SC · Signal basics

Operations on a Signal

Learn how the three operations that shift, stretch and flip a signal along the time axis act on the independent variable t. The heart of this unit is the order in which x(at − b), with a shift and a scaling together, has to be read for it to match the figure.

Lay the signal onto the target with two sliders

The faint gray curve is the original signal x(t), which jumps up at t = 0 and dies away exponentially. The gold dashed curve is the target to match, and as you move the scale a and the shift b, the blue curve y(t) = x(at − b) changes with them. Lay the blue curve exactly over the target and it turns gold; finding the a and b at that moment is the task of this screen.

original xtarget

scale a1.00

shift b0.00

Current transform

y(t) = x(1.00 t)

Both the start and the width differ from the target. a sets the width and b the starting position, but the start b/a depends on both handles together, so moving just one does not get you there.

Far off

A time shift slides the whole thing left or right

x(t − t₀) shifts the whole signal to the right by t₀. The reason a minus sign moves it right is that the original value at t = 0 only appears once t = t₀. In other words the signal arrives late, a delay; conversely, x(t + t₀) pulls the signal to the left so that it arrives early.

Time scaling compresses or dilates

In x(at), if a > 1 time runs faster and the signal is compressed, if 0 < a < 1 it is stretched, and the width becomes 1/a times as large. The key point is that the quantity multiplying the time axis is a, while the resulting width is its reciprocal. The original signal in this unit dies away with a time constant of 1/1.4 ≈ 0.71, so with a = 2 that time is halved to about 0.36.

Observey(t) = x(t − t₀)

Subtracting t₀ moves it right (a 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(?)

Compressed by a, then moved right by t₀.

Reflection and the order of reading

x(−t) flips the signal left to right. When a shift and a scaling come together, the order needs care: grouping at − b = a(t − b/a) shows that x(at − b) is the signal compressed by a, like x(at), and then moved right by b/a. The same result also comes from shifting by b first and then compressing by a, but shifting by b/a first and then compressing puts the start at b/a², which is wrong. That is why, in the figure, the start landed at b/a and the width at 1/a.

Back to the first screen

To lay the curve over the target you had to set a and b together, and at the moment they matched, the blue curve's start was at b/a = 1 and its width was 1/2 of the original. That is what reading x(at − b) as a signal compressed by a and then moved by b/a means. The two sliders moved separately, but the screen showed plainly that the start is set by their ratio b/a, while the width is set by a alone.

Operations on the independent variable move the time axis the opposite way. A time shift x(t − t₀) moves the signal right (a delay) when t₀ > 0 and left when t₀ < 0. Time scaling x(at) compresses when |a| > 1 and stretches when |a| < 1, and the width becomes 1/|a| times as large. Reflection x(−t) flips left and right. The combined form x(at − b) = x(a(t − b/a)) is read as compressing by a and then shifting by b/a, or as shifting by b and then compressing by a.

On to the next units

Time shift and reflection are used right away in the next chapter. Convolution is the operation that flips the impulse response (reflection), slides it (shift) and adds up the overlapping area. Time scaling reappears in sampling and in frequency scaling, where time and frequency stretch and shrink as reciprocals of each other. The next unit, even and odd parts of a signal, splits a signal by symmetry and, as a bonus, measures its size as energy and power.