Commonly used signals
Sorry, no match for the embedded content.

### 1. Real Exponential Signals

These signals are given by:

(1)
$$x(t)=Be^{at}$$

When $a<0$: Decaying Exponential

When $a>0$: Growing Exponential

Sorry, no match for the embedded content.

### 2. Real Discrete-time Exponential Signal

The Continuous-time Exponential Signal is given by-

$x(t)=Be^{at}$

So, assuming the sampling time as $T_s$, it's discrete-time version is-

$x[n]=Be^{aT_sn}$

Let $e^{aT_sn}=r$
So,

(2)
$$x[n]=Br^n$$

Sorry, no match for the embedded content.

### 3. Continuous-time Sinusoidal Signals

These signals are given by:

(3)
\begin{align} x(t)=Acos(\omega t+\theta) \hspace{0.2cm} \text{or} \hspace{0.2cm} Asin(\omega t+\theta) \end{align}

where,

A:Amplitude
$\theta$:Phase Shift
$\omega$:Angular Frequency (rad/s)$=\omega = 2\pi f$
f:frequency in Hz

$x(t+T)=Acos(\omega t+\omega T+\theta)$

For this to be same as $x(t)$, $\omega T=k.2\pi$.For the fundamental period, $k=1$ $\Rightarrow T=\frac{2\pi}{\omega}$.So, $x(t+T)=x(t)$.Therefore, T is the Time period.

Sorry, no match for the embedded content.

### 4. Discrete-time Sinusoidal Signal

The Continuous-time Sinusoidal Signal is given by-

$x(t)=Acos(\omega t+\theta)$

So, assuming the sampling time as $T_s$ , it's discrete-time version is-

$x[n]=Acos(\omega n T_s+\theta)$

Let $\Omega=\omega T_s$ So,

(4)
\begin{align} x[n]=Acos(\Omega n+\theta) \end{align}

Now, $x[n+N]=Acos(\Omega n +\Omega N+\theta)$. For this to be same as $x[n]$, $\Omega N=2\pi m$ , $m$ and $N$ are integers.

$\Rightarrow$ $\Omega = 2\pi \frac{m}{N}$
So,

$x[n+N]=x[n]$

Remark:
• $x(t)=Acos(6t)$ is periodic but $x[n]=Acos(6n)$ is not periodic.

Sorry, no match for the embedded content.

### 5. Complex Exponential Signal

These signals are given by-

(5)
$$x(t)=Ae^{st}$$

A,S are complex, $s=\sigma +j\omega$ and $A=|A|e^{j\theta}$
Substituting these,

$x(t)=|A|.e^{(\sigma+j\omega)t}.e^{j\theta}$

$x(t)=e^{st}=e^{(\sigma+j\omega)t}=e^{\sigma t}.e^{j\omega t}=e^{\sigma t}\cos \omega t + j e^{\sigma t}\sin \omega t$

In the above equation,$e^{\sigma t}\cos \omega t$ is real and $e^{\sigma t}\sin \omega t$ is imaginary.

### 6. Unit Step Function

The continuous time unit step function is given by-

(10)
\begin{align} u(t)= \begin{cases} 1 & \text{if}\ t>0 \\ 0 & \text{if}\ t<0 \end{cases} \end{align}

The discrete time unit step function is given by-

(11)
\begin{align} u[n]= \begin{cases} 1 & \text{if}\ n\geq0 \\ 0 & \text{if}\ n<0 \end{cases} \end{align}

### 7. Rectangular Function

The rectangular function is given by-

(12)
\begin{align} rect(t)= \begin{cases} 1 & \text{if}\ \frac{-1}{2}\geq t \geq \frac{1}{2} \\ 0 & \text{otherwise} \end{cases} \end{align}
The general form of the rectangular function both in continuous and discrete form is shown in the following figure.

### 8. Ramp Function

The continuous time ramp function is given by-

(13)
\begin{align} ramp(t)= \begin{cases} t & \text{if}\ t \geq 0 \\ 0 & \text{if}\ t<0 \end{cases} \end{align}

The discrete time ramp function is given by-

(14)
\begin{align} ramp[n]= \begin{cases} n & \text{if}\ n \geq 0 \\ 0 & \text{if}\ n<0 \end{cases} \end{align}

* Relation between Unit step Function and Rectangular Function
The rectangular function can be expressed in the form of unit step functions as show below-

$rect(t)=u(t+\frac{1}{2})-u(t-\frac{1}{2})$

In the same way,in discrete time-

$rect[n]=u[n+N]-u[n-(N+1)]$

* Relation between Ramp function and Unit Step Function
The ramp function can be expressed in the form of unit step functions as show below-

(15)
\begin{align} rampt(t)= \int_{-\infty}^{t} u(\tau) d\tau \end{align}

But when $t<0$

$\int_{-\infty}^{t} u(\tau) d\tau= 0$

So,

$rampt(t)= \int_{-\infty}^{t} u(\tau) d\tau=0+\int_{0}^{t} u(\tau) d\tau=\int_{0}^{t} 1.d\tau=t$

In the same way, in discrete time-

(16)
\begin{align} ramp[n]= \sum_{-\infty}^{n} u[m] \end{align}

### 9. Discrete time Impulse or Delta Function" hide="Hide the example" hideLocation="both"]]

The unit impulse function is also known as Kronecker delta function

(17)
\begin{align} \delta[n]= \begin{cases} 1 & \text{if}\ n = 0 \\ 0 & \text{otherwise} \end{cases} \end{align}
(18)
\begin{align} \delta[n].x[n]= \begin{cases} x[0] & \text{if}\ n = 0 \\ 0 & \text{otherwise} \nonumber \end{cases} \end{align}

So,

$x[n].\delta[n]=x[0].\delta[n]$

Therefore,

(19)
\begin{align} \sum_{-\infty}^{-\infty} x[n].\delta[n]=x[0] \end{align}

This is an important property called as Shifting Property used in many applications. So generalizing this we get,

(20)
\begin{align} \sum_{-\infty}^{-\infty} x[n].\delta[n-n_0]=x[n_0] \end{align}

10. Continuous Time Impulse Function or Dirac Delta Function:

(21)
\begin{align} \delta(t)= \begin{cases} 1 & \text{if}\ t = 0 \\ 0 & \text{otherwise} \end{cases} \end{align}

But this does not work.
Following Equation 1.45, the same thing can be written in continuous form as follows-

$\int_{-\infty}^{-\infty} x(t).\delta(t)=x(0)$

This will be non-zero only for one value of $t$

So a formal definition can be-

$\delta(t)=\lim_{\Delta\to 0} w_\Delta(t)$

Now,

$\lim_{\Delta\to 0} \int_{-\infty}^{-\infty} x(t).w_\Delta(t)= \lim_{\Delta\to 0} \int_{\frac{-\Delta}{2}}^{\frac{\Delta}{2}} x(t).\frac{1}{\Delta} dt = x(0).\lim_{\Delta\to 0} \int_{\frac{-\Delta}{2}}^{\frac{\Delta}{2}} \frac{1}{\Delta} dt=1$

$\delta(t)$ as the $\lim_{\Delta\to 0} w_\Delta(t)$ then,

$\int_{-\infty}^{-\infty} x(t).\delta(t)dt=x(0)$

Therefore,the two take aways are:

(22)
\begin{align} \delta(t)x(t)=x(0)\delta(t) \text{\hspace{3mm}(function of time)} \end{align}
(23)
\begin{align} \int_{-\infty}^{-\infty} x(t).\delta(t)dt=x(0) \text{\hspace{3mm}(scalar)} \end{align}
So,Impulse function is a generalized function and singularity function.

Similarly, in continuous time also the shifting property can be applied-

$x(t).\delta(t-t_0)=x(t_0).\delta(t-t_0)$

(24)
\begin{align} \int_{-\infty}^{-\infty} x(t).\delta(t-t_0)dt=x(t_0) \end{align}