- #1
muzialis
- 166
- 1
Let us write a convolution
$$\int_{0}^{t} A(t-\tau) \mathrm{d}x(\tau)$$ as
$$A \star \mathrm{d}x$$
I would like to write down the expression for the double convolution
$$A \star \mathrm{d}x \star \mathrm{d}x $$
Following the definition I obtain
$$ \int_{0}^{t} \int_{0} ^{t-\tau} A(t-\tau-s) \mathrm{d}x(s) \mathrm{d}x(\tau)$$
Can this be given a more compact form, especially in reference to the upper limit of integration in the inner integral?
I would like to perform the change of variable $$t-\tau = w$$ but unsure as to how to proceed, any hint would be the most appreciated, thanks
$$\int_{0}^{t} A(t-\tau) \mathrm{d}x(\tau)$$ as
$$A \star \mathrm{d}x$$
I would like to write down the expression for the double convolution
$$A \star \mathrm{d}x \star \mathrm{d}x $$
Following the definition I obtain
$$ \int_{0}^{t} \int_{0} ^{t-\tau} A(t-\tau-s) \mathrm{d}x(s) \mathrm{d}x(\tau)$$
Can this be given a more compact form, especially in reference to the upper limit of integration in the inner integral?
I would like to perform the change of variable $$t-\tau = w$$ but unsure as to how to proceed, any hint would be the most appreciated, thanks