eqn =
結果:
Christmas is coming, here are two dynamic Christmas tree drawing codes:
Crystal XMas Tree
function XmasTree2024_1
fig = figure('Units','normalized', 'Position',[.1,.1,.5,.8],...
'Color',[0,9,33]/255, 'UserData',40 + [60,65,75,72,0,59,64,57,74,0,63,59,57,0,1,6,45,75,61,74,28,57,76,57,1,1]);
axes('Parent',fig, 'Position',[0,-1/6,1,1+1/3], 'UserData',97 + [18,11,0,13,3,0,17,4,17],...
'XLim',[-1.5,1.5], 'YLim',[-1.5,1.5], 'ZLim',[-.2,3.8], 'DataAspectRatio', [1,1,1], 'NextPlot','add',...
'Projection','perspective', 'Color',[0,9,33]/255, 'XColor','none', 'YColor','none', 'ZColor','none')
%% Draw Christmas tree
F = [1,3,4;1,4,5;1,5,6;1,6,3;...
2,3,4;2,4,5;2,5,6;2,6,3];
dP = @(V) patch('Faces',F, 'Vertices',V, 'FaceColor',[0 71 177]./255,...
'FaceAlpha',rand(1).*0.2+0.1, 'EdgeColor',[0 71 177]./255.*0.8,...
'EdgeAlpha',0.6, 'LineWidth',0.5, 'EdgeLighting','gouraud', 'SpecularStrength',0.3);
r = .1; h = .8;
V0 = [0,0,0; 0,0,1; 0,r,h; r,0,h; 0,-r,h; -r,0,h];
% Rotation matrix
Rx = @(V, theta) V*[1 0 0; 0 cos(theta) sin(theta); 0 -sin(theta) cos(theta)];
Rz = @(V, theta) V*[cos(theta) sin(theta) 0;-sin(theta) cos(theta) 0; 0 0 1];
N = 180; Vn = zeros(N, 3); eval(char(fig.UserData))
for i = 1:N
tV = Rz(Rx(V0.*(1.2 - .8.*i./N + rand(1).*.1./i^(1/5)), pi/3.*(1 - .6.*i./N)), i.*pi/8.1 + .001.*i.^2) + [0,0,.016.*i];
dP(tV); Vn(i,:) = tV(2,:);
end
scatter3(Vn(:,1).*1.02,Vn(:,2).*1.02,Vn(:,3).*1.01, 30, 'w', 'Marker','*', 'MarkerEdgeAlpha',.5)
%% Draw Star of Bethlehem
w = .3; R = .62; r = .4; T = (1/8:1/8:(2 - 1/8)).'.*pi;
V8 = [ 0, 0, w; 0, 0,-w;
1, 0, 0; 0, 1, 0; -1, 0, 0; 0,-1,0;
R, R, 0; -R, R, 0; -R,-R, 0; R,-R,0;
cos(T).*r, sin(T).*r, T.*0];
F8 = [1,3,25; 1,3,11; 2,3,25; 2,3,11; 1,7,11; 1,7,13; 2,7,11; 2,7,13;
1,4,13; 1,4,15; 2,4,13; 2,4,15; 1,8,15; 1,8,17; 2,8,15; 2,8,17;
1,5,17; 1,5,19; 2,5,17; 2,5,19; 1,9,19; 1,9,21; 2,9,19; 2,9,21;
1,6,21; 1,6,23; 2,6,21; 2,6,23; 1,10,23; 1,10,25; 2,10,23; 2,10,25];
V8 = Rx(V8.*.3, pi/2) + [0,0,3.5];
patch('Faces',F8, 'Vertices',V8, 'FaceColor',[255,223,153]./255,...
'EdgeColor',[255,223,153]./255, 'FaceAlpha', .2)
%% Draw snow
sXYZ = rand(200,3).*[4,4,5] - [2,2,0];
sHdl1 = plot3(sXYZ(1:90,1),sXYZ(1:90,2),sXYZ(1:90,3), '*', 'Color',[.8,.8,.8]);
sHdl2 = plot3(sXYZ(91:200,1),sXYZ(91:200,2),sXYZ(91:200,3), '.', 'Color',[.6,.6,.6]);
annotation(fig,'textbox',[0,.05,1,.09], 'Color',[1 1 1], 'String','Merry Christmas Matlaber',...
'HorizontalAlignment','center', 'FontWeight','bold', 'FontSize',48,...
'FontName','Times New Roman', 'FontAngle','italic', 'FitBoxToText','off','EdgeColor','none');
% Rotate the Christmas tree and let the snow fall
for i=1:1e8
sXYZ(:,3) = sXYZ(:,3) - [.05.*ones(90,1); .06.*ones(110,1)];
sXYZ(sXYZ(:,3)<0, 3) = sXYZ(sXYZ(:,3) < 0, 3) + 5;
sHdl1.ZData = sXYZ(1:90,3); sHdl2.ZData = sXYZ(91:200,3);
view([i,30]); drawnow; pause(.05)
end
end
Curved XMas Tree
function XmasTree2024_2
fig = figure('Units','normalized', 'Position',[.1,.1,.5,.8],...
'Color',[0,9,33]/255, 'UserData',40 + [60,65,75,72,0,59,64,57,74,0,63,59,57,0,1,6,45,75,61,74,28,57,76,57,1,1]);
axes('Parent',fig, 'Position',[0,-1/6,1,1+1/3], 'UserData',97 + [18,11,0,13,3,0,17,4,17],...
'XLim',[-6,6], 'YLim',[-6,6], 'ZLim',[-16, 1], 'DataAspectRatio', [1,1,1], 'NextPlot','add',...
'Projection','perspective', 'Color',[0,9,33]/255, 'XColor','none', 'YColor','none', 'ZColor','none')
%% Draw Christmas tree
[X,T] = meshgrid(.4:.1:1, 0:pi/50:2*pi);
XM = 1 + sin(8.*T).*.05;
X = X.*XM; R = X.^(3).*(.5 + sin(8.*T).*.02);
dF = @(R, T, X) surf(R.*cos(T), R.*sin(T), -X, 'EdgeColor',[20,107,58]./255,...
'FaceColor', [20,107,58]./255, 'FaceAlpha',.2, 'LineWidth',1);
CList = [254,103,110; 255,191,115; 57,120,164]./255;
for i = 1:5
tR = R.*(2 + i); tT = T+i; tX = X.*(2 + i) + i;
SFHdl = dF(tR, tT, tX);
[~, ind] = sort(SFHdl.ZData(:)); ind = ind(1:8);
C = CList(randi([1,size(CList,1)], [8,1]), :);
scatter3(tR(ind).*cos(tT(ind)), tR(ind).*sin(tT(ind)), -tX(ind), 120, 'filled',...
'CData', C, 'MarkerEdgeColor','none', 'MarkerFaceAlpha',.3)
scatter3(tR(ind).*cos(tT(ind)), tR(ind).*sin(tT(ind)), -tX(ind), 60, 'filled', 'CData', C)
end
%% Draw Star of Bethlehem
Rx = @(V, theta) V*[1 0 0; 0 cos(theta) sin(theta); 0 -sin(theta) cos(theta)];
% Rz = @(V, theta) V*[cos(theta) sin(theta) 0;-sin(theta) cos(theta) 0; 0 0 1];
w = .3; R = .62; r = .4; T = (1/8:1/8:(2 - 1/8)).'.*pi;
V8 = [ 0, 0, w; 0, 0,-w;
1, 0, 0; 0, 1, 0; -1, 0, 0; 0,-1,0;
R, R, 0; -R, R, 0; -R,-R, 0; R,-R,0;
cos(T).*r, sin(T).*r, T.*0];
F8 = [1,3,25; 1,3,11; 2,3,25; 2,3,11; 1,7,11; 1,7,13; 2,7,11; 2,7,13;
1,4,13; 1,4,15; 2,4,13; 2,4,15; 1,8,15; 1,8,17; 2,8,15; 2,8,17;
1,5,17; 1,5,19; 2,5,17; 2,5,19; 1,9,19; 1,9,21; 2,9,19; 2,9,21;
1,6,21; 1,6,23; 2,6,21; 2,6,23; 1,10,23; 1,10,25; 2,10,23; 2,10,25];
V8 = Rx(V8.*.8, pi/2) + [0,0,-1.3];
patch('Faces',F8, 'Vertices',V8, 'FaceColor',[255,223,153]./255,...
'EdgeColor',[255,223,153]./255, 'FaceAlpha', .2)
annotation(fig,'textbox',[0,.05,1,.09], 'Color',[1 1 1], 'String','Merry Christmas Matlaber',...
'HorizontalAlignment','center', 'FontWeight','bold', 'FontSize',48,...
'FontName','Times New Roman', 'FontAngle','italic', 'FitBoxToText','off','EdgeColor','none');
%% Draw snow
sXYZ = rand(200,3).*[12,12,17] - [6,6,16];
sHdl1 = plot3(sXYZ(1:90,1),sXYZ(1:90,2),sXYZ(1:90,3), '*', 'Color',[.8,.8,.8]);
sHdl2 = plot3(sXYZ(91:200,1),sXYZ(91:200,2),sXYZ(91:200,3), '.', 'Color',[.6,.6,.6]);
for i=1:1e8
sXYZ(:,3) = sXYZ(:,3) - [.1.*ones(90,1); .12.*ones(110,1)];
sXYZ(sXYZ(:,3)<-16, 3) = sXYZ(sXYZ(:,3) < -16, 3) + 17.5;
sHdl1.ZData = sXYZ(1:90,3); sHdl2.ZData = sXYZ(91:200,3);
view([i,30]); drawnow; pause(.05)
end
end
I wish all MATLABers a Merry Christmas in advance!
I have a problem with the movement of a pawn by two fields in its first move does anyone have a suggestion for a solution
function chess_game()
% Funkcja główna inicjalizująca grę w szachy
% Inicjalizacja stanu gry
gameState = struct();
gameState.board = initialize_board();
gameState.currentPlayer = 'white';
gameState.selectedPiece = [];
% Utworzenie GUI
fig = figure('Name', 'Gra w Szachy', 'NumberTitle', 'off', 'MenuBar', 'none', 'UserData', gameState);
ax = axes('Parent', fig, 'Position', [0 0 1 1], 'XTick', [], 'YTick', []);
axis(ax, [0 8 0 8]);
hold on;
% Wyświetlenie planszy
draw_board(ax, gameState.board);
% Obsługa kliknięcia myszy
set(fig, 'WindowButtonDownFcn', @(src, event)on_click(ax, src));
end
function board = initialize_board()
% Inicjalizuje planszę z ustawieniem początkowym figur
board = {
'R', 'N', 'B', 'Q', 'K', 'B', 'N', 'R';
'P', 'P', 'P', 'P', 'P', 'P', 'P', 'P';
'', '', '', '', '', '', '', '';
'', '', '', '', '', '', '', '';
'', '', '', '', '', '', '', '';
'', '', '', '', '', '', '', '';
'p', 'p', 'p', 'p', 'p', 'p', 'p', 'p';
'r', 'n', 'b', 'q', 'k', 'b', 'n', 'r';
};
end
function draw_board(~, board)
% Rysuje szachownicę i figury
colors = [1 1 1; 0.8 0.8 0.8];
for row = 1:8
for col = 1:8
% Rysowanie pól
rectColor = colors(mod(row + col, 2) + 1, :);
rectangle('Position', [col-1, 8-row, 1, 1], 'FaceColor', rectColor, 'EdgeColor', 'k');
% Rysowanie figur
piece = board{row, col};
if ~isempty(piece)
text(col-0.5, 8-row+0.5, piece, 'HorizontalAlignment', 'center', ...
'FontSize', 20, 'FontWeight', 'bold');
end
end
end
end
function on_click(ax, fig)
% Funkcja obsługująca kliknięcia myszy
pos = get(ax, 'CurrentPoint');
x = floor(pos(1,1)) + 1; % Zaokrąglij współrzędne w poziomie i dopasuj do indeksów
y = 8 - floor(pos(1,2)); % Dopasuj współrzędne w pionie (odwrócenie osi Y)
% Pobranie stanu gry z figury
gameState = get(fig, 'UserData');
if x >= 1 && x <= 8 && y >= 1 && y <= 8
disp(['Kliknięto na pole: (', num2str(x), ', ', num2str(y), ')']);
if isempty(gameState.selectedPiece)
% Wybór figury
piece = gameState.board{y, x};
if ~isempty(piece)
if (strcmp(gameState.currentPlayer, 'white') && any(ismember(piece, 'RNBQKP'))) || ...
(strcmp(gameState.currentPlayer, 'black') && any(ismember(piece, 'rnbqkp')))
gameState.selectedPiece = [y, x];
disp(['Wybrano figurę: ', piece, ' na pozycji (', num2str(x), ', ', num2str(y), ')']);
else
disp('Nie możesz wybrać tej figury.');
end
else
disp('Nie wybrano figury.');
end
else
% Sprawdzenie, czy kliknięto ponownie na wybraną figurę
if isequal(gameState.selectedPiece, [y, x])
disp('Anulowano wybór figury.');
gameState.selectedPiece = [];
else
% Ruch figury
[sy, sx] = deal(gameState.selectedPiece(1), gameState.selectedPiece(2));
piece = gameState.board{sy, sx};
if is_valid_move(gameState.board, piece, [sy, sx], [y, x], gameState.currentPlayer)
% Wykonanie ruchu
gameState.board{sy, sx} = ''; % Usuwamy figurę z poprzedniego pola
gameState.board{y, x} = piece; % Umieszczamy figurę na nowym polu
gameState.selectedPiece = [];
% Przełącz gracza
gameState.currentPlayer = switch_player(gameState.currentPlayer);
% Odśwież planszę
cla(ax);
draw_board(ax, gameState.board);
else
disp('Ruch niezgodny z zasadami.');
end
end
end
% Zaktualizowanie stanu gry w figurze
set(fig, 'UserData', gameState);
end
end
function valid = is_valid_move(board, piece, from, to, currentPlayer)
% Funkcja sprawdzająca, czy ruch jest poprawny
[sy, sx] = deal(from(1), from(2));
[dy, dx] = deal(to(1), to(2));
dy_diff = dy - sy;
dx_diff = abs(dx - sx);
targetPiece = board{dy, dx};
% Sprawdzenie, czy ruch jest w granicach planszy
if dx < 1 || dx > 8 || dy < 1 || dy > 8
valid = false;
return;
end
% Nie można zbijać swoich figur
if ~isempty(targetPiece) && ...
((strcmp(currentPlayer, 'white') && ismember(targetPiece, 'RNBQKP')) || ...
(strcmp(currentPlayer, 'black') && ismember(targetPiece, 'rnbqkp')))
valid = false;
return;
end
% Zasady ruchu dla każdej figury
switch lower(piece)
case 'p' % Pion
direction = strcmp(currentPlayer, 'white') * 2 - 1; % 1 dla białych, -1 dla czarnych
startRow = strcmp(currentPlayer, 'white') * 2 + 1; % Rząd startowy dla białych i czarnych
if isempty(targetPiece)
% Ruch o jedno pole do przodu
if dy_diff == direction && dx_diff == 0
valid = true;
% Ruch o dwa pola do przodu z pozycji startowej
elseif dy_diff == 2 * direction && dx_diff == 0 && sy == startRow
if isempty(board{sy + direction, sx}) && isempty(board{dy, dx})
valid = true;
else
valid = false;
end
else
valid = false;
end
else
% Zbijanie na ukos
valid = (dx_diff == 1) && (dy_diff == direction);
end
case 'r' % Wieża
valid = (dx_diff == 0 || dy_diff == 0) && path_is_clear(board, from, to);
case 'n' % Skoczek
valid = (dx_diff == 2 && abs(dy_diff) == 1) || (dx_diff == 1 && abs(dy_diff) == 2);
case 'b' % Goniec
valid = (dx_diff == abs(dy_diff)) && path_is_clear(board, from, to);
case 'q' % Hetman
valid = ((dx_diff == 0 || dy_diff == 0) || (dx_diff == abs(dy_diff))) && path_is_clear(board, from, to);
case 'k' % Król
valid = max(abs(dx_diff), abs(dy_diff)) == 1;
otherwise
valid = false;
end
end
function clear = path_is_clear(board, from, to)
% Sprawdza, czy ścieżka między polami jest wolna od innych figur
[sy, sx] = deal(from(1), from(2));
[dy, dx] = deal(to(1), to(2));
stepY = sign(dy - sy);
stepX = sign(dx - sx);
y = sy + stepY;
x = sx + stepX;
while y ~= dy || x ~= dx
if ~isempty(board{y, x})
clear = false;
return;
end
y = y + stepY;
x = x + stepX;
end
clear = true;
end
function nextPlayer = switch_player(currentPlayer)
% Przełącza aktywnego gracza
if strcmp(currentPlayer, 'white')
nextPlayer = 'black';
else
nextPlayer = 'white';
end
end
Speaking as someone with 31+ years of experience developing and using imshow, I want to advocate for retiring and replacing it.
The function imshow has behaviors and defaults that were appropriate for the MATLAB and computer monitors of the 1990s, but which are not the best choice for most image display situations in today's MATLAB. Also, the 31 years have not been kind to the imshow code base. It is a glitchy, hard-to-maintain monster.
My new File Exchange function, imview, illustrates the kind of changes that I think should be made. The function imview is a much better MATLAB graphics citizen and produces higher quality image display by default, and it dispenses with the whole fraught business of trying to resize the containing figure. Although this is an initial release that does not yet support all the useful options that imshow does, it does enough that I am prepared to stop using imshow in my own work.
The Image Processing Toolbox team has just introduced in R2024b a new image viewer called imageshow, but that image viewer is created in a special-purpose window. It does not satisfy the need for an image display function that works well with the axes and figure objects of the traditional MATLAB graphics system.
I have published a blog post today that describes all this in more detail. I'd be interested to hear what other people think.
Note: Yes, I know there is an Image Processing Toolbox function called imview. That one is a stub for an old toolbox capability that was removed something like 15+ years ago. The only thing the toolbox imview function does now is call error. I have just submitted a support request to MathWorks to remove this old stub.
The int function in the Symbolic Toolbox has a hold/release functionality wherein the expression can be held to delay evaluation
syms x I
eqn = I == int(x,x,'Hold',true)
which allows one to show the integral, and then use release to show the result
release(eqn)
Maybe it would be nice to be able to hold/release any symbolic expression to delay the engine from doing evaluations/simplifications that it typically does. For example:
x*(x+1)/x, sin(sym(pi)/3)
If I'm trying to show a sequence of steps to develop a result, maybe I want to explicitly keep the x/x in the first case and then say "now the x in the numerator and denominator cancel and the result is ..." followed by the release command to get the final result.
Perhaps held expressions could even be nested to show a sequence of results upon subsequent releases.
Held expressions might be subject to other limitations, like maybe they can't be fplotted.
Seems like such a capability might not be useful for problem solving, but might be useful for exposition, instruction, etc.
Watt's Up with Electric Vehicles?EV modeling Ecosystem (Eco-friendly Vehicles), V2V Communication and V2I communications thereby emitting zero Emissions to considerably reduce NOx ,Particulates matters,CO2 given that Combustion is always incomplete and will always be.
Reduction of gas emissions outside to the environment will improve human life span ,few epidemic diseases and will result in long life standard
I want to build a neural network that takes a matrix A as input and outputs a matrix B such that a constant C=f(A,B)is maximized as much as possible.(The function f()is a custom complex computation function involving random values,probability density,matrix norms,and a series of other calculations).
I tried to directly use 1/f(A,B)or-f(A,B)as the loss function,but I encountered an error stating:"The value to be differentiated is not tracked.It must be a tracked real number dlarray scalar.Use dlgradient to track variables in the function called by dlfeval."I suspect this is likely because f(A,B)is not differentiable.
However,I've also seen people say that no matter what function it is,the dlgradient function can differentiate it.
So,I'm not sure whether it's because the function f()is too complex to be used as a loss function to calculate gradients,or if there's an issue with my code.
If I can't directly use its reciprocal or negative as the loss function,how should I go about training this neural network?Currently,I only know how to implement:providing target values and using functions like mse or huber as loss functions.
Objectif : Etude d'une chaine de transmission numérique avec des turbo-codes combiné avec la technique HARQ (Hybrid Automatic Repeat reQuest):
* Mettre en place une chaîne de transmission numérique avec des turbo codes intégrant la technique HARQ.
* Évaluer les performances de cette chaîne, en termes de taux d'erreur et de débit sous diverses conditions de canal.
En structurant ainsi votre étude, vous pourrez mener une analyse approfondie des turbo codes et de la technique HARQ.
Outils : Utilisez des outils comme MATLAB, Simulink ou Python (avec des bibliothèques comme Scipy pour la modélisation des canaux et NumPy pour la gestion des calculs).
Simulations : Créez une série de simulations en variant les paramètres comme le SNR, le type de HARQ utilisé, etc.
Compétences développées : Maîtrise des techniques de détection et de correction d’erreurs pour améliorer la fiabilité des transmissions
Hello,
Anyone has basic MATLAB code upon RSMA i.e. to understand basic implementation of RSMA, how to generate common and private messages in MATLAB, etc...?
Always and almost immediately!
26%
Never
30%
After validating existing code
15%
Y'all get the new releases?
29%
1843 票
Many of my best friends at MathWorks speak Spanish as their first language and we have a large community of Spanish-speaking users. You can see good evidence of this by checking out our relatively new Spanish YouTube channel which recently crossed the 10,000 subscriber mark
For a project that i am working on, I have to deal with a set of 36 superheated steam tables in one of 3 sheets in an excel file. Since the tables are organized as a set of 5 columns (T,V,U,H,S) with a Pressure "title", but Pressure is a variable I need to be able to read, I want to go about converting the pressure into the first column (unless there is an easier way) of each table where for the length of that table (27 rows) the pressure is the same even as the other variables change (like a matrix). However when I did that I faced a number of issues:
1. the tables didn't separate properly so it combined into 3 tables which is incorrect
2. The pressure column was created but didn't fill with numbers so all showed NaN
3. The column names for the variables are different than my inputs (The columns are listed with units like T_C_ or U_kJ_kg_ and so on, but I need to be able to input any 2 properties and get the rest and I need to be able to input them in the form of T and U and have it understand what columns I am looking for.
3a. When I tried to replace the variable names in the code, Since there are 3 tables it started giving me an error message that there are duplicates of properties when there shouldn't be because they should all be different tables and I want it to recognize that.
Here is a bit of what my table looks like: Please help
Apart optimize angle by Lagrange equation, how many methodes have we about anti-swing for the stabilisation of gantry crane while loading and unloading contenainers? thank you a lot.
hello !! Can we design a control method in backstepping for second order? and how to design it? thank you for all.
I've always used MATLAB with other languages. In the early days, C and C++ via mex files were the most common ways I spliced two languages together. Other than that I've also used MATLAB with Java, Excel and even Fortran.
In more recent years, Python is the language I tend to use most alongside MATLAB and support for this combination is steadily improving. In my latest blog post, I show how easy it has become to use Python's Numpy with MATLAB.
Have you used this functionality much? If so, what for? How well did it work for you?
I am inspired by the latest video from YouTube science content creator Veritasium on his distinct yet thorough explanation on how rainbows work. In his video, he set up a glass sphere experiment representing how light rays would travel inside a raindrop that ultimately forms the rainbow. I highly recommend checking it out.
In the meantime, I created an interactive MATLAB App in MATLAB Online using App Designer to visualize the light paths going through a spherical raindrop with numerical calculations along the way. While I've seen many diagrams out there showing the light paths, I haven't found any doing calculations in each step. Hence I created an app in MATLAB to show the calculations along with the visualizations as one varies the position of the incoming light ray.
Demo video:
For more information about the app and how to open it and play around with it in MATLAB Online, please check out my blog article:
Our MathWorks Usability Team is working on an accessibility project and they want to interview people who use MATLAB and also have experience with screen readers.
If you fit the criteria and are interested, sign up here https://www.mathworks.com/products/usability.html?tfa_30=A11Y
I wish I knew more about the intended evolution of the capabilities of the function arguments block. I love implementing function syntaxes using this relatively new form, but it doesn't yet handle some function syntax design patterns that I think are valuable and worth keeping.
For example, some functions take an input quantity that can something numeric, or it can be an option string that descriptively names a particular value of that quantity. One example is dateshift(t,"dayofweek",dow), where dow can be an integer from 1 to 7, or it can be one of the option strings "weekday" or "weekend".
Another example is Image Processing Toolbox that take a connectivity specifier as input. The function bwconncomp is one particular case. Connectivity can be specified using certain scalars, certain arrays, or the option string "maximal".
I think this is a worthwhile function design pattern, but I don't think the arguments block validation functionality supports it well (unless you use a lot of extra code that duplicates standard MATLAB behavior, which undermines the value of the arguments block).
MathWorkers - believe me, I know that it is not in your DNA to discuss future features. But would anyone care to offer a hint about directions for the arguments block functionality?
Hi! My name is Mike McLernon, and I’m a product marketing engineer with MathWorks. In my role, I look at the trends ongoing in the wireless industry, across lots of different standards (think 5G, WLAN, SatCom, Bluetooth, etc.), and I seek to shape and guide the software that MathWorks builds to respond to these trends. That’s all about communicating within the Mathworks organization, but every so often it’s worth communicating these trends to our audience in the world at large. Many of the people reading this are engineers (or engineers at heart), and we all want to know what’s happening in the geek world around us. I think that now is one of these times to communicate an important milestone. So, without further ado . . .
Bluetooth 6.0 is here! Announced in September by the Bluetooth Special Interest Group (SIG), it’s making more advances in its quest to create a “world without wires”. A few of the salient features in Bluetooth 6.0 are:
- Channel sounding (for accurate distance measurements)
- Decision-based advertising filtering (for more efficient channel scanning)
- Monitoring advertisers (for improved energy efficiency when devices come into and go out of range)
- Frame space updates (for both higher throughput and better coexistence management)
Bluetooth 6.0 includes other features as well, but the SIG has put special promotional emphasis on channel sounding (CS), which once upon a time was called High Accuracy Distance Measurement (HADM). The SIG has said that CS is a step towards true distance awareness, and 10 cm ranging accuracy. I think their emphasis is in exactly the right place, so let’s learn more about this technology and how it is used.
CS can be used for the following use cases:
- Keyless vehicle entry, performed by communication between a key fob or phone and the car’s anchor points
- Smart locks, to permit access only when an authorized device is within a designated proximity to the locks
- Geofencing, to limit access to designated areas
- Warehouse management, to monitor inventory and manage logistics
- Asset tracking for virtually any object of interest
In the past, Bluetooth devices would use a received signal strength indicator (RSSI) measurement to infer the distance between two of them. They would assume a free space path loss on the link, and use a straightforward equation to estimate distance:
where
received power in dBm,
transmitted power in dBm,
propagation loss in dB,
distance between the two devices, in m,
Bluetooth signal wavelength, in m,
Bluetooth signal frequency, in Hz,
speed of light (3 x 1e8 m/s).So in this method, 
. But if the signal suffers more loss from multipath or shadowing, then the distance would be overestimated. Something better needed to be found.
. But if the signal suffers more loss from multipath or shadowing, then the distance would be overestimated. Something better needed to be found.Bluetooth 6.0 introduces not one, but two ways to accurately measure distance:
- Round-trip time (RTT) measurement
- Phase-based ranging (PBR) measurement
The RTT measurement method uses the fact that the Bluetooth signal time of flight (TOF) between two devices is half the RTT. It can then accurately compute the distance d as
, where c is again the speed of light. This method requires accurate measurements of the time of departure (TOD) of the outbound signal from device 1 (the Initiator), time of arrival (TOA) of the outbound signal to device 2 (the Reflector), TOD of the return signal from device 2, and TOA of the return signal to device 1. The diagram below shows the signal paths and the times.
If you want to see how you can use MATLAB to simulate the RTT method, take a look at Estimate Distance Between Bluetooth LE Devices by Using Channel Sounding and Round-Trip Timing.
The PBR method uses two Bluetooth signals of different frequencies to measure distance. These signals are simply tones – sine waves. Without going through the derivation, PBR calculates distance d as
, where
distance between the two devices, in m,
speed of light (3 x 1e8 m/s),
phase measured at the Initiator after the tone at 
completes a round trip, in radians,
phase measured at the Initiator after the tone at 
completes a round trip, in radians,
frequency of the first tone, in Hz,
frequency of the second tone, in Hz.The mod() operation is needed to eliminate ambiguities in the distance calculation and the final division by two is to convert a round trip distance to a one-way distance. Because a given phase difference value can hold true for an infinite number of distance values, the mod() operation chooses the smallest distance that satisfies the equation. Also, these tones can be as close as 1 MHz apart. In that case, the maximum resolvable distance measurement is about 150 m. The plot below shows that ambiguity and repetition in distance measurement.
If you want to see how you can use MATLAB to simulate the PBR method, take a look at Estimate Distance Between Bluetooth LE Devices by Using Channel Sounding and Phase-Based Ranging.
Bluetooth 6.0 outlines RTT and PBR distance measurement methods, but CS does not mandate a specific algorithm for calculating distance estimates. This flexibility allows device manufacturers to tailor solutions to various use cases, balancing computational complexity with required accuracy and adapting to different radio environments. Examples include simple phase difference calculations and FFT-based methods.
Although Bluetooth 6.0 is now out, it is far from a finished version. The SIG is actively working through the ratification process for two major extensions:
- High Data Throughput, up to 8 Mbps
- 5 and 6 GHz operation
See the last few minutes of this video from the SIG to learn more about these exciting future developments. And if you want to see more Bluetooth blogs, give a review of this one! Finally, if you have specific Bluetooth questions, give me a shout and let’s start a discussion!
