Pulse Code Modulation

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Pulse Code ModulationEE 442 – Spring SemesterLecture 9m(t)Analog signalPulse Amplitude ModulationPulse Width ModulationPulse Position ModulationPulse Code Modulation(3-bit 1--pulse-code-modulationPulse Code Modulation1

Advantages of Digital Over Analog For Communications1. Digital is more robust than analog to noise and interference†2. Digital is more viable to using regenerative repeaters3. Digital hardware more flexible by using microprocessors and VLSI4. Can be coded to yield extremely low error rates with error correction5. Easier to multiplex several digital signals than analog signals6. Digital is more efficient in trading off SNR for bandwidth7. Digital signals are easily encrypted for security purposes8. Digital signal storage is easier, cheaper and more efficient9. Reproduction of digital data is more reliable without deterioration10. Cost is coming down in digital systems faster than in analog systemsand DSP algorithms are growing in power and flexibility† Analog signals vary continuously and their value is affected by all levels of noise.Pulse Code Modulation2

Typical PCM Communication SystemPCM signaltransmittedto channelTransmitterDistorted PCMsignal fromimperfect channelRegeneratedPCM signal sentalong channelTransmission .com/pulse-code-modulation-and-demodulation/Pulse Code Modulation3

Analog to Digital Conversion Process (ADC) SampleAnalogSignalSamplingAnalog signalis continuousin time &litudeselects thedata pointswe use tocreate thedigital datatime timeamplitudetimeamplitudeamplitudeThree Step Process QuantizeEncodeCapturedQuantizedSampled DataSampledValuesDataQuantizingDiscretetime values:few amplitudesfrom analogsignalchooses theamplitudevalues usedto encodeNow havediscreteValues inboth time nalEncodingassigns binarynumbers tothoseamplitudevaluesNow have thedigitaldata whichis the finalresultNote: “Discrete time” corresponds to the timing of the sampling.Pulse Code Modulation4

Next Topic – Pulse Code ModulationPulse-code modulation (PCM) is used to digitally representsampled analog signals. It is the standard form of digital audioin computers, CDs, digital telephony and other digital audioapplications. The amplitude of the analog signal is sampled atuniform intervals and each sample is quantized to its nearestvalue within a predetermined range of digital levels.Four-bit coding(16 discrete levels)https://en.wikipedia.org/wiki/Pulse-code modulationPulse Code Modulation5

Second Step – Quantization IQuantization is the process of changing a continuous-amplitudesignal into on with discrete amplitudes.mpQuantized samples of mq(t)Allowed quantization levelsm(t) m pt2m pL After Lathi & Ding, 4th ed., 2009; page 6.L 16 levels 4 bitsMaximum value mp Pulse Code Modulation6

Four-Bit Binary Pulse Code (Example)Table 5.1 on page 249To communicatesampled values, wesend a sequence of bitsthat represents thequantized value.For 16 quantization levels,4 bits are required.PCM can use abinaryrepresentation ofvalue.The PSTN uses PCMAfter Lathi & Ding, 4th ed., 2009; page 8.Pulse Code Modulation7

Quantization IIWe start with a sampled signal {call it m(t)} and now we want to quantize it.The quantized amplitude is limited to a range, say from –mp to mp.(Note: the range of m(t) may extend beyond (-mp, mp) in some cases.)Divide the range (-mp, mp) into L uniformly spaced intervals. The numberintervals is L and the separation between quantized levels is kth2 mpLThesample point of m(t) is designated as m(kTS) and is assigned a valueequal to the midpoint between two adjacent levels. Define:m(kTS) kth sample’s value, andmq(kTS) kth quantized sample’s value.Then the quantization error q(kTS) is equal to mq(kTS) - m(kTS)Pulse Code Modulation8

Error Generated by Quantization (Quantization Noise)Quantization noise q(t ) m(t ) mq (t )After Haykin & Moher, 5th ed., 2009; page 258. Quantization fluctuation or “noise”Pulse Code Modulation9

Quantization IIIThe quantized levels are separated by 2 mpLThe maximum error for any sample point’s quantized value is at most ½ .The “time average” mean-square quantization error isq2 q2 mp23L2( ) 122.Let Nq q2. Thus, Nq is proportional to the fluctuation of the error signal.This is usually called the quantization noise power.Next we define m(t) mq(t) q(t).The signal (or message) power S0 is proportional to the mean square of m(t),S0 2 m2 (t ) ; and if m(t ) is sinusoidal, S0 Note:mp22denotes a time average.Pulse Code Modulation10

Quantization IVWe want a measure of the quality of received signal (that is, the ratio ofthe strength of the received signal power S0 relative to the strength of thenoise power Nq due to quantization).This is the Signal-to-Quantization Noise Ratio (SQNR) and is given by m 2 (t ) m 2 (t )S0 SQNR 3L2 22 N q mpmp 3L2 2 m (t )Note: mp2 It is usually expressed in decibels, S0SQNRdB 10 log 10 Nq 1 2 3L2 10 log 10 2 Conclusion:To reduce the quantization error relative to the message signal level, usesmaller quantization steps .Pulse Code Modulation11

The Dilemma of Strong Signals versus Weak SignalsStrong SignalWeak Signal(a) Linear encoding(b) With non-linear encodingLinearCompressedNote different encoding levels on each yer-part-2-data-encoding-techniquesPulse Code Modulation12

Use Compression and Expansion p://www.slideshare.net/91pratham/unit-ipcmvshPulse Code Modulation13

Companding Laws -Law Companding (North America)Output (y/ymax)Output (y/ymax)A-Law Companding (Europe)Input (m/mp)Input (m/mp)y m A 1 log e A mp y Am A 1 log e mp 1 log e A for 0 m 1 mp Afor1 m 1A mp m 1y log e 1 log e (1 )mp for 0 m 1mpAfter Lathi & Ding, 4th ed., 2009; page 327.Pulse Code Modulation14

A-law Compandor (Europe) Shown GraphicallyA law.phpPulse Code Modulation15

Flattening of the S/N Ratio Using the -LawFor optimal S0/Nq ratio in North America 255 is used.An approximately constant S0/Nq ratio is the most desirable. S0 Nq (8 bits)Relative signal power S0 (dB)After Lathi & Ding, 4th ed., 2009; page 328.Pulse Code Modulation16

Transmission BandwidthIn binary PCM, we have a group of n bits corresponding to L levels with nbits. Thus,L 2n or n log2(L)Signal m(t) is band-limited to B Hz, which requires 2B samples per second.For 2nB elements of information, we must transfer 2nB bits/second. Thus,the minimum bandwidth BT needed to transmit 2nB bits/second isBT nB HzPractically speaking, we usually choose the transmission bandwidth to bea little higher than the minimum bandwidth required.Pulse Code Modulation17

Summary of Binary PCM With Linear QuantizationGiven message signal m(t) with range from –mp to mp. For n levels thequantization step is given by 2 mpLThe mean square error (MSE) is found fromThe means signal power isNq q2 mp23L2( ) 212S0 2 m2 (t )The signal-to-noise ratio (SQNR) is m 2 (t )S0SQNR 3L2 mp2Nq m 2 (t ) 12 2 For binary PCM, L 2n ( with n bits/sample)For a message signal of bandwidth B Hz, then the Nyquist rateis 2B samples/second. Thus, the bit rate is 2nB bits/second with achannel bandwidth is now nB Hz.Pulse Code Modulation18

ExampleProblem: A band-limited signal m(t) of 3 kHz bandwidth is sampled at rate of33⅓ % higher than the Nyquist rate. The maximum allowable error in thesample amplitude (i.e., the maximum quantization error) is 0.5% of the peakamplitude mp. Assume binary encoding. Find the minimum bandwidth of thechannel to transmit the encoded binary signal.Solution:The Nyquist rate is RN 2 x 3000 Hz 6000 Hz (samples/second), but the actualrate is 33⅓ % higher, so the sample rate is 6000 Hz (⅓ x 6000) 8000 Hz.The quantization step is and the maximum quantization error is plus/minus /2. Hence, we can write mp 0.5 mp L 2002L 100For binary coding, L, must be a power of two; therefore, knowing that L 27 128 and 28 256, we must choose n 8 to guarantee better than a 0.5% error.Pulse Code Modulation19

Example ContinuedSolution (continued):Having chosen n 8 to guarantee 0.5% error, to find the bandwidth requiredwe start withTotal number of bits per second C 8 bits 8000 Hz 64,000 bits/secondHowever, we know we can transmit 2 bits/Hz of bandwidth¶, so it requires abandwidth BT ofBT C/2 32,000 Hz 32 kHzIf 24 such signals are multiplexed onto a single line (known as a T1 Line in theBell telephone system), thenCT1 24 x 64 kb/s 1.536 Mb/s; so the bandwidth 768 kHz ¶A maximum of 2B independent elements of information per second can betransmitted, error-free, over a noiseless channel of bandwidth B Hz.Pulse Code Modulation20

Exponential Increase of the Output SNR (S/N Ratio)We start with the SNR (signal-to-noise ratio) equation from slide 11 above: m 2 (t )S0 3 mp2Nq L2 denotes time averageThe number of levels L can be expressed as L2 22n where n log2(L) and isthe number of bits to generate L levels. The SNR can now be expressed as m 2 (t ) 2 nS0 3 2 mp2 ( )Nq Using the expression for bandwidth, BT nB, then we arrive at m 2 (t )S0 3 mp2Nq Taking the logarithm gives S0 Nq S0 10 log 10 Nq dB 2 B /B (2) T m 2 (t ) 10 log 10 3 mp2 10 2 n log 10 ( 2 ) ( 6n ) dB Pulse Code Modulation21

SNR ExampleGiven a sinusoidal modulating signal m(t) of amplitude Am into aload resistance R 1 ohm, find the signal-to-quantization noise ratio(sometimes called SNR):Am2Pave 2and set mmax AmSetting mmax Am m 2 (t )S0 3 mp2Nq S010 log 10 Nq 3P 3 (2)2 n 2ave (2)2 n (2)2 n 2 mmax ( 1.76 6n ) dB LnSNR32531.8 dB64637.8 dB128743.8 dB256849.8 dBPulse Code Modulation22

Bell System’s T1 Carrier System (1962)The T-carrier is a member of the group of carrier systems developed by AT&TBell Laboratories for digital transmission of multiplexed telephone calls usingPulse Code Modulation and Time Division Multiplexing.The first version, the Transmission System 1 (T1), was introduced in 1962 inthe Bell System, and could transmit up to 24 telephone calls simultaneouslyover a single transmission line consisting of copper wire.193 bit frame – 122 sec/frame1.544 Mbit/s data rateshttps://sun.iwu.edu/ jhaefner/CS390/Lecture9/lec9.htmPulse Code Modulation23

T1 Carrier – Time Division MultiplexingSynchronization is required between the transmitter and receiver.After Lathi and Ding, 4th edition, 2009; page 333.Pulse Code Modulation24

Comparison of T-Carrier (North America) and E-Carrier (Europe)Carrier LevelT-Carrier Data RatesE-Carrier Data RatesZero-level64 kbits/s (DS-0)64 kbits/sFirst-level1.544 Mbits/s (DS-1)T1 – 24 channels2.048 Mbits/s (E1)32 user channelsSecond-level6.312 Mbits/s (DS-2)T2 – 96 channels8.448 Mbits/s (E2)128 channelsThird-level44.736 Mbits/s (DS3)T3 – 672 channels34.368 Mbits/s (E3)512 channelsFourth-level274.176 Mbits/s (DS4)T4 – 4032 channels139.264 Mbits/s (E4)2048 channelsFifth-level400.352 Mbits/s (DS5)T5 – 5760 channels565.148 Mbits/s (E5)8192 channelsPulse Code Modulation25

Worked Example for PCMWe are given a signal m(t) 2 cos(2 250 t) as a single-tone signal input.(a) Find the SNR with 8-bit PCM.For 8-bit encoding, L 2n where n 8, therefore, the number of levels 256.The amplitude Am of the sinusoidal waveform means that mp 2 volts. Thetotal signal swing possible (- mp to mp) will be 2mp 4 volts, therefore,the average signal power is Pave [(Am)2/2] [22/2] 2 watts. (See slide 10)The interval [2mp/L] 4 volts/256 levels 1.5625 10-2 volt. (See slide 17)Now we can find the SNR (signal-to-quantized noise ratio) (See side 22)Using for the quantization noise Nq [ 2/12], and taking Pave 2 W, theSNR is given by SO Pave 2 24 12 98, 304 N q N q 2 (1.5625 10 2 )2 SNRdB 10 log 10 ( 98, 304 ) 49.93 dB Pulse Code Modulation26

Worked Example for PCM (continued)We are given a signal m(t) 2 cos(2 250 t) as the signal input.(b) If the minimum SNR is to be at least 36 dB, how many bits n are neededto encode the signal (i.e., find n)? Other parameters such as signal power remainthe same as in part (a) on previous slide.Note that 36 dB is numerically equivalent to 3,981.Remembering that the interval is [2mp /L] and 2mp 4 volts.3, 981 2 mp 2 442; 0.001005 and 0.0317 volt2 3981Therefore, we can determine the number of levels L and then find n.L 2 mp 4 31.50.0317The lowest integer number of bits n that will give at least 31.5 levels is n 5because 25 32 levels. So the answer is 5 bits.Pulse Code Modulation27

Differential Pulse Code Modulation (DPCM)PCM is not really efficient because it generates so many bits taking up a lotof bandwidth. Can we improve on this? YES.Suppose we have a slowly varying signal m(t), then we exploit this byusing the difference between two adjacent samples. This will form the basisof differential pulse code modulation (DPCM).Let m[k] be the kth sample reading of signal m(t).Then we can express the difference between two adjacent samples asd[k] m[k] – m[k-1]Principle: Instead of transmitting m[k], we transmit d[k].Pulse Code Modulation28

Differential Pulse Code Modulation (continued)At the receiver knowing d[k] and the previous value of m[k-1] allows us toconstruct the value of m[k].How do we benefit from doing this?The difference of successive samples almost always is much smaller thanthe full range of the sample values of m(t) (full range covers -mp to mp).We use this fact to improve upon the efficiency of PCM by requiringfewer bits.Furthermore, we can make use of the estimate of m[k], denoted by mest[k].We use previous sample values of m(t) to make this estimate.Suppose mest[k] is the estimate of the kth sample, then the difference d[k]is defined byd[k] m[k] – mest[k]and it is the difference d[k] that is transmitted.Pulse Code Modulation29

Differential Pulse Code Modulation (continued)Receiver Concept:At the receiver we determine the estimate mest[k] from previous samplevalues, and then generate m[k] by adding the received d[k] values to theestimate mest[k]. Thus, the reconstruction of the samples is doneiteratively.How do we carry out such an estimation?Pulse Code Modulation30

Digression on Signal PredictionStarting with a Taylor series (with time step Ts),d(m(t )) TS2 d 2 (m(t )) TS3 d 3 (m(t ))m[t TS ] m(t ) TS .dt2! dt 23! dt 3dm(t )m[t TS ] m(t ) TSfor small TSdtWe denote the kth sample of m(t) by m[k], that is, m[kTS] m[k], andm[kTS TS] m[k 1], and so on. This is a first-order predictor.In handling the derivatives, we writeThus,m( kTS ) m( kTS TS )d( m( kTS )) dtTS m[ k ] m[ k 1] m[ k 1] m[ k ] TS TS m[ k 1] 2 m[ k ] m[ k 1]So we get an approximation of the (k 1)th sample, m[k 1], from the twoprior samples, namely m[k] and m[k-1].Pulse Code Modulation31

Signal Prediction (continued)But we can do even better than this. In general,m[ k ] a1m[ k 1] a2 m[ k 2] . . . aN m[ k N ] mq [ k ]The set of {ai} are the predictor coefficients.This is the predicted value of m[k]. It is an Nth order predictor.Note that the input consists of the weighted previous samples m[k-1],m[k-2], etc. We say that input m[k] gives output mest[k].For a first-order prediction, mest[k] m[k-1].The next slide shows how to implement this prediction of m[k].Pulse Code Modulation32

Linear Predictor Implemented With Transversal Filtermest [ k ] a1m[ k 1] a2 m[ k 2] . . . aN m[ k N aNa3After Lathi and Ding, 4th edition,2009; page 343. Output mest[k]Transversal filter is a tapped delay line (with required weights {ai} )Pulse Code Modulation33

DPCM TransmitterInputm[k] Outputdq[k]d[k]Quantizer mest[k]4thAfter Lathi and Ding,edition, 2009; page 344. Predictormq[k]d[ k ] m[ k ] mest [ k ] and is quantized to yield,dq [ k ] d[ k ] q[ k ] where q[ k ] is the quantization errorThe predictor output mest[k] is fed back to the input so the predictor inputmq[k] is given bymq [ k ] mq [ k ] dq [ k ] m[ k ] d[ k ] dq [ k ] m[ k ] q[ k ]This shows that mq[k] is the quantized version of m[k].Pulse Code Modulation34

DPCM ReceiverInputdq[k] Outputmq[k] mest[k]PredictorThe receiver’s output (which is the predictor’s input) is also the same,mq[k] m[k] q[k].Hence, we are able to receive the desired signal m[k] plus the quantizationnoise, q[k]. It is important to note that from the difference signal d[k] ismuch smaller that the noise associated with m[k].After Lathi and Ding, 4th edition, 2009; page 344.Pulse Code Modulation35

DPCM SNR ImprovementHow much better is DPCM with regard to SNR?To determine this, define mp and dp as the peak amplitudes of m(t) andd(t), respectively. Assuming the same number of steps L for both, thenthe quantization step in DPCM is reduced in magnitude by dp/mp.The quantization noise is proportional to ( )2 – the quantization noisepower is reduced by a factor (dp/mp)2 and the SNR is therefore increased by(mp/dp)2.Maintaining the same SNR, the number of bits can be reduced.Example:The AT&T telephone system sometimes operates at 32 kbits/s(or even 24 kbits/s) when using DPCM. [The telephone system was initiallydesigned to use a 64 kbits/second data rate.]Pulse Code Modulation36

Adaptive Differential PCMAdaptive differential PCM (ADPCM) can further improve upon DPCM byIncorporating an adaptive quantizer (variable ) at encoding.The quantized prediction error dq[k]is a good measure of the predictederror size – it can be used to change which minimizes dq[k]. When dq[k]fluctuates around large positiveor negative values, the predictionerror is large and needs toincrease, but when dq[k] fluctuatesaround zero (small values), then needs to decrease.m[k] AdaptiveQuantizer dq[k]ToChannelnth orderPredictor4thAfter Lathi and Ding,edition, 2009; page 345.Example: An 8-bit PCM sequence can be encoded into a 4-bit ADPCMsequence at the same sampling rate. This reduces the channel bandwidthby one-half with no loss in quality.Pulse Code Modulation37

Example of Adaptive Differential PCM Output Waveform (Signal)time1100000101111101000011100010101 .Reference: ?Pulse Code Modulation38

stions/Next Topic is Delta ModulationPulse Code Modulation39

Illustrating PAM, PWM, PPM, PFM sand PCMPulseamplitudemodulationPulseduration -modulation-PAM-and-pulse-code-modulation-PCMPulse Code Modulation40

Comparing PCM, DM, ADM and -pcm-dm-adm-and-dpcm/Pulse Code Modulation41

Error Generated by Quantization (Quantization Error or Noise)https://en.wikipedia.org/wiki/Quantization (signal processing)Pulse Code Modulation42

Comparing Pulse Width Modulation with Pulse Frequency pic.php?id 21Pulse Code Modulation43

m-ppm/Pulse Code Modulation44

https://slideplayer.com/slide/7352323/Pulse Code Modulation45

Pulse-code modulation (PCM) is used to digitally represent sampled analog signals. It is the standard form of digital audio in computers, CDs, digital telephony and other digital audio appli

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