analog-to-digital conversion pam(pulse amplitude modulation) pcm(pulse code modulation)
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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