SC · Signal basics

Operations on a Signal

Shift, stretch, and flip a signal along the time axis. Learn how these three operations act on the independent variable t, and why you read them from the inside out.

Lay the signal onto the target with two sliders

The faint curve is the original signal x(t); the dashed curve is the target to match. Move the scale a and the shift b so the transformed y(t) = x(at − b) lands exactly on the target.

original xtarget

scale a1.00

shift b0.00

Current transform

y(t) = x(1.00 t − 0.00)

Both the onset and the width differ from the target. You must set the width with a and the onset with b together.

Far off

A time shift slides the whole thing left or right

x(t − t₀) slides the whole signal right by t₀. It is a minus yet moves right because the value originally at t = 0 only appears when t = t₀ — it arrives late (a delay). Conversely x(t + t₀) pulls it left, arriving early.

Time scaling compresses or dilates

In x(at), a > 1 makes time run faster so the signal compresses, while 0 < a < 1 dilates it. The width becomes 1/a. The key is that a acts on the time axis but the resulting width is its reciprocal.

Observey(t) = x(t − t₀)

Minus t₀ is a shift to the right (delay).

Choosey(t) = x(?)

a > 1 compresses; the width is 1/a.

Fill iny(t) = x(?)

Flipping the sign reflects left-to-right.

On your owny(t) = x(?)

Factor and read inside out: shift, then scale.

Reflection and the order of reading

x(−t) flips the signal left-to-right. When shift and scale come together, read x(at − b) from the inside out. Factoring at − b = a(t − b/a) shows it equals a shift by b/a first, then a scaling by a. The order the inner t undergoes is the order the signal undergoes.

Back to the first screen

To overlap the target you had to fix the width with a and the onset with b. The transformed signal’s onset moved to b/a and its width to 1/a. This is what reading x(at − b) from the inside out means: the order in which the inner t is first shifted and then scaled is exactly the order the curve moved on screen.

Operations on the independent variable move the time axis in reverse. A time shift x(t − t₀) slides right (delay) when t₀ > 0 and left when t₀ < 0. Time scaling x(at) compresses when |a| > 1 and dilates when |a| < 1, with width 1/|a|. Reflection x(−t) flips left-to-right. The combined form x(at − b) = x(a(t − b/a)) is read inside out: shift by b/a first, then scale by a.

On to the next units

Time shift and reflection are used right away in the next chapter. Convolution flips the impulse response (reflection) and slides it (shift), summing the overlapping area. Time scaling reappears in sampling and frequency scaling, where time and frequency stretch and shrink as reciprocals. The next unit, even/odd parts and energy versus power, splits a signal by symmetry and by size.