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SQNR with and Without Companding Sampling PAM- Pulse Amplitude Modulation (continued) EELE445-14 Lecture 17 SQNR with and without Companding EELE445-13 Lecture 17 (continuation of lecture 16) 1 SQNR - summary V 2 P = max quantization noise power nq 3M 2 2 n Px 3M Px 3× 4 Px SQNR = = 2 = 2 Pnq Vmax Vmax • M is the number of quantization levels • n is the number of bits • Vmax is ½ the A/D input range SQNRdB 2 Px ≤ Vmax The SQNR decreases as Px The input dynamic range 2 ≤ 1 increases Vmax ⎛ P ⎞ SQNR | ≅ 10log ⎜ x ⎟ + 6n + 4.8 dB 10 ⎜ 2 ⎟ ⎝Vmax ⎠ 1 • V is the peak to peak range of the quantizer max 2 • n is the number of bits in the full scale quantizer range 2 Summary of SQNR|dB for linear, μ−law, A-law SQNRdB = 6.02n + α (3 - 25) ⎛Vmax ⎞ α = 4.77 − 20log⎜ ⎟ (uniform quantizing) (3 - 26a) ⎝ xrms ⎠ α ≅ 4.77 − 20log[]ln()1+ μ (μ - law companding) (3 - 26b) α ≅ 4.77 − 20log[]1+ ln A (A - law companding) (3 - 26c) 2 Px = xrms is the input signal power Vmax is the peak design level of the quantizer Exam 1 Wednesday February 26 6 3 Line Codes: Baseband Digital Signals ELE445-14 Lecture 17 7 PCM Signal Transmission PCM Signal Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 4 PCM Waveforms Shaped Polar NRZ Polar NRZ Unipolar NRZ Digital Signaling – Signal Vectors N w(t) = w ϕ (t) 0 < t < T ∑k=1 k k o Where wk is the digital data and φk(t) is the set of basis waveforms Example: ⎛ t ⎞ T ⎜ ⎟ s ϕ(t) = Π⎜ ⎟ To = for RZ codes ⎝ To ⎠ 2 To = Ts for NRZ codes w ∈ ()0,1 for Unipolar w ∈ ()−1,1 for polar or bipolar w ∈ ()− .75,−.25,.25,.75 for 4 state multilevel 5 Figure 3–11 Representation for a 3-bit binary digital signal. Vector lengths are in units of E Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Vector Signaling Tb T0 * w1 = w(t)φ 1 (t) N ∫0 ∫ 0 w(t) = ∑wkφk (t) k=1 T Tb 0 * w = w(t)φ k (t) k ∫ ∫0 0 The φk are orthogonal basis functions, wk are weights that correspond to the digital data. 6 Figure 3–13 Binary-to-multilevel signal conversion. ⎛ t ⎞ T ⎜ ⎟ s ϕ(t) = Π⎜ ⎟ To = for RZ codes ⎝ To ⎠ 2 To = Ts for NRZ codes w ∈ ()− 3,−1,1,3 for 4 state multilevel Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–12 Binary signaling (computed). Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 7 Figure 3–14 L = 4-level signaling (computed). Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Desired Properties of Line Codes • Self-synchronization: There is enough timing information built into the code so that bit synchronizers can be designed to extract the clock signal. A long series of 1’s or 0’s should not cause a problem. • Low probability of bit error: Receiver can be designed that will recover the binary data with a low probability of bit error when the input data signal is corrupted by noise or ISI (intersymbol interference) • Good Spectral Properties: If the channel is ac coupled, the PSD of the line code signal should be negligible at frequencies near zero. The signal bandwidth needs to be sufficiently small compared to the channel bandwidth, so that ISI will not be a problem. 8 Desired Properties of Line Codes •Transmission bandwidth: As small as possible •Error detection capability: It should be possible to implement this feature easillyby the addition of channel encoders and decoders, or the feature should be incorporated into the line code. • Transparency: The data protocol and line code are designed so that every possible sequence of data is faithfully and transparently received. (The sequence of bits does not matter.) Figure 3–15 Binary signaling formats. NRZ: non-return to zero has smallest occupied bandwidth but must constrain consecutive 0’s or 1’s Good: simple Bad: DC Good: Lowest BER Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 9 Figure 3–15 Binary signaling formats. Bipolar Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Line Codes: Power Spectral Density (PSD) ELE445-14 Lecture 18 20 10 PSD of Line Codes F( f ) 2 ∞ j2πkfTs Py ( f ) = ∑ R(k)e Ts k=−∞ l R(k) = ∑(anan+k )i Pi for random data i=1 Binary Signals: Ts =Tb Multilevel Signals: Ts = l Tb Pi is the probability of anan+k occurring R(k) example for a deterministic data file Data 1 1 0 1 0 Unipolar A= 0,1 1 1 0 1 0 Polar A= -1,1 1 1 -1 1 -1 Bipolar A = 0, (1,-1) 1 -1 0 1 0 1 N R(k) = ∑(anan+k ) N k=− N where N is the number of bits 11 R(k) example Polar , k=0 Data 1 1 0 1 0 an 1 1 -1 1 -1 an+0 1 1 -1 1 -1 2 (an) 1 1 1 1 1 2 (1/Ν)Σ(an) 1 1 N R(k) = ∑(anan+k ) N k=−N R(k) example Polar , k=1 Data 1 1 0 1 0 an 1 1 -1 1 -1 an+1 -1 1 1 -1 1 Shift Right 1 ana+1 -1 1 -1 -1 -1 (1/Ν)Σ ana+1 -0.6 See Mathcad file bpsk_psd.xmcd and linecodepsd.xmcd 12 R(k) example Polar , k=1 R(k) 1 k -0.6 PSD of Line Codes 13 PSD of Line Codes PSD of Line Codes Check the website for Matlab and mathcad files to plot the psd of line codes 14 Figure 3–16 PSD for line codes (positive frequencies shown). 2 A2T ⎛ sinπfT ⎞ ⎡ 1 ⎤ b ⎜ b ⎟ Punipolar NRZ ( f ) = ⎜ ⎟ ⎢1+ δ ( f )⎥ (3 - 39b) 4 ⎝ πfTb ⎠ ⎣ Tb ⎦ Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–16 PSD for line codes (positive frequencies shown). 2 ⎛ sinπfT ⎞ 2 ⎜ b ⎟ Ppolar NRZ ( f ) = A Tb ⎜ ⎟ (3 - 41) ⎝ πfTb ⎠ Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 15 Figure 3–16 PSD for line codes (positive frequencies shown). 2 A2T ⎛ sinπfT / 2 ⎞ ⎡ 1 n=∞ n ⎤ b ⎜ b ⎟ Punipolar RZ ( f ) = ⎜ ⎟ ⎢1+ ∑ δ ( f − )⎥ (3 - 43) 16 ⎝ πfTb / 2 ⎠ ⎣ Tb n=−∞ Tb ⎦ Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–16 PSD for line codes (positive frequencies shown). Bipolar 2 A2T ⎛ sinπfT / 2 ⎞ b ⎜ b ⎟ Pbipolar RZ ( f ) = ⎜ ⎟ ()1− cos(2πfTb (3 - 45) 8 ⎝ πfTb / 2 ⎠ Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 16 Figure 3–16 PSD for line codes (positive frequencies shown). 2 ⎛ sinπfT / 2 ⎞ 2 ⎜ b ⎟ 2 PManchester NRZ ( f ) = A Tb ⎜ ⎟ sin ()πfTb / 2 (3 - 46c) ⎝ πfTb / 2 ⎠ Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Eye Diagrams Clock Recovery Regenerative Repeater Spectral Efficiency EELE445-14 Lecture 19 34 17 Exam in Class EELE445-14 Lecture 20 35 EYE Diagrams Figure 3–18 Distorted polar NRZ waveform and corresponding eye pattern. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 18 EYE Diagrams Eye diagram of a 20 Gbps data stream The time scale is set at 10 picoseconds/division. 19 Figure 3–19 Regenerative repeater for unipolar NRZ signaling. •The ability to use a regenerative repeater is one of the major advantages of a digital binary system over an analog system Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–20 Square-law bit synchronizer for polar NRZ signaling. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 20 Figure 3–20 Square-law bit synchronizer for polar NRZ signaling. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–21 Early–late bit synchronizer for polar NRZ signaling. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 21 Figure 3–22 Binary-to-multilevel polar NRZ signal conversion. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 Figure 3–22 Binary-to-multilevel polar NRZ signal conversion. Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0 22 Spectral Efficiency Spectral Efficiency Definition: The spectral efficiency of a digital signal is given by the number of bits per second that can be supported by each hertz of bandwidth. R η = in ()bits / sec / Hz B where R is the data rate in bits per second and B is the bandwidth in Hz Spectral Efficiency Limits C ⎛ S ⎞ ηmax = = log2 ⎜1+ ⎟ (3 − 56) B ⎝ N ⎠ for multilevel signaling : η = l(bit / s) / Hz l is the number (3 − 57) of bits used in the A/ D • Shannon’s Law is the best we can do •We are a long way from this •Multiple bits/symbol get us closer (l- multilevel PCM) • η is in (bits/sec)/Hz 23 Spectral Efficiency Table 3-6 SPECTRAL EFFICIENCIES OF LINE CODES Code Type First null Bandwidth Spectral Efficiency (Hz) η=R/B Unipolar NRZ R 1 Polar NRZ R 1 Unipolar RZ 2R 1/2 Bipolar RZ R 1 Manchester NRZ 2R 1/2 Multilevel polar NRZ R/l l 24.
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