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Analog-to-Digital Conversion

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					Analog-to-Digital Conversion

  PAM(Pulse Amplitude Modulation)
    PCM(Pulse Code 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 sampling
 W(t)                          Ws(t)


                    t                                  t



 S(t)                              Analog bilateral switch
            Ts
                                                 Ws(t)
                            W(t)
                                                  =W(t)S(t)
                        t
Duty Cycle D=/Ts=1/3                   S(t)
       Spectrum of PAM with natural sampling
                                                               |W(f)|
          Spectrum of input analog signal
          Spectrum of PAM
                                                           1
              D=1/3, fs=4B
                                                                                 f
              BT= 3fs = 12B                                   -B     B
                                          
                                                     sin  f
                                                D
                                                        f
                                                              W ( f  nf s )
                               |Ws(f)|   n 


                                                                sin  f
                     D=1/3                                  D
                                                                   f


-3fs       -2fs       -fs      -B   B      fs                   2fs        3fs
PAM with flat-top sampling
W(t)                Ws(t)


            t                          t

                        
S(t)                        Ts


                     Sample and Hold
                t
       Spectrum of PAM with flat-top sampling
                                               |W(f)|
          Spectrum of Input
          Spectrum of PAM
              /Ts=1/3, fs=4B             1
                                                                       f
              BT= 3fs = 12B
                                               -B      B
                                                        
                                           1
                                 |Ws(f)|   Ts
                                              H( f )    W ( f  nf )
                                                       n 
                                                                           s


                                                                sin  f
                     D=1/3
                                                               Ts  f


-3fs       -2fs       -fs    -B       B        fs                2fs           3fs
Summary 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)
   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 ?
   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/receiver
                    Bandlimited                   Flat-top
Analog     LPF      Analog signal     Sampler     PAM signal
signal    BW=B                         & Hold

                                                   Quantizer
                          Encoder
                 PCM                Quantized    No. of levels=M
                 signal             PAM signal
         Channel, Telephone lines with regenerative repeater

                                                 Reconstruction
                          Decoder
                 PCM                                  LPF
                                    Quantized
                 signal             PAM signal            Analog
                                                          Signal
                                                          output
  Waveforms in PCM
  Uniform quantizer

                      Error signals




Waveform of signals
                       PCM signal
                                PCM word
Encoder
   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 PCM
                                              Uniform
                                              distribution
    Let M=2n is large enough
Xmax         =2Xmax/M
                                                   x

                                       xi
                                              Distortion
                                                     2
                                               Di 
                                                    12M
                                                       M
                                                          2
                                               D   Di 
                                                   i 1   12
-Xmax                                              x
                                -/2         /2
         x                              xi
SQNR of PCM
   Distortion
                2 xmax 2
                   (  )
             2            2
                           xmax      2
                                    xmax       2
                                              xmax
        D     M             2
                                      n 2
                                           
           12     12       3M      3(2 )     3(4n )

   SQNR
                                            E[ X 2 ]
       Let normalized input : X  xmax
              E[ X 2 ] 3M 2 E[ X 2 ] 3(4n ) E[ X 2 ]
        SQNR                                        3(4n ) X 2
                 D         xmax           xmax

       SQNR dB  10log10 SQNR  4.77  6.02n  10log10 X 2
            SQNR dB _ pk  4.77  6.02n
Bandwidth of PCM
   Hard to analyze because PCM is nonlinear
   Bandwidth of PCM
       If sinc function is used to generate PCM
                      1    1
            BPCM      R  nf s     , where R is bit rate
                      2    2
       If rectangular pulse is used
             BPCM  R  nf s      , first null bandwidth
       If fs=2B (Nyquist sampling rate)
            Lower bound of BW: BPCM  nB
            In practice, BPCM  1.5nB is closer to reality
Performance of PCM
  Quantizer   n bits   Bandwidth   SQNR|dB_PK
  Level, M    M=2n     >nB         4.8+6n
  2           1        2B          10.8
  4           2        4B          16.8
  8           3        6B          22.8
  16          4        8B          28.9
  32          5        10B         34.9
  64          6        12B         40.9
  128         7        14B         46.9
  256         8        16B         52.9
  512         9        18B         59.0
  1024        10       20B         65.0
  2048        11       22B         71.0
  4096        12       24B         77.0
  8192        13       26B         83.0
  16384       14       28B         89.1
  32768       15       30B         95.1
  65536       16       32B         101.1
PCM 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
             channels
PCM 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
Homework
   Illustrative Problems
       4.9, 4.10, 4.11, 4.12
   Problems
       4.14
Nonuniform 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        PCM with
Analog                                       PCM
             (Nonlinear)        Uniform
Input                                        output
                filter        Quantization
Nonuniform Quantization
   Two types according to compression filter
       -law : used in US
                   ln(1   x )
         
              y                  sgn( x)
                    ln(1   )

            See Figure 4.9, Page 155
       A-law : used in Europe
               Ax
                        sgn( x),      0 x  1
               1  ln A                        A
             y
               1  ln( A x ) sgn( x), 1  x  1
                1  ln A
                                       A
Nonuniform Quantization
   Compandor = Compressor + Expandor
       Compressor: Compression filter in transmitter
       Expander: Inverse Compression filter in receiver
             -law : x    (1   )  1
                                    y
         
                                          sgn( y )
                                
   SQNR
        SQNR dB    6.02n
            Uniform quantizing:   4.77  10log10 X 2
            -law:   4.77  20log10 (ln(1  ))
            A-law:   4.77  20log10 (1  ln A)
Homework
   Illustrative Problems
       4.13, 4.14
   Problems
       4.17

				
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posted:9/20/2011
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