9/13/2012

1

Digital Communications

Lecture-3

Dr. Sarmad SohaibAssistant Professor

Faculty of Electrical and Electronics Engineering

University of Engineering and Technology, Taxila

sarmad.sohaib@uettaxila.edu.pk

9/13/2012

2

Downconverter ≈ Quad Demodulator

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

2

9/13/2012

3

Condition for Equivalence:

• As long as input BW < 2fc:

x( t ) = sI( t ) and y( t ) = sQ( t ).

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

3

9/13/2012

4

Complex Model of Quad Demod:

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

4

9/13/2012

5

Quad Demodulator

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

5

9/13/2012

6

Filtering

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

6

s(t)h(t)

r(t)

If this is given,

What will be the C.E. of r(t) or what happens if we pass C.E. of s(t) rather than s(t)?

9/13/2012

7

Euclidean Space: Rn

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

7

9/13/2012

8

Inner Product for Rn

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

8

9/13/2012

9

Geometric Interpretation

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

9

9/13/2012

10

Complex Euclidean Space: Cn

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

10

9/13/2012

11

Inner Product for Cn

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

11

9/13/2012

12

Geometric Interpretation for Cn

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

12

9/13/2012

13

Linear Space = Vector Space

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

13

9/13/2012

14

Other Linear Spaces (Besides Rn & Cn)

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

14

9/13/2012

15

Another Vector Space

• {[0 0], [0 1], [0 2], [1 0], [1 1], [1 2], [2 0], [2 1], [2 2]} with

mod-3 addition and multiplication, and with the mod-3 field

{0, 1, 2}:

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

15

9/13/2012

16

Inner Product

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

16

A mapping from a pair of vectors and to a scalar , , satisfying axioms:

, , , , and ,

, 0,equality iff 0

, ,

, , , ,

This leads

a a a a

∗

∗

< >

< + > = < > + < > < + > = < + > + < + >

< > ≥ =

< > = < >

< > = < > ⇒ < > = < >

x y x y

x y z x z y z x y z x y x z

x x x

y x x y

x y x y x y x y

�

�

�

�

{

2 to the defination of the norm: by ,

Normsquared

x x=< >xx

9/13/2012

17

Inner Product for Waveforms

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

17

2 2

( ), ( ) ( ) ( ) " "

( ) ( ) ( ) ( ) " "

x t y t x t y t dt Correlation

x t x t x t dt x t dt Energy

∞

∗

−∞

∞ ∞

∗

−∞ −∞

< >=

= =

∫

∫ ∫

9/13/2012

18

Inner Product for Sequences

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

18

2 2

, "Correlation"

"Energy"

k k k k

k

k k

k

x y x y

x x

∞

∗

=−∞

∞

=−∞

< >=

=

∑

∑

9/13/2012

19

Triangle Inequality

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

19

Equality iff (i.e. and are in same direction)

where is a +ve real scalar

a

a

+ ≤

=

+

x y x y

x y x y

9/13/2012

20

Cauchy-Schwarz Inequality

9/13/2012Digital Communication (Dr. Sarmad Sohaib)

20

2 2 2

2

2 2

2

,

( ) ( ) ( ) ( )

Equality iff

x y

x t y t dt x t dt y

a

t dt

Correlation E E

∞ ∞ ∞

∗

−∞ −∞ −∞

< > ≤

≤

≤

=

∫ ∫ ∫

x y x

x

y

y