Question

A causal and stable LTI system with impulse response \(h(t)\) produces an output \(y(t)\) for an input signal \(x(t)\). A signal \(x(0.5t)\) is applied to another causal and stable LTI system with impulse response \(h(0.5t)\). The resulting output is:

• A. \(2y(0.5t)\)
• B. \(4y(0.5t)\)
• C. \(0.25y(2t)\)
• D. \(0.25y(0.25t)\)

Hint:

For LTI systems, if both the input and impulse response are scaled by a factor of \(a\), the output is scaled by \(\frac{1}{|a|}\) and compressed in time by the same factor. Use this principle for scaling problems.

Answer 1

The Correct Option is A

Step 1: Understand the scaling property of LTI systems.
In a causal and stable LTI system, the output \(y(t)\) is derived from the convolution of the input \(x(t)\) with the impulse response \(h(t)\): \[ y(t) = x(t) * h(t). \] When the input is scaled to \(x(at)\) and the impulse response is scaled to \(h(at)\), the output transforms according to the scaling property: \[ y(t) \to \frac{1}{|a|} y\left(\frac{t}{a}\right). \] Step 2: Apply the specific scaling factors.
Given that the input is \(x(0.5t)\) and the impulse response is \(h(0.5t)\), substitute \(a = 0.5\) into the scaling property: \[ y(t) \to \frac{1}{|0.5|} y\left(\frac{t}{0.5}\right) = 2y(0.5t). \] Step 3: Interpret the result.
The scaling factor of \(a = 0.5\) causes the output to be scaled by a factor of 2 and compressed in time by a factor of \(0.5\), resulting in: \[ \boxed{{(1) } 2y(0.5t)}. \]