"
+ ],
+ "text/markdown": [
+ "-3.19687642514905e-13"
+ ],
+ "text/plain": [
+ "[1] -3.196876e-13"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "data": {
+ "text/html": [
+ "-2.93520443410758e-12"
+ ],
+ "text/latex": [
+ "-2.93520443410758e-12"
+ ],
+ "text/markdown": [
+ "-2.93520443410758e-12"
+ ],
+ "text/plain": [
+ "[1] -2.935204e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "hprod(signal,v1,times)\n",
+ "a1*Qmat[1,1] + a2*Qmat[1,2] + a3 * Qmat[1,3] * a4 * Qmat[1,4]\n",
+ "a1*hprod(v1, v1, times) + a2 * hprod(v1,v2,times) + a3*hprod(v1,v3,times) + a4*hprod(v1,v4,times)\n",
+ "\n",
+ "\n",
+ "Qmat[1,1] - hprod(v1,v1,times)\n",
+ "Qmat[1,2] - hprod(v1,v2,times)\n",
+ "Qmat[1,3] - hprod(v1,v3,times)\n",
+ "Qmat[2,3] - hprod(v2,v3,times)\n",
+ "\n",
+ "Qmat[3,4] - hprod(v3,v4,times)\n",
+ "Qmat[1,4] - hprod(v1,v4,times)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 329,
+ "id": "3371254a-6c0f-4cfa-a53b-152a1e5d81c8",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "-9.73994026911055e-12"
+ ],
+ "text/latex": [
+ "-9.73994026911055e-12"
+ ],
+ "text/markdown": [
+ "-9.73994026911055e-12"
+ ],
+ "text/plain": [
+ "[1] -9.73994e-12"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "hprod(signal_single, v1, times) - a1 * Qmat[1,1] - a2 * Qmat[2,1]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 331,
+ "id": "996af80c-8ff7-4d47-a5e2-ef8dfd7977dc",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<table class=\"dataframe\">\n",
+ "<caption>A matrix: 4 × 4 of type dbl</caption>\n",
+ "<tbody>\n",
+ "\t<tr><td> 5.000274e-01</td><td>-4.656294e-05</td><td> 1.144497e-03</td><td> 3.944598e-04</td></tr>\n",
+ "\t<tr><td>-4.656294e-05</td><td> 4.999726e-01</td><td>-5.148489e-04</td><td> 9.811628e-04</td></tr>\n",
+ "\t<tr><td> 1.144497e-03</td><td>-5.148489e-04</td><td> 5.001754e-01</td><td>-4.123781e-05</td></tr>\n",
+ "\t<tr><td> 3.944598e-04</td><td> 9.811628e-04</td><td>-4.123781e-05</td><td> 4.998246e-01</td></tr>\n",
+ "</tbody>\n",
+ "</table>\n"
+ ],
+ "text/latex": [
+ "A matrix: 4 × 4 of type dbl\n",
+ "\\begin{tabular}{llll}\n",
+ "\t 5.000274e-01 & -4.656294e-05 & 1.144497e-03 & 3.944598e-04\\\\\n",
+ "\t -4.656294e-05 & 4.999726e-01 & -5.148489e-04 & 9.811628e-04\\\\\n",
+ "\t 1.144497e-03 & -5.148489e-04 & 5.001754e-01 & -4.123781e-05\\\\\n",
+ "\t 3.944598e-04 & 9.811628e-04 & -4.123781e-05 & 4.998246e-01\\\\\n",
+ "\\end{tabular}\n"
+ ],
+ "text/markdown": [
+ "\n",
+ "A matrix: 4 × 4 of type dbl\n",
+ "\n",
+ "| 5.000274e-01 | -4.656294e-05 | 1.144497e-03 | 3.944598e-04 |\n",
+ "| -4.656294e-05 | 4.999726e-01 | -5.148489e-04 | 9.811628e-04 |\n",
+ "| 1.144497e-03 | -5.148489e-04 | 5.001754e-01 | -4.123781e-05 |\n",
+ "| 3.944598e-04 | 9.811628e-04 | -4.123781e-05 | 4.998246e-01 |\n",
+ "\n"
+ ],
+ "text/plain": [
+ " [,1] [,2] [,3] [,4] \n",
+ "[1,] 5.000274e-01 -4.656294e-05 1.144497e-03 3.944598e-04\n",
+ "[2,] -4.656294e-05 4.999726e-01 -5.148489e-04 9.811628e-04\n",
+ "[3,] 1.144497e-03 -5.148489e-04 5.001754e-01 -4.123781e-05\n",
+ "[4,] 3.944598e-04 9.811628e-04 -4.123781e-05 4.998246e-01"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "Qmat"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 341,
+ "id": "5414caa5-f263-4860-89a5-f1468bac7ec1",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ "Loading required package: pracma\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "require(pracma)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 342,
+ "id": "1b34c634-3a2c-4818-a549-61bb30e0c9ec",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<dl>\n",
+ "\t<dt>$Q</dt>\n",
+ "\t\t<dd><table class=\"dataframe\">\n",
+ "<caption>A matrix: 4 × 4 of type dbl</caption>\n",
+ "<tbody>\n",
+ "\t<tr><td> 9.999971e-01</td><td> 9.392831e-05</td><td>-2.288700e-03</td><td>-7.892456e-04</td></tr>\n",
+ "\t<tr><td>-9.312051e-05</td><td> 9.999975e-01</td><td> 1.029699e-03</td><td>-1.962417e-03</td></tr>\n",
+ "\t<tr><td> 2.288862e-03</td><td>-1.029324e-03</td><td> 9.999968e-01</td><td> 8.223247e-05</td></tr>\n",
+ "\t<tr><td> 7.888741e-04</td><td> 1.962576e-03</td><td>-8.201804e-05</td><td> 9.999978e-01</td></tr>\n",
+ "</tbody>\n",
+ "</table>\n",
+ "</dd>\n",
+ "\t<dt>$R</dt>\n",
+ "\t\t<dd><table class=\"dataframe\">\n",
+ "<caption>A matrix: 4 × 4 of type dbl</caption>\n",
+ "<tbody>\n",
+ "\t<tr><td>0.5000288</td><td>-9.352492e-05</td><td> 0.002289341</td><td> 7.885716e-04</td></tr>\n",
+ "\t<tr><td>0.0000000</td><td> 4.999739e-01</td><td>-0.001029663</td><td> 1.962184e-03</td></tr>\n",
+ "\t<tr><td>0.0000000</td><td> 0.000000e+00</td><td> 0.500170644</td><td>-8.212482e-05</td></tr>\n",
+ "\t<tr><td>0.0000000</td><td> 0.000000e+00</td><td> 0.000000000</td><td> 4.998213e-01</td></tr>\n",
+ "</tbody>\n",
+ "</table>\n",
+ "</dd>\n",
+ "</dl>\n"
+ ],
+ "text/latex": [
+ "\\begin{description}\n",
+ "\\item[\\$Q] A matrix: 4 × 4 of type dbl\n",
+ "\\begin{tabular}{llll}\n",
+ "\t 9.999971e-01 & 9.392831e-05 & -2.288700e-03 & -7.892456e-04\\\\\n",
+ "\t -9.312051e-05 & 9.999975e-01 & 1.029699e-03 & -1.962417e-03\\\\\n",
+ "\t 2.288862e-03 & -1.029324e-03 & 9.999968e-01 & 8.223247e-05\\\\\n",
+ "\t 7.888741e-04 & 1.962576e-03 & -8.201804e-05 & 9.999978e-01\\\\\n",
+ "\\end{tabular}\n",
+ "\n",
+ "\\item[\\$R] A matrix: 4 × 4 of type dbl\n",
+ "\\begin{tabular}{llll}\n",
+ "\t 0.5000288 & -9.352492e-05 & 0.002289341 & 7.885716e-04\\\\\n",
+ "\t 0.0000000 & 4.999739e-01 & -0.001029663 & 1.962184e-03\\\\\n",
+ "\t 0.0000000 & 0.000000e+00 & 0.500170644 & -8.212482e-05\\\\\n",
+ "\t 0.0000000 & 0.000000e+00 & 0.000000000 & 4.998213e-01\\\\\n",
+ "\\end{tabular}\n",
+ "\n",
+ "\\end{description}\n"
+ ],
+ "text/markdown": [
+ "$Q\n",
+ ": \n",
+ "A matrix: 4 × 4 of type dbl\n",
+ "\n",
+ "| 9.999971e-01 | 9.392831e-05 | -2.288700e-03 | -7.892456e-04 |\n",
+ "| -9.312051e-05 | 9.999975e-01 | 1.029699e-03 | -1.962417e-03 |\n",
+ "| 2.288862e-03 | -1.029324e-03 | 9.999968e-01 | 8.223247e-05 |\n",
+ "| 7.888741e-04 | 1.962576e-03 | -8.201804e-05 | 9.999978e-01 |\n",
+ "\n",
+ "\n",
+ "$R\n",
+ ": \n",
+ "A matrix: 4 × 4 of type dbl\n",
+ "\n",
+ "| 0.5000288 | -9.352492e-05 | 0.002289341 | 7.885716e-04 |\n",
+ "| 0.0000000 | 4.999739e-01 | -0.001029663 | 1.962184e-03 |\n",
+ "| 0.0000000 | 0.000000e+00 | 0.500170644 | -8.212482e-05 |\n",
+ "| 0.0000000 | 0.000000e+00 | 0.000000000 | 4.998213e-01 |\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n"
+ ],
+ "text/plain": [
+ "$Q\n",
+ " [,1] [,2] [,3] [,4]\n",
+ "[1,] 9.999971e-01 9.392831e-05 -2.288700e-03 -7.892456e-04\n",
+ "[2,] -9.312051e-05 9.999975e-01 1.029699e-03 -1.962417e-03\n",
+ "[3,] 2.288862e-03 -1.029324e-03 9.999968e-01 8.223247e-05\n",
+ "[4,] 7.888741e-04 1.962576e-03 -8.201804e-05 9.999978e-01\n",
+ "\n",
+ "$R\n",
+ " [,1] [,2] [,3] [,4]\n",
+ "[1,] 0.5000288 -9.352492e-05 0.002289341 7.885716e-04\n",
+ "[2,] 0.0000000 4.999739e-01 -0.001029663 1.962184e-03\n",
+ "[3,] 0.0000000 0.000000e+00 0.500170644 -8.212482e-05\n",
+ "[4,] 0.0000000 0.000000e+00 0.000000000 4.998213e-01\n"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "gramSchmidt(Qmat)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 343,
+ "id": "b388c8ee-721d-41de-ac4f-3cf6b0643c16",
+ "metadata": {},
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+ "data": {
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+ "phi1\n",
+ "phi2"
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}
],
"metadata": {
--
GitLab