Suppose for input x(t) a linear time-invariant system with impulse response h(t) produces output y(t), so that x(t) * h(t) = y(t). Further, if |x(t)| * |h(t)| = z(t), which of the following statements is true?

The correct option is: For all t ∈ (-∞, ∞), z(t) ≥ y(t)

To understand why this statement is true, let us look at the mathematical definition of convolution for continuous-time signals.

The output $y\left(t\right)$ of a linear time-invariant (LTI) system with input $x\left(t\right)$ and impulse response $h\left(t\right)$ is given by the convolution integral:

$$y\left(t\right)=x\left(t\right)*h\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right)d\tau$$

Similarly, the signal $z\left(t\right)$ is defined as the convolution of the absolute values of the input and the impulse response:

$$z\left(t\right)=\left|x\left(t\right)\right|*\left|h\left(t\right)\right|={\int }_{-\infty }^{\infty }\left|x\left(\mathrm{��}\right)\right|\left|h\left(t-\tau \right)\right|d\tau$$

Now, we can apply a fundamental property of integrals. For any real-valued or complex-valued functions, the value of an integral is always less than or equal to the integral of its absolute value. This is a continuous version of the triangle inequality:

$$y\left(t\right)={\int }_{-\infty }^{\infty }x\left(\tau \right)h\left(t-\tau \right)d\tau \le {\int }_{-\infty }^{\infty }\left|x\left(\tau \right)h\left(t-\tau \right)\right|d\tau$$

Using the property of absolute values that $\left|a·b\right|=\left|a\right|·\left|b\right|$, we rewrite the integrand on the right-hand side:

$${\int }_{-\infty }^{\infty }\left|x\left(\tau \right)h\left(t-\tau \right)\right|d\tau ={\int }_{-\infty }^{\infty }\left|x\left(\tau \right)\right|\left|h\left(t-\tau \right)\right|d\tau$$

Since the expression on the right-hand side is exactly the definition of $z\left(t\right)$, we obtain:

$$y\left(t\right)\le z\left(t\right)$$

This inequality holds for all values of $t$ in the interval $\left(-\infty ,\infty \right)$. Therefore, we have established that $z\left(t\right)\ge y\left(t\right)$ for all $t\in \left(-\infty ,\infty \right)$.