How to generate multi-dimensional matrix (3 dimension and above) without nested loops to reduce overall run time?

My code contains a lot of multi-dimensional matrices (3 dim and above) which are increasing code complexity and run time. I want to replace nested loops with simple matrix multiplication to generate a 4-dimensional matrix as shown in the following code.

L=10; NF=512; Nr=4, Nt=4

for r=1:Nr

for t=1:Nt

h_T(:,:,r,t) = normrnd(0,sqrt(1/(2*2*L)),L,1)+1i*normrnd(0,sqrt(1/(2*2*L)),L,1);

h_F(:,:,r,t) = dft([h_T(:,:,r,t);zeros(NF-L,1)]);

end

end

%% the dft function is given as

function ft=dft(x)

N=length(x);

if size(x,2)~=1
    x=x';
end

ft=sum(bsxfun(@times,x(1:N),exp(-2*pi*1i*([1:N]-1)'*([1:N]-1)/(8*N))),1)'; 

end

[Sajid Sarwar]

Can anyone help??

Answer 1

[Pavan Sahith]

Hello Sajid,

I observe that you're looking to optimize your MATLAB code by replacing nested loops with efficient matrix operations to generate a 4-dimensional matrix. In your DFT function , I assume that 'x' is a matrix where each column is a separate signal to transform.

You can utilize MATLAB's built-in functions and array operations to handle multi-dimensional matrices more efficiently.

Here's how you can achieve this while potentially reducing the runtime:

• Pre-allocate the matrices to avoid dynamic resizing during loop execution.
• Generate the complex Gaussian random variables for all 'r' and 't' values without loops.
• Reshape 'h_T' to a 2D matrix for Discrete Fourier Transform (DFT).
• Vectorize the DFT function to apply it to all columns of the input matrix.

L = 10; NF = 512; Nr = 4; Nt = 4;

% Pre-allocate h_T and h_F for efficiency
h_T = zeros(L, 1, Nr, Nt);
h_F = zeros(NF, 1, Nr, Nt);

% Generate the complex Gaussian random variables for all r and t at once
realPart = normrnd(0, sqrt(1/(2*2*L)), L, Nr, Nt);
imagPart = 1i * normrnd(0, sqrt(1/(2*2*L)), L, Nr, Nt);
h_T = realPart + imagPart;

%We reshape h_T to a 2D matrix, where each column represents a different combination of r and t. 
% This allows us to apply the DFT to all columns simultaneously.
h_T_reshaped = reshape(h_T, L, Nr*Nt);

% Apply DFT to each column
h_F_reshaped = dft([h_T_reshaped; zeros(NF-L, Nr*Nt)]);

% Reshape the result back to a 4D matrix
h_F = reshape(h_F_reshaped, NF, 1, Nr, Nt);

%% Vectorized dft function
function ft = dft(x)
    N = size(x, 1);
    n = (0:N-1)';
    k = n;
    W = exp(-2 * pi * 1i * k * n' / (8 * N));
    ft = W * x;
end

To know more about the 'reshape' function, you can refer to the following MathWorks documentation.

reshape-https://www.mathworks.com/help/matlab/ref/reshape.html

Hope this will help.