Modulation and coding course- lecture 8

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Some bandpass modulation schemes -M-PAM, M-PSK, M-FSK, M-QAM -How to perform coherent and non-coherent detection -How to calculate the average probability of symbol error for different modulation schemes that we studied?

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Nội dung Text: Modulation and coding course- lecture 8

Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 8
Last time we talked about: Some bandpass modulation schemes M-PAM, M-PSK, M-FSK, M-QAM How to perform coherent and non- coherent detection Lecture 8 2
Example of two dim. modulation ψ 2 (t ) 16QAM 8PSK s3 “011” “010” “001” ψ 2 (t ) s4 s2 “0000” “0001” “0011” “0010” s1 s2 3 s3 s4 Es “000” “110” s1 s5 ψ 1 (t ) “1000” “1001” “1011” “1010” s5 s6 s7 s8 1 “111” “100” -3 -1 1 3 ψ 1 (t ) s6 s8 s9 s10 -1 s11 12 s “101” s 7 ψ 2 (t ) “1100” “1101” “1111” “1110” QPSK “00” s 2“01” s1 s13 s14 -3 s15 16s “0100” “0101” “0111” “0110” Es ψ 1 (t ) Lecture 8 s3 “11” “10” s4 3
Today, we are going to talk about: How to calculate the average probability of symbol error for different modulation schemes that we studied? How to compare different modulation schemes based on their error performances? Lecture 8 4
Error probability of bandpass modulation Before evaluating the error probability, it is important to remember that: Type of modulation and detection ( coherent or non- coherent), determines the structure of the decision circuits and hence the decision variable, denoted by z. The decision variable, z, is compared with M-1 thresholds, corresponding to M decision regions for detection purposes. ψ 1 (t ) T r1 ∫ 0 ⎡ r1 ⎤ Decision r (t ) ⎢M⎥ =r r Circuits ˆ m ψ N (t ) ⎢ ⎥ Compare z T ⎢rN ⎥ ⎣ ⎦ with threshold. ∫ 0 rN Lecture 8 5
Error probability … The matched filters output (observation vector= r ) is the detector input and the decision variable is a z = f (r) function of r , i.e. For MPAM, MQAM and MFSK with coherent detection z = r For MPSK with coherent detection z = ∠r For non-coherent detection (M-FSK and DPSK), z =| r | We know that for calculating the average probability of symbol error, we need to determine Pr(r liesinside Zi | si sent)≡ Pr(z satisfies conditionCi | si sent) Hence, we need to know the statistics of z, which depends on the modulation scheme and the detection type. Lecture 8 6
Error probability … AWGN channel model: r = s i + n Signal vector s i = (ai1 , ai 2 ,..., aiN ) is deterministic. Elements of noise vector n = (n1 , n2 ,..., nN ) are i.i.d Gaussian random variables with zero-mean and variance N 0 / 2 . The noise vector pdf is 1 ⎛ n2⎞ pn (n) = exp⎜ − ⎟ (πN 0 )N /2 ⎜ N0 ⎟ ⎝ ⎠ The elements of observed vector r = (r1 , r2 ,..., rN ) are independent Gaussian random variables. Its pdf is 1 ⎛ r − si 2 ⎞ pr (r | s i ) = exp⎜ − ⎟ (πN 0 )N / 2 ⎜ N 0 ⎟ ⎝ ⎠ Lecture 8 7
Error probability … BPSK and BFSK with coherent detection: ⎛ s1 − s 2 / 2 ⎞ PB = Q⎜ ⎟ ⎜ N /2 ⎟ ψ 2 (t ) ⎝ 0 ⎠ ψ 1 (t ) “0” “1” s1 s2 “0” s1 − s 2 = 2 Eb BPSK BFSK s1 Eb − Eb Eb ψ 1 (t ) s 2“1” ψ 2 (t ) s1 − s 2 = 2 Eb Eb ⎛ 2 Eb ⎞ ⎛ Eb ⎞ PB = Q⎜ ⎟ PB = Q⎜ ⎟ ⎜ N ⎟ ⎜ N ⎟ ⎝ 0 ⎠ ⎝ 0 ⎠ Lecture 8 8
Error probability … Non-coherent detection of BFSK Decision variable: 2 / T cos( ω 1 t ) Difference of envelopes r11 ( )2 z = z1 − z2 T ∫ 0 z1 = r11 + r12 2 2 2 / T sin( ω 1 t ) T r12 r (t ) ∫ 0 ( )2 + z Decision rule: ˆ m 2 / T cos( ω 2 t ) if z (T ) > 0, m = 1 ˆ if z (T ) < 0, m = 0 ˆ T r21 ( )2 ∫ - 0 z 2 = r21 + r22 2 2 2 / T sin( ω 2 t ) T r22 ∫ 0 ( )2 Lecture 8 9
Error probability – cont’d Non-coherent detection of BFSK … 1 1 PB = Pr( z1 > z 2 | s 2 ) + Pr( z 2 > z1 | s1 ) 2 2 = Pr( z1 > z 2 | s 2 ) = E [Pr( z1 > z 2 | s 2 , z 2 )] ∞ = ∫ Pr( z1 > z 2 | s 2 , z 2 ) p ( z 2 | s 2 )dz 2 = ∫ ∞ ⎡ ∞ p ( z | s )dz ⎤ p( z | s )dz 0 0 ⎢ ∫z2 ⎣ 1 2 1⎥ ⎦ 2 2 2 1 ⎛ Eb ⎞ PB = exp⎜ − ⎜ 2N ⎟⎟ Rayleigh pdf Rician pdf 2 ⎝ 0 ⎠ Similarly, non-coherent detection of DBPSK 1 ⎛ Eb ⎞ PB = exp⎜ − ⎜ N ⎟ ⎟ 2 ⎝ 0 ⎠ Lecture 8 10
Error probability …. Coherent detection of M-PAM Decision variable: z = r1 “00” “01” “11” “10” s1 s2 s3 s4 4-PAM ψ 1 (t ) − 3 Eg − Eg 0 Eg 3 Eg ψ 1 (t ) T r1 ∫ ML detector r (t ) (Compare with M-1 thresholds) ˆ m 0 Lecture 8 11
Error probability …. Coherent detection of M-PAM …. Error happens if the noise, n1 = r1 − s m , exceeds in amplitude one-half of the distance between adjacent symbols. For symbols on the border, error can happen only in one direction. Hence: ( Pe (s m ) = Pr | n1 |=| r1 − s m |> E g ) for 1 < m < M ; P (s ) = Pr (n = r − s e 1 1 1 1 > Eg ) ( and Pe (s M ) = Pr n1 = r1 − s M < − E g ) M −2 ( ) ( ) ( ) M 1 1 1 PE ( M ) = M ∑ Pe (s m ) = m =1 M Pr | n1 |> E g + Pr n1 > E g + Pr n1 < − E g M M 2( M − 1) ⎛ 2 E g ⎞ = 2( M − 1) M ( Pr n1 > E g = M ) 2( M − 1) ∞ ∫ Eg pn1 (n)dn = M Q⎜ N 0 ⎜ ⎟ ⎟ ⎝ ⎠ ( M 2 − 1) Es = (log 2 M ) Eb = Eg 3 Gaussian pdf with 2( M − 1) ⎛ 6 log 2 M Eb ⎞ ⎟ zero mean and variance Q⎜ N0 / 2 PE ( M ) = ⎜ M 2 −1 N ⎟ M ⎝ 0 ⎠ Lecture 8 12
Error probability … Coherent detection ψ 2 (t ) of M-QAM “0000” “0001” s1 s2 s 3“0011”4 “0010” s “1000” s “1001” s s 7“1011” 8 “1010” s 5 6 16-QAM ψ 1 (t ) s9 s10 s11 s12 “1100” “1101” “1111” “1110” ψ 1 (t ) s13 s14 s15 s16 T r1 ML detector “0100” “0101” “0111” “0110” ∫ 0 (Compare with M − 1 thresholds) r (t ) Parallel-to-serial ˆ m converter ψ 2 (t ) T r2 ML detector ∫ 0 (Compare with M − 1 thresholds) Lecture 8 13
Error probability … Coherent detection of M-QAM … M-QAM can be viewed as the combination of two M − PAM modulations on I and Q branches, respectively. No error occurs if no error is detected on either I and Q branches. Hence: Considering the symmetry of the signal space and orthogonality of I and Q branches: PE ( M ) = 1 − PC ( M ) = 1 − Pr(no error detected on I and Q branches) Pr(no error detected on I and Q branches) = Pr(no error on I)Pr(no error on Q) ( = Pr(no error on I)2 = 1 − PE ( M )) 2 ⎛ 1 ⎞ ⎛ 3 log 2 M Eb ⎞ PE ( M ) = 4⎜1 − ⎟Q⎜ ⎜ M −1 N ⎟ ⎟ Average probability of ⎝ M ⎠ ⎝ 0 ⎠ symbol error for M − PAM Lecture 8 14
Error probability … Coherent detection of MPSK ψ 2 (t ) s 3 “011” “010” s4 “001” s2 Es “110” s“000” 1 8-PSK s5 ψ 1 (t ) ψ 1 (t ) “111” s6 s8“100” T r1 s “101” 7 ∫ r1 φ 0 ˆ r (t ) ˆ m arctan Compute Choose ψ 2 (t ) r2 | φi − φ | ˆ smallest T ∫ 0 r2 Decision variable z = φ = ∠r ˆ Lecture 8 15
Error probability … Coherent detection of MPSK … The detector compares the phase of observation vector to M-1 thresholds. Due to the circular symmetry of the signal space, we have: M π /M 1 PE ( M ) = 1 − PC ( M ) = 1 − M ∑ Pc (s m ) = 1 − Pc (s1 ) = 1 − ∫ m =1 −π / M pφˆ (φ )dφ where 2 Es ⎛ E ⎞ π pφˆ (φ ) ≈ cos(φ ) exp⎜ − s sin 2 φ ⎟; ⎜ N ⎟ | φ |≤ π N0 ⎝ 0 ⎠ 2 It can be shown that ⎛ 2 Es ⎛ π ⎞⎞ ⎛ 2(log 2 M )Eb ⎛ π ⎞⎞ ⎜ PE ( M ) ≈ 2Q⎜ sin ⎜ ⎟ ⎟ or ⎜ PE ( M ) ≈ 2Q⎜ sin ⎜ ⎟ ⎟ ⎝ N0 ⎝ M ⎠⎟ ⎠ ⎝ N0 ⎝ M ⎠⎟ ⎠ Lecture 8 16
Error probability … Coherent detection of M-FSK ψ 1 (t ) T r1 ∫ 0 ⎡ r1 ⎤ ML detector: r (t ) ⎢M⎥ =r r Choose the largest element ˆ m ψ M (t ) ⎢ ⎥ in the observed vector T ⎢rM ⎥ ⎣ ⎦ ∫ 0 rM Lecture 8 17
Error probability … Coherent detection of M-FSK … The dimensionality of signal space is M. An upper bound for average symbol error probability can be obtained by using union bound. Hence ⎛ Es ⎞ PE ( M ) ≤ (M − 1)Q⎜ ⎜ N ⎟ ⎟ ⎝ 0 ⎠ or, equivalently ⎛ PE ( M ) ≤ (M − 1)Q⎜ (log 2 M )Eb ⎞ ⎟ ⎜ N0 ⎟ ⎝ ⎠ Lecture 8 18
Bit error probability versus symbol error probability Number of bits per symbol k = log 2 M For orthogonal M-ary signaling (M-FSK) PB 2 k −1 M /2 = k = PE 2 − 1 M − 1 PB 1 lim = k →∞ P 2 E For M-PSK, M-PAM and M-QAM PE PB ≈ for PE
Probability of symbol error for binary modulation Note! • “The same average symbol energy for different sizes of PE signal space” Eb / N 0 dB Lecture 8 20