SQNR with and Without Companding

Home , 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

3 Line Codes: Baseband Digital Signals

ELE445-14 Lecture 17

PCM Signal Transmission

PCM Signal

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

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

Figure 3–12 Binary signaling (computed).

7 Figure 3–14 L = 4-level signaling (computed).

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

9 Figure 3–15 Binary signaling formats.

Bipolar

Line Codes: Power Spectral Density (PSD)

ELE445-14 Lecture 18

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)

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 ⎦

Figure 3–16 PSD for line codes (positive frequencies shown).

2 ⎛ sinπfT ⎞ 2 ⎜ b ⎟ Ppolar NRZ ( f ) = A Tb ⎜ ⎟ (3 - 41) ⎝ πfTb ⎠

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 ⎦

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 ⎠

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 ⎠

Eye Diagrams Clock Recovery Regenerative Repeater Spectral Efficiency

EELE445-14 Lecture 19

17 Exam in Class

EELE445-14 Lecture 20

EYE Diagrams

Figure 3–18 Distorted polar NRZ waveform and corresponding eye pattern.

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

Figure 3–20 Square-law bit synchronizer for polar NRZ signaling.

20 Figure 3–20 Square-law bit synchronizer for polar NRZ signaling.

Figure 3–21 Early–late bit synchronizer for polar NRZ signaling.

21 Figure 3–22 Binary-to-multilevel polar NRZ signal conversion.

Figure 3–22 Binary-to-multilevel polar NRZ signal conversion.

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