Signal transformations

The four basic transformations of a signal $g(t)$ are amplitude scaling, time shifting, time scaling, and combinations (sum and product). Together they’re the basic grammar for manipulating signals in this course.

• Amplitude scaling: $g(t)\to A\cdot g(t)$ multiplies every value by $A$. Horizontal axis unchanged. $A>1$ taller, $0<A<1$ shorter, $A<0$ flips upside down.
• Time shifting: $g(t)\to g(t-t_0)$ slides the signal right by $t_0$ seconds (if $t_0>0$) or left by $|t_0|$ (if $t_0<0$). Catches every student at least once: $g(t-t_0)$ with positive $t_0$ shifts right. The reason is that the value of the new signal at $t$ is what $g$ used to have at time $t-t_0$, so what used to be at $t=0$ is now at $t=t_0$.
• Time scaling: $g(t)\to g(t/a)$ stretches horizontally if $|a|>1$, compresses if $|a|<1$. Negative $a$ also reflects about the vertical axis. The value of the new signal at $t$ is what $g$ had at $t/a$, so what was at $t=1$ is now at $t=a$.
• Combinations: sum or product of two signals, pointwise.

The first three are simple individually. The trouble starts when you combine them, because the order matters.

The standard form

The course standardizes how combined transformations are written:

$g(t)\to A\cdot g\left(\frac{t-t_0}{a}\right).$

Three parameters: amplitude $A$, time scaling $a$, time shift $t_0$. Whenever we’re going from $g(t)$ to $Ag((t-t_0)/a)$ step by step, we apply the steps in this fixed order:

1. Amplitude scaling first: $g(t)\to Ag(t)$.
2. Time scaling second: $Ag(t)\to Ag(t/a)$.
3. Time shifting third: $Ag(t/a)\to Ag((t-t_0)/a)$.

Why this order

Because in any other order, you’ll usually get a different signal. Suppose we wanted $Ag((t-t_0)/a)$ but tried time-shifting before time-scaling. We’d first replace $t$ by $t-t_0$: $Ag(t-t_0)$. Then we’d replace $t$ by $t/a$: $Ag(t/a-t_0)$. But the target was $Ag((t-t_0)/a)=Ag(t/a-t_0/a)$. The two differ in where the shift ends up.

The recommended order works because each step modifies a different part of the expression cleanly: amplitude scaling doesn’t touch the argument, time scaling replaces $t$ with $t/a$, and time shifting replaces $t$ with $t-t_0$ inside that scaling.

Non-standard form

Sometimes a problem hands us $Ag(bt-t_0)$ — scaling factor $b$ multiplies $t$ without dividing it. This is not in standard format. Convert by factoring $b$ out:

$g(bt-t_0)=g\left(b\left(t-\frac{t_0}{b}\right)\right)=g\left(\frac{t-t_0/b}{1/b}\right).$

So the standard-form parameters are $a^{\prime }=1/b$ and $t_0^{\prime }=t_0/b$.

Concrete: to plot $3g(-2t-1)$, factor out $-2$:

$3g(-2t-1)=3g(-2(t+1/2))=3g\left(\frac{t-(-1/2)}{-1/2}\right),$

so $A=3$, $a=-1/2$, $t_0=-1/2$. Then apply amplitude → scaling → shift in order.

If you do it any other way, you’ll end up with the wrong shift amount and a sign error. The bookkeeping looks tedious until you’ve made the mistake once.

Reading off parameters from two plots

The reverse problem: given $g_1(t)$ and $g_2(t)$, write $g_2$ as $A{g}_1((t-t_0)/w)$. Read off the three parameters in the order $(A,w,t_0)$ — they uncouple in that order. Compare peak values for $A$. Compare widths of nonzero regions for $w$. Plot $A{g}_1(t/w)$ and slide it until it matches $g_2$ to read off $t_0$.