Analogy between Vectors and Signals

we have already defined the signal as any ordinary function of time. To understand more about signal we consider it as a problem. A problem is better understood (or) better remembered if it can be associated with some familiar phenomenon.

we always search for analogies while studying a new problem.

i.e, In the study of abstract problems analogies are very helpful. Particularly if the problem can be shown to be analogous to some concrete phenomenon.

It is then easier to gain some insight into the new problem from the knowledge of the analogous phenomenon.

There is a perfect analogy that exists between vectors and signals which leads to a better understanding of signal analysis. we shall now briefly review the properties of vectors.

Vectors:-

A vector is specified by magnitude and direction $\overrightarrow{A}$ .

Let us consider two vectors $\overrightarrow{V_{1}}$ and $\overrightarrow{V_{2}}$ . It is possible to find out the component of one vector along the other vector.

In order to find out the component of vector $\overrightarrow{V_{1}}$ along $\overrightarrow{V_{2}}$ . Let us assume it as $C_{12}V_{2}$ , which is only the magnitude.

how do we represent physically the component of one vector $\overrightarrow{V_{1}}$ along $\overrightarrow{V_{2}}$ ? This is possible by finding the projection of one vector on to the other.

i.e, by drawing a perpendicular from $\overrightarrow{V_{1}}$ to $\overrightarrow{V_{2}}$

$\overrightarrow{V_{1}} = C_{12}\overrightarrow{V_{2}} + \overrightarrow{V_{e}}$ .

There exists two other possibilities.

but these are not suitable. $\because$ the error vectors are more in these cases.

$\overrightarrow{V_{1}}.\overrightarrow{V_{2}}=V_{1}V_{2}\cos \theta$ .

If $\theta$ is the angle between two vectors $\overrightarrow{V_{1}}$ and $\overrightarrow{V_{2}}$ , the component of $\overrightarrow{V_{1}}$ along $\overrightarrow{V_{2}}$ is

$\frac{\overrightarrow{V_{1}}.\overrightarrow{V_{2}}}{\left | V_{2} \right |}=V_{1}\cos \theta-----EQN(1)$ .

The component of $\overrightarrow{V_{1}}$ along $\overrightarrow{V_{2}}$ is $C_{12}V_{2}----EQN(2)$ .

$\therefore (1) = (2)$ .

$\frac{\overrightarrow{V_{1}}.\overrightarrow{V_{2}}}{\left | V_{2} \right |} = C_{12}V_{2}$ .

$C_{12}=\frac{\overrightarrow{V_{1}}.\overrightarrow{V_{2}}}{V_{2}^{2}}$ .

If two vectors are orthogonal $\overrightarrow{V_{1}}.\overrightarrow{V_{2}} =0$ .

i.e, $C_{12} =0$ .

Signals:-

The concept of vector comparison & orthogonality can be extended to signals.

i.e, a signal is nothing but a single-valued function of independent variable. Assume two signals $f_{1}(t)$ and $f_{2}(t)$ , now to approximate $f_{1}(t)$ in terms of $f_{2}(t)$ over $t_{1}< t<t_{2}$ .

$f_{1}(t) \approx C_{12}f_{2}(t)$ .

$\therefore f_{1}(t) \approx C_{12}f_{2}(t)+f_{e}(t)$ .

$f_{e}(t) = f_{1}(t)-C_{12}f_{2}(t)$ .

Now, we choose in order to achieve the best approximation.

i.e, which keeps the error as minimum as possible.

One possible way for minimizing error $f_{e}(t)$ is to choose minimize the average value of $f_{e}(t)$ .

i.e, as $\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}dt$ .

But the process of averaging gives a false indication.

i.e, for example while approximating a function $\sin t$ with a null function $f(t)=0$ is

$f_{1}(t) =C_{12}f_{2}(t)$ .

$\sin t =0. \ \ 0\leq t\leq 2\pi$ .

indicates that $\sin t =0$ during $0$ to $2\pi$ without any error

i.e, $f_{e}(t) = f_{1}(t)-C_{12}f_{2}(t)$ .

$f_{e}(t) = \sin t$ .

Average value of error is $= \frac{1}{2 \pi } \int_{0}^{2\pi } \sin t \ dt =0$ .

This seems to be error is zero but actually there exists some error.

To avoid this false indication, we choose to minimize the average of the square of the error

i.e, Mean Square Error $\epsilon = \frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}f_{e}^{2}(t) \ dt$ .

$\epsilon = \frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}(f_{1}(t)-C_{12}f_{2}(t))^{2} \ dt$ .

To find value which keeps error minimum $\frac{d\epsilon }{dC_{12}}=0$ .

$C_{12} = \frac{\int_{t_{1}}^{t_{2}}f_{1}(t)f_{2}(t) \ dt}{\int_{t_{1}}^{t_{2}}f_{2}^{2}(t) \ dt}$ .

$C_{12}$ Which is similar to $C_{12}=\frac{\overrightarrow{V_{1}}.\overrightarrow{V_{2}}}{V_{2}^{2}}$ where $\int_{t_{1}}^{t_{2}}f_{1}(t)f_{2}(t) \ dt$ denotes the inner product between two Real signals