Can someone explain to me this solution :
$$\int_{-\infty}^{\infty}\delta (\tau -\frac{1}{4}t)e^{-j2\pi f_dt}dt=e^{-j2\pi f_d4\tau }$$
When I have solved, I got $e^{-j2\pi f_d4t }$
But the answer is: $e^{-j2\pi f_d4\tau }$
Where I have made the mistake?
If we want linear changes of variables to be the same for regular and for singular functionals, we get $$\int \delta {\left( \tau - \frac t 4 \right)} e^{-2 \pi j t f_d} dt = 4 \int \delta {\left( \tau - u \right)} e^{-8 \pi j u f_d} du = 4 e^{-8 \pi j \tau f_d }.$$ Also note that this is just a notation (say, for the Fourier transform), because typically $e^{j t}$ is not a test function to which we can directly apply the functional $\delta$.