Question

For the figure below, \( x(t) \) and \( y(t) \) are related as _______.

 

• A. \( y(t) = x(4(t + 1)) \)
• B. \( y(t) = x((3t - 1)) \)
• C. \( y(t) = 2x(t + 1) \)
• D. \( y(t) = x(2(t + 1)) \)

Hint:

To analyze time transformations, always check for compression/stretch (scaling) and left/right shift in the signal's shape and time axis.

Answer 1

The Correct Option is D

Step 1: Understand the transformation
From the graph, we observe that the time axis of \( y(t) \) has been compressed (shrinking width) and shifted left. This suggests a time-scaling and a time-shift operation.
Step 2: Time-scaling
The signal is compressed by a factor of 2, implying a time scaling: \[ x(2t) \] Step 3: Time-shifting
The graph also shows a leftward shift by 1 unit, which implies: \[ x(2(t + 1)) \] Therefore, the correct transformation is:
\[ y(t) = x(2(t + 1)) \]