# Unit impulse response of a cascade interconnection of three discrete-time systems

I am nearly at the end of finishing a problem in my textbook but I couldn't understand something in the answer;

I did everything to the point I found the overall response of the system in terms of $$h_{1}[n]$$ which is $$h[n]=h_{1}[n]+2h_{1}[n-1]+h_{1}[n-2]$$ Now while comparing each $$n$$ value with the given graph for overall response, the answer is saying that for $$n=0$$; $$h[0]=h_{1}[0]$$ but I didn't get why it isn't like this: $$h[0]=h_{1}[0]+2h_{1}[-1]+h_{1}[-2]$$ How do we know $$h_{1}[n]$$ is zero for $$n$$ values smaller than zero?

What does that imply for the impulse response $$h_1[n]$$?
What is more, since the total impulse response is causal, and you know from its definition that $$h_2[n]$$ is causal, $$h_1[n]$$ must also be causal.
• I thought about that too actually but from the definition I remember for causality, the output should only be effected from present or past values, the $h_{1}[-1]$ and $h_{1}[-2]$ are past values and I thought they should be included. Mar 24, 2021 at 13:58
• Oh, okay, I checked the convolution sum with the conditions of causality, the value of $h[n-k]$ should be zero for $n-k<0$ since for $y[n]$ and $x[k]$ we know that it should be $k>!n$ therefore $h[n]=0$ while $n<0$, thanks for pointing out! Mar 24, 2021 at 14:55