= + 2 cos (2 ) 0.5 (1 cos(2 2 )) 0.5( ) ft dt - PDF document

Backgorund: Digital Communication Systems In this note, we will review the basic transmitter and receiver principles in digital communication systems. The discussions will provide useful background for this course. Binary phase shift keying (BPSK) based on a cos function With BPSK modulation, the transmitted signal in ( kT , ( k+ 1) T ) is given by s ( t ) = ax k cos(2  f c t ). (1) where a is used for power control and x k represents a bit of information 0 or 1. For example: x k = +1 represents information 0, and x k = − 1 represents information 1. This is called a bipolar format of a binary bit. The following is an example for four bits in time duration [0, 4T]. Pay attention to the phase jumps at time instances t = 2 T and t = 3 T . This is caused by the change of sign of x k at these points. x 0 =+ 1 x 1 =+ 1 x 2 =- 1 x 3 =+ 1 t t= 0 t=T t= 2 T t= 3 T t= 4 T Assume that the channel has an attenuation factor  and a delay factor  . The received signal is r ( t ) =  ax k cos(2  f c ( t-  )) + n ( t ) =  ax k [cos(2  f c t )cos(2  f c  ) + sin(2  f c t )sin(2  f c  )]+ n ( t ). (2) where n ( t ) is an additive white Gaussian noise (AWGN) with zero mean. The following is an illustration of n ( t ). t AWGN noise n ( t ) t The following illustrates the received signal affected by AWGN noise. 0 1 1 0 1 1 1 0

Some useful equations: The following equations are useful in understading the pricniples of BPSK and QPSK mentioned later. In all cases, we assume that f is very large. In practice, f represents carrier frequency. A typical range of f is 1GHz ~ 5GHZ (i.e., 10 9 Hz to 5×10 9 Hz) for the 4G systems. b 1 2   =  −    cos(2 ) | sin(2 ) sin(2 ) | 0 ft dt fb fa (3a)   2 2 f f a b 1 2 ( ) ( )   = −  − −    sin(2 ) | cos(2 ) cos(2 ) | 0 ft dt fb fa (3b)   2 2 f f a b b     =    sin(2 )cos(2 ) 0.5 sin(2 2 ) 0 ft ft dt ft dt (3c) a a The following graphic illustration helps to understand the above relationships. Furthermore, we have the following useful approximations: b b    = −    − 2 sin (2 ) 0.5 (1 cos(2 2 )) 0.5( ) ft dt ft dt b a (3d) a a b b    = +    − 2 cos (2 ) 0.5 (1 cos(2 2 )) 0.5( ) ft dt ft dt b a (3e) a a Again, the above relationships can be understood using the following graph.

Correlation receiver The following shows a correlation receiver to estimate x k . s ( t ) = ax k cos(2  f c t ) r ( t ) =  ax k cos(2  f c ( t-  )) + n ( t ) y k channel +  ( k 1) T  kT reference cos(2  f c t ) We repeat (2) as: r ( t ) =  ax k [cos(2  f c t )cos(2  f c  ) + sin(2  f c t )sin(2  f c  )]+ n ( t ). (4) =  + ( 1) k T   ( )cos(2 ) n t f t dt Define . (5) k c kT From (3), we have + ( 1)  k T   2 cos (2 ) 0.5 f t dt T , c kT 1) sin(2 +  ( k T    )cos(2 ) 0 f t f t dt , c c kT The output y k in the above receiver can then be expressed as ( ) (6) =    +  0.5 cos 2 y aTx f k k c k Assume that  is known. We estimate x k using y k and the following rule: ( )    (7a) x k = +1 if y k has the same sign as 0.5 cos 2 aT f c (7b) x k = − 1 otherwise. Note that  k is unknown, which may cause detection error. We model  k as Gaussian distributed with zero mean and variance = 0.5 2 TN 0 , where N 0 is called single-sided channel noise density. N 0 is a measurement of channel noise level. The performance of the above receiver is determined by the following signal to noise ration (SNR) ( ) ( ) ( ) 2       2 2 2 2 2 0.5 cos 2 aT f 0.5 cos 2 a f T = = c c SNR  2 variance of 0.5 N T 0 k ( )( ) =    2 2 2 cos 2 / . (8) f E N 0 c b where E b =0.5 a 2 T is the transmitted energy per bit (based on (3b)). Note that SNR is affected by E b , path gain   and delay  .

In-phase and quadrature signal components The signal above is modulated by cos(2  f c t ). This is usually referred to as the in-phase component. If sin(2  f c t ) is used for modulation, as discussed below, the related signal is called the quadrature compoenent. If the channel delay  is not zero, then part of the in-phase componenet is turned into a quadrature componenet as (see (2))  ax k sin(2  f c t )sin(2  f c  ). Such component is orthogonal to the reference cos(2  f c t ) (see the figure and (3)) since b     sin(2 )cos(2 ) 0 ft ft dt . a The related energy is not utilized, which reduces energy efficiency. Binary phase shift keying (BPSK) based on a sin function We can modulate a signal using a sin function as s ( t ) = ax k sin(2  f c t ). (9) The received signal is r ( t ) =  ax k sin(2  f c ( t-  )) + n ( t ) =  ax k [sin(2  f c t )cos(2  f c  ) − cos(2  f c t )sin(2  f c  )]+ n ( t ). (10) The following shows a correlation receiver to estimate x k . Note that the reference is sin(2  f c t ) now . s ( t ) = ax k sin(2  f c t ) y k r ( t ) =  ax k sin(2  f c ( t-  )) + n ( t ) + channel ( 1)  k T  kT reference sin(2  f c t ) =  + ( 1) k T   ( )sin(2 ) n t f t dt Define . (11) k c kT We can write the output y k in the above receiver as ( ) =    +  0.5 cos 2 y aTx f . (12) k k c k We can again estimate x k based on y k . The rule is similar to (7). Again, part of the received signal, i.e.,  ax k cos(2  f c t )sin(2  f c  ), is not utilized.

Quadrature phase shift keying (QPSK) We can increase transmission speed as well as energy efficiency using both in-phase and quadrature componenets. This is referred to as quadrature phase shift keying (QPSK). The transmitted signal is =  −  Re Im ( ) cos(2 ) sin(2 ) s t ax f t ax f t . (13) k c k c The received signal is ( ) ( ) ( ) ( ) =   −  −   −  + Re Im ( ) cos 2 sin 2 ( ) r t ax f t ax f t n t k c k c =     +    Re Re [ cos(2 )cos(2 ) sin(2 )sin(2 )] a x f t f x f t f k c c k c c +  −    +    + Im Im [ sin(2 )cos(2 ) cos(2 )sin(2 )] ( ) a x f t f x f t f n t k c c k c c (14) The following receiver is used for detection. y k Re   in-phase branch cos(2  f c t ) r ( t ) y kIm   quadrature branch − sin(2  f c t ) Similar to the discussions earlier, we can shown the following. ( ) ( ) ( ) =    +   +  Re Re Im Re 0.5 cos 2 sin 2 y aT x f x f , (15a) k k c k c k ( ) ( ) ( ) =  −   +   +  Im Re Im Im 0.5 sin 2 cos 2 y aT x f x f . (15b) k k c k c k Now we introduce notations: + Re Im = x x jx , (16a) k k k + Re Im = y y jy , (16b) k k k ( ) ( ) ( )    −   =0.5 cos 2 sin 2 h T f j f , (16c) c c  =  +  Re Im j . (16d) k k k Then (15) and (16) lead to a very simple expression: (17) = +  y ahx k k k

Detection for QPSK Im jy Im received y k jx k k with noise + j Re x × ah -1 +1 k Re y - j k four possibilities for y k without noise We can carry out detection using the above graphis illustration. Assume that h k is know. Without noise  k , we can find four noiseless possibilities for y k . With noise  k , the actual y k is not on these four points. We can use the minimum Euclidian distance to find the best estimate. For example, for the received signal shown by the “star” above, the best est imate is the “diamond” since it is closest to the “star”. Notes: • The transmitted and received signals are all real. The complex = + Re Im x x jx notations are used only for simplicity. We call a k k k QPSK modulated symbol. It is a simplified notation for the actual =  −  Re Im ( ) cos(2 ) sin(2 ) s t ax f t ax f t signal . k c k c Re Im • The received signal consists of two parts, i.e., y y and . They are k k both real signals. Complex notation is again used only for simplicity. • Since exp( jz ) = cos( z ) + j sin( x ). We can write h in a phasor form: ( )  −   =0.5 exp 2 . h T j f c Here  is refered to as channel magnitude, |  | 2 as channel power gain (or simply channel gain) and  as channel delay. Both  and  are determined by channel only. A constant factor 0.5 T is introduced by the intergration operation at the receiver. • In the QPSK receiver, both in-phase and quadrature compoenets can be fully utilized. Such a structure can also be used to detect a BPSK signal with improved power efficiency.