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Rasaki Badaru

Pulse Code Modulation and Companding

(Voice Communications – TEL500)

(Fall 2010)

Supervised by: Prof. John Marsh

Written by: Rasaki Badaru

Introduction Companding



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The project is to demonstrate the process of digitization of analog voice

signal using Pulse Code Modulation process. To perform this demonstration, spread

sheet application is used to simulate voice signal to be used, define and compute

various functions used at different stages of the process demonstration. The

graphical function of the program is used to depict visual comparison and analysis

of results from the demonstration.

Pulse Code Modulation is the process of converting analog voice signal into

digital form that can be easily transmitted using either digital or analogue

transmission method. The PCM process starts with a sampling stage that discretized

the analog voice signal into a series of sampled values that is quantized based on a

predefined interval. The quantized values are then converted to binary digits to

form a digitized signal that is transmitted. The digitized signal is easier to multiplex

and less susceptible to noise problem than the original signal.

Companding is an intermediary processing stage that is used to optimize the

PCM process. It is a technique that transforms a given raw analog signal into anintermediary form that is more suitable for sampling when the original signal

composes of both low and high signal power. Companding is carried out in two

stages, compression and expansion. The original signal is first compressed using a

specific companding algorithm. The compressed output signal is sampled and

quantized and then expanded back for reconstruction. Companding is used to avoid

poor quantization of signal with low power content. This prevents loss of data and

misrepresentation of information.



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Procedure

The spread sheet application is used to define a sine wave to be used as a

working analogue voice signal. It is a time varying function with the time range of

0ms to 200ms with a step of 1ms with amplitude of 1. Though for some certain

section in the lab, the amplitude of the voice signal will be varied in step of .05 atsome stage of the lab. The sine wave signal is used as the best representative of an

analog voice signal because it has a similar characteristic features. The sine

function to be defined is as followed:

yt=A*sin(2πft+ θ) ,

where A is the amplitude, f is the frequency and t is the time with phase angle that

is always equal to zero.

For the time range, the signal level of the analog voice signal will be

computed and stored in a cell column. The signal level, ranging from -1 to 1, will bemapped onto a new interval based on a mapping function that will be defined in the

spread sheet. This mapping process is analogous to the sampling stage of the PCM.

Two forms of mapping will be performed: static and variable range mapping. In the

static range mapping, the upper and lower bound of the mapping function are

specified as number values while in the variable range mapping, the lower bound of

the mapping function is set to 0.5 while the upper bound value is based on a

function N+0.5 with variable N. The values from the mapping process will be the

new signal levels that will be processed to a digitized voice signal.

The next stage of the exercise is a companding process that is introduced to

enhance the sampling signal for a much better sampling points. Mu law compandingalgorithm will be used in the exercise. The process is in two stages: expansion and

compression. The mu-law compading composes of compression and expansion

functions that are defined with the inbuilt functions of the spread sheet. The

computed values from the compression function will now be used for the PCM

process and the final quantized values are expanded with the expansion function to

form the signal levels for digitized signal.

The computed results in various columns will be used to investigate the

digital voice signal vis-a-vis the analog voice signal. Some of the properties to be

investigated are amplitude effect on quality of a digitized signal, signal-to-quantized

noise ratio (SQR), the sampling bit depth and the companding practice. Graphical

plots of the various column results will be drawn to give a pictorial analysis of the

process and comparison.



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Pulse Code Modulation

The signal power of analog voice signal attenuated during transmission. The

loss of power makes it very impossible to detect the attenuated signal level in a

very noisy situation. In order to boost the analog signal power, amplifiers are used

at some predefined interval along the line of transmission. However, the amplifier

does not only increase the signal, but also increases the noise content of the

transmitting signal.

Also, analog signal transmission is known to be prone to noise generated in

the transmitting line and environment. When the signal source is very far away from

the receiving end, the noise power that builds up along the line of transmission

could rise to an equal level to the transmitting analog signal. Because of this, thesignal level will be hard to detect and the information with the signal is

misinterpreted. The noise content is very difficult to remove from the transmitting

analog signal and the only alternative to counteract the build up noise is to increase

the signal. But this approach as mentioned above compromised further the fidelity

of the transmitting analog signal.

Because of the problems associated with the analog voice signal

transmission, a digital voice transmission is developed. In digital voice transmission,

the transmitting analog voice signal is converted to a digital form that is

transmitted instead. Instead of a continuous rise and fall of signal level found in

analogue voice signal, the digitized voice signal has a signal level that is discretized

in time and value represented with binary digits.

The digitized voice signal is less prone to noise compare to analog signal

because the signal levels in the form of the binary digits can always be replace with

signal power that will always be above the prevailing noise power. Thus, digitized

voice signal is inherently immune to noise that builds along the line of transmission.

Also when noise builds up in the transmitting digitized voice signal, it does so

evenly across the spectrum of the transmitting signal as an additive to the signal

level whereby the actual signal level can easily be recovered.

During a long distance transmission that attenuated the transmitting digitized

voice signal power, repeaters are used to amplify the transmitting digitized voice

signal. Amplification is done by reconstructing the digitized signal. The process

involves extraction of unwanted signal contents and rebuilding of the transmitting

digitized signal. Unlike the analogue amplification process that amplifies both the

noise and the signal levels, the digital signal is amplified without the noise content.

Because of inherent immunity to noise, clean signal amplification and flexibility to



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multiplex, voice signal is predominantly preferred to be transmitted in digitized

form.

Pulse Code Modulation process is used to transformed analog voice signal

into digitized voice signal. It is implemented in three stages: sampling, quantization

and coding. The sampling stage is a very important part of the process because itforms the signal elements that determine the quality of digitized voice signal that is

generated at the end of the process. Sampling rate is the amount of sampled values

from the analog signal level taken in a second. It determines the quality of the voice

signal reconstructed from the sampled values. Based on Nyquist theorem, the

sampling rate must be twice the highest frequency component of the analog signal

for reconstruction of signal to be free of alliance effect and be of good quality. It is

the minimum optimal rate for sampling.

The analog voice signal consists of a frequency range between 0 Hz to 4 kHz

that combines together to form the continuous analog signal in the time domain. At

a rate twice the highest frequency component - 4 kHz - sample values of the voicesignal are taken for digital transformation. Thus, the ideal sampling rate for voice

signal is 8 kHz. The sampled values at this rate will yield a good reproduction of the

original signal. However, far more voice reproduction could be achieved with

sampling rate above 8 kHz. Thus, the sampling stage discretized the analog voice

signal in time with series of continuous sampled at the rate 8000 samples per

second.

To ensure that the sampled values are always above the noise level, each of

the continuous value is mapped onto an integer interval to form discrete set of

value. This mapping process is known as quantization. During the quantization

process, a mapping function is used to map each sampled value to a nearestinteger in a predefined interval. The mapping function is many-to-one type of

function; more than one continuous sampled value could be mapped onto an

integer. At the end of the process, new set of values called quantized values are

generated. These values are discrete in time and in value unlike the sampled that is

only discrete in time. Due to the fact that some of the sampled values are rounded

up to an integer, the signal reproduce from the quantized values are less in

dynamism when compared to the one from sampled values. In order to minimize

the effect of this rounding off, the range of the mapping interval is always chosen to

be as high as possible. The number of possible mapping that can be done within an

interval is referred to as quantization levels.

The next step taken is the representation of each quantized values with

equivalent binary digits in 1s and 0s. Equal number of binary digits is used to

represent each quantized value and the number digits represent a binary number is

referred to as the bit depth. The higher the quantization levels the more the bit

depth. The two digits constituting the binary number are indicated with two distinct



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voltage levels that are always chosen to be above noise level. These series of bits

represented with voltage level form the digitized voice signal.

The steps above analyze the PCM process of converting analog voice signal

to a digital signal with the assurance of having a signal power above the prevailing

noise. By this, the transmission of the signal is done with the assurance of powerthat is always above the noise level. The quality of the digitized signal is dependent

on both the sampling rate and the quantization levels. As mentioned earlier, the

minimum requirement for sampling rate is the rate that is twice the highest

frequency component of the analog voice signal and the quantization levels, though

has no minimum requirement, but must always be kept to an optimum number that

minimizes the loss of dynamism in the digitized voice signal.

The sampling rate and quantized level are always kept to the optimum

minimum in order to reduce the cost of transmission. When the two parameters are

set as low as possible, less number of bits are generated for transmission. Likewise,

the higher the sampling rate, the more the sampled values to be quantized andtherefore, there is need for large quantization levels in order to minimize the loss in

dynamism. Thus more bits are needed for transmission of a single quantized value

of the digitized voice signal. For an optimum digitized voice signal, the sampling

rate of the analog voice signal is kept at 8 KHz with quantization levels of 256 with 8

bit depth. As a result of this, digitized voice signal is transmitted at the rate of 64

Kbps that forms the DS0 signal.

The lab experiment is used to demonstrate the PCM process and the effect of

changing parameters at different stages of the process on generated digitized voice

signal. The following result below provides explanation with diagrams from the lab

exercise.

Strong signal with large quantization levels : In Figure 1.1 below, the original signal

amplitude is 1 and the quantization levels is 128; the digitized voice signal

generated from the PCM process has a high fidelity because there is large number

of possible integer values that the sampled values could be mapped onto. Due to

this, there is a great dispersal of sampled values among the integers thereby

creating a good dynamic range in the digitized voice signal generated.



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-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time [sec]

original signal

digitized signal

Figure 1.1 Strong signal of amplitude 1 with large quantization levels of N = 128

Weak signal with large quantization levels : In the Figure 1.2 below with sampling

level remained the same as above but with low amplitude of 0.1, the fidelity of the

digitized voice signal generated is not as much as it in the strong signal above.

However, the quality of the voice signal reconstructed from this digitized voice

signal is good enough to hearing due to large quantization levels. Thus, the quality

of a digitized voice signal is dependent on the strength of the original signal and

large quantization levels is more appropriate to be used when original signal power

is low.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time [sec]

original signal

digitized signal

Figure 1.2 Weak signal of amplitude 0.1 with large quantization levels of N=128.



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In the two previous observations, the quantization level is high enough to

yield a reasonable good digitized voice signal at a low signal power of the original

analog voice signal. The next two studies show how the quality of digitized voice

signal differed from one another when quantization level is low at different

amplitudes of original signal power.

Strong signal with low quantization level: In Figure 1.3 below, the signal power is

low with quantization levels of 8 to give the digitized voice signal that is

superimposed on the original signal as in the image. It can be seen that the quality

of the digitized voice is barely good enough to replace the original signal with loss

of quality. Though the quality of sound from this process might be impaired but not

to the point of not being intelligible to hearing. Thus, it is important that the original

signal power be high enough in order to get a good digitized voice signal. However,

at a low analog signal power with a large quantization levels, a digitized voice signal

that is good enough to hearing can still be generated.

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time [sec]

original signal

digitized signal

Figure 1.3 Strong signal amplitude 1.0 with low quantization levels of 8

L ow signal with low quantization level: When a low analog signal power is

quantized with few quantization levels, the process yields a very poor digitized

signal that barely resemble the original signal as depicted in Figure 1.4 below. The

digitized voice signal is completely different from the original signal. It is impossible

to reconstruct the voice signal from such digitized signal. The digitized signal is

almost a square with just only two signal levels. This form of digitized signal is acomplete distortion of the original signal. Unlike the case above that has low

amplitude but still yield a reconstructable digitzed voice signal, this is completely

out of order. Thus, it is important that the quantization levels should be as high as

possible when the signal power is too low.



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-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time[sec]

original signal

digitized signal

Figure 1.4 Weak signal of amplitude 0.1 with quantization levels of 8.

As demonstrated in the plots above it can be seen that at a constantsampling rate, the quality of the digitized voice signal is dependent on both the

quantization levels and the original signal power.

Companding

When the transition rate of analogue voice signal is highly non-uniform from

point to point, certain segments of the signal range tend to be quantized at a

disproportionate level. In such a situation, a high dynamic range region is likely tobe quantized with more levels while the low rate region is inadequately quantized.

This situation leads to misrepresentation and low quality of the voice signal in a

digitized form common with sampling done a linear scale. To avoid this situation, a

different approach that adequately sampled every segment of the signal is

introduced.

The common technique used in handling this situation is companding. It is a

process that transforms a given analog signal into an intermediate form which can

be sampled to yield adequate sampled values for every segment during digitization

of the original signal. During the process, the analog signal is compressed at one

point and expands at the last stage. The process ensures that high priority is givento the low energy portion of the signal during the sampling process with a minimal

sampling preference to the high energy portion.The two main companding

techniques used are A-law and the mu-law. The A-law compading is commonly used

in Europe and other part of the World while the µ-law companding is used in United

States and Japan.



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The µ -law companding is based on a logarithmic functions used to

compressed and its inverse that expands the processed signal. The two functions

are defined as followed:

Compression function,

y(x)=sgn(x)*ln1+µxln1+µ , -1 ≤x≤1; where x is the analog signal

level that

forms the input into the compression function.

Expansion function,

y-1(y)=sgn(y)* 1µ*(1+µ)|y|-1 , -1 ≤y≤1; where y is the quantized

value from the compressed PCM process.

The mu parameter determines the quality of the digitized signal derived from

the process; the most commonly used parameter number for mu is 255. The

following results from the lab illustrate how mu-law companding improve s the

quality of digitized signal even at a very low signal power when the mu parameter is

set to 255.

Strong signal and large sampling levels : With mu parameter of 255, Figure 2.1

shows the result of the digitized signal when companding is used. It can be seen

from the plot that the less dynamic region of the original signal is more represented

with quantized levels than the high dynamic sections. Also the quantized values are

lower in range and less dense when compared to the initial result in Figure 1.1. Due

to this, the digitized signal based on companding has less bits to be transmitted andthis reduces the cost of transmission. However, the digitized voice signal

reconstructed is less in quality when compared to what is obtained from Figure 1.1

above.

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time[sec]

original signal

digitizedsignal

Figure 2.1 Strong signal of amplitude 1.0, 256 quantization levels with mu

parameter of 255



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Weak signal and large sampling levels: The companding effect is more pronounce

on weak signal than in the strong signal. This is illustrated in the Figure 2.2 where

the digitized voice signal has almost the same signal reconstruction quality as when

in strong signal. This happens because the dynamic rage of the weak signal is less

and as such, the companding process tends to compensate for this less off energy

content with high density distribution of sampling point across the whole signalrange. Based on the same reason, the digitized voice signal derived from

companding is better representation of the original signal the digitized voice signal

from Figure 1.2 above with no companding. Thus, there is a great improvement in

quality of digitized signal derived from weak signal when companding is used.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time[sec]

original signal

digitized signal

Figure

Figure 2.2 Weak signal of amplitude 0.1, large sampling level with companding mu-

parameter of 255.

Strong signal with low sampling levels: As mentioned earlier, companding tends to

have little effect on strong signal as again repeated in the Figure 2.3 below where

the signal is strong but the sampling level is low. The resulting digitized voice signal

is always poor in quality but always has a signal reconstruction that is similar in

structure with the original. This is as a result of the fact that sampling is done on

appropriation that based on proper representation of low energy portion of the

sampling signal. The important thing is that despite the fact that the digitized signal

is poor quality, it can still retain the basic structure of the original signal with very

few bits.



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-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time[sec]

original signal

digitizedsignal

Figu

re 2.3 Strong signal, strong level with mu=255

Weak signal with low sampling level: In any situation where the analog signal

quality is poor, companding is more efficient in yielding a reasonable digitized voice

signal that is good enough for reconstruction of the original signal. However, thequality of the digitized signal may be poor but it will still be acceptable as a

replacement for the original signal. This is exactly the outcome of the lab as shown

in Figure 2.5 below when companding is used along with weak signal at a very low

sampling level. It can be seen that despite the poor quality of the signal, it is still

possess structural form that is similar to the original signal. This is contrarily to the

result in Figure 1.4 above that is entirely different in structure to the original

signal.Thus, companding is more beneficial when used along with weak signal.

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.05 0.1 0.15 0.2

s g n a a m p u e

time[sec]

original signal

digitized signal

Figure 2.4. Weak signal, low quantization with mu=255

Quantization Noise



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The quality of digitized voice signal is measured by the value of the signal-to-

quantization- noise ratio SQR calculated based on the ratio of the root-mean-square

value of sampled values of the analog voice signal and the quantization error

formed from the difference between the sampled values and the quantized values.

The SQR is calculated with the following equation

SQR=iNyi2NiN(yi-xi)2N=iNyi2(yi-xi)2, where y is the sampled value and x is the

quantized value.

The quantization error is inversely proportional to the sampling levels; the

higher the sampling levels the less the quantization error and the better the SQR.

The SQR does not only vary with sampling levels; it changes when never any single

factor of PCM process varies. How SQR varies with various factors of PCM is best

studied on individual basis. This is done by setting every other factor that is

involved in the PCM process to constant value with the exception of the interested

factor which is varied accordingly.

The graph of Figure 3.1 shows how SQR is affected by varying the amplitude

of analog voice signal with and without companding. When companding is not used

in PCM process, the SQR plots shows direct proportionality of SQR values as the

signal power increases. At a very low signal power, the SQR value is low resulting in

a poor quality digitized signal that is depicted in Figure 1.3 above. Also, when the

signal power is high, the SQR values is high and the digitized signal is expected to

be good as depicted in Figure 1.1 above. Thus, the SQR value is a good indicator of

nature of digitized signal from a PCM processing.

On the other hand, as indicated in the Figure 3.1 below, when companding is

used, the SQR maintains a relatively constant value that every SQR values jitteraround as the amplitude increases. However, the average SQR jittered value is

gives an indication of relatively good digitized signal. This can be seen from the

results above under the companding studies. Thus, unlike the case where SQR

increases with increases in signal power when compading is not used, the SQR

values, with companding, maintains an average value that other values jitter

around as amplitude increases.



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0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2

signal amplitude

SQRDBmu =0

SQRDBmu =225

Figure 3.1. The graph of SQR with variable amplitudes at mu = 0 and 255

Another important factor studies along with SQR is the bit depth. As indicated

above, digitized signal improves with increasing in sampling levels which ultimately

improves the SQR with reduction in quantization error. This relationship is depicted

in Figure 3.2 where the SQR values plotted against the varying bit depths is directly

proportional to the incrementing bit depth. It can be seen that SQL values at every

bit depths where companding is used is less than the corresponding values without

companding. This is because companding optimizes the sampling by minimizing thenumber of bits needed to number that is just good enough to reconstruct the voice

signal. Also from the graph, the SQR increases as the bit depth increases when

companding is used unlike the case of increasing amplitude. It can also be deducted

from the graph that considerable increase in SQL only occurred with the bit depths

above 50.



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-10

-5

0

5

10

15

20

25

30

0 50 100 150 200 250 300

bit depth

SQR [DB] mu =0

SQR [DB] mu =255

Figure 3.2. The graph SQR with variable bit depths at mu=0 and 255

Conclusion

PCM process digitizes analog voice signal through the process of sampling

and quantization. The digitized signal from PCM process improves in quality as thesignal power and bit depth increases. With companding, analog signal with low

signal power can still be digitized to give signal quality above poor level. Also,

companding minimize the cost of transmitting digitized signal by optimizing the

PCM process to the lowest minimal number of bits requires for reconstruction

without distortion.

Appendix



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A sample screenshot from the lab depicting of analog voice signal with

corresponding digitized voice signal.

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