MATLAB: Vectorization time-varying recursive linear function

iir filterrecursiontime varying

I try to vectorize this simple recursive relation (all quantities are scalars)
x_{0} = 0;
x_{n} = x_{n-1}*a_{n} + b_{n} for n=1,2,...,N
In MATLAB code it can be carried out by for loop
% test inputs
b=rand(1,10);
a=0.9+zeros(size(b));
xk=0;
x=zeros(size(b));
for k=1:length(x)
    xk = a(k)*xk+b(k);
    x(k) = xk;
end
For a(:) constant this can be vectorized by IIR filter
ac = unique(a);
if length(ac)==1
    x = filter(1, [1 -ac], b);
end
I would though it could have some time-varying IIR filter that I can use to vectorize the case where a is time-dependent.
But I couldn't find anywhere such stock function. anyone have an idea?

Best Answer

Hi Bruno,
a = rand(1,50);
b = rand(1,50);
% method 1
xk = 0;
x = zeros(1,50);
for k = 1:50
    xk = a(k)*xk + b(k);
    x(k) = xk;
end
% method 2
cpa = cumprod([1 a(2:end)])
x1 = filter(1,[1 -1],b./cpa).*cpa;
max(abs(x1-x))
  ans =  4.4409e-16
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