Principle of Communications, Fall 2017 Lecture 05 Wideband Communication I-Hsiang Wang ihwang@ntu.edu.tw National Taiwan University 2017/11/23
Recap • Lecture 02 (Digital Modulation) ‣ introduce the interface between the digital and the physical world ‣ discrete-time sequence ⟷ continuous-time waveform ‣ channel: ideal ‣ main challenge: representing a waveform by a sequence ‣ resource: bandwidth • Lecture 03 (Optimal Detection under Noise) ‣ optimal detection of symbols from noisy observations ‣ MAP , ML, MD; performance analysis ‣ channel: additive Gaussian noise channel ‣ main challenge: optimally combat the noise at the receiver ‣ resource: energy 2
Recap • Lecture 04 (Reliable Communication) ‣ coding to achieve reliable communication ‣ orthogonal coding, linear block code, convolutional code ‣ channel: additive Gaussian nose channel (soft decision) ‣ channel: binary symmetric channel (hard decision), erasure channel ‣ main challenge: introduce redundancy to combat noise so that probability of error can be arbitrarily small with positive rate and finite energy per bit, by using good encoder and decoder ‣ resource: time and energy • So far, the physical model of noise is the white Gaussian process ‣ Physical channel: Y ( t ) = x ( t ) + Z ( t ) , S Z ( f ) = N 0 2 ‣ Without noise, output is just the input go through a filter with flat frequency response ‣ Particularly good model for narrowband communication 3
This Lecture Channel Coding Binary Interface x ( t ) ECC Symbol Pulse Up Encoder Mapper Shaper Converter Information coded discrete baseband passband Noisy LTI filter bits bits sequence waveform waveform Channel + noise Filter + ECC Symbol Down Sampler + Decoder Demapper Converter Detection Y ( t ) • Physical channel model for wideband communication Y ( t ) = ( h ∗ x )( t ) + Z ( t ) , S Z ( f ) = N 0 2 ‣ Intuition: when the band is wide, signals in di ff erence band will experience di ff erent frequency response of the channel ‣ Use an LTI filter to model the channel 4
This Lecture Channel Coding Binary Interface x ( t ) ECC Symbol Pulse Up Encoder Mapper Shaper Converter Information coded discrete baseband passband Noisy LTI filter bits bits sequence waveform waveform Channel + noise Filter + ECC Symbol Down Sampler + Decoder Demapper Converter Detection Y ( t ) ISI happens and deteriorate detection! • New challenge: inter-symbol interference (ISI) ‣ Detect each symbol individually is no longer optimal • Our focus: mitigate ISI in the digital world (after sampling) ‣ HW1 tells us that dealing with ISI in the analog world is a pretty bad idea ‣ Receiver-side solution, transmitter-side solution, and Tx-Rx solution 5
Outline • LTI filter channel and inter-symbol interference (ISI) • Optimal Rx-side solution: MLSD • Rx-side solution: linear equalizations • Tx-Rx-side solution: OFDM 6
Part I. LTI Filter Channel and Inter-Symbol Interference Equivalent Discrete-Time Baseband Channel; Inter-Symbol Interference; MLSD 7
Physical Channel Model Y ( t ) = ( h ∗ x )( t ) + Z ( t ) , S Z ( f ) = N 0 2 Z ∞ = h ( τ ) x ( t − τ ) d τ + Z ( t ) −∞ • Use LTI filter to model wireline channels ‣ Examples: telephone lines, Ethernet cables, cable TV wires, optical fibers ‣ Operating bandwidth range from 1~2MHz to 250~500 MHz. • Why use LTI filter to model wireline channels? ‣ Frequency responses are no longer flat ‣ Channel is rather stationary compared to wireless channels ‣ Within the interest of time, can be assumed to be time-invariant 8
Features of the LTI Filter Channel h ( τ ) Z T d = ⇒ Y ( t ) = h ( τ ) x ( t − τ ) d τ + Z ( t ) 0 τ time dispersion (delay spread) T d • Causal : naturally, impulse response should be causal. h ( τ ) = 0 , ∀ τ < 0 • Dispersive : naturally, input signal cannot “stay” in the channel for too long, and hence most energy of the impulse response of the channel should be contained in an interval [0 , T d ] h ( τ ) = 0 , ∀ τ > T d 9
Derivation of the Discrete-Time Model Step 1: real passband ⟷ complex baseband (ignore noise) Pulse shaping: x b ( t ) � � m u m p ( t − mT ) Up conversion: x ( t ) � Re √ � � x b ( t ) 2 exp(j2 π f c t ) check! √ LTI channel: � � y ( t ) = ( h ∗ x )( t ) = Re ( h b ∗ x b )( t ) 2 exp(j2 π f c t ) h b ( τ ) � h ( τ ) exp( − j2 π f c τ ) Down conversion: y b ( t ) = ( h b ∗ x b )( t ) 10
������ Step 2: continuous-time ⟷ discrete-time x b ( t ) � � k u k p ( t − kT ) Demodulation: u m = ( y b ∗ q )( mT ) = ( x b ∗ h b ∗ q )( mT ) ˆ g ( t ) � ( p ∗ q )( t ) Z T d X = h b ( τ ) g ( mT − kT − τ ) d τ u k 0 k X = u k h m − k = ( u ∗ h d ) m k h d [ ℓ ] � ( h b ∗ g )( ℓ T ) = ( p ∗ h b ∗ q )( ℓ T ) Step 3: adding noise back ∼ CN (0 , N 0 ) V m = � ℓ h d [ ℓ ] u m − ℓ + Z m , Z m 11
������ ������ Number of Taps ∼ CN (0 , N 0 ) V m = � ℓ h d [ ℓ ] u m − ℓ + Z m , Z m • What is the range of in the summation of the discrete-time ℓ convolution in the equivalent discrete-time model? ‣ Recall: h d [ ℓ ] � ( h b ∗ g )( ℓ T ) h b ( τ ) � h ( τ ) exp( − j2 π f c τ ) h ( τ ) g ( t ) T p T d t τ ‣ The overall “spread” of the digital filter is hence T p + T d T • The equivalent discrete-time filter has finite impulse response , L ≈ T p + T d that is, the number of taps is finite: T L − 1 ∼ CN (0 , N 0 ) V m = � h d [ ℓ ] u m − ℓ + Z m , Z m ℓ =0 12
������ Discrete-Time Complex Baseband Model • With a little abuse of notation, identifying , the equivalent h d [ ℓ ] ≡ h ℓ discrete-time baseband channel model is given as { Z m } L − 1 X V m = h ` u m − ` + Z m , ↓ { u m } → FIR L -tap LTI filter − → { V m } → ⊕ − ` =0 h , [ h 0 h 1 ... h L − 1 ] ∈ C L ∼ CN (0 , N 0 ) Z m • The filter tap coe ffi cients depends on { h ` } ‣ one-sided bandwidth (or symbol time ) 1 T = W 2 W ‣ carrier frequency f c ‣ modulation pulse g ( t ) ‣ channel impulse response h ( τ ) • In practice, these taps are measured via training : sending known pilot symbols to estimate the tap coe ffi cients. • Total # of taps is proportional to bandwidth : L ≈ T p + T d ∝ W T 13
������ ������ Inter-Symbol Interference Narrowband channel (no ISI) ∼ CN (0 , N 0 ) V m = h 0 u m + Z m , Z m Wideband channel (with ISI) L − 1 X ∼ CN (0 , N 0 ) V m = h ` u m − ` + Z m , Z m ` =0 = h 0 u m + ( h 1 u m − 1 + ... + h L − 1 u m − L +1 ) + Z m I m inter-symbol interference • With ISI, it is no longer optimal to detect each symbol from the u m single observed only. V m • ISI introduces memory, and hence one needs to detect the entire sequence jointly ⟹ Maximum Likelihood Sequence Detection 14
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