Absolutely! Let's break down each transformation step by step, including what each one **means** and **how to sketch it** based on the original signal \( x(t) \). --- ## **Original Signal \( x(t) \)** - The plot shows a piecewise function: - For \( -2 \leq t < -1 \), it goes from (−2, ) to (−1, −1) - For \( -1 \leq t < \), it goes from (−1, −1) to (, 1) - For \( \leq t < 1 \), it goes from (, 1) to (1, 2) - For \( 1 \leq t \leq 2 \), it goes from (1, 2) to (2, ) --- ## **a) \( x(t-1) \) — **Delay by 1 unit** ### **Explanation** - \( x(t-1) \) **shifts the signal to the right** by 1 unit. - Every feature at time \( t_ \) in \( x(t) \) moves to \( t_ + 1 \). ### **How to Sketch** - The segment from (−2, ) to (−1, −1) now goes from (−1, ) to (, −1). - The segment from (−1, −1) to (, 1) now goes from (, −1) to (1, 1). - The segment from (, 1) to (1, 2) now goes from (1, 1) to (2, 2). - The segment from (1, 2) to (2, ) now goes from (2, 2) to (3, ). --- ## **b) \( x(2-t) \) — **Time Reversal and Compression** ### **Explanation** - \( x(2-t) \) applies **time reversal** (flip over the y-axis), then shifts **right by 2 units** (since the minus sign is inside). - Alternatively: Flip x(t) about \( t=1 \). ### **How to Sketch** - Replace \( t \) with \( 2-t \): To find the new value at \( t=a \), look at the value \( x(2-a) \). - E.g., for \( t= \), plot \( x(2-) = x(2) \). - The shape is flipped (mirrored) and shifted. --- ## **c) \( x(2t+1) \) — **Time Scaling and Shift** ### **Explanation** - \( x(2t+1) \) is compressed by a factor of 2 (since 2 is multiplied inside), **and shifted left by .5** (since \( 2t+1= \) at \( t=-.5 \)). ### **How to Sketch** - To get the point at \( t = a \), calculate \( x(2a+1) \). - The signal is compressed horizontally by 2 and shifted left by .5 units. --- ## **d) \( x\left(1-\frac{t}{2}\right) \) — **Time Reversal, Scaling, and Shift** ### **Explanation** - \( x\left(1-\frac{t}{2}\right) \) reverses time (because of the minus), shifts right by 1 (since \( t=2 \) at \( x() \)), and stretches by a factor of 2. ### **How to Sketch** - Set \( s = 1-\frac{t}{2} \), solve for \( t \): \( t = 2(1-s) \). - So, the signal is **flipped**, **stretched**, and **shifted**. --- ## **Summary Table** | Transformation | Operation(s) | Effect | |----------------------|---------------------------------------------------------|--------------------------------------------| | \( x(t-1) \) | Shift right by 1 | All features move right by 1 | | \( x(2-t) \) | Time reversal & shift right by 2 | Flip over y-axis, shift right by 2 | | \( x(2t+1) \) | Compress by 2, shift left by .5 | Features halve in width, move left by .5 | | \( x(1-\frac{t}{2}) \) | Time reversal, stretch by 2, shift right by 1 | Flip, double width, shift right by 1 | --- ## **Sketching Tips** - **Shifts**: Move all points left/right by the shift amount. - **Scaling**: Divide all \( t \)-values by the scaling factor. - **Reversal**: Replace \( t \) with \( -t \), flipping over the y-axis. --- Would you like **step-by-step sketches** for each, or a specific transformation explained in more detail?