lecture 2-done.pdf

8/13/2019 Lecture 2-Done.pdf

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Remote Sensing LaboratoryDept. of Information Engineering and Computer Science

University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy

Remote Sensing LaboratoryDept. of Information Engineering and Computer Science

University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy

Begm Demir

Digital Signal Processing

Lecture 2

Digital Signal Processing

Lecture 2

E-mail: [email protected]

Web page: http://rslab.disi.unitn.it


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University of Trento, Italy

Discrete Time Signals

2B. Demir 2

A discrete time signal is a sequence of numbers

The length Nof the signal is the total number of samples in the sequence.


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Operations on Sequences

3B. Demir

Addition/Subtraction

x3[n]=x1[n]+x2[n] x4[n]=x1[n]-x2[n]


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4B. Demir

Multiplication

How will you apply multiplication if the sizes are not the same?

x5[n]=x1[n]x2[n]


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5B. Demir

Multiplication & Addition with constants


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6B. Demir

Shifting y[n]=x[n-n0] If n0 >0time delay If n0


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7B. Demir

Time Reverse: y[n]= x[-n]


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University of Trento, Italy 8B. Demir

[ ] is given below. Estimate [2 ] and [ / 2]x n x n x n

Signal compression

Signal stretching


[ ] [ ]

[ ]

y n x n

ny n x

=

=


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Linear Interpolation


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Elementary Digital Signals

10B. Demir

Unit Impulse (Digital Impulse, Discrete Time Impulse):

1 0[ ]

0 0

nn

n

==

1[ ]0

n mn mn m

= =


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Exercises

11B. Demir

Write the signals given below in terms of

if [ ] ( ) ( 1) ( 2) 4 ( 3) 5 ( 4)

[ 2] ?

x n n n n n n

x n

= + + + +

+ =

[ ]n


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12B. Demir

Unit Step Sequence:

Unit Impulse and Unit Step Sequences are related as follows:

1 0[ ]

0 0

nu n

n

=


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Exercises


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14B. Demir

Real exponential sequence:

[ ] nx n a=[ ] nx n a=


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15B. Demir

Real exponential sequence:

( ) nx n a=( ) nx n a=


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16B. Demir

Sinusoidal sequence:

[ ] cos( )x n A n = +[ ] cos( )x n A n = + cos( ) 2

sin( )2

j n j n

j n j n

e en

e en

j

+

=

=


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17B. Demir

Complex exponential sequence:

[ ] j nx n e =[ ] j n

x n e = ( )1,j

z re r n = = = ( )1,jz re r n = = =


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18B. Demir

Complex exponential sequence:


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Properties of Sinusoidal/Complex Signals

19B. Demir


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Properties of Sinusoidal Signals

20B. Demir

[ ] )cos()2(cos nnr =+


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Examples

22B. Demir

[ ]

[ ] 4 / 7

Is cos periodic?

11

Is 5 periodic?j n

x n n

x n e

=

=


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Examples

23B. Demir

Let us assume that we have 3 sinusoidal signals x1[n], x2[n] and x3[n] thatare periodic with N1, N2 and N3, respectively. The signal x4[n] given belowis periodic?

[ ] [ ] [ ] [ ]

[ ]

4 1 2 3

4

;

is periodic?

x n ax n bx n cx n

x n

= + +

4 1 2 3( , , )N LCM N N N=

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