Summary: Digital Carrier Modulation (with corrections)
Modulated
Signal
Binary
ASK
Mathematical Expression
12-09-2002
Bandwidth (Hz)
Demodulation
Probability of Error
2Rb
Coherent
 Eb 

Pb  Q
 N 


2Rb
Non-Coherent
 A cos( c t ),
 ASK (t )  
0,
for bit 1
Binary
ASK
 A cos( c t ),
 ASK (t )  
0,
for bit 1
Binary
PSK
 A cos( c t ), for bit 1
 PSK (t )  
- A cos( c t ), for bit 0
2Rb
Coherent
Differential
PSK
 A cos( c t ), for bit 1
 DPSK (t )  
- A cos( c t ), for bit 0
2Rb
Non-Coherent
Pb = 0.5exp(-Eb/N)
2Rb+(f2-f1)
Coherent
 Eb 

Pb  Q
 N 


2Rb+(f2-f1)
Non-Coherent
Pb = 0.5exp(-Eb/2N)
2R
Non-Coherent
2R
Coherent
 M  1   6 log 2 M
PeM  2
Q
2
 M   M  1
2R
Coherent
 2 E b log 2 M

PeM  2Q
sin( / M ) ,


N


E b / N  1, M  2
2R+(fM-f1)
f1<f2<…<fM
2B
(B is the bandwidth of p(t))
Coherent or
Non-Coherent
Coherent
for bit 0
for bit 0
Pb 
 E b 
1   Eb / 2 N

e
 Q
 N 
2



 2Eb
Pb  Q
 N





Binary sequence is differentially encoded
Binary
FSK
Binary
FSK
M-ary
ASK
M-ary
ASK
M-ary PSK
 A cos(1t ),
 FSK (t )  
 Acos( 2 t ),
for bit 0
for bit 1
 A cos(1t ), for bit 0
 FSK (t )  
 Acos( 2 t ), for bit 1
 MASK (t )  Ai cos(c t ), Ai  0, A,2 A,..., (M  1) A
i  1,2,..., M
 MASK (t )  Ai cos(c t ), Ai   A,3 A,..., (M  1) A
i  1,2,..., M
 MPSK (t )  A cos( c t  2(i  1) / M )
 A cos( c t ) cos(2(i  1) / M )  A sin( c t ) sin(2(i  1) / M )
i  1,2,..., M
M-ary FSK
M-ary
QAM
or APK
 MFSK (t )  A cos(i t ), i  1,2,... M
 MQAM (t )  a i p (t ) cos( c t )  bi p (t ) sin( c t )
 ri p (t ) cos( c t   i ), i  1,2,..., M
ri  a i2  bi2 ,
 i  tan 1 (bi / a i )
 2Eb
PeM  4Q

 N M

 Eb

 N




M is an efficiency factor
 

 
Notations
 In binary case there are two symbols (bits) only: 0 and 1
 In M-ary case there are M different symbols, where M=2m. Each symbol carries m bits.
 c=2fc is the carrier frequency in rad/s ( fc  B), where B is the bandwidth of the base-band pulse. The carrier frequency in digital carrier modulation is
always selected as c=2n/Tb, where n is a positive integer. This simply means that we should have an integer number of carrier cycles during each Tb.
 Since PSK signals can only be detected coherently, a modified version of it called DPSK is used, which can be detected non-coherently. The binary
sequence has to be differentially encoded before being PSK-modulated.
 Binary PSK and QPSK (Quadri-Phase Shift Keying) are used in wireless communications, whereas MQAM is used in modems.
 When coherent detection is used for binary FSK, the two frequencies 1 and 2 must be chosen so that cos(1t) and cos(2t) are orthogonal over the
interval [0,Tb], i.e. 2 - 1 = 2(f2 –f1) = k/Tb, where k is a positive integer.
 When non-coherent detection is used for binary FSK, there is no necessity to have cos(1t) and cos(2t) orthogonal.
 When 2 - 1 = /Tb, which is the minimum difference for which cos(1t) and cos(2t) are orthogonal, we have the so-called "Minimum Shift Keying" or
MSK. MSK is considered as a special case of FSK with the minimum bandwidth. MSK with a Gaussian pulse (called GMSK) is applied in GSM (Global
System for Mobile) wireless communications. (GSM is the wireless standard used in Europe.)
 M-ary FSK uses M different frequencies 1, 2, …, M. Each frequency corresponds to a given symbol. Again if coherent detection is used then those
frequencies must be spaced as i+1 - i = k/T, where k is a positive integer and T is the symbol interval. Since each symbol represents m bits, T=mTb.
 For binary communication systems the data rate or bit rate is Rb=1/Tb, where Tb is the bit interval.
 For M-ary communication systems the data rate or symbol rate is R=1/T, where T is the symbol interval. Thus R=1/(mTb)=Rb/m, where m is the number of
bits per symbol. This shows that with M-ary system the bandwidth has been reduced by a factor m=log2(M) compared to the bandwidth of a binary system.
 M-ary signaling allows us to exchange or trade the transmission rate, transmission bandwidth, and transmitted power.
 The performance of a digital communication system is measured in terms of probability of error, i.e. the bit-error rate (BER) for binary case and symbolerror rate (SER) for M-ary case. Usually the probability of error is computed under the assumption that the channel noise is white and Gaussian.
 In the table given above, the BER is given as Pb and SER is given as PeM. The quantity N/2 is the PSD (or the variance) of the channel white noise.

From the fifth column of the table given above, the BER (or SER) is given in terms of the Q function, which is defined as Q(u ) 
1

e
2
 x2 / 2
dx . In some
u
books the complementary error function erfc (u ) 
2

e

x2
dx is used instead. We can deduce the relationship between these two functions:
u
Q(u ) 

 u 
1
 or erfc (u)  2Q( 2u) .
erfc 

2
 2
There are no closed or simple forms of probability of error for non-coherent MASK and MFSK. For example the probability of symbol error for coherent

1
  y  2 E log M / N  / 2
1  Q( y)M 1 dy . Of course in this case, some numerical methods should be used to perform the
MFSK is given by P  1 
e
eM

2

2
b
2

integration and calculate PeM.
MQAM combines amplitude and phase shift keying. The probability of symbol error for MQAM is given as an upper bound in terms of a parameter M
called efficiency factor. This factor is less than 1 and depends on the size and shape of the MQAM constellation.