# Breaking down a higher order filter to simple cascade of averages

I am having the following filter structure, which is a second-order decimation:

Which actually implements the following equation/code: (ans-0.2727) $$\frac{2}{N(N+1)}\sum_{i=1}^{N} (N+1-i).d(i)$$

d=[1 0 1 0 1 0 1 0 1 0 1];
N=10;
sum2=0;
for i=1:1:N
sum2=sum2+((N+1-i)*d(i));
end
second_order_ans=sum2/(N*(N+1));


However, I wanted to break this to two separate for loops (or) a cascade of two first-order decimations.

The first-order decimation corresponds to the following equation and code: (ans-0.5) $$\frac{1}{N}\sum_{i=1}^{N} d(i)$$

d=[1 0 1 0 1 0 1 0 1 0 1];
N=10;
sum=0;
for i=1:1:N
sum=sum+d(i);
end
first_order_ans=sum/N;


I tried the following for cascade but I get the output as 0.6182 and far away from 0.2727.

%second order - cascade
for i=2:1:(N-2)
sum2_c1(i)=sum2_c1(i-1)+d(i); %First counting
end

for i=2:1:(N-2)
sum2_c2(i)=sum2_c1(i)+sum2_c2(i-1); %Second counting
end
sum2_c=0;

for i=2:1:(N-2)

• Is this line OK : second_order_ans=sum2/(N*(N+1)) (equ is 2/(N*(N+1))*sum2) ? – Juha P Feb 25 at 15:42