Regarding t: Your time intervals don't need to change length just the step size.
Regarding x_a: you need to keep only the samples from the orignal signal other samples should be zero
Regarding y_a: you can just use colon indexing to downsample ever other 'M_a' smaples
Also, typically you need to do low pass filtering. When you interpolate first upsample then low pass using the fs_original/2. When you decimate, low pass first with fs_final/2.
To get your new signals you can use loops or some built in indexing methods
[N,D]=rat(1.66)
t_original = 0:1/fs:1
t_interpol = 0:1/fs/N:1;
t_decimate = t_interpol(1:D:end) % OR 0:1/fs/N*D:1, note you downsample the upsampled signal
x_original = % evaluate at t_original
x_interpol = % you can evaluate at t_interpol, BUT you should only keep the samples at times in t_original
x_decimate = x_interpol(1:D:end) % look at x_interpol and keep samples at times t_decimate
% for x_interpol and x_decimate you can use loops, or check out
doc ismember % to see when t_interpol and t_original are the same ex. x2(~ismember(t2,t))=0
doc intersect % similar use to ismember ex. [~,idx] = intersect(t2,t)
% i recommend you compare your results to the built in methods
doc upsample
doc downsample
doc resample % note resample has special filter to remove artifacts
