Bandlimited communication systems
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Bandlimited communication systems Lecture 3 Vladimir Stojanović 6.973 Communication System Design – Spring 2006 Massachusetts Institute of Technology Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Passband channel example Two-ray wireless channel (multi-path – 1+0.9D) Multi-path creates notching in frequency domain Just slide the frequency window to bb Add single-sided noise Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 2 0 pulse response Attenuation [dB] Bandlimited channel example -10 -20 -30 -40 -50 0.8 0.6 Tsymbol=160ps 0.4 0.2 -60 0 1 0 2 4 6 8 10 frequency [GHz] 0 1 2 3 ns Low-pass channel causes pulse attenuation and dispersion Notches cause ripples in time domain Makes it hard to transmit successive messages Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 3 Inter-Symbol Interference (ISI) 1 Error! Amplitude 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 Symbol time 14 16 18 Middle sample is corrupted by 0.2 trailing ISI (from the previous symbol), and 0.1 leading ISI (from the next symbol) resulting in 0.3 total ISI As a result middle symbol is detected in error Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 4 Bandlimited communication systems Block detector vs. symbol-by-symbol φ1(KT - t) yp(t) Block Detector φ2(KT - t) . . . X φk(KT - t) t = KT yp(t) ϕp(T - t) R SBS detector t = kT, k = 0,...,K - 1 Xk k = 0,...,K -1 yp(t) Figure by MIT OpenCourseWare. Block of K symbols – MK messages MAP/ML detector complexity grows exponentially MK basis functions (branches in the matched filter) Sequence detection can bound that growth Simpler detector is “Symbol-By-Symbol” Optimal for AWGN channel Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 5 Symbol-by-symbol detection n (t) x (t) xk Input symbol at time k mod h(t) Bandlimited channel + yk Matched filter Sampler zk R Receiver SBS detector xk Estimate of input symbol at time k Figure by MIT OpenCourseWare. Suffers significantly from Intersymbol-interference (channel memory), so need to remove ISI to get almost AWGN channel Need to adapt basis functions to the particular channel, to avoid ISI Alternatively, use equalization to remove ISI Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 6 Vector channel - revisited np (t) xp (t) x(t) h(t) + yp (t) np (t) xkn ϕn(t) xn (t) xp (t) + h(t) yp (t) np (t) xp (t) xkn pn(t) + yp (t) Figure by MIT OpenCourseWare. Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 7 ISI impact xk pulse response p(t) xp,k xp(t) ||p|| jp(t) np(t) S yp(t) jp (-t) y(t) yk discrete time sample at receiver times kT xk q(t) = jp(t)*jp(-t) Figure by MIT OpenCourseWare. Mean-distortion Treat ISI as noise Peak-distortion Treat worst-case ISI as constellation offset Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 8 Matched Filter Bound You can’t do better with successive transmissions than with one-shot Matched filter collects the pulse energy ||p||2 Then calculate performance as on AWGN Example – binary transmission Will use MFB to compare different ISI compensation techniques Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 9 Nyquist criterion – 6.011 revisited A channel specified by pulse response p(t) is ISI free if Nyquist frequency: w=pi/T or f=1/2T Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 10 Raised-cosine pulses Can have “excess” bandwidth as long as there is symmetry that “fills” the aliased spectrum flat Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 11 Basic equalization concepts Zero-forcing equalization xk xˆk W(D) Flattens equalized channel transfer function = X Equalizer W(w) Channel Q(w) Q(D) yk H(D)=Q(D)*W(D) Equalized Wzfe(D)=1/(Q(D)||p||) => Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 12 Linear equalization Zero-forcing not good on channels with nulls Equalizer enhances noise Remember, Pe depends on both noise and ISI Balance noise and ISI in the mean-square sense Minimizing MMSE wrt. Wk Same as using the orthogonality principle Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 13 ZFE vs. MMSE - LE Q(e-jωT ) WZFE (e-jωT ) SNR ||p|| WMMSE-LE (e-jωT ) 0 π T ω 0 π ω T Figure by MIT OpenCourseWare. Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 14 Example: ZFE vs. MMSE LE 1+0.9D channel Equalizer response zfe mmse Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 15 MIT OCW. Fractional equalizers np(t) xk p xp,k ϕp(t) xp(t) yp(t) + anti-alias filter gain T y(t) yk Fractionally-spaced equalizer Wk zk sample at times k(T/l) Figure by MIT OpenCourseWare. Oversampling in the receiver Can merge matched filter and equalizer Can reconstruct original signal from oversampled signal (as long as original is band-limited) Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 16 ISI channel model Oversampled channel representation (3x e.g) Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 17 Finite length equalizer formulation xk p yk w zk Write convolution as multiply with Toeplitz matrix zk = wPxk Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 18 ZFE and MMSE solution Zero forcing equalizer (ZFE) zk = xk −∆ = wPxk ⇒ 1∆ = wzfe P ⇒ wzfe = 1 P T ∆ T ( PP ) T −1 , 1∆ = [0 0...1...0 0]T Minimum-mean square error (MMSE) equalizer => Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 19 Decision feedback equalizer p yk w zk xk-∆ b Feed-forward equalizer 1 Matched + whitening filter + remove pre-cursor ISI Feedback equalization 0.8 Amplitude xk 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 Symbol time Feed-back equalizer Removes trailing ISI To get w, first puncture the channel matrix to emulate the effect of feedback on the equalized pulse response wP Then, get b from the causal taps of equalized pulse response wP Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 20 18 MMSE DFE Selects the feedback taps Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 21 Basic multitone modulation Bits/chan Gain Bits/chan = Frequency Bits/chan Frequency Gain Frequency Bits/chan RF xtalk Frequency Frequency Frequency Figure by MIT OpenCourseWare. Best performance if basis functions are tailored to the channel Use each tone as a basis function Each tone transmits narrow QAM signal and satisfies Nyquist criterion – i.e. no ISI per tone Put less energy where channel is bad or where there is more noise Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 22 A bit of history 1948 Shannon constructs capacity bounds Analog multi-tone AWGN channel with linear ISI – effectively uses multi-tone modulation 1958 Collins Kineplex modem (first voiceband modem) – analog multitone 1964 Holsinger’s MIT thesis – modem that approximates Shannon’s “water-filling” 1967 Saltzberg, 1973 Bell Labs, 1980 IBM … Digital multi-tone ~ 1990s DMT for DSL - Major push by prof. Cioffi’s group at Stanford Use DSP power to improve the robustness and algorithms for discrete multi-tone modulation We will mostly focus on this type of modulation Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 23 Basic multitone transmission X0 ϕ0(-t) ϕ0(t) e-j2πf1t e j2πf1t X1 E N bits C O D X N-1 E R ϕ1(t) . . .  + h(t) p h a s e + s p l i t N = 2N X ϕN (t) ϕn(t) = X e-j2πfN-1t X (( D E C bits T=kNT' =kT O D YN-1 E ϕN-1(-t) R e-j2πfN t  1 . sinc t T T Y1 Yn = Hn . Xn+ Nn  e j2πfN t XN ϕ1(-t) X n(t) X e j2πfN-1t ϕN-1(t) Y0 ϕp,n(t) = ϕn(t) * h(t) bn = X ( 1 log 1+ SNRn 2 Γ 2 ϕN (-t) YN ( Figure by MIT OpenCourseWare. Each tone sees AWGN channel (no ISI) N QAM-like symbols (complex) 1 PAM symbol Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 24 The effect of the channel X( f ) input X0 N = 2N X1 X2 f1 f2 XN-1 fN-1 XN fN H( f ) Yn » Hn . Xn Y( f ) output Y0 Y1 Y2 f1 f2 YN-1 fN-1 YN fN Figure by MIT OpenCourseWare. Each channel can be treated as AWGN With only one basis function – hence simple symbol-by-symbol detector is optimal Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 25 Gap review 3 bits/dimension 2.5 2 0 dB 1.5 3 dB 6 dB 1 9 dB 0.5 b -6 SNR for Pe = 10 (dB) 22b 1 (dB) (dB) .5 1 2 8.8 13.5 20.5 26.8 32.9 38.9 0 4.7 11.7 18.0 24.1 30.1 8.8 8.8 8.8 Table of SNR Gaps for Pe = 10-6 . 3 8.8 4 8.8 5 8.8 0 2 4 6 8 10 12 SNR (dB) Achievable bit rate for various gaps 14 16 Illustration of bit rates versus SNR for various gaps. Figure by MIT OpenCourseWare. Figure by MIT OpenCourseWare. Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 26 Example – simplified multitone 1+0.9D channel Put unit energy per dimension (simply guessed) Same as baseband DFE Data rate 1bit/dimension With gap 4.4 dB Re-calculate the necessary SNR – margin SNRmfb=10dB SNRmultitone=8.8dB (with more tones to better approx no-ISI case) SNRdfe=7.1dB Can do even better with multitone, if allocated energy properly Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 27 Water-filling derivation Find optimum energy allocation that maximizes b for given total energy constraint b is a convex function in energy/dimension Use Lagrange multipliers to solve for εn d dε n => Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 28 Water-filling spectrum Flip the channel and pour in energy like water Energy n + g = constant n n . + σn2 = constant Hn 2 constant Channel E0 E1 E2 E3 L L L L L L g0 g1 g2 g3 g4 g5 0 1 2 3 4 Subchannel index 5 Figure by MIT OpenCourseWare. Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 29 Water-fill loading algorithms Rate maximization Margin maximization Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 30 Rate-adaptive loading Solve through matrix inversion Solve iteratively Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 31 Water-filling example (rate-adaptive) 1+0.9D again (Gap=1, so calculating capacity) Energy Try 8 dim first 1.29 1.29 1.29 1.29 1.29 1.29 1.29 E0 E1 E1 E2 E2 E3 E3 1.24 1.23 1.23 1.19 1.19 .96 .96 1 19.94 1 17.03 1 17.03 1 10 1 10 2.968 2.968 1 .0552 0 1 2 3 4 5 6 7 n Figure by MIT OpenCourseWare. => Try 7 dim next Capacity=1.55bits/dim Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 32 Summary Bandlimited communication Try to make bandlimited channel look AWGN ZFE removes ISI but enhances noise Trade-off by designing MMSE equalizer DFE removes trailing ISI without noise enhancement Multitone Use complex block detectors to orthogonalize basis functions (MAP) Simplify with equalization+sbs detector Generate basis functions that don’t loose orthogonality when passing through frequency selective channle (multitone modulation) Equalization Block vs. symbol-by-symbol detector Optimal transmission with proper allocation of energy/dimension (waterfilling) Next – practical loading algorithms and DMT/OFDM, Vector coding Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6.973 Communication System Design 33
