As a signal pass through a system, a number of transformation can takes place. In this article, we look at few of these transformations which involves only the change in the independent variables. Further, we provide a quiz at the end of the article to test the concepts learned.
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2. Signal Transformations
Signal transformations are also known as signal operations. As a signal can be a function of one or more independent variables. For simplicity, we consider only signals with one independent variable. The independent variable in a signal is usually referred as time and denoted as $t$. The following types of transformations are possible on a signal:
- Time shifting
- Time scaling
- Time advance
We study them with an example. For that, consider a signal $x(t)$ as follows:
$x(t) = \begin{cases} 1, & 0 \leq t \leq 1, \\
2-t, \quad & 1 \leq t \leq 2, \\
0, & \text{otherwise}.
\end{cases}
$
A graphical representation of this signal is shown below.
We now see the effect of various signal transformations on this signal.
2.1 Time shifting
A time shift results in the shifting of the signal along the time-axis. This shift can happen in two days: time advance and time delay.
Time Advance
We denote it as $x(t+a), a > 0$, where $a$ is the amount of time advance. For example, let us have a time advance by $a = 2$ units. The resulting signal $x (t+2)$ is shown below.
We can observe that the signal has shifted by $a = 2$ units on the left side.
Time delay
It is represented as $x(t-a), a > 0$, where $a$ is the amount of time delay. For example, a time delay by $a = 2$ results in a signal $x (t-2)$ shown below.
2.2 Time Scaling
The transformation results in a signal given by $x(at), a > 0$, where $a$ is the amount of scaling. We will see two examples of it with different values of $a$. For $a = 2$, we can observe that signal has been compressed by half.
On the other hand, with $a = \frac{2}{3}$, we can see that the resulting signal is an expanded version of the original signal.
We can deduce that with $a > 1$, signal compression takes place while with $0 < a < 1$, signal expansion takes place. You may have noted that $a = 1$ results in no change. Further, the case of $a < 0$ comes under time reversal which we will see next.
2.3 Time Reversal
This operation results in a signal $x(-t)$, shown below.
We can observe that reversal of a signal can be obtained by taking a mirror image of the signal with respect to the $y$-axis.
3. Composition of Transformations
Till now, we have seen individual applications of each transformation. Now, we consider a case where all transformations are together possible on a signal. This case is better understood with the help of an example. Please consider the same signal $x(t)$ taken earlier. Now, we have a signal transformation given by $x(-2t+3)$. The resulting signal can be obtained in two ways which we see one by one.
3.1 Method 1: Right Hand Side (RHS) to Left Hand Side (LHS)
In this method, we carry out transformation $(-2t+3)$ by moving from RHS to LHS, symbolically $ (-2t+3_{\leftarrow})$. So, we will have these steps
- Step 1: Time advance by 3 to obtain $x(t+3)$
- Step 2: Scaling by 2 to obtain $x(2t+3)$
- Step 3: Time reversal to get finally $x(-2t+3)$
3.2 Method 2: LHS to RHS
This method is performed from LHS to RHS, symbolically $ (_\rightarrow-2t+3)$ . However, it has an additional step than method 1, which we see next.
- Step 1: Write the transformation in a factored form such that the variable $t$ has a unit coefficient. For that, take out common factor $(-2)$ from the equation to obtain $-2(t-\frac{3}{2})$.
- Step 2: Time reversal to get $x(-t)$
- Step 3: Time scaling by 2 to get $x(-2t)$
- Step 4: Time delay by $\frac{3}{2}$ to finally get $x \left(-2\left(t-\frac{3}{2}\right)\right)$
We can observe that both methods result in the same result. Hence, we can use any of the methods depending on our individual preference.
4. Summary
We have seen a number of transformations of a signal in the time-domain. The learning from the lesson is summarized below
- Time shifting
- Time advance by $a$ units results in shifting of the signal on the left-side by $a$ units.
- Delay by $a$ units means shifting of the signal on the right-side by $a$ units.
- Time shifting operation does not change the magnitude or width of the signal.
- Time scaling
- Scaling by $a > 1$ units results in compression of the signal by a factor $a$.
- Time scaling by $0 < a < 1$ units results in expansion of the signal by a factor $\frac{1}{a}$.
- Time scaling affects only the width of the signal; magnitude remains the same.
- Time reversal
- Reversal does not affect the amplitude and width of the signal.
- Composition of transformations
- The composition of reversal and scaling is commutative, i.e., we can interchange the two operations.
- The composition of shifting with reversal/scaling is not commutative.