Analog-to-Digital ConversionAnalog-to-Digital Conversion

PAM(Pulse Amplitude Modulation)

PCM(Pulse Code Modulation)

PAM(Pulse Amplitude Modulation)PAM(Pulse Amplitude Modulation)

Conversion of analog signal to a pulse type signal where the amplitude of signal denotes the analog information

Two class of PAM signals Natural sampling (gating)

Easier to generate Instantaneous sampling

Flat-top pulse More useful to conversion to PCM

PAM with natural samplingPAM with natural sampling

W(t)

t

S(t)

t

Ts

Duty Cycle D=/Ts=1/3

Ws(t)

t

W(t)

S(t)

Ws(t)=W(t)S(t)

Analog bilateral switch

Spectrum of PAM Spectrum of PAM with natural samplingwith natural sampling

Spectrum of input analog signal Spectrum of PAM

D=1/3, fs=4B BT= 3fs = 12B

|W(f)|

-B Bf

1

|Ws(f)|

-3fs -2fs -fs -B B fs 2fs 3fs

D=1/3 sin fD

f

sin( )s

n

fD W f nf

f

PAM with flat-top samplingPAM with flat-top sampling

W(t)

t

S(t)

t

Ws(t)

t

Ts

Sample and Hold

Spectrum of PAM Spectrum of PAM with flat-top samplingwith flat-top sampling

Spectrum of Input Spectrum of PAM

/Ts=1/3, fs=4B BT= 3fs = 12B

|W(f)|

-B Bf

1

|Ws(f)|

-3fs -2fs -fs -B B fs 2fs 3fs

D=1/3 sin

s

f

T f

1( ) ( )s

ns

H f W f nfT

Summary of PAMSummary of PAM

Require very wide bandwidth Bad noise performance

Not good for long distance transmission Provide means for converting a analog signal to

PCM signal Provide means for TDM(Time Division Multiplexing)

Information from different source can be interleaved to transmit all of the information over a single channel

PCM(Pulse Code Modulation)PCM(Pulse Code Modulation)

Definition PCM is essentially analog to digital conversion of a

signal type where the information contained in the instantaneous samples of an analog signal is represented by digital words in a serial bit stream

Analog signal is first sampled at a rate higher than Nyquist rate, and then samples are quantized Uniform PCM : Equal quantization interval Nonuniform PCM : Unequal quantization interval

Why PCM is so popular ?Why PCM is so popular ?

PCM requires much wider bandwidth But,

Inexpensive digital circuitry PCM signal from analog sources(audio, video, etc.) may be

merged with data signals(from digital computer) and transmitted over a common high-speed digital communication system (This is TDM)

Regeneration of clean PCM waveform using repeater. But, noise at the input may cause bit errors in regenerated PCM output

signal The noise performance is superior than that of analog

system. Further enhanced by using appropriate coding techniques

PCM transmitter/receiverPCM transmitter/receiver

LPFBW=B

Sampler& Hold

QuantizerNo. of levels=M

Encoder

Analogsignal

BandlimitedAnalog signal

Flat-topPAM signal

QuantizedPAM signal

PCMsignal

Channel, Telephone lines with regenerative repeater

DecoderPCMsignal

QuantizedPAM signal

ReconstructionLPF

AnalogSignaloutput

Waveforms in PCMWaveforms in PCMUniform quantizer

Waveform of signals

Error signals

PCM signal

PCM word

EncoderEncoder

Usually Gray code is used Only one bit change for each step change in

quantized level Single errors in received PCM code word will

cause minimum error if sign bit is not changed In text, NBC(Natural Binary Coding) is used Multilevel signal can be used

Much smaller bandwidth than binary signals Requires multilevel circuits

Uniform PCMUniform PCM

Let M=2n is large enough

Xmax

-Xmax

x

=2Xmax/Mx

Uniform distribution

ix

Distortion

x

ix

-/2 /2

2

2

1

12

12

i

M

ii

DM

D D

SQNR of PCMSQNR of PCM

Distortion

SQNR Let normalized input :

2max2 2 22max max max

2 2

2( )

12 12 3 3(2 ) 3(4 )n n

xx x xMDM

2

max

[ ]E XX

x

2 2 2 22

max max

[ ] 3 [ ] 3(4 ) [ ]3(4 )

nnE X M E X E X

SQNR XD x x

210 1010log 4.77 6.02 10log

dBSQNR SQNR n X

_4.77 6.02

dB pkSQNR n

Bandwidth of PCMBandwidth of PCM

Hard to analyze because PCM is nonlinear Bandwidth of PCM

If sinc function is used to generate PCM , where R is bit rate

If rectangular pulse is used , first null bandwidth

If fs=2B (Nyquist sampling rate) Lower bound of BW: In practice, is closer to reality

1 1

2 2PCM sB R nf

PCM sB R nf

PCMB nB

1.5PCMB nB

Performance of PCMPerformance of PCM

QuantizerLevel, M2481632641282565121024204840968192163843276865536

n bitsM=2n

12345678910111213141516

Bandwidth>nB2B4B6B8B10B12B14B16B18B20B22B24B26B28B30B32B

SQNR|dB_PK

4.8+6n10.816.822.828.934.940.946.952.959.065.071.077.083.089.195.1101.1

PCM examplesPCM examples

Telephone communication Voice frequency : 300 ~ 3400Hz

Minimum sampling frequency = 2 x 3.4KHz = 6.8KHz In US, fs = 8KHz is standard

Encoding with 7 information bits + 1 parity bit Bit rate of PCM : R = fs x n = 8K x 8 = 64 Kbits/s Buad rate = 64Ksymbols/s = 64Kbps

Required Bandwidth of PCM If sinc function is used: B > R/2 = 32KHz If rectangular is used: B = R = 64KHz

SQNR|dB_PK = 46.9 dB (M = 27) Parity does not affect quantizing noise but decrease errors caused by ch

annels

PCM examplesPCM examples

CD (Compact Disk) For each stereo channel

16 bit PCM word Sampling rate of 44.1KHz Reed-Solomon coding with interleaving to correct burst

errors caused by scratches and fingerprints on CD High quality than telephone communication

HomeworkHomework

Illustrative Problems 4.9, 4.10, 4.11, 4.12

Problems 4.14

Nonuniform quantizationNonuniform quantization

Example: Voice analog signal Peak value(1V) is less appears while weak

value(0.1V, 20dB down) around 0 is more appears (nonuniform amplitude distribution)

Thus nonuniform quantization is used Implementation of nonuniform quantization

Compression(Nonlinear)

filter

PCM withUniform

Quantization

AnalogInput

PCMoutput

Nonuniform QuantizationNonuniform Quantization

Two types according to compression filter -law : used in US

See Figure 4.9, Page 155 A-law : used in Europe

ln(1 )sgn( )

ln(1 )

xy x

1sgn( ), 01 ln1 ln( ) 1sgn( ), 1

1 ln

A xx x AAy

A xx xAA

Nonuniform QuantizationNonuniform Quantization

Compandor = Compressor + Expandor Compressor: Compression filter in transmitter Expander: Inverse Compression filter in receiver

-law : SQNR

Uniform quantizing: -law: A-law:

(1 ) 1sgn( )

y

x y

6.02dB

SQNR n 2

104.77 10log X

104.77 20log (ln(1 ))

104.77 20log (1 ln )A

HomeworkHomework

Illustrative Problems 4.13, 4.14

Problems 4.17