COMMUNICATION SYSTEM EEEB453Chapter 5 (Part II)

DIGITAL TRANSMISSION

Intan Shafinaz MustafaDept of Electrical Engineering

Universiti Tenaga Nasionalhttp://metalab.uniten.edu.my/~shafinaz

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PCM Quantization

Quantization is a process of rounding off the amplitudes of flat-top samples to a manageable number of levels.

With quantization, the total voltage range is subdivided into smaller number of sub ranges.

Called folded binary code – mirror image

Table1 shown a PCM code with a three-bit sign magnitude together with eight possible combinations.

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PCM Quantization

Magnitude difference between adjacent steps is called quantization interval or quantum or step size or resolution.

Resolution = the magnitude of the minimum step size or, = magnitude of Vlsb .

From the Table 1, the quantization interval = 1V The smaller the magnitude of the minimum step size, the

better (smaller) the resolution and the more accurate the quantized signal will resembles the original signal.

Quantization noise, Qn ≈ Quantization error, Qe is due to

any round-off errors (quantization) in the transmitted signal, and the error would be reproduced at the Rx.

Mathematically, Qn,e = ½ quantum = Vlsb/2

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(a) Analog input signal (b) sample pulse (c) PAM signal (d) PCM code

At t1, Vi = +2V, PCM code = 110, Qe = 0

At t2, Vi = -1V, PCM code = 001, Qe = 0

At t3, Vi = +2.6V, magnitude of the sample is rounded off to the nearest valid code, i.e +3V, PCM code = 111. The rounding off process results in a quantization error, Qe = 0.4V

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Amplitudes of the signal m(t) lie in the range (-mp.mp), which is partitioned into L intervals.

Then each magnitude, v = 2mp /L

Where L = 2n, and n = number of bits.

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An example of 3-bit quantization.

An example of 2-bit quantization.

An example of 3-bit quantization with increased sample rate.

The quality of the PCM signal can be improved by:

i. Using more bits for PCM code

ii. Reduce the magnitude of quantum

iii. Improve the resolution

iv. Sampling at higher rate

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Dynamic Range (DR)

Define as the ratio of largest possible magnitude to the smallest possible magnitude (other than 0V) that can be decoded by DAC in the Rx.

Mathematically, =

where Vmax = max voltage magnitude, Vmin = resolution (quantum value)

Dynamic range is generally expressed in dB, therefore,

DRdB = 20log

min

max

V

VDR

resolution

Vmax

min

max

V

V

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Dynamic Range (DR)

The number of bits used for a PCM code depends of the dynamic range i.e

2n – 1 ≥ DR and for a minimum number of bits

2n – 1 = DR or 2n = DR + 1

where n = number of bits in a PCM code, excluding the sign bit

DR = absolute value of dynamic range

Then DRdB = 20log (2n – 1),

and for n > 4,

DRdB ≈ 20log (2n) ≈ 6n,

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Coding Efficiency

Coding efficiency is a numerical indication of how efficiently a PCM code is utilized.

It is the ratio of the minimum number of bits required to achieve a certain dynamic range to the actual number of PCM bits used.

Mathematically,

Coding efficiency 100

bit)sign (including bits ofnumber actual

bit)sign (including bits ofnumber minimum

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Example 3 - For a PCM system with the following parameters, Maximum analog input frequency = 4kHz Maximum decoded voltage at the Rx = 2.55V Minimum dynamic range = 46dB, determine:

a. minimum sample rateb. minimum number of bits used in the PCM codec. resolutiond. quantization errore. coding efficiency

Example 4 – In a digital PCM system, the maximum quantization error is 0.6% of the peak amplitude of the modulating signal of 10 Khz. Determine:

a. The number of quantization levelsb. The number of bits per samplec. Total number of samplesd. Total number of bits.

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Signal to Quantization Noise Ratio (SQR) Linear codes – the magnitude change between any two successive

codes is the same i.e the quantum/quantization interval is equal, thus the magnitude of the quantization errors are also equal.

Recall, maximum quantization noise, Qe = ½ quantum = Vlsb/2,

Then worst-case (minimum) voltage SQR (occurs when input signal is at its minimum amplitude) is

Maximum SQR occurs at the maximum signal amplitude, i.e

From previous example,

22min

lsb

lsb

e V

V

Q

resolutionSQR

dBSQR dB 62log20min,

65.0

3maxmax

eQ

VSQR

dBSQR dB 6.156log20max,

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Signal to Quantization Noise Ratio (SQR) From the example, even though the magnitude of the Qe remains constant, the

percentage of error decreases as the magnitude of the sample increase.Thus, SQR is not constant.

For linear PCM code i.e all quantization intervals have equal magnitude, SQR or SNR is defined as

Generally,SQRdB = 10.8 + 20 log v/q

where v = rms signal voltageq = quantization interval

or SQRdB = 6.02n + 1.76 where n = no. of bits

powernoiseonquantizatiavg

powersignalavgSQRdB log10

Rq

RvSQRdB

/12log10

2

2

12log10

2

2

q

v (assume equal R)

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Linear versus Nonlinear PCM Codes. For linear coding, accuracy of the higher amplitude analog signal is the

same as for the lower amplitude signal.

SQR for lower amplitude signal is less than the higher amplitude signal.

For voice transmission, low amplitude signals are more likely to occur than large amplitude signals.

Thus a nonlinear encoding is the solution.

With non-linear coding, the step size increases with the amplitude of the input signal.

Nonlinear encoding gives larger dynamic range.

SQR is sacrificed for higher amplitude signals to achieve more accuracy for the lower amplitude signals.

However, it is difficult to fabricate nonlinear ADC.

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SQR at lower amplitude < SQR at higher amplitude

Lower amplitude values are relatively more distorted

More of voice signal is at lower amplitude

Reduce step size at lower amplitude

More accuracy at lower amplitude

Sacrifices SQR at higher amplitude

Provides higher dynamic range