From the definition of convolution, we know:
( x ∗ h ) ( n ) = ∑ m = − ∞ ∞ x [ m ] h [ n − m ] {\displaystyle {\begin{aligned}(x*h)(n)&=\sum _{m=-\infty }^{\infty }{x[m]h[n-m]}\\\end{aligned}}}
where h [ n ] {\displaystyle h[n]} is the shifted delta, δ [ n − n 0 ] {\displaystyle \delta [n-n_{0}]} . Let us substitute this in to obtain:
∑ m = − ∞ ∞ x [ m ] δ [ ( n − m ) − n 0 ] {\displaystyle \sum _{m=-\infty }^{\infty }{x[m]\delta [(n-m)-n_{0}]}}
Let k = n − m − n 0 {\displaystyle k=n-m-n_{0}} . Then we have:
( x ∗ h ) ( n ) = ∑ m = − ∞ ∞ x [ m ] δ [ ( n − m ) − n 0 ] = ∑ m = − ∞ ∞ x [ ( n − n 0 ) − k ] δ [ k ] = ∑ k = − ∞ ∞ x [ ( n − n 0 ) − k ] δ [ k ] = ( x ∗ δ ) ( n − n 0 ) {\displaystyle {\begin{aligned}(x*h)(n)&=\sum _{m=-\infty }^{\infty }{x[m]\delta [(n-m)-n_{0}]}\\&=\sum _{m=-\infty }^{\infty }{x[(n-n_{0})-k]\delta [k]}\\&=\sum _{k=-\infty }^{\infty }{x[(n-n_{0})-k]\delta [k]}\\&=(x*\delta )(n-n_{0})\end{aligned}}}
i.e., the result of ( x ∗ h ) {\displaystyle (x*h)} where h [ n ] {\displaystyle h[n]} is the shifted delta is simply the convolution of x {\displaystyle x} and δ {\displaystyle \delta } , shifted by the same amount.
![{\displaystyle {\begin{aligned}(x*h)(n)&=\sum _{m=-\infty }^{\infty }{x[m]h[n-m]}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3894a9c70dd2340657feceec38d7ef547f14617f)
![{\displaystyle h[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46)
is the shifted delta, ![{\displaystyle \delta [n-n_{0}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdb0265027e056f16fce87ab282b57cb03c4f8c)
. Let us substitute this in to obtain:
![{\displaystyle \sum _{m=-\infty }^{\infty }{x[m]\delta [(n-m)-n_{0}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88626e95c89aacde3e136e3d9c04420d579cf8c6)
. Then we have:
![{\displaystyle {\begin{aligned}(x*h)(n)&=\sum _{m=-\infty }^{\infty }{x[m]\delta [(n-m)-n_{0}]}\\&=\sum _{m=-\infty }^{\infty }{x[(n-n_{0})-k]\delta [k]}\\&=\sum _{k=-\infty }^{\infty }{x[(n-n_{0})-k]\delta [k]}\\&=(x*\delta )(n-n_{0})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5a5bbee7885daa81a5a23ebd57495006f6c00f)
where is the shifted delta is simply the convolution of 
and 
, shifted by the same amount.
