MODULASI PSK (PHASE SHIFT KEYING)

MODULASI PSK (PHASE

SHIFT KEYING)

Binary Shift Keying (BPSK)

 Perubahan Parameter Fasa dari sinyal pembawa sesuai dengan sinyal informasi.

 Menggunakan alternatif-alternatif fasa gelombang sinus utk mengkodekan bit-bit dimana Fasa dipisahkan 180 derajat  Sederhana utk diimplementasikan, tidak efisien dalam

penggunaan bandwidth_.

Blok Sistem BPSK

















"

0

"

)

(

;

)

2

cos(

"

1

"

)

(

;

)

0

2

cos(

)

(

t

b

t

f

A

t

b

t

f

A

t

s

c c





Data

Carrier

Carrier+ 

BPSK waveform

1 1 0 1 0 1

Konstelasi sinyal BPSK



Fasa dipisahkan 180 derajat.

'0'

binary

)

2

cos(

)

(

'1'

binary

)

2

cos(

)

(

2 1













t

f

A

t

s

t

f

A

t

s

c c

c c

Q

0

f

S(f)

f

c

f

c

+ R

f

c

- R

f

c

+ 2R

f

c

- 2R

BW

min

BW

min

=2B

N

= 2.Rb/2

B_N=Bandwidth Nyquist

Quadrature Phase Shift Keying

•

Teknik modulasi multilevel : 2 bit per symbol

•

Lebih efisien spektrum, lebih kompleks

receiver.

•

Dua kali lebih efisien bandwidth daripada

BPSK

Q

00 State

11 State _{10 State} 01 State

4 bentuk gelombang berbeda:

-1.5 -1 -0.5 0 0.5 1 1.5

0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5

0 0.2 0.4 0.6 0.8 1

-1.5 -1 -0.5 0 0.5 1 1.5

0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5

0 0.2 0.4 0.6 0.8 1

00 ₀₁

11 10

cos+sin

-cos+sin

cos-sin -cos-sin

Bandpass modulation: Signal Space & Vector

 Bandpass modulation:

The process of converting data

signal to a sinusoidal waveform where its amplitude,

phase or frequency, or a combination of them, is

varied in accordance with the transmitting data.



Bandpass signal (

General Condition

):

where is the baseband pulse shape with energy .



We assume here (otherwise will be stated):

 is a rectangular pulse shape with unit energy.

 Gray coding is used for mapping bits to symbols.

 denotes average symbol energy given by



t

i

t

t



t

T

T

E

t

h

t

s

_i

(

)



(

)

2

i

cos



_c



(



1

)









_i

(

)

0





)

(

t

h

E

_h

s

E

s





_iM_

E

i

M

E

1

1

Phase Shift Keying (PSK)

I.

PSK signal waveform (transmitted signal):



Phase (examples):



Symbol energy and symbol interval

0 , phase: ,

2

( ) cos[ ( )] _i( ) 2 0 , 1, , ,

i i

E

s t t t t i

T



M t T i M









   



 

7 / 2

1/ 2

3

5

3

, ,

, 0

2

7

,

,

,

BPSK(

2) :

, 0

QPSK(

4

4

4

4

) :

o

2

r

4

i

i

i

M

M















  

















 















2 0

is symbol energy. is symbol inter

.

Can you show that

va

(

?

l

)

T i

E

s t dt

T

II. Signal space representation

Note: decomposition of PSK signal waveform:

1) Orthonormal basis:

2) Signal vector:

3) Constellations:

2 2

cos , si

( ) : n , 1, ,

i i

i i

E E

s t i

M M M





 

 _  _

  

s

1 0 2 0

2

2

( )

t

cos(

t

),

( )

t

sin(

t

)

T

T













0 0

2

2

( )

cos[ ( )]cos(

)

sin[ ( )]sin(

)

i i i

E

E

s t

t

t

t

t

T





T









Signal space representation (cont.)

4)

A proof of signal space representation

 Bases are orthonormal

 Signal space vector for each waveform s_i(t)

15 2 0 2 2 2 2

0 0 0

1 _{0 0}

1 2 _{0 0}

0

0 0

0

0

( ) 2 sin ( ) 2 [1 cos(

( ) 2 2

cos( )sin( ) sin(2 ) /

1.

.

( ) ( ) 2

cos ( ) [1 cos

2

2 )]/ 2 1

(2 )] 2 0. / 2 T T T T T T

t dt T t dt T t

t dt T dt T

t t t

dt

t t dt T

t t

d

d

d

t

t T t























                



















1 1 ₀ 0 0

2 2 ₀ 0 0

0

0

( ) ( ) 2 T

T c cos[ ( )]

( ) ( ) 2 T cos[ ( )]sin( ) T sin[ ( )] sin[2 (

cos[ ( )]cos( )

os[ ( )] cos[2 (

] n ) ) i ] s [ T i i T

i i i

T

i i i

T

i

a s t t dt

a s t t dt E dt

E dt

E t t t dt

E t t t d

E t

t

t E

t t

t t t

                                   













Signal space representation (cont.)

5) What happens if baseband pulse-shaping h(t) is considered?

 Signal waveform:

 Use basis (note that h(t) can be assumed normalized):

 Signal space vector is still:

 Conclusion: there is no difference in signal space whether pulse-shaping is considered. We can study only PSK instead of the more general PM.

1 0 2 0

2

2

( )

t

h t

( ) cos(

t

),

( )

t

h t

( )sin(

t

)

T

T













2

2

cos

,

si

( ) :

n

,

1,

,

i i

i

i

E

E

s t

i

M

M

M























s

0

2

( )

( ) cos[

( )]

i i

E

s t

h t

t

t

T





PSK modulator



Special case: BPSK modulator



General case: M-ary PSK modulator

i

a

 

E

( )

h t

0

2

T

cos(



t

)

( )

i

s t

binary

bits

1

i

a

2

i

a

( )

h t

( )

h t

0

2

T

cos(



t

)

0

2

T

sin(



t

)

( )

i

s t

Note:

Inputs are signal-space vector.

Carriers are in basis form.





1 0 2 0

1 2

( ) 2 cos( ) 2 sin( )

( , ) cos(2 ), sin(2 )

i i i

i i i

s t a T t a T t

a a E i M E i M

 

 

 

  

s

Bandwidth of PSK signal waveform



Just like DSB modulation:



Exercise :

Consider QPSK transmission with date rate

2000 bps. The magnitude of the signal si

(t) is √2E/T

=1 volt.

a) What is the minimum PSK signal bandwidth? b) Find the signal space points

c) Draw the constellation

d) Find signal waveform for transmitting {1001}.

18

PSK

2

baseband

W



W

2 PSK baseband,min 3

1 0

a) (log ) 2000 / 2 1000. 2 2 2 1000Hz. b) ( cos 2 4, sin 2 4), where / 2 0.5 10 , 1, , 4 d) Define mapping as: {00:0, 01: , 10: 2, 11: 3 2}.

Then {10} ( ) cos(

s b s

i

R R M W W R

E i E i E T i

s t t

 

  





     

     

 

s

2 0

2). {01} s t( ) cos( t )

  

   

Phase ( ) in ( ) is different from phase of

i t s ti



Demodulation and detection

Demodulation: The receiver signal is converted to baseband, filtered and sampled.

Detection: Sampled values are used for detection using a decision rule such as ML detection rule.





















N

z

z



1

z





T

0

)

(

1

t





T

0

)

(

t

N



)

(

t

r

1

z

N

z

z

Decision

circuits

(ML detector)

m

ˆ

Demodulations type:



Some notations

 Carrier:

 Modulation types with respect to carrier parameters

Modulatio n

Varying parameter

Demodulation

PSK

Coherent or

noncoherent

QAM

Coherent or

noncoherent

FSK

Coherent or

Noncoherent

( )

t



0



both ( ) and ( )

A t



t

0 0 0

( )

( ) cos[

( )],

2

Two dimensional modulation,

demodulation and detection (M-PSK)

 M-ary Phase Shift Keying (M-PSK)

21

















M

i

t

T

E

t

s

_i

(

)

2

s

cos



_c

2



Two dimensional

mod…

(MPSK)

)

(

1

t



2

s

1

s

b

E

“0” “1” b

E



)

(

2

t



3

s

7

s

“110”

)

(

1

t



4

s

s

₂

s

E

_“000”

)

(

2

t



6

s

s

₈

1

s

5

s

“001” “011” “010” “101” “111” _“100”

)

(

1

t



2

s

s

₁

Demodulation BPSK

 BPSK with coherent _detection:



T

0

)

(

1

t



)

(t

r Decision

Circuits

Compare z with threshold.

m

ˆ

z

)

(

1

t



2

s

1

s

b

E

“0” “1”

b

E



b

E

2

2 1



s



Error probability …

 BPSK with coherent _{detection (with perfect carrier}

synchronization):

2

0



















N

E

Q

P

_B b

2

/

2

/

0 2 1





















N

Q

P

_B

s

s

)

(

1

t



b

E

b

E



0

1

s

2

s

)

|

(

m

₁

p

_z

z

)

|

Demodulation M-PSK

 Coherent detection of Q-PSK

25

Decision Region QPSK

s₂

s₄

decide s₁

s₁ decide s₂

decide s₃

decide s₄

s₃



₁

(

t

)

Power Spectra of M-Ary PSK

 

 

 

f

E

M

c



T

f

M



S

Tf

c

E

f

S

b b

B B

2 2

2

log

sin

log

2

sin

2

QPSK vs. BPSK

 Let’s compare the two based on BER and bandwidth

BER Bandwidth

BPSK QPSK BPSK QPSK

R

b

R

b

/2

EQUAL

2

0



















N

E

Q

P

_B b

2

0



















N

E

Q

Error probability …



Coherent

detection

of M-PSK

28 8-PSK 3

s

7

s

“110” ) ( 1 t  4

s

s

₂

s E “000” ) ( 2 t  6

s

s

₈

1

s

5

s

“001” “011” “010” “101” “111” _“100” Decision variable r   ˆ z

Compute _Choose smallest

X

Y

arctan



ˆ

|

ˆ

|



_i







T

0

)

(

1

t





T

0

)

(

2

t



)

(

t

r









X

T

r

t

t

dt

r

0

1 1

(

).

(

)

m

ˆ









Y

T

r

t

t

dt

r

(

).

(

)

Error probability …

 Coherent detection of MPSK …



The detector compares the phase of observation vector

to M-1 thresholds.



Due to the circular symmetry of the signal space, we

have:

where



It can be shown that (

for M > 4

)

29









p



d

P

P

M

M

P

M

P

M M c M m m c C

E

(

)

1

(

)

1

(

)

1

1

)

(

1

)

(

/ / ˆ 1 1





_ 

















s

s

               M N E Q M P s E



sin 2 2 ) ( 0



































M

N

E

M

Q

M

P

_E

(

)

2

2

log

2 b

sin



or

2

|

|

;

sin

exp

)

cos(

2

)

(

2 0 0 ˆ

































N

E

N

E

Probability of symbol error for

M-PSK

E

P

Note!

• M = 2k

• _“The same average symbol

energy for different sizes of