Fund. of Digital Communications Chapter 5: Demodulation ...

GRAZ UNIVERSITY OF TECHNOLOGY

Fund. of Digital Communications Chapter 5: Demodulation, Detection Klaus Witrisal

[email protected] Signal Processing and Speech Communication Laboratory

www.spsc.tugraz.at Graz University of Technology January 23, 2014

Signal Processing and Speech Communications Lab

Fund. of Digital CommunicationsChapter 5: Demodulation, Detection – p. 1/46


Outline 5-1 Correlation-Type Demodulator and Matched Filter 5-2 Optimal Detection 5-3 Error Probability 5-4 Equalization (and Channel Distortions)

Reference: (Figures taken from these books.) J. G. Proakis and M. Salehi, “Communication System Engineering,” 2nd Ed., Prentice Hall, 2002 J. G. Proakis and M. Salehi, “Grundlagen der Kommunikationstechnik,” 2. Aufl., Pearson, 2004 (in German – Achtung! tw. fehlerhafte Übersetzung) Signal Processing and Speech Communications Lab



Correlation-type demodulator Channel: Additive white Gaussian noise (AWGN) is added r(t) = sm (t) + n(t) transmitted signal {sm (t)}, m = 1, 2, ...M is represented by N basis functions {ψk (t)}, k = 1, 2, ...N received signal r(t) is projected onto these basis functions {ψk (t)}

T 0

r(t)ψk (t)dt =

T 0

[sm (t) + n(t)]ψk (t) dt

rk = smk + nk ,

k = 1, 2, ..., N

Output vector in signal space: r = sm + n Signal Processing and Speech Communications Lab



Correlation demod. (cont’d)

[Proakis 2002] Signal Processing and Speech Communications Lab



Correlation demod. (cont’d) Received signal N N r(t) = smk ψk (t) + nk ψk (t) + n (t) k=1

=

N

k=1

rk ψk (t) + n (t)

k=1

Correlator outputs r = [r1 , r2 , ...rN ]T are sufficient statistik for the decision i.e.: there is no additional info in n (t) n (t) is part of n(t) that is not representable by {ψk (t)} Interpretation of r: noise cloud in signal space Signal Processing and Speech Communications Lab



Correlation demod. (cont’d)




Correlation demod. (cont’d) Characterization of the noise component Mean values T E{nk } = ... = E{n(t)}ψk (t) dt = 0 0

Correlations (= covariances, since means = 0) E{nk nm } = ... =

N0 δ[m − k] 2

i.e.: N noise components {nk } are zero-mean, uncorrelated, Gaussian random variables their variances σn2 = N0 /2 (because ||ψk (t)||2 = 1) Signal Processing and Speech Communications Lab



Correlation demod. (cont’d) Statistics of the correlator outputs {rk } Mean values E{rk } = E{smk + nk } = smk Variances (identical) σr2 = σn2 = N0 /2 conditional PDF; likelihood function 2 1 f (rk |smk ) = √ e−(rk −smk ) /N0 , k = 1, 2, ..., N πN0 f (r|sm ) =

N

f (rk |smk )

m = 1, 2, ..., M

k=1 Signal Processing and Speech Communications Lab



Matched Filter Demodulator Uses N Filters (instead of the correlators) with impulse responses hk (t) = ψk (T − t), 0 ≤ t ≤ T Filter outputs

yk (t) =

0

t

r(λ)hk (t − λ) dλ = ...

Sampling at t = T T r(λ)ψk (λ) dλ = rk , yk (T ) =

k = 1, 2, ..., N

0

identical to correlator outputs! Signal Processing and Speech Communications Lab



Matched Filter Demodulator (cont’d)




Matched Filter Demodulator (cont’d) Property of the matched filter: maximization of the output SNR at sampling time t = T Derivation: arbitrary impulse response h(t), 0 ≤ t ≤ T y(t) = ... y(T ) = ... Signal-to-noise ratio (SNR) S ys2 (T ) = N 0 E{yn2 (T )} Maximum SNR is found with Cauchy-Schwarz inequ. S 2Es = h(t) = Cs(T − t) und N 0 N0 Signal Processing and Speech Communications Lab



Matched Filter Demodulator (cont’d) Derivation of the Matched Filter Demodulator (cont’d) SNR of the sampled filter output T 2 2 [ S ys (T ) 0 h(λ)s(T − λ) dλ] = ... = = N0 T 2 N 0 E{yn2 (T )} h (T − t) dt 2

0

denominator is energy of the filter IR → hold constant maximization of the numerator using Cauchy-Schwarz: 2 ∞ ∞ ∞ g1 (t)g2 (t) dt ≤ g12 (t) dt g22 (t) dt −∞

−∞

−∞

Equality (i.e. maximum) holds for g1 (t) = Cg2 (t) This yields h(t) = Cs(T − t) to obtain max. SNR Signal Processing and Speech Communications Lab



5-2 Optimum Detection (= Entscheidung) Signal model: received signal vector with i.i.d. noise r = sm + n sm ... points in N -dimensional signal space n ... random vector with i.i.d. components ∼ N (0, N0 /2) Find: decision minimizing the error probability, based on the observed vector r Solution: decision rule based on a posteriori probabilities: P (signal sm was transmitted |r) m = 1, 2, ..., M choose sm that maximizes P (sm |r) for m = 1, 2, ..., M Signal Processing and Speech Communications Lab



Optimum Detection (cont’d) Maximizing the posterior probability: Maximum a posteriori probability (MAP) criterion finding the posterior probability using Bayes’ rule: f (r|sm )P (sm ) P (sm |r) = f (r) f (r|sm ) ... conditional PDF of r given sm was transmitted (likelihood function) P (sm ) ... a priori probability

PDF f (r) = M m=1 f (r|sm )P (sm ) is independent of sm , hence irrelevant Metrik for MAP criterion: P M (r, sm ) = f (r|sm )P (sm ) Signal Processing and Speech Communications Lab



Optimum Detection (cont’d) Maximum Likelihood (ML) Criterion equivalent to MAP; for equal a priori probabilities 1 P (sm ) = , ∀m = 1, 2, ..., M M decision means finding the largest (the maximum) likelihood function f (r|sm ) (at the observed point r by choosing sm )




Optimum Detection (cont’d) ML criterion for AWGN channel (logarithm of f (r|sm )): N N 1 ln[f (r|sm )] = − ln(πN0 ) − (rk − smk )2 2 N0 k=1

note: first term is irrelevant; 1/N0 too: ML criterion means minimizing the distance! N D(r, sm ) = (rk − smk )2 = ||r − sm ||2 k=1

D(r, sm ) ... distance metric Equivalently: maximizing the correlation metric: C(r, sm ) = 2rT sm − ||sm ||2 Signal Processing and Speech Communications Lab



Optimum Detection – Example Given: binary, antipodal PAM s1 (t) = −s2 (t) = gT (t) √ s1 = −s2 = Eb (Eb ... energy per bit) P (s1 ) = p, P (s2 ) = 1 − p Find: metrics for MAP detection in AWGN Result: threshold detection can be performed! r ≷ss12 γ




Optimum Detection – Example (cont’d) Likelihood functions for binary antipodal PAM:

[Proakis 2002]




5-3 Error Probability Assumption: binary antipodal signals with √ P (s1 ) = P (s2 ) = 0.5 and s1 = −s2 = Eb therefore we can use a threshold detector with threshold γ = 0 error probability for s1 (t)

0 2Eb P (e|s1 ) = f (r|s1 ) dr = ... = Q N0 −∞ Q(x) =

√1 2π

∞ x

2

e−x

/2 dx

... integral over Gaussian

function




Error Probability (cont’d) average error probability per bit: Pb = P (s1 )P (e|s1 ) + P (s2 )P (e|s2 ) 1 1 = P (e|s1 ) + P (e|s2 ) 2 2 2Eb /N0 =Q using the distance between signal constellations: ⎛ ⎞ d212 ⎠ ⎝ Pb = Q 2N0 holds for any binary signals with P (s1 ) = P (s2 ) = 0.5 √ √ e.g. bin. antipodal: d12 = 2 Eb ; orthogonal: d12 = 2Eb Signal Processing and Speech Communications Lab



Error Probability (cont’d) −1

10

for orthogonal binary Signals

−2

for antipodal, binary Signals

−3

10

3 dB

−4

10

b

P ; Bit−Error−Rate (BER)

10

−5

10

−6

10

0

2

4

6 8 SNR/Bit (E /N ) [dB] b

10

12

14

0

Pb for antipodal and orthogonal binary signals Signal Processing and Speech Communications Lab



Error Probability (cont’d) Approximate symbol error rate (SER) for higher-order modulation schemes ⎛ ⎞ d2min ⎠ ⎝ PM ≈ Ne Q 2N0 where Ne ... average number of nearest neighbors (in signal space) dmin ... (min.) distance to those neighboring signal vectors




Error Probability (cont’d) for M -ary PAM M −1 Ne =2 M 3 log2 M dmin =2 Eb M2 − 1 M −1 PM ≈2 M 6 log2 M Eb ×Q M 2 − 1 N0

BER for Gray coding Pb ≈ PM /k Signal Processing and Speech Communications Lab



Error Probability (cont’d) for M -ary PSK Ne =2 π dmin =2 Es sin M π =2 kEb sin M

π 2kEb PM ≈2Q sin N0 M

BER for Gray coding Pb ≈ PM /k Signal Processing and Speech Communications Lab



Error Probability (cont’d) for M -ary orthogonal sign. Ne =M − 1 dmin = 2Es = 2kEb

kEb PM ≤(M − 1)Q N0

Upper bound: PM < e−k(Eb /N0 −2 ln 2)/2

i.e. PM → 0 for k → ∞ if Eb /N0 > 2 ln 2 = 1.4 dB in fact: Eb /N0 > −1.6 dB sufficient: Shannon limit Signal Processing and Speech Communications Lab



Comparison of modulation schemes: shows R/W : transmission rate normalized with bandwidth as a function of SNR per bit: Eb /N0 to reach SER of 10−5 Shannon limit is approached for orthogonal signals if k → ∞ [Proakis 2002] Signal Processing and Speech Communications Lab



Channel Capacity [Proakis02, Sec. 9.3] Channel capacity for AWGN channel (Shannon) P C = W log2 1 + N0 W

C ... channel capacity [bit/s] W ... bandwidth [Hz] P ... signal power [W] N0 ... noise PSD [W/Hz]

increase power P → (slow) increase of capacity no limit but lots of power required Signal Processing and Speech Communications Lab



Channel Capacity (cont’d) increase bandwidth W at fixed power P C = W log2 1 +

P N0 W

consider limit W → ∞ lim C =

W →∞

P P log2 e = 1.44 N0 N0

there is a hard limit need to increase P too Signal Processing and Speech Communications Lab



Channel Capacity (cont’d) in practice we must have R < C P R < W log2 1 + N0 W define r = R/W “spectral rate” Eb = P/R energy per bit we obtain Eb r < log2 1 + r N0 Signal Processing and Speech Communications Lab



5-4 Equalization [Proakis 2002, Section 8.6] Nyquist criterion must be fulfilled for ISI-free transmission Cascade of linear filters GT (f )C(f )GR (f ) = Xrc (f )

Xrc (f ) ... Fourier transform of raised-cosine pulse (ISI free!)




Channel Distortion (cont’d)

Design considerations; assume channel C(f ) known Cascade GT (f )C(f )GR (f ) = Xrc (f ) fulfills Nyquist Noise n(t) is AWGN (flat spectrum) → Receiver filter GR (f ) should be matched to cascade GT (f )C(f ) Xrc (f ) −j2πf t0 GT (f ) = GR (f ) = Xrc (f )e−j2πf tr e C(f ) Signal Processing and Speech Communications Lab



Equalization Channel is unknown or changes with time e.g. telephone channel – routing changes e.g. mobile radio – multipath propagation → design GT (f ) and GR (f ) as for AWGN channel: root raised cosine frequency response GT (f ) = Xrc (f )e−j2πf t0 |GR (f )| · |GT (f )| = Xrc (f )

Output of receiver filter r(t) = y(t) + n(t) =

∞

a[k]x(t − kT ) + n(t)

k=−∞

a[k] ∈ {am }M m=1 ... PAM (or QAM) symbols x(t) = gT (t) ∗ c(t) ∗ gR (t) ... cascaded system response Signal Processing and Speech Communications Lab



Equalization (cont’d) Equivalent output with periodic sampling:

2 y[k] = y(kT ) = L l=−L1 xl a[k − l] (noise-free) xk = x(kT ) ... equiv. FIR discrete-time channel filter




Equalization (cont’d) Maximum likelihood sequence detection (MLSD) optimum detector (for AWGN: r[k] = y[k] + n[k]): choose information sequence a := {a[k]}K k=1 based on the received sequence r := {r[k]}K+L−1 k=1

each a[k] ∈ {am }M m=1 , L hence there are M K possible sequences {a(m) }M m=1 joint likelihood function: (w/ y (m) [k] = a(m) [k] ∗ xk ) K+L−1 (m) f (r|a ) = f (r[k]|a(m) [k]) k=1

=

K+L−1


k=1

√

(m) 2 1 e(r[k]−y [k]) /N0 πN0 Fund. of Digital CommunicationsChapter 5: Demodulation, Detection – p. 34/46


Maximum likelihood sequence detection (MLSD) (cont’d) joint likelihood function (cont’d): K+L−1 1 1 f (r|a(m) ) = exp − (r[k] − y (m) [k])2 K+L N0 (πN0 ) 2 k=1

equivalent distance metric (squared distance) D(r, a(m) ) =

K+L−1

(r[k] − y (m) [k])2 ;

y (m) [k] =

k=1

L2

xl a(m) [k − l]

l=−L1

MLSD: channel {xk } is known; find for m = 1, 2, ..., M K m ˆ = arg max f (r|a(m) ) = arg min D(r, a(m) ) m


m



Viterbi (MLSD) can be done sequentially using the Viterbi algorithm (m)

distances are computed recursively: D(r(1:k) , a(1:k) ) =

L2 (m) D(r(1:k−1) , a(1:k−1) ) + (r[k] − l=−L1 xl a(m) [k − l])2 Complexity: symbols a[k] are M -ary PAM symbols FIR channel has memory L − 1 = L1 + L2 → compute M L branch metrics at each stage k , yielding M L−1 surviving sequences (trellis diagram) → complexity: (K + L − 1)M L (scales linearly with K ) while M L M K , still M and L must be small! Signal Processing and Speech Communications Lab



Viterbi example (MLSD) Channel model; L = 2 (1 memory); M = 2 (binary) r[k] = x0 a[k] + x1 a[k − 1] + n[k];

a[k] ∈ {−1, +1}

state variable (memory) a[k − 1] ∈ {−1, +1} information sequence: {a[k]}K k=1 = a; each a[k] ∈ {±1}; i.e. 2K possible sequences denoted a(m) observation sequence: {r[k]}K+1 k=1 = r likelihood functions (for m = 1, ..., M K ) K+1 1 (m) f (r|a ) ∝ exp − (r[k] − x0 a(m) [k] − x1 a(m) [k − 1])2 N0 k=1




Viterbi example (MLSD) (cont’d) MLSD maximizes likelihood fct. by testing any a(m) , which is equivalent to minimizing:

K+L−1 D(r, a(m) ) = k=1 (r[k] − x0 a(m) [k] − x1 a(m) [k − 1])2 at stage k

k−1 (m) D(r(1:k) , a(1:k) ) = l=1 (r[l] − x0 a(m) [l] − x1 a(m) [l − 1])2 +(rk − x0 a(m) [k] − x1 a(m) [k − 1])2 (m)

= D(r(1:k−1) , a(1:k−1) ) + branch metric

Viterbi requires: M L = 22 = 4 branch metrics at each state M L−1 = 21 = 2 surviving paths at each state → (K + 1)4 M K = 2K metric computations Signal Processing and Speech Communications Lab



Linear Equalizers Use linear filters to compensate for channel distortion filter has adjustable parameters preset equalizers: measure channel; set parameters adaptive equalizers: update parameters during data transmission




Zero-Forcing Linear Equalizer How to select equalizer GE (f )? GR (f ) is matched to GT (f ) to fulfill zero-ISI Hence GE (f ) must compensate channel distortion 1 1 GE (f ) = = e−jΘc (f ) C(f ) |C(f )| → inverse channel filter forces ISI to zero (at sampling times) detector input: r[k] = y[k] + n[k] noise variance (for AWGN); noise gain! ∞ W N |Xrc (f )| 0 σn2 = Sn (f )|GR (f )|2 |GE (f )|2 df = df 2 2 |C(f )| −∞ −W Signal Processing and Speech Communications Lab



Linear transversal filter Practical implementation of the linear equalizer use FIR filter with adjustable taps {cn } delays τ : τ = T ... symbol spaced; aliasing, if 1/T < 2W τ < T ... fractionally-spaced equalizer (often: τ = T /2)




Zero-Forcing Equalizer (cont’d) How to determine taps {cn } for the zero-forcing equalizer? Equalized output pulse N q(t) = cn x(t − nτ ) n=−N

for x(t) = gT (t) ∗ c(t) ∗ gR (t); number of taps 2N + 1 ≥ L enforce zero-ISI at sampled output: N 1, k = 0 q(kT ) = cn x(kT − nτ ) = 0, k = ±1, ±2, ..., ±N n=−N

system of 2N + 1 linear equations: q = Xc for coefficients {cn } → approx. solution c = X† q Signal Processing and Speech Communications Lab



Minimum mean-square-error equalizer ZF equalizer ignores noise! → noise gain solution: relax zero-ISI condition consider residual ISI plus noise at equalizer output → use minimum mean-square-error (MMSE) criterion

sampled equalizer output N z(kT ) = cn r(kT − nτ ) = cT r[k] n=−N

mean-square-error (MSE) MSE = E{(z(kT ) − a[k])2 } = E{(cT r[k] − a[k])2 } Signal Processing and Speech Communications Lab



MMSE equalizer (cont’d) Find the equalizer c that minimizes the MSE E{[cT r[k] − a[k]]2 } = E{[cT r[k] − a[k]][rT [k]c − a[k]]} = E{cT r[k]rT [k]c} − 2E{cT r[k]a[k]} + E{a[k]2 } = cT Kr c − 2cT kar + E{a[k]2 }

with Kr = E{r[k]rT [k]} and kar = E{r[k]a[k]} minimization: find derivative with c; set to zero Kr c − kar = 0

→

c = Kr −1 kar

correlation matrix Kr and correlation sequence kar must be estimated → training sequences Signal Processing and Speech Communications Lab


data and distorted data

Impulse Response channel impulse reponse zero−forcing equalizer MMSE equalizer

1 0.8

6000OF TECHNOLOGY GRAZ UNIVERSITY

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received signal samples

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channel frequency reponse −2 0.2 0.4 0.6 1 −3 zero−forcing 0.8 equalizer 0 Normalized Frequency (×π rad/sample) MMSE equalizer

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Signal Processing

0.2 0.4 0.6 0.8 (×π rad/sample) and Normalized Speech Frequency Communications Lab

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0 20Fund.30 40 50 −4 −2 0 2 4 of Digital CommunicationsChapter 5: Demodulation, Detection – p. 45/46


Adaptive equalizers perform the MMSE equalization recursively search along gradient for MSE (e.g. LMS algorithm) discussed in detail in VL: Adaptive Systems

