formula list ex tc

7/26/2019 Formula List Ex Tc

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Semester-IV Signals and Systems Formulae List

1

Energy and power

1.

Energy:A Signal is said to be an energy signal if it has finiteenergy.

2.

Power: A Signal is said to be and power signal if it has finite

power.

Energy ||

E= ||2=

1 ||

1

21 ||2

Note

1.

exponential- energy

2.periodic- power

3.

x(t)->E=finite,P=0

x(t)->P=finite,E=


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2

II. Symmetric (even) or Asymmetric (odd)Symmetric: e.g

Asymmetric: x(-t)=-x(t)

e.g

Every s/g can be expressed as

x(t)=xe(t)+xo(t)

wherexe(t)=

+

xo(t)=


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3

NOTE:

1 1 1 1


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4

Precidance Rule:

A x ( a)i. Perform time reversal ( if t isve)

x(t)x(-t)ii.

Perform time shifting

x(t)x(-t)x(-t+a)

iii.

Perform time scaling

x(t)x(-t)x(-t+a)x ( a)iv.

Finally amplitude scaling

x (

a)A x (

a)


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5

Sampling Theorem:

x(t) x(t) x(t)

0 1 2 3 4 5

Nyquist criteria:-

> 2 Steps to convert x(t) into x(n)

i. Consider the given x(t)

ii.

Put t =

in x(t) to get x(n)

iii.

Obtain the expression of x(n)


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6

CLASSIFICATION & ANALYSIS OF SYSTEM

Linear or Non-linear

A system is said to be a linear system if it satisfies the principle of

superposition theorem

Following steps are to be use

1.

Apply the i/p x1(t),let o/p be y1(t)

2.

Apply the i/p x2(t),let o/p be y2(t)

3.

Find y3(t) using the formulay3(t)=a1x1(t)+a2x2(t)

4.

Apply the composite i/p

[a1x1(t)+a2x2(t)],let the o/p be y3*(t)

5. If y3(t)= y3*(t),then system is linear otherwise non-linear

Time Variant and Time Invariant:i. Apply the delayed i/p x(t-k),let the o/p be y1(t)

ii.

Delay the normal o/p by k.

i.e y(t-k),let the new o/p be y2(t).

iii.

If y1(t)= y2(t),then system is time invariant, otherwise time

variant.


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7

Casual or Non-casual:

A system is said to be a casual system if its present o/p depends on

either past i/p or present i/p.

Otherwise it is called as non-casual.

Convolution

Convolution is used to find o/p of the Linear Time Invarient system

when i/p to the system and impulse response of the system are known.

Output=input*imp.response

1.

Continious Time System.

y(t)=x(t)*h(t)

.

=

2.Discrete time system

y[n]=x[n] * h[n]

y[n]= =

Steps for circular convolution:

1.

x[n]->outer

2.

h[n]->Inner

3.

To get y(0),multiply the amplitudes of x[n] and h[n] and addthem.

4.

To get y(1),keep outer as it is,rotate inner and clockwise by 1

position and repeat step 3.

5.

To get y(2),y(3),.Repeat step 4.


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8

Z-TRANSFORMS

DefinitionIf []is a discrete time signal then its z-Transform is

defined as and given by

[]

=

= Two sided (double Z-TransformOne-sided/Unilateral Z-Transform is defined as

.

=

Parallel IZT is defined as

[] 12 .

Geometric Series

1)

= if < 1

2) = if < 1

3) =


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9

Properties of Z-Transform

1) Linearity:

If [ ] and [ ] []Then [] 2)

Time shifting:

If []Then [ ]

3)

Time Reversal:

If []Then

[ ]

4)

Scaling in Z-domain:

If [][](or multiplication by )then [ ]

5)

Multiplication by n in T.D OR Differentiation in ZD

If []Then [][]

6)

Convolution in Time Domain:If [] []then [] [] .

7)

Correlation in T.D:

(1)8)Initial value theorem:

0 lim

9)

Final value theorem:

lim


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10

STANDARD ZT PAIRS

x(n) x(z) ROC

1 Entire z-plane

u(n) 1

|z|>1

anu(n)

|z|>a

nanu(n) |z|>a

-u(-n-1) 1

|z|


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11

IZT

X(z)=

If ROC:|z|>a If ROC:|z|


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12

Difference equation without Initial condition:

y(n)y(z)

y(n-1)z-1y(z)

y(n-2)z-2y(z)

x(n)z-1x(z)

x(n-1)x(z)

1.Response means o/p

2. Impulse response means IZT of [H(z)]

3.

Step response means o/p when i/p is step s/g i.e x(n)=u(n)

4.

Transfer function means H(z)=

Difference Equation with Initial Conditions

y(n)y(z)

y(n-1)z-1y(z)+y(-1)

y(n-2)z-2y(z)+ z-1y(-1)+y(-2)

y(n-3)z-3y(z)+ z-2y(-1)+ z-1y(-2)+y(-3)

x(n)x(z)

Note

1.

x(-1)=put n=-1 in i/p x(n)

2.y(-1),y(-2) will be given in the question.


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13

Interconnected System

1.

Series (Cascade) connection.2.

Parallel connection.

1.Series(Cascade):-

Now its o/p is

y(n)=x(n)*heq(n)

where heq(n)= h1(n)* h2(n)

2.Parallel:-

Now its o/p is

y(n)=x(n)*heq(n)

where heq(n)= h1(n)+h2(n)


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14

Laplace Transforms

If x(t) is a continuos time signal,then its Laplace transform isdefined as and given by

L.T[x(t)]=x(s)= [] Range of t in x(t)

And s= x(t) (L.T) x(s)

Time Frequency

Domain Domain


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15

Properties of L.T

1.

Linearity: If x1(t) (L.T)x1(s)

And x2(t) (L.T)x2(s)

Then a1x1(t)+ a2x2(t)a1x1(s)+ a2x2(s)2.

Time Shifting: If x(t)x(s)

then x(t-a)e-asx(s)

3.

Frequency Shifting:

If x(t)x(s)

Then e-atx(t)x(s+a)

4. Integration in T.D

If x(t)x(s)

Then 5.Differentiation in T.D

If x(t)x(s)

Then

.6.Multiplication by tn

If x(t)x(s)

Then tnx(t)(-1)n dn/dsn[x(s)]

7.

Convolution in T.D

If x(t)x(s)

H(t)H(s)

Then x(t)*h(t)x(s).H(s)

8.

Initial value theorem

X(0)= lim .9.Final value theorem

X()=lim .


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16

STD L.T Pairs

SR.No x(t) x(s) R.O.S

1. S(t) 1 Entire s plane

2. u(t) Re(s) > 0

3. ut +a Re(s) > -a

4.

ut

a Re(s) > a

5. ut Re(s) < 0

6. ut a Re(s) < -a

7. ut a Re(s) < a

8.

! u(t)

Entire s plane

9. t.u(t)= r(t) Entire s plane

10. . u(t) + Re(s) > -a

11. cos.u(t) + Re(s) > 0

12. sin.u(t) + Re(s) > 0

.

S

L

ultiplication

byt


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17

Inverse Laplace Transforms:

X(z)=

+

If Re(s)>a

If Re(s)


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18

Solving differential equations without initial conditions

y(t)(L.T)y(s)

y|(t) (L.T)s.y(s)

y||(t) (L.T)s2.y(s)

y|||(t) (L.T)s3.y(s)

Differential equation with initial condition

y(t)(L.T)y(s)

y|(t) (L.T)s.y(s) -y(0)

y||(t) (L.T)s2.y(s)s.y(0)- y|(0)

y|||(t) (L.T)s3.y(s) -s2.y(0)s.y|(0)- y||(0)

Note:

1.

If Initial conditions of y(t) are not given then assume it as

zero

2.x(0)=x(t)|t=0


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19

Laplace transform for periodic signal:

If x(t) is a periodic signal with Time period T,then its laplace transform

is given by x(s)

x(s)=(1-e-st)-1 * x1(s)

where T-Time period

x1(s)=Laplace Transform of one cycle

=


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20

CTFT & DTFT

CTFT

If is a continuous time signal thenIts CTFT is given by

. [] []

Its IFT is given by

12

|=


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21

Properties of Fourier series

1.Scaling property:

If x(t) --> (FT) x(w)

Then x(a.t) --> (FT) (1/a).X(w/a)

2.Time shifting property:

If x(t) --> (FT) X(w)

Then x(t-t0) --> (FT) e-jwt0X(w+w0)

3.Frequency shifting:

If x(t) --> (FT) X(w)

Then e-jwt0x(t) --> (F.T)X(w+w0)

4.Time diff:

/[x(t)] --> (FT) j.w.X(w)5.Convolution property:

If x(t) --> (FT) X(w)

h(t) --> (F.T) H(w)

Then x(t).h(t) --> (FT)X(w).H(w)


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22

FOURIER SERIES

Trigonometric F.S

[cos]=

[sin]=

Where,

1

1

/

/

2 cos

2 cos/

/

2

sin 2

/

/sin

2021


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Special Case:

Even Symmetric:

Ifx (t)has even symmetric then &

.

Odd Symmetric:

, &

.


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Exponential (Complex) F.S :

. = Where,

OR

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