Recursive vector operation without for loops

Sometimes one want to create a vector from some recursive expression, for example when we have fibonacci:

x(0)=0, x(1)=1, x(i)=x(i-1)+x(i-2)

Is there a function in matlab which creates arrays recursively, for example:

fib = recursivevector(start=[0,1], expr=@(i,x) x(1)+x(2), len=20)

Or, a bit more complicated, we already have some vector and want to change it recursively, as is done in a exponential smoothing filter:

x2(1) = x(1), x2(i) = a*x(i) + (1-a)*x2(i-1)

Which we could then execute as:

denoised = recursivevector(start=[noised(1)], expr=@(i,x) a*x(1)+(1-a)*noised(2), len=length(noised))

In for-loops this would go like this:

function y = exponentialdenoise(x,a)
    y=zeros(size(x));
    y(1)=x(1);
    for i=2:length(x)
        y(i)=a*x(i)+(1-a)*y(i-1);
    end
end

I strongly dislike the for-loops ;) of course and prefer a version without them.

Answer 1

[Azzi Abdelmalek]

Your equation can be written as:

%x(i)-x(i-1)-x(i-2)=u(i) with u(i)=0
D=[1 -1 -1];
N=1
y=filter(N,D,zeros(1,10),[0 1])

[Herbert]

Could this also solve

a*x(i)+(1-a)*y(i-1)-y(i) = u(i) with u(i)=0 ? Should I say  (1-a)*y(i-1)-y(i) = -a*x(i)

and hence:

y=filter(1,[-1, (1-a)],-a*x,[0 1]) (DOES NOT WORK, see correct version below.)

What does N and [0 1] do in your expression?

[Azzi Abdelmalek]

%-y(i)+(1-a)*y(i-1) = -a*x(i)+0*x(i-1)
N=[-a 0]
D=[-1  1-a]
y0=[1]  %initial conditions
y=filter(N,D,x,y0)

[Herbert]

I resolved it as:

exponentialdenoise = @(x, a) filter([a 0], [1, -(1-a)], x, 0)

Which is equivalent to what you say (with a factor -1), only my initial condition (which I determined empirically) needs to be 0 to set y(1)=x(1).

[Jan]

And if run-time matters, you can try: FEX: FilterM.

Answer 2

[Jan]

Why do you dislike loops? They solve problems efficiently. And yes, sometimes there are even faster methods.

function y = exponentialdenoise(x,a)
  y    = zeros(size(x));
  y(1) = x(1);
  n    = length(x);
  y(2:n) = a * x(2:n) - (1-a) * x(1:n-1);
end

[Herbert]

I made a mistake in my maltab code, sorry, this is not what I am looking for. I want y(i-1) at the right side of the =.