Principles of Communications Lecture 8: Baseband Communication Systems. Chih-Wei Liu 劉志尉 National Chiao Tung University
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1 Principles of Communications Lecture 8: Baseband Communication Systems Chih-Wei Liu 劉志尉 National Chiao Tung University
2 Outlines Introduction Line codes Effects of filtering Pulse shaping toward zero ISI Zero-forcing equalization Eye diagrams Synchronization Commun.-Lec8 2
3 Introduction Digital vs. analog signals Analog sampling quantization digital Baseband vs. passband Channel distortion ISI (Channel) bandwidth limitation Commun.-Lec8 3
4 Line Codes Baseband data format used to represent digital data (for transmission purpose). Examples are given on the next page Operation: time or frequency shaping Purposes: usually to cope with the channel limitations (or provide extra function such as synchronization) Needed for certain applications Commun.-Lec8 4
5 Non-return-to-zero (NRZ) change NRZ mark (data1-> change in level; data0 -> no change) Unipolar return-to-zero (URZ) <1/2 width pulses> Polar RZ Bipolar RZ ( 0 -> 0 level; 1 -> alternate sign Split phase (Manchester) ( 1 -> level A to level A at 1/2 interval; 0 -> level A to level A at 1/2 interval) Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 5
6 Purposes of Line Codes Self synchronization Proper power spectrum Transmission bandwidth Transparency Error detection capability Good error probability performance Commun.-Lec8 6
7 Power Spectra (I) The transmitted signal is a pulse train: X() t = a p( t kt Δ). k= The amplitudes can be viewed as random variables with R = a a m = 0, ± 1, ± 2, m k k+ m The autocorrelation function of the waveform is R X ( τ ) = R r( τ m= k m mt ), K 1 in which r( τ) = p( t+ τ) p( t) dt. T Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 7
8 Power Spectra (II) The power spectral density is the Fourier transofrm of R ( τ ): S ( f) = FT[ R ( τ)] = FT[ R r( τ mt)] X X m m= = R FT[( r τ mt)] = R S ( f) e = m= m m r m= j2 mtf r( ) π m. m= S f R e j2π mtf 2 1 P( f) Note that Sr( f ) = FT[ r( τ )] = FT[ p( t) p( t)] =. T T X Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 8
9 Example 1: NRZ Assume the message m[n] is random (white noise) with equally 0 and 1 values. Step 1: Compute R m based on the above assumption and format Step 2: Compute r(t) based on the pulse shape. 1. NRZ, 50% "0" and 50% "1". R = A + ( A) = A, m= 0; m R = A + A( A) + ( A) A+ ( A) = 0, m 0. m pt () =Π( t/ T) P( f) = Tsinc( Tf) S f = A S f = A T Tf Therefore, NRZ ( ) r( ) sinc ( ). Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 9
10 Example 2: Unipolar RZ 2. Unipolar RZ, 50% 1-level and 50% 0-level A + (0) = A, m= Rm = A A+ A + A+ = A m pt () =Π(2 t/ T) P( f) = sinc( f) , 0 T T 2 2 T T 1 1 SURZ ( f) = sinc ( f) A + A T T 1 1 = sinc ( f ) 4 2 A + A e 4 4 m= m= j2π mtf m=, m 0 j2π mtf 2 n= m= n= T 2 T A n j2π mtf 1 n = sinc ( f) A ( f ). e ( f ) δ δ 4 4 T n= T Q = m= T n= T m= e Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 10
11 Commun.-Lec8 11
12 Inter-symbol Interference (ISI) Intersymbol interference refers to one specific type of distortion. It is typically due to insufficient bandwidth of the channel. Note: Narrow BW Long time-duration Example: Rectangular waveforms through a lowpass RC filter. The neighboring pulses smear out and interfere with each other. Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 12
13 BW= 1bit/sec Commun.-Lec8 13
14 BW= 0.5bit/sec Commun.-Lec8 14
15 ISI Reduction Pulse shaping: Assume the channel is ideal (flat) with the minimum BW (W). What shape of pulse warrants ISI-free transmission? Equalization: If the channel is non-ideal (not ideally flat LPF), an equalizer helps in reducing ISI. Commun.-Lec8 15
16 Pulse Shaping Consider 2W independent samples per second are transmitted through a channel with bandwidth W Hz. The output is as follows. n yt () = yn() t = ansinc 2 W t. n n 2W = = The samples at t = m/ 2 W are ISI-free, Q sinc( m n) = 0. Are there other pulse shapes warrant ISI-free? m p(t) -T 0 T 2T t a1 p( t T ) Sampling of p(t-2t) a2 p( t 2T ) a3 p( t 3T ) Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 16 t
17 Raised Cosine Family Raised Cosine: a example of ISI-free pulses raised cosine spectra at BW edges. 1 β T, f 2T T πt 1 β 1 β 1+ β PRC ( f) = 1+ cos 2 f f β 2T < 2T 2T 1+ β 0 f > 2T Roll-off factor β: β=0 rectangular pulse (sinc) β=1 cosine in freq; time pulse has narrow main lobe with very low sidelobes Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 17
18 Time pulse of the raised cosine: p RC cos( πβt/ T) t ( t) = sinc( ), β is called the roll-off factor. 2 1 (2 βt/ T) T cosine curves Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 18
19 Nyquist s Pulse Shaping Criterion A pulse shape p( t) with a Fourier transform P( f k= k 1 P( f + ) = T, f T 2T 1, n = 0 then its sampled values pnt ( ) =. 0, n 0. ) satisfying Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 19
20 Proof of Nyquist's Pulse Shaping Criterion j2π ft pt () = P( f) e df j2π fnt nt p nt = P f e df Sampling at, ( ) ( ) 2k+ 1 2 T j 2 π fnt pnt ( ) = P( f) e df k= 2k 1 2 T 1 2T k j2π f ( unt+ kn) = Pu ( + ) e du; u= f k= 1 2 T T 1 k j2π funt = 2 T Pu ( + ) e du 1 2 T T k= 1 2 T j 2 π funt 1, n = 0 = Te du = 1 2 T 0, n 0 k T Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 20
21 Pulse Transmission System xt () = ah( t kt). k= k T v() t = y() t h () t = [ x() t h ()] t h (). t R C R Consider the combined effect of three filters, we would like to have vt ( ) to be ISI-free. Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 21
22 Transmitter and Receiver Filters Let vt ( ) = A ap ( t kt t) k= k RC d where A represents a scale factor and t a possible delay. Ap ( t t ) = h () t h () t h (), t or in frequency domain, RC d T C R j2π ftd RC ( ) = T ( ) C ( ) R( ). AP f e H f H f H f If H ( f) is given, then H C T d ( f) H ( f) = 1 H ( f) The overall spectrum should satisfy the Nyquist criterion to avoid ISI. A filter design problem. Practically, H c (f) is unknown of time-varying R C Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 22
23 Let H ( f) = H ( f) = 1 H ( f) T R C 1/2 Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 23
24 Zero-Forcing Equalization Now, assume digital processing at the receiver. Purpose: Design an FIR filter that compensates for the channel distortion zero-isi. Show an example of zero-isi equalizer design. Commun.-Lec8 24
25 Let the combined impulse response after the channel be pc( t), Pass such a pulse through our finite-length equalizer, then n peq() t = αn pc( t nδ). n= N If the resulting pulse satisfies the zero-isi condition, ISI-free transmission is achieved. Assume that Δ= T, the condistion is n 1, m = 0 peq( mt) = αn pc[( m n) T] = m= 0, ± 1, K, ± N. n= N 0, m 0 Notice that we only enforce the zero-isi condition on the 2N + 1 samples. Why? We only have 2N + 1 coefficients (variables) and can only achieve this much. Solve the 2N + 1 equations and you obtain the zero-forcing equalizer. Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 25
26 0 M pc(0) pc( T) L pc( 2 NT) a N 0 pc( T) pc(0) pc(( 2N 1) T) a N+ 1 1 L + = M M M 0 pc(2 NT) pc(0) an M 0 Example: A 3-tap equalizer The equalizer cannot eliminate ISI beyond its span. Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 26
27 Eye Diagrams Eye diagram: Constructed by overlapping a number of segments of the base-band signals A qualitative measure of the system performance. (Lee et al., Digital Comm., 1994) Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 27
28 Increasing bandwidth may mitigate ICI, but will also allows more noise to enter. Another example of trade-offs in commu systems design Amplitude jitter: ICI causes the amplitude of each symbol fluctuate. eye eye Timing jitter: ICI causes the timing of each symbol fluctuate. The best place to sample the signal. Commun.-Lec8 28
29 Symbol, Bit, Word, and Frame Bits Symbol (a single pulse in transmission) --- carrier sync., symbol timing Bits Word --- word sync. Words Frame --- frame sync. (Benedetto et al., Digital Tansmission Theory, 1987) Commun.-Lec8 29
30 Synchronization Methods : (1) external sync signals; (2) self-synchronization: derivation from the modulated signals Example 1: squaring the received NRZ signals and PLL Sync signal Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 30
31 Example 2 4 Sync signal Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 31
32 Carrier Modulation Simple modulation applied to baseband digital signals Example: Let dt ( ) be the NZR waveform. Amplitude Shift-Keying (ASK): x ( t) = A[1 + d( t)]cos(2 π f t) ASK c c π Phase Shift-Keying (PSK): xpsk ( t) = Ac cos[2 π fct+ d( t)] 2 Frequency Shift-Keying (FSK): x ( t) = A cos[2π f t+ k d( α) dα] FSK c c f t Commun.-Lec8 cwliu@twins.ee.nctu.edu.tw 32
33 Commun.-Lec8 33
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