Binary Phase Shift Keying (BPSK), On/Off Keying (OOK), and Quadrature Phase Shift Keying (QPSK)
Published:
This post is a project from my Detection and Estimation course. The project is an analysis of the performance of three communication schemes in the presence of noise. The bit error rate is simulated with a monte-carlo simulation, for each scheme and at various SNR’s. The details can be found in ee526-project-2-mackkv.pdf.
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EE526 Project 2
Kevin Mack, Clarkson University 03/08/2018
Problem Definition:
This project inspects three communication schemes (BPSK, OOK, and QPSK) in order to de-
termine the achievable BER for each scheme under different noise conditions. The theoretical
BER for each SNR value will be compared to the simulated results (Monte-Carlo Simulation) for
each scheme. In the end, all schemes will be compared over the same range of SNR values.
Binary Phase Shift Keying (BPSK):
For BPSK the message, X ∈ [−1, 1] and N ∼ N
0, σ
2
n
. This means that, assuming equal prior
probabilities for each symbol, the Bayesian Decision Rule gives threshold at x = 0. The signal
is given by,
H
0
: X = N, (1)
H
1
: Y = S + N. (2)
In this case, the theoretical value for the BER is given by,
P
e
= Q
p
2γ
b
(3)
where Q(.) is the q-function, and γ
b
is the SNR per bit. In order to test this theoretical value,
a Monte-Carlo simulation is used to give an experimental result. Figure 1 represents the results
of the Monte-Carlo solution, and Listing 1 below provides the code to generate these results.
Listing 1: Matlab Code – BPSK Simulation
1 %% BPSK Monte - Carlo Si m u l a t i o n
2 snr_max = 9 . 0 ; %dB
3 snr_min = 2 . 0 ; %dB
4 snr_step = 0 . 2 ;
5
6 snr_range = snr_min : snr_step : snr_max ;
7
8 thresh_BPSK = 0 ;
9
10 num_transmissions = 1 e6 ;
11
12 % s e t up f i g u r e
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13 f i g u r e
14 hold a l l
15 g r i d on
16 g r i d minor
17
18 erro r _the o = z e r o s ( s i z e ( snr_range , 2 ) , 1) ;
19 error_sim = z e r o s ( s i z e ( snr_range , 2) , 1) ;
20 f o r i = 1 : s i z e ( snr_range , 2)
21 rece ived_mes sage = z e r o s ( num_transmissions , 1) ;
22
23 %c a l c u l a t e v a r i a n c e from SNR
24 N_0 = 1/ snr_range ( i ) ;%1;% n o i s e va r i a n c e
25 sigma_sq = N_0/2 ;
26
27 % ge n er a t e r andom s i g n a l o f 1 ’ s and 0 ’ s (P( 0 )=P( 1 )
28 b i t s = ra n d i ( [ 0 1 ] , num_transmissions , 1) ;
29
30 % f o r BPSK, the b i t s sh ould be -1 and +1, so 0 -> -1
31 X = b i t s ;
32 X(X == 0 ) = - 1 ;
33
34 % ge n er a t e random g a u s s i a n n o i s e
35 Z = s q r t ( sigma_sq ) .* randn ( s i z e (X, 1) , 1) ;
36
37 % add n o i s e to s i g n a l
38 Y = X + Z ;
39
40 % check s i g n a l a g a i n s t t h r e s h o l d
41 ind = f i n d (Y> thresh_BPSK ) ;
42 rece ived_mes sage ( ind ) = 1 ;
43
44 % g e t s i m u l a t i o n b i t e r r o r r a t e (BER)
45 e r r o r s = re cei ved_m essag e ( r eceived_ messa ge ~= b i t s ) ;
46 error_sim ( i ) = s i z e ( e r r o r s , 1 ) / s i z e (X, 1 ) ;
47
48 % c a l c u l a t e t h e o r e t i c a l BER
49 err o r_th e o ( i ) = qfunc ( s q r t (2* snr_range ( i ) ) ) ;
50 end
51
52 p l o t ( snr_range , lo g 1 0 ( error_sim ) , ’ ko ’ )
53 p l o t ( snr_range , lo g 1 0 ( e rror_ t heo ) , ’ k - - ’ )
54 t i t l e ( ’ De t e c t o r Performance with Varying SNR (BPSK) ’ )
55 x l a b e l ( ’SNR [ dB ] ’ )
56 y l a b e l ( ’BER ’ )
57 l e g e nd ( ’ Simulated BER ’ , ’ Th e o r e t i c a l BER ’ )
EE526 Project 2 Page 2
Figure 1: This figure depicts the results of a Monte-Carlo simulation (1e
6
iterations) to determine
the relationship between SNR and BER the BPSK communication scheme. It is clear from the
graph that the theoretical BER is comparable to the simulated BER.
On/Off Keying (OOK):
For the OOK communication scheme, the initial setup is the same as for BPSK. However, the
only difference is that X ∈ [0, 1]. This means that the threshold for detection is at x = 0.5, and
that there is less ’space’ between the on and off state. This will result in a degradation of BER
performance as compared with BPSK, as can be seen in 2. The theoretical BER is given by,
P
e
= Q
√
γ
b
, (4)
The code to simulate the OOK setup is given in Listing 2 below.
Listing 2: Matlab Code – OOK Simulation
1 snr_max = 1 2 . 0 ; %dB
2 snr_min = 5 . 0 ; %dB
3 snr_step = 0 . 2 ;
4
5 range = snr_min : snr_step : snr_max ;
6
7 thresh_OOK = 0 . 5 ;
8
9 num_transmissions = 1 e6 ;
10
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11 % s e t up f i g u r e
12 f i g u r e
13 hold a l l
14 g r i d on
15 g r i d minor
16
17 erro r _the o = z e r o s ( s i z e ( range , 2 ) , 1 ) ;
18 f o r i = 1 : s i z e ( range , 2)
19 rece ived_mes sage = z e r o s ( num_transmissions , 1) ;
20
21 %c a l c u l a t e v a r i a n c e from SNR
22 N_0 = 0 . 5 / range ( i ) ;% n o i s e v a r i a n c e
23 sigma_sq = N_0/2 ;
24
25 % th r e s h = some_equation_based_on_noise_variance ;
26 % ge n er a t e random s i g n a l o f 1 ’ s and 0 ’ s (P( 0 )=P( 1 )
27 b i t s = ra n d i ( [ 0 1 ] , num_transmissions , 1) ;
28
29 % f o r OOK, the b i t s shoul d be 0 and +1
30 X = b i t s ;
31
32 % ge n er a t e random g a u s s i a n n o i s e
33 Z = s q r t ( sigma_sq ) .* randn ( s i z e (X, 1) , 1) ;
34
35 % add n o i s e to s i g n a l
36 Y = X + Z ;
37
38 % check s i g n a l a g a i n s t t h r e s h o l d
39 ind = f i n d (Y > thresh_OOK ) ;
40 rece ived_mes sage ( ind ) = 1 ;
41
42 % g e t s i m u l a t i o n b i t e r r o r r a t e (BER)
43 e r r o r s = re cei ved_m essag e ( r eceived_ messa ge ~= b i t s ) ;
44 error_sim ( i ) = s i z e ( e r r o r s , 1 ) / s i z e (X, 1 ) ;
45
46 % c a l c u l a t e t h e o r e t i c a l BER
47 err o r_th e o ( i ) = qfunc ( s q r t ( range ( i ) ) ) ;
48 end
49
50 p l o t ( range , l o g 1 0 ( error_sim ) , ’ ko ’ )
51 p l o t ( range , l o g 1 0 ( e rror_ t heo ) , ’ k - - ’ )
52 t i t l e ( ’ De t e c t o r Performance with Varying SNR (OOK) ’ )
53 x l a b e l ( ’SNR [ dB ] ’ )
54 y l a b e l ( ’ l o g (BER) ’ )
55 l e g e nd ( ’ Simulated BER ’ , ’ Th e o r e t i c a l BER ’ )
EE526 Project 2 Page 4
Figure 2: This figure depicts the effect of SNR on the BER for the OOK communication scheme.
In this case, it is shown that the theoretical results agrees with the results of a Monte-Carlo
simulation (1e
6
iterations)
.
Quadrature Phase-Shift Keying (QPSK):
In this section, the communication scheme will be identical to that of BPSK except that X ∈ C.
This means that it is possible to send two bits of information per transmission (a real part and an
imaginary part). Theoretically, the bit error rate is the same as BPSK, but this type modulation
can allow for twice as much information to be sent. In order to test this, a Monte-Carlo simulation
was done just as with BPSK and OOK. In this case, the noise added to this system was N ∼
CN
0, σ
2
n
. The results are given by Figure 3 and 4, and the code to generate them is given
below in Listing 3. Figure 4 illustrates the data that the receiver would be processing in a noisy
QPSK system.
Listing 3: Matlab Code – OOK Simulation
1 %% QPSK Monte - Carlo S i m u l a t i o n
2 %
3 % |
4 % 10 x | x 00 ( odd b i t , even b i t )
5 % |
6 % - - - - - - -+ - - - - - - - > r e a l part ( I ch anne l )
7 % |
8 % 11 x | x 01
9 % |
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10
11 snr_max = 9 . 0 ; %dB
12 snr_min = 2 . 0 ; %dB
13 snr_step = 0 . 2 ;
14
15 snr_range = snr_min : snr_step : snr_max ;
16
17 num_transmissions = 1 e6 ;
18
19 % s e t up f i g u r e
20 f i g u r e
21 hold a l l
22 g r i d on
23 g r i d minor
24
25 l e n = 1/ s q r t (2 ) ;
26 t h et a = z e r o s ( num_transmissions , 1) ;
27 erro r _the o = z e r o s ( s i z e ( snr_range , 2 ) , 1) ;
28 s i n g l e _ e r r o r s = z e r o s ( s i z e ( snr_range , 1) , 1) ;
29 d o u b l e _ e r r o r s = z e r o s ( s i z e ( snr_range , 1) , 1) ;
30 f o r i = 1 : s i z e ( snr_range , 2)
31 rece ived_mes sage = z e r o s ( num_transmissions , 1) ;
32 r e c e i v e d = z e r o s ( num_transmissions , 1) ;
33 % c a l c u l a t e v a r i a n c e from SNR
34 N_0 = 1/ snr_range ( i ) ;% n o i s e va r i a n c e
35 sigma_sq = N_0/4 ;
36 \ beg i n { f i g u r e } [ h ! ]
37 \ b egin { minipage } [ b ] { 1 . 0 \ l i n e w i d t h }
38 \ c e n t e r i n g
39 \ c e n t e r l i n e {\ i n c l u d e g r a p h i c s [ width=15cm , h e i g h t =10cm] { EE526_Project2_ OOK←-
. png }}
40 % \ vspa ce {2. 0cm}
41 \ end{ minipage }
42 %
43 \ ca p t i o n {}
44 \ l a b e l { f i g : ook_ber }
45 \ end{ f i g u r e }
46 % ge n er a t e random s i g n a l o f 0 ’ s , 1 ’ s , 2 ’ s , 3 ’ s o f eq u a l p r o b a b i l i t y
47 % t h a t r e p r e s e n t the 4 p o s i t i o n s f o r symbos in QPSK
48 b i t s = ra n d i ( [ 0 3 ] , num_transmissions , 1) ;
49
50 % f o r QPSK, the b i t s shoul d be
51 X = b i t s ;
52
53 X(X == 0) = l e n + l e n *1 i ; % quadrant 1 = 00
54 X(X == 1) = - l e n + l e n *1 i ; % qudrant 2 = 10
55 X(X == 2) = - l e n + - l e n *1 i ;% quadrant 3 = 11
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56 X(X == 3) = l e n - l e n *1 i ;% quadrant 4 = 01
57
58 % ge n er a t e random complex g a us s i a n n o i s e
59 Z = s q r t ( sigma_sq ) .* randn ( s i z e (X, 1) , 1) + s q r t ( sigma_sq ) . * randn ( s i z e (←-
X, 1) , 1) *1 i ;
60
61 % add n o i s e to s i g n a l
62 Y = X + Z ;
63
64 % check s i g n a l a g a i n s t t h r e s h o l d
65
66 f o r j = 1 : num_transmissions
67 re c _ b i t s = [ r e a l (Y( j ) ) imag (Y( j ) ) ] ;
68 i f ( r e a l (Y( j ) ) > 0 && imag (Y( j ) ) > 0 )
69 % quadrant 1
70 rece ived_mes sage ( j ) = 0 ;
71 e l s e i f ( r e a l (Y( j ) ) < 0 && imag (Y( j ) ) > 0 )
72 % quadrant 2
73 rece ived_mes sage ( j ) = 1 ;
74 e l s e i f ( r e a l (Y( j ) ) < 0 && imag (Y( j ) ) < 0 )
75 % quadrant 3
76 rece ived_mes sage ( j ) = 2 ;
77 e l s e
78 %quadrant 4
79 rece ived_mes sage ( j ) = 3 ;
80 end
81 end
82
83 r e c e i v e d ( rec eived _message == 0) = le n + le n *1 i ; % quadrant 1 = 00
84 r e c e i v e d ( rec eived _message == 1) = - l e n + l e n *1 i ; % qudrant 2 = 10
85 r e c e i v e d ( rec eived _message == 2) = - l e n + - l e n *1 i ;% quadrant 3 = 11
86 r e c e i v e d ( rec eived _message == 3) = le n - l e n *1 i ;% quadrant 4 = 01
87
88 %get s i m u l a t i o n b i t e r r o r r a t e (BER)
89
90 d i f f = abs ( r e c e i v e d - X) ;
91 d o u b l e _ e r r o r s ( i ) = s i z e ( d i f f ( d i f f > 1 . 5 ) , 1) ;
92 s i n g l e _ e r r o r s ( i ) = s i z e ( d i f f ( d i f f > 1 & d i f f < 1 . 5 ) , 1) ;
93 error_sim ( i ) = ( s i n g l e _ e r r o r s ( i ) + 2* d o u b l e _ e r r o r s ( i ) ) /(2* s i z e (X, 1 ) ) ;
94
95 %c a l c u l a t e t h e o r e t i c a l BER
96 err o r_th e o ( i ) = qfunc ( s q r t (2* snr_range ( i ) ) ) ;
97 end
98
99 %p l o t
100 p l o t ( snr_range , lo g 1 0 ( error_sim ) , ’ ko ’ )
101 p l o t ( snr_range , lo g 1 0 ( e rror_ t heo ) , ’ k - - ’ )
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102 t i t l e ( ’ De t e c t o r Performance with Varying SNR (QPSK) ’ )
103 x l a b e l ( ’SNR [ dB ] ’ )
104 y l a b e l ( ’ l o g (BER) ’ )
105 l e g e nd ( ’ Simulated BER ’ , ’ Th e o r e t i c a l BER ’ )
Figure 3: This figure represents the results of a Monte-Carlo simulation (1e
6
iterations) to verify
the theoretical BER based on a given SNR value. As can be seen by the graph, the theoretical
results agree with the experimental results.
EE526 Project 2 Page 8
Figure 4: This figure represents the noisy data that is received by the detector. The distribution
around the true signal (red circles) is due to the noise added by the system. Here it can be seen
how too much of an error will confuse the detector and cause an error.
Summary and Conclusions:
In closing, the most efficient scheme is QPSK. This is due to the fact that it can transmit twice
as much information as OOK and BPSK, while achieving the same BER as BPSK (which is
better than OOK). The BER for all schemes over a range of SNR values is given by Figure 5.
The figure clearly depicts that the OOK has the worst performance (in terms of transfer rate vs.
BER), while QPSK and BPSK are the same.
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Figure 5: This figure represents the relationship between SNR and BER. Here it is clear that the
BPSK and QPSK schemes outperform the OOK scheme in terms of BER. Also, it is clear that
the BPSK and QPSK schemes have the same performance per bit.
EE526 Project 2 Page 10
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