Commonly used signals
Introduction Energy and Power Basic Operations Practice Problems Transformation of signals defined piecewise Even and Odd Signals Commonly encountered signals
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1. Real Exponential Signals

These signals are given by:

(1)
\begin{equation} x(t)=Be^{at} \end{equation}

When $a<0$: Decaying Exponential

When $a>0$: Growing Exponential
ch1.1.7.1.jpg

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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)
\begin{equation} x[n]=Br^n \end{equation}

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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.


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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.

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5. Complex Exponential Signal

These signals are given by-

(5)
\begin{equation} x(t)=Ae^{st} \end{equation}

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}
ch1.1.7.4.jpg
ch1.1.7.4%282%29.jpg

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}
ch1.1.7.5.jpg

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}
ch1.1.7.6.jpg
The general form of the rectangular function both in continuous and discrete form is shown in the following figure.
ch1.1.7.7.jpg

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}
ch1.1.7.8.jpg

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}
ch1.1.7.9.jpg

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

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

ch1.1.7.10.jpg

In the same way,in discrete time-

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

ch1.1.7.11.jpg

* 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}
ch1.1.7.12.jpg
(18)
\begin{align} \delta[n].x[n]= \begin{cases} x[0] & \text{if}\ n = 0 \\ 0 & \text{otherwise} \nonumber \end{cases} \end{align}
ch1.1.7.13.jpg

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}
ch1.1.7.14.jpg

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 $t$
ch1.1.7.16.jpg

So a formal definition can be-

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

ch1.1.7.17.jpg

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.
ch1.1.7.18.jpg

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}

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