What is the linear convolution between x(n) and v(2n)
So far my professor was doing the convolution only with the tabular method and I don't know how to do this using delta functions
2026-03-28 03:51:41.1774669901
linear convolution using delta functions
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We want the convolution of $\delta(x+1)+2\delta(x)+\delta(x-1)$ with $\delta(x+2)+\delta(x-2)$. Since these respectively integrate to $4,\,2$, the problem is equivalent to determining the distribution of $X+Y$ in terms of Dirac spikes, with independent $X,\,Y$ where$$P(X=1)=P(X=-1)=\tfrac14,\,P(X=0)=P(Y=2)=P(Y=-2)=\tfrac12,$$then multiplying all weights by $8$. So now you don't even need calculus. You're welcome to determine the full result from first principles, but for a multiple choice question we have a shortcut. All weights must be $\ge0$ (this is an advantage of recasting the problem into probabilities), which eliminates B, C and D, and $X+Y=-3$ is achievable, which eliminates E, so A is right.