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In computer engineering, we are interested in two different kinds of signals: Continuous signals and discrete signals. A continuous signal is defined at all times. A discrete signal is defined only at integer times. For example, daily sales of a company is a discrete signal. Atmospheric pressure at a given location is a continuous signal. Signals can also have more than one dimension. A digital photograph is a two dimensional discrete signal, formed of pixels. Total atmospheric pressure P(x,y,z,t) is a four dimensional continuous signal (we can measure the pressure at every latitude, longitude, height, and time). Some signals are inherently discrete, like the daily sales of a company, mentioned above. Some other signals are inherently continuous, but we convert them into discrete format for storing and processing as our computers (and our brains) are not capable to deal with the infinite amount of information associated with a continuous signal. In this and the next chapters, we will consider some important Continuous Signals.

### 1. Continuous Exponential Signals

Dividing $Ae^{\lambda t}$ where $A$ , $\lambda$ $\in$ $\mathbb{R}$. If $\lambda < 0$ the signal will decay as $t \rightarrow \infty$ and it will decay faster as $|\lambda|$ becomes larger. Conversely, If $\lambda > 0$ , the signal will grow as $t \rightarrow \infty$, and it will grow faster as $|\lambda|$ becomes larger. At $\lambda = 0$ , the signal will become constant (ie, $A$) , A useful concept in understanding exponential signals is the concept of half life. A decaying exponential function’s value will be divided by half at every $\frac{1}{|\lambda|}ln(\frac{1}{2}$) units of time.

### 2. Continuous Trigonometric Signals

Continuous trigonometric signals has three properties: frequency, amplitude and phase.

$x(t) = Asin(\omega t + \varphi) = Asin(2 \pi ft + \varphi)$
$y(t) = Acos(\omega t + \varphi) = Acos(2 \pi ft + \varphi)$

Here $A$ is the amplitude, $\omega$ is the angular frequency,  $f$ is the frequency, and $\varphi$ is the phase.

The frequency $f$ denotes how many times the signal $x(t)$  oscillates per second. There is a related quantity, called period (T), which indicates how long it takes for  $x(t)$ to complete one oscillation. Naturally

T = $\frac{1}{f}$

The unit of $f$ is 1/second, which is known as “Hertz”.

— Explain orthogonality relations here..

### 3. Indicator Function

Plot of (a)$X_{(-2,3)}(t)$ (above), (b)$X_{(2,4)}(t)$ (middle), (c)$U(t)$ (below).

### 4. Unit Step Function

Unit step function $U(t)$ is defined by

$U(t) = \begin{cases} 0, & t < 0 \\ 1, & 0 < t \end{cases}$ Note that unit step function is a special case of the indicator function

$U(x) = \chi_{(0,\infty)}(t)$

Conversely, indicator function can be written in terms of unit step functions:

$\chi_{(a,b)}(t) = U(a) – U(b)$

### 5. Transformations of Continuous Signals

We will first consider two different kinds of continuous signal transformations separately: shifting (delaying or advancing a signal) and scaling. Then we will consider the cases when these two transformations are applied together to the same signal.

#### – Shifting of continuous signal

Let us have a signal $f(t)$ . It is easy to see that if $t_0 > 0$ is any constant number, $f(t – t_0)$ is the $t_0$ seconds delayed version of $f(t)$. Or, expressed in words, whatever value $f(t)$ takes, that value will also be taken by $f(t – t_0)$ in exactly $t_0$ seconds later. This delay will appear as a “shift to the right” in plots.

Similarly $f(t + t_0)$ will appear as $t_0$ seconds earlier than $f(t)$ This earliness will appear as a “shift to the left” in plots.

Example 1: Consider the signal $y = 3t + 2$, $y = 3(t-1) + 2$ shifts that signal 1 units to left, while $y = 3(t + 1.5) + 2$ shifts it 1.5 units to right.

Example 2: Consider the signal $f(t)$, shown in Fig ?. Fig ? depicts $f(t-4)$, which is $f(t)$ shifted to the left by 4. Fig ? depicts $f(t + 6)$, which is $f(t)$ shifted to the right by 6.

Example 3: When we shift the indicator functions, the rule is

$\chi_{(a,b)}{(t-k)} = \chi_{(a+k,b+k)}(t)$

Example 4Unit step functions can also be shifted to right and left. Fig ? shows $U(t)$, $U(t-2)$ and $U(t+3)$ Plot of (a)$f(t)$ (above) (b)$f(t-4)$ (middle), (c)$f(t+6)$ (below Coming Soon)

#### – Scaling of continous signal

Given a signal $f(x)$, $f(ax)$ is its scaled version. For $|a| > 1$ , the signal will “expand”. For $|a| < 1$ , the signal will “contract”. For $a$ negative, the signal will also “reflect around the origin”. For examples, see Fig. scale