Signals & Systems Lecture 4

Index

1. Even and Odd Part of the Signal
   • Question 1
     • Answer
   • Question 2
     • Answer
   • Question 3
2. Explain Standard Continuous Time Signal
   • Impulsing
   • Step Signal
   • Ramping
   • Exponential
3. Explain Standard Discrete Time Signal
   • Impulsing
   • Step Signal
   • Ramping
   • Exponential

Main note

Even And Odd Part of the signal

Question 1

$x(t)=e^t$

Answer

$x(-t)=e^{-t}$

Even

The even part of a signal ( x(t) ), denoted as ( X_e(t) ), is given by: $X_e(t)=\frac{1}{2}[x(t)+x(-t)]$ $X_e(t)=\frac{1}{2}[e^t+e^{-t}]$

Odd

For an odd signal it will be $X_e(t)=\frac{1}{2}[x(t)-x(-t)]$ $X_e(t)=\frac{1}{2}[e^t-e^{-t}]$

Question 2

$x(t)=3+2t+5t^2$

Answer

$x(-t)=3+2(-t)+5(-t^2)$ $x(-t)=3-2t+5t^2$

Even

The even part of a signal ( x(t) ), denoted as ( X_e(t) ), is given by: $X_e(t)=\frac{1}{2}[x(t)+x(-t)]$ $X_e(t)=\frac{1}{2}[3+2t+5t^2+3-2t+5t^2]$ $X_e(t)=\frac{1}{2}[6+10t^2]$ $X_e(t)=2+5t^2$

Odd

For an odd signal it will be $X_e(t)=\frac{1}{2}[x(t)-x(-t)]$

$X_o(t)=\frac{1}{2}[(3+2t+5t^2)-(3-2t+5t^2)]$ $X_o(t)=\frac{1}{2}[x(t)-x(-t)]$ $X_o(t)=\frac{1}{2}[3+2t+5t^2-3+2t-5t^2]$ $X_o(t)=\frac{1}{2}[2t+2t]$ $X_o(t)=\frac{1}{2}[4t]$ $X_o(t)=2t$

Question 3

Explain Standard Continuous Time Signal

Impulsing

It is a signal with infinite magnitude with zero duration But with an area of A.

SNS 1.excalidraw

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Excalidraw Data

Text Elements

T=0

Area = A

∂(t) = Infinite ; t = 0 = 0 ; t != 0

Link to original

$\delta (t)=\infty ;t=0$ $\delta (t)=0;t\ne 0$ ${\int }_{-\infty }^{\infty }d(t)dt=a$

Step Signal

$x(t)=A;t\ge 0$ $x(t)=0;t<0$

SNS 3.excalidraw

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Text Elements

x(t)

t

Link to original

Ramping

$x(t)=At;t\ge 0$ $x(t)=0;t>0$

Drawing 2024-07-25 09.47.41.excalidraw

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Excalidraw Data

Text Elements

0

1

A

2a

2

Link to original

Exponential

$x(t)=A{e}^{bt};t\ge 0$ $x(t)=0;t>0$ When b is positive then it is exponentially increasing and when b is negative it is exponentially decaying.

SNS 6.excalidraw

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b is Positive

x(t)

A

t

Link to original

Explain Standard Discrete Time Signal

Impulsing

Equation

$\delta (n)=1;n=0$ $\delta (n)=0;n\ne 0$

Graphical Representation

SNS 2.excalidraw

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Text Elements

n

d(n)

Link to original

Step Signal

Equation

$x(n)=A;n\ge 0$ $x(n)=0;n<0$

Graphical Representation

SNS3.excalidraw

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Text Elements

0

1

2

3

4

5

Link to original

Ramping

Equation

$x(n)=An;n\ge 0$ $x(n)=0;n<0$

Graphical Representation

SNS 5.excalidraw

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Text Elements

x(n)

n

0

1

2

3

Link to original

Exponential

Equation

$x(n)=A{e}^{bn};n\ge 0$ $x(n)=0;n<0$

Graphical Representation

SNS 7.excalidraw

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Text Elements

x(t)

t

Link to original

References

Information

• date:: 2024.07.25
• time:: 09:07