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II. Modulation & Coding
© Tallal Elshabrawy Design Goals of Communication Systems 1.Maximize transmission bit rate 2.Minimize bit error probability 3.Minimize required transmission power 4.Minimize required system bandwidth 5.Minimize system complexity, computational load & system cost 6.Maximize system utilization 2
© Tallal Elshabrawy Some Tradeoffs in M-PSK Modulaion 1Trades off BER and Energy per Bit 2Trades off BER and Normalized Rate in b/s/Hz 3Trades off Normalized Rate in b/s/Hz and Energy per Bit m=4 m=3 m=1, 2
© Tallal Elshabrawy Shannon-Hartley Capacity Theorem C:System Capacity (bits/s) W:Bandwidth of Communication (Hz) S:Signal Power (Watt) N:Noise Power (Watt) 4 System Capacity for communication over of an AWGN Channel is given by:
© Tallal Elshabrawy Shannon-Hartley Capacity Theorem 5 Practical Systems Unattainable Region
© Tallal Elshabrawy Shannon Capacity in terms of E b /N 0 Consider transmission of a symbol over an AWGN channel 6
© Tallal Elshabrawy Shannon Limit 7 Let
© Tallal Elshabrawy Shannon Limit 8 Shannon Limit=-1.6 dB
© Tallal Elshabrawy Shannon Limit No matter how much/how smart you decrease the rate by using channel coding, it is impossible to achieve communications with very low bit error rate if E b /N 0 falls below -1.6 dB
© Tallal Elshabrawy Shannon Limit 10 Shannon Limit=-1.6 dB BPSK Uncoded P b = QPSK Uncoded P b = PSK Uncoded P b = PSK Uncoded P b =10 -5 Room for improvement by channel coding Normalized Channel Capacity b/s/Hz E b /N 0
© Tallal Elshabrawy 1/3 Repetition Code BPSK Is this really purely a gain? No! We have lost one third of the information transmitted rate 11 Coding Gain= 3.2 dB
© Tallal Elshabrawy E b /N 0 P b BPSK Uncoded 8 PSK 1/3 Repitition Code 1/3 Repetition Code 8 PSK 12 Coding Gain= -0.5 dB When we don’t sacrifice information rate 1/3 repetition codes did not help us
© Tallal Elshabrawy The waveform generator converts binary data to voltage levels (1 V., -1 V.) The channel has an effect of altering the voltage that was transmitted Waveform detection performs a HARD DECISION by mapping received voltage back to binary values based on decision zones Channel Encoder Waveform Generator Waveform Detection Channel Decoder Channel v r x y v = [v 1 v 2 … v i … v n ] e = [e 1 e 2 … e i … e n ] r = [r 1 r 2 … r i … r n ] x = [x 1 x 2 … x i … x n ] y = [y 1 y 2 … y i … y n ] 0 T 0 T +1 V. -1 V. vivi v i =1 v i =0 xixi 0 y i >0 y i <0 r i =1 r i =0 riri + zizi ]-∞, ∞[ yiyi Hard Decision Decoding
© Tallal Elshabrawy The waveform generator converts binary data to voltage levels (1 V., -1 V.) The channel has an effect of altering the voltage that was transmitted The input to the channel decoder is a vector of voltages rather than a vector of binary values Channel Encoder Waveform Generator Channel Decoder Channel v r x v = [v 1 v 2 … v i … v n ] e = [e 1 e 2 … e i … e n ] r = [r 1 r 2 … r i … r n ] x = [x 1 x 2 … x i … x n ] 0 T 0 T +1 V. -1 V. vivi v i =1 v i =0 xixi + zizi ]-∞, ∞[ riri Soft Decision Decoding
© Tallal Elshabrawy Hard Decision -Each received bit is detected individually -If the voltage is greater than 0 detected bit is 1 -If the voltage is smaller than 0 detected bit is 0 -Detection information of neighbor bits within the same codeword is lost Channel Encoder Waveform Generator Waveform Detection Channel Decoder Channel r y v = [v 1 v 2 … v i … v n ] e = [e 1 e 2 … e i … e n ] r = [r 1 r 2 … r i … r n ] x = [x 1 x 2 … x i … x n ] y = [y 1 y 2 … y i … y n ] Hard Decision: Example 1/3 Repetition Code BPSK
© Tallal Elshabrawy Soft Decision -If the accumulated voltage within the codeword is greater than 0 detected bit is 1 -If the accumulated voltage within the codeword is smaller than 0 detected bit is 0 -Information of neighbor bits within the same codeword contributes to the channel decoding process Channel Encoder Waveform Generator Channel Decoder Channel r v = [v 1 v 2 … v i … v n ] e = [e 1 e 2 … e i … e n ] r = [r 1 r 2 … r i … r n ] x = [x 1 x 2 … x i … x n ] y = [y 1 y 2 … y i … y n ] Accumulated Voltage = =-0.7<0 Soft Decision: Example 1/3 Repetition Code BPSK
© Tallal Elshabrawy 1/3 Repetition Code BPSK Soft Decision Channel Coding (1/3 Repetition Code) Waveform Representation Channel Soft Decision Decoding r Important Note
© Tallal Elshabrawy BER Performance Soft Decision 1/3 Repetition Code BPSK Select b*=0 if Note that r 0 r 1 and r 2 are independent and identically distributed. In other words Therefore Similarly
© Tallal Elshabrawy Select b*=0 if BER Performance Soft Decision 1/3 Repetition Code BPSK
© Tallal Elshabrawy where BER Performance Soft Decision 1/3 Repetition Code BPSK n is Gaussian distributed with mean 0 and variance 3N 0 /2
© Tallal Elshabrawy Hard Vs Soft Decision: 1/3 Repetition Code BPSK Coding Gain= 4.7 dB
© Tallal Elshabrawy 1/3 Repetition Code 8 PSK Hard Decision 22 Coding Gain= 1.5 dB
© Tallal Elshabrawy Shannon Limit and BER Performance 23 Shannon Limit=-1.6 dB BPSK Uncoded P b = QPSK Uncoded P b = PSK Uncoded P b = PSK Uncoded P b =10 -5 BPSK 1/3 Rep. Code Hard Decision P b = BPSK 1/3 Rep. Code Sodt Decision P b = Normalized Channel Capacity b/s/Hz E b /N 0 1/3 8PSK 1/3 Rep. Code Hard Decision P b = PSK 1/3 Rep. Code Soft Decision P b = 10 -5
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