II. Modulation & Coding

Design Goals of Communication Systems Maximize transmission bit rate Minimize bit error probability Minimize required transmission power Minimize required system bandwidth Minimize system complexity, computational load & system cost Maximize system utilization

Some Tradeoffs in M-PSK Modulaion 2 1 m=4 m=3 m=1, 2 3 Trades off BER and Energy per Bit Trades off BER and Normalized Rate in b/s/Hz Trades off Normalized Rate in b/s/Hz and Energy per Bit

Shannon-Hartley Capacity Theorem System Capacity for communication over of an AWGN Channel is given by: C: System Capacity (bits/s) W: Bandwidth of Communication (Hz) S: Signal Power (Watt) N: Noise Power (Watt)

Shannon-Hartley Capacity Theorem Unattainable Region Practical Systems

Shannon Capacity in terms of Eb/N0 Consider transmission of a symbol over an AWGN channel

Shannon Limit Let

Shannon Limit Shannon Limit=-1.6 dB

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 Eb/N0 falls below -1.6 dB

Shannon Limit Room for improvement by channel coding 16 PSK Uncoded Pb=10-5 8 PSK Uncoded Pb=10-5 QPSK Uncoded Pb = 10-5 Normalized Channel Capacity b/s/Hz BPSK Uncoded Pb = 10-5 Shannon Limit=-1.6 dB Eb/N0 10

1/3 Repetition Code BPSK Is this really purely a gain? Coding Gain= 3.2 dB Is this really purely a gain? No! We have lost one third of the information transmitted rate

1/3 Repetition Code 8 PSK 1 2 3 4 5 6 7 8 9 10 -6 -5 -4 -3 -2 -1 E b /N P BPSK Uncoded 8 PSK 1/3 Repitition Code Coding Gain= -0.5 dB When we don’t sacrifice information rate 1/3 repetition codes did not help us

Hard Decision Decoding v = [v1 v2 … vi … vn] e = [e1 e2 … ei … en] r = [r1 r2 … ri … rn] x = [x1 x2 … xi … xn] y = [y1 y2 … yi … yn] Channel Encoder Waveform Generator Detection Channel Decoder Channel v r x y T +1 V. -1 V. vi vi=1 vi=0 xi yi>0 yi<0 ri=1 ri=0 ri + zi ]-∞, ∞[ yi 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

Soft Decision Decoding v = [v1 v2 … vi … vn] e = [e1 e2 … ei … en] r = [r1 r2 … ri … rn] x = [x1 x2 … xi … xn] v x r Channel Encoder T +1 V. -1 V. vi vi=1 vi=0 xi Waveform Generator Channel Decoder Channel + zi ]-∞, ∞[ ri 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

Hard Decision: Example 1/3 Repetition Code BPSK v = [v1 v2 … vi … vn] e = [e1 e2 … ei … en] r = [r1 r2 … ri … rn] x = [x1 x2 … xi … xn] y = [y1 y2 … yi … yn] y r Channel Encoder Waveform Generator Waveform Detection Channel Decoder Channel 0 0 0 -1 -1 -1 0.1 -0.9 0.1 1 0 1 1 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

Soft Decision: Example 1/3 Repetition Code BPSK v = [v1 v2 … vi … vn] e = [e1 e2 … ei … en] r = [r1 r2 … ri … rn] x = [x1 x2 … xi … xn] y = [y1 y2 … yi … yn] r Channel Encoder Waveform Generator Channel Decoder Channel 0.1 -0.9 0.1 0 0 0 -1 -1 -1 Accumulated Voltage = 0.1-0.9+0.1=-0.7<0 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

1/3 Repetition Code BPSK Soft Decision Channel Coding (1/3 Repetition Code) Waveform Representation Channel r Soft Decision Decoding Important Note

BER Performance Soft Decision 1/3 Repetition Code BPSK Select b*=0 if Note that r0 r1 and r2 are independent and identically distributed. In other words Therefore Similarly

BER Performance Soft Decision 1/3 Repetition Code BPSK Select b*=0 if

BER Performance Soft Decision 1/3 Repetition Code BPSK where n is Gaussian distributed with mean 0 and variance 3N0/2

Hard Vs Soft Decision: 1/3 Repetition Code BPSK Coding Gain= 4.7 dB

1/3 Repetition Code 8 PSK Hard Decision 1 2 3 4 5 6 7 8 9 10 -6 -5 -4 -3 -2 -1 E b /N P BPSK Uncoded 8PSK 1/3 Repetition Code Hard Decision 8PSK 1/3 Repetition Code Soft Decision Coding Gain= 1.5 dB 22

Shannon Limit and BER Performance 8 PSK Uncoded Pb=10-5 16 PSK Uncoded Pb=10-5 QPSK Uncoded Pb = 10-5 8PSK 1/3 Rep. Code Soft Decision Pb = 10-5 8PSK 1/3 Rep. Code Hard Decision Pb = 10-5 Normalized Channel Capacity b/s/Hz BPSK Uncoded Pb = 10-5 1/3 BPSK 1/3 Rep. Code Sodt Decision Pb = 10-5 BPSK 1/3 Rep. Code Hard Decision Pb = 10-5 Shannon Limit=-1.6 dB Eb/N0 23