Pulse Code Modulation

Pulse Code ModulationPCM is a method of converting an analog signal into a digital signal. (A/D conversion)The amplitude of Analog signal can take any value over a continuous range i.e. it can take on an infinite values.Digital signal amplitude can take on finite values.Analog signal can be converted into digital by sampling and quantizing.Cont.

Cont.The amplitude of analog signal m(t) lie in the range (-mp, mp) and is partitioned into L sub-intervals each of magnitude 2mp/L

Binary pulse codes

QuantizingDigital signals come from variety of sources e.g. computerSome sources are analog but are converted into digital form by variety of techniques such as PCM and DMFor quantizing , we limit the amplitude of m(t) to a range(-mp, mp) as shown in the previous slides This amplitude is uniformly divided into L subintervals and each interval is ,

Cont.A sample value is approximated by the mid point of the intervalThe quantized samples are coded and transmitted as binary pulsesAt the receiver some pulses will be detected incorrectlyThere are two types of errorsQuantization errorPulse detection errorCont.In almost all practical schemes, the pulse detection error is very small compared to the quantization error and can be ignoredNow we analyze the quantization errorCont.

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Cont.The integral of the cross product terms is zero and we obtain,

Because the sampling rate is 2B, hence the total number of samples over the averaging interval is 2BTThis is called the mean of the quantization error

Cont.The quantized levels are separated by 2mp/LSince sample value is approximated by the midpoint of the subinterval in which the sample fallsThe maximum quantization error is The mean square quantizing error is

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Non-uniform quantizationSNR is an indication of the quality of the received signalIdeally we would like to have constant SNRUnfortunately, the SNR is directly proportional to the signal power, which varies from talker to talkerThe signal power can also vary because of the connecting circuitsSNR vary even for the same talker, when the person speaks softlySmaller amplitudes pre-dominate in speech and larger amplitude much less frequent.This means the SNR will be low most of the timeCont.The root of this difficulty is that the quantization steps are of uniform valueThe quantization noise is directly proportional to the square of the step size.The problem can be solved by using smaller steps for smaller amplitudes as shown in fig. on the next slideCont.

Cont.The same result can be obtained by first compressing a signal and then using uniform quantizationThe input-output characteristics of compressor are shown in fig.

Cont.The horizontal axis is normalized input signal and the vertical axis is the output signal y.The compressor maps the input signal into larger increments Hence the interval delta(m) contains large number of steps when m is smallThe quantization noise is small for smaller input signalThus loud talker and stronger signals are penalized with higher noise steps in order to compensate the soft talker and weak signalsCompression LawsThere are two laws regarding compressions(1)This law is used in North America and Japan

(2) A-LawThis law is used in Europe and the rest of the word

Cont.The compressed samples are restored to their original values at receiver by using an expanderThe compressor and expander together are called compandor.Compression of a signal increases its bandwidth but in PCM, we are not compressing the signal but its samples the number of samples does not change, therefore bandwidth does not riseWhen meu-law compandor is used then output SNR is

Transmission BW and output SNRFor binary PCM, we assign distinct group of n binary digits to each of the L quantization levels

Each quantized level is encoded into n-bitsMinimum channel BW isThis is the theoretical minimum transmission bandwidth required to transmit the PCM signal

Example 6.2A signal m(t) band-limited to 3kHz is sampled at a rate 33.33% higher than Nyquist rate, a maximum acceptable error in the sample amplitude is 0.5% of the peak amplitude. The quantized samples are binary coded. Find the minimum channel BW required to transmit the coded signal. If 24 such channels are time-division multiplexed, determine the minimum transmission BW required to transmit the multiplexed signalSolution

Exponential Increase of output SNR

SNR in decibel scale

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Example 6.3

Comments on Logarithmic UnitsVery small and very large values are expressed in logarithmic units

T1 carrier systemA schematic of T1-system is shown in fig.

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Differential Pulse Code Modulation

Taylor's series

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Analysis of DPCM

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Delta Modulation

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Delta Modulator

Delta Demodulator

Delta Modulator output

Working of DM