Eeng 360 1

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Chapter 3 PCM Noise and Companding

• Quantization Noise
• Signal to Noise Ratio
• PCM Telephone System
• Nonuniform Quantization
• Companding

Huseyin Bilgekul Eeng360 Communication Systems
I Department of Electrical and Electronic
Engineering Eastern Mediterranean University
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Quantization Noise

• The process of quantization can be interpreted as
  an additive noise process.
• The signal to quantization noise ratio (SNR)QS/N
  is given as

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Effects of Noise on PCM

• Two main effects produce the noise or distortion
  in the PCM output
• Quantizing noise that is caused by the M-step
  quantizer at the PCM transmitter.
• Bit errors in the recovered PCM signal, caused by
  channel noise and improper filtering.
• If the input analog signal is band limited and
  sampled fast enough so that the aliasing noise on
  the recovered signal is negligible, the ratio of
  the recovered analog peak signal power to the
  total average noise power is
• The ratio of the average signal power to the
  average noise power is
• M is the number of quantized levels used in the
  PCM system.
• Pe is the probability of bit error in the
  recovered binary PCM signal at the receiver DAC
  before it is converted back into an analog
  signal.

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Effects of Quantizing Noise

• If Pe is negligible, there are no bit errors
  resulting from channel noise and no ISI, the Peak
  SNR resulting from only quantizing error is
• The Average SNR due to quantizing errors is
• Above equations can be expresses in decibels as,

Where, M 2n a 4.77 for peak SNR a 0 for
average SNR
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DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS

• Assume that an analog audio voice-frequency(VF)
  telephone signal occupies a band from 300 to
  3,400Hz. The signal is to be converted to a PCM
  signal for transmission over a digital telephone
  system. The minimum sampling frequency is 2x3.4
  6.8 ksample/sec.
• To be able to use of a low-cost low-pass
  antialiasing filter, the VF signal is oversampled
  with a sampling frequency of 8ksamples/sec.
• This is the standard adopted by the Unites States
  telephone industry.
• Assume that each sample values is represented by
  8 bits then the bit rate of the binary PCM
  signal is

8

• This 64-kbit/s signal is called a DS-0 signal
  (digital signal, type zero).
• The minimum absolute bandwidth of the binary
  PCM signal is

This B is for a sinx/x type pulse sampling
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DESIGN OF A PCM SIGNAL FOR TELEPHONE SYSTEMS

• If we use a rectangular pulse for sampling the
  first null bandwidth is given by

• We require a bandwidth of 64kHz to transmit this
  digital voice PCM signal, whereas the bandwidth
  of the original analog voice signal was, at most,
  4kHz.
• We observe that the peak signal-to-quantizing
  noise power ratio is

• Note
• Coding with parity bits does NOT affect the
  quantizing noise,
• However coding with parity bits will improve
  errors caused by channel or ISI, which will be
  included in Pe ( assumed to be 0).

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Nonuniform Quantization

• Many signals such as speech have a nonuniform
  distribution.
• The amplitude is more likely to be close to zero
  than to be at higher levels.
• Nonuniform quantizers have unequally spaced
  levels
• The spacing can be chosen to optimize the SNR for
  a particular type of signal.

Output sample XQ
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Example Nonuniform 3 bit quantizer
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2
4
6
8
-2
-4
-6
-8
Input sample X
-2
-4
-6
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Uniform and Nonuniform Quantization
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Companding

• Nonuniform quantizers are difficult to make and
  expensive.
• An alternative is to first pass the speech signal
  through a nonlinearity before quantizing with a
  uniform quantizer.
• The nonlinearity causes the signal amplitude to
  be Compressed.
• The input to the quantizer will have a more
  uniform distribution.
• At the receiver, the signal is Expanded by an
  inverse to the nonlinearity.
• The process of compressing and expanding is
  called Companding.

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?-Law Companding

• Telephones in the U.S., Canada and Japan use
  ?-law companding
• Where ? 255 and x(t) lt 1

Output x(t)
Input x(t)
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Non Uniform quantizing

• Voice signals are more likely to have amplitudes
  near zero than at extreme peaks.
• For such signals with non-uniform amplitude
  distribution quantizing noise will be higher for
  amplitude values near zero.
• A technique to increase amplitudes near zero is
  called Companding.

Effect of non linear quantizing can be can be
obtained by first passing the analog signal
through a compressor and then through a uniform
quantizer.
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Example m-law Companding
xnspeech /song/
ynC(xn) Companded Signal
Close View of the Signal
Segment of xn
Segment of yn Companded Signal
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m-law Encoder Transfer Characteristics
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A-law and m-law Companding

• These two are standard companding methods.
• u-Law is used in North America and Japan
• A-Law is used elsewhere to compress digital
  telephone signals

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SNR of Compander

• The output SNR is a function of input signal
  level for uniform quantizing.
• But it is relatively insensitive for input level
  for a compander

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SNR Performance of Compander

• The output SNR is a function of input signal
  level for uniform quantizing.
• But it is relatively insensitive for input level
  for a compander.
• a 4.77 - 20 Log ( V/xrms) for Uniform
  Quantizer
• V is the peak signal level and xrms is the rms
  value
• a 4.77 - 20 logLn(1 µ) for µ-law
  companding
• a 4.77 - 20 log1 Ln A for A-law
  companding

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V.90 56-Kbps PCM Computer modem

• The V.90 PC Modem transmits data at 56kb/s from a
  PC via an analog signal on a dial-up telephone
  line.
• A µ law compander is used in quantization with a
  value for µ of 255.
• The modem clock is synchronized to the 8-ksample/
  sec clock of the telephone company.
• 7 bits of the 8 bit PCM are used to get a data
  rate of 56kb/s ( Frequencies below 300Hz are
  omitted to get rid of the power line noise in
  harmonics of 60Hz).
• SNR of the line should be at least 52dB to
  operate on 56kbps.
• If SNR is below 52dB the modem will fallback to
  lower speeds ( 33.3 kbps, 28.8kbps or 24kbps).