Continuous-Time Signal Transformation Explorer

Interactive visualization of time transformations x(at + b): shift, scale, and reverse a continuous-time signal one step at a time and watch where every point goes.

This interactive signal transformation visualization shows how the continuous-time operation x(at + b) reshapes a signal, built up one clear operation at a time so you can see exactly where every point moves. Pick a signal (a custom piecewise shape, the unit step, the unit impulse, a gate, a single-period sinusoid, or a sinc), then set the scale factor a and the shift b, or turn on time reversal, and step through the animation.

The lesson follows the order that makes the single expression x(at + b) come out right: shift first, then scale, then reverse. First the whole signal is shifted by b. A common mistake is to read x(t + 3) as a shift to the right; it is actually a shift to the left, because the value that was at t = 0 now needs t + 3 = 0, so t = -3. Next the time axis is scaled by the size of a: when |a| is greater than one the signal is compressed toward t = 0, and when |a| is less than one it is stretched, with every width multiplied by 1/|a|. Finally, a negative a reflects the signal about the line t = 0.

You can watch the transformation play smoothly, step through it one operation at a time, or scrub across it. A running filmstrip records the sequence x(t), x(t+3), x(2t+3), x(-2t+3), and a live panel confirms where the origin lands, t' = (t - b)/a. The impulse case highlights a subtlety many courses gloss over: time-scaling changes the impulse area to 1/|a|, not just its position. Use the presets to compare a pure shift, a compression, a stretch, a reversal, and a full mix, and build a solid mental model of how transforming the independent variable of a signal works.