Optimum Receiver for CPM over AWGN channel

Optimum Receiver for CPM over AWGN channel

special case: GMSK Receiver Mohsen Jamalabdollahi

Overview of MLSD

• The optimum receiver for modulation with memory is Maximum Likelihood Sequence Detection/Estimation (MLSD/MLSE)

Viterbi Algorithm and Trellis diagram

• For a sequence of N received samples the trellis diagram has N steps• Each steps has states where is the length of the applied filter in transmitter

( is the length of memory)• Each state has output branches • Each branch indicates an outcome (transmission of )• Each state has input branches but we can accept only one (winner branch) and

delete the others• Each states at each step has a weigh

• Index of state• : Index of step

• For

CPM Modulation

• Where • In GMSK: and , therefore:

• General form considering :

𝜃𝑛𝜃(𝑡 ; 𝐼 )

Total States?

• Possible values of

• It is function of {} • For total number of all possible

value is • For GMSK and

• Possible value of

• Infinite?• Consider and 1,-1}?

• Consider and 1,-1}?

Total States?

• Therefore the possible values of determine by the possible values of and • This value is the total number of states at each step for trellis diagram

of Viterbi algorithm• GMSK over AWGN?• has 8 possible values and has 4. This means 32 states.• Different states number for different • The GMSK trellis diagram will have all 32 state for • for calculate the weight for each state as follow

𝑘=1,2• Weight Calculation:

• and • (lower case is NOT from state table)•

•

• , • ,

𝑘=3•

•

•

•

• • • •

𝑘=4• Upper Part:• All have a • • Upper case from state table• For (from state table) • • For

Lower Part:All have a • • For (from state table)

𝑘≥5• We have all 32 states available

(state table)• Each state has two incoming and

two outgoing branches (state table)

• Each state belongs to a and goes to a state of if 1 comes and if -1 comes

• State will be transferred to if 1 comes and to if -1 comes

• Example • 1 comes: • -1 comes:

• 1 comes: • -1 comes:

𝑘≥5• For k=5:N do

• For m = 1:32• Calculate

• Save the winner state number for state m at path matrix

• Find min{,.., } • Follow the path of the final winner

state to the first state

GMSK MLSD Performance

0 2 4 6 8 10 1210-7

10-6

10-5

10-4

10-3

10-2

10-1

100

SNR

BER