Consider a $1$D convolution following by sub$-$sampling,

$z[j]={\displaystyle \underset{k}{\overset{?}{}}}w[k]x[j+k],u[m]=z[sm],$

for some stride parameter $s>0.$ That is$,u$ takes every $s$ samples of $z.$

$($a$)$ If $x$ is $250$ milliseconds of single channel audio sampled at $20kHz,$ how many samples are in $x?$

$($b$)$ If $x$ has length $1000,w$ has length $10$ and $s=4,$ how many output samples are in $u$ assuming the convolution is only computed on the valid samples.

$($c$)$ Given a gradient $$J$/$u$[$m$],$ how do you compute $$J$/$w$[$k$]?$