Let $X(\omega)$ be the Fourier transform of the signal

$x(t) = e^{-t^4} \cos t, \quad -\infty < t < \infty$.

The value of the derivative of $X(\omega)$ at $\, \omega = 0$ is ______ (rounded off to 1 decimal place).

The input $x(t)$ and the output $y(t)$ of a system are related as

$$ y(t) = e^{-t} \int\limits_{-\infty}^{t} e^{\tau} x(\tau) d\tau, \quad - \infty < t < \infty. $$

The system is

Consider the discrete-time systems $T_1$ and $T_2$ defined as follows:

{ $T_1 x[ n ] = x[ 0 ] + x[ 1 ] + \cdots + x[ n ] $}

{ $T_2 x[ n ] = x[ 0 ] + \frac{1}{2} x[ 1 ] + \cdots + \frac{1}{2^n} x[ n ] $}

Which one of the following statements is true?

If the Z-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the Z-transform of the signal $y[n] = x[2n]$ is