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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              Eeng 360 1
Quantization Noise
 The process of quantization can be interpreted as an additive noise
process.
Signal                    Quantized Signal
X                             XQ

Quantization Noise
nQ

•   The signal to quantization noise ratio (SNR)Q=S/N is given as:

Average Power{ X }
( SNR )Q 
Average Power{nQ }

Eeng 360 2
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.

Eeng 360 3
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
α = 4.77 for peak SNR
α = 0 for average SNR

Eeng 360 4
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

R nf s
BPCM    
2  2
This B is for a sinx/x type pulse sampling

Eeng 360 5
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:
1.   Coding with parity bits does NOT affect the quantizing noise,
2.   However coding with parity bits will improve errors caused by channel
or ISI, which will be included in Pe ( assumed to be 0).
Eeng 360 6
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      6

4

2       Example: Nonuniform 3 bit quantizer

-8   -6   -4   -2        2    4    6      8

-2
Input sample
X
-4

-6

Eeng 360 7
Uniform and Nonuniform Quantization

Eeng 360 8
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.

Eeng 360 9
-Law Companding

1

• Telephones in the U.S., Canada and
Japan use -law companding:
Output |x(t)|

ln(1   | x (t )|)
| y (t ) |
ln(1   )
– Where  = 255 and |x(t)| < 1

0                    1
Input |x(t)|

Eeng 360 10
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.

x                  x’     x’                 y
Q(.)
C(.)
Compressor          Uniform Quantizer

Eeng 360 11
Example: -law Companding
1

0.5

x[n]=speech /song/     0

-0.5

-1
0   1000    2000   3000    4000   5000   6000    7000   8000    9000   10000

1

0.5

y[n]=C(x[n])           0

-0.5
Companded Signal       -1
0   1000    2000   3000    4000   5000   6000    7000   8000    9000   10000

1

0.5
Close View of the Signal
Segment of x[n]       0

-0.5

-1
2200       2300      2400       2500    2600    2700      2800       2900    3000

1

0.5
Segment of y[n]       0

Companded Signal    -0.5

-1
2200       2300      2400       2500    2600    2700      2800       2900    3000

Eeng 360 12
-law Encoder Transfer Characteristics

Eeng 360 13
A-law and -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

Eeng 360 14
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

Eeng 360 15
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.

• α = 4.77 - 20 Log ( V/xrms)                 for Uniform Quantizer
V is the peak signal level and xrms is the rms value

• α = 4.77 - 20 log[Ln(1 + μ)]                for μ-law companding
• α = 4.77 - 20 log[1 + Ln A]                for A-law companding

Eeng 360 16
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).

Eeng 360 17

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