Download 5 Data Conversion III - Resolution, Quality and Intelligibility

5. Data Conversion III: Resolution,
Quality & Intelligibility
5.1 Design Examples
Example 2
The output of a pressure transducer is to be digitised for display after
signal conditioning. The digitised value must have an accuracy of
±0.05% FS. If the maximum magnitude of the error voltage is to be 1
mV, determine the full-scale output voltage of the transducer and the
number of bits required in the analogue-to-digital converter.
Solution:
The digitised value will have an error of ± ½ level or ± ½ LSB
Therefore Δ = 0.05% FS or 1 part in 2000
Then 1 level = 2Δ = 0.1% FS or 1 part in 1000 ( 2 parts in 2000 )
This means that the full output signal range occupies 1000 parts and
hence the number of levels is given as:
L = 1000 levels
The number of bits is given by the power of the radix 2 which gives a
decimal value of greater than or equal to 1000. From the previously
given table, it can be seen that 210 = 1024. Therefore, the number of
bits required is:
N = 10 bits.
If the error is ± ½ level = ± 1mV then the quantum step voltage is:
VQ = 2mV
If there are 1000 levels then the full scale output voltage from the
transducer is:
VFS = (L – 1)VQ = 999 x 2mV = 1998 mV = 1.998V
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Example 3:
A Compact Disc has music stored in 16-bit binary format. The
maximum signal level is 2.5 V peak when recovered from the disk.
Determine the maximum error voltage in the decoded analogue signal
and the percentage error.
Solution:
The number of quantisation levels corresponding to 16 bits is:
L  2N  216  65536
The quantisation step is then:
VQ 
VFS
2.5

L  1 65535
This is evaluated as:
VQ 
2.5
 3.8x10 5  38μV
65535
The absolute error voltage is half the quantisation step:
V   0.5VQ  0.5 x 38μV  19μV
This can be expressed as a percentage of full scale as:
% Error  
19V
19 x 10 -6
x100 %  
x102 %
2.5V
2.5
19 x 10 -4
% Error 
%  7.6 x 10 4 %  .00076%
2.5
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5.2 Effects of Quantization
Pure Audio Tones:
Purely sinusoidal waveforms illustrating the effects of
quantization are shown in Fig. 5.1. The difference between the
quantised staircase-like waveform and the original sinewave is the
quantisation error. This is indicated in the figure as a signal in itself as
a function of time with a saw-tooth like profile as illustrated. It can be
seen that the peak value of the error signal is directly proportional to
the number of bits or digital resolution applied in the quantisation.
Fig. 5.1
Error in Quantisation of a Sinewave with 4 and 5 Bits
The following audio files give examples of a fixed tone at 1Kz
quantized at different resolutions.
1 kHz Tone quantised with 8 bits resolution: Tone 1kHz 8 bits.wav
1 kHz Tone quantised with 4 bits resolution: Tone 1kHz 4 bits.wav
1 kHz Tone quantised with 2 bits resolution: Tone 1kHz 2 bits.wav
A pure tonal quality can be heard at high resolutions of greater than 8
bits. Below this the tone can clearly be heard but takes on a kind of
‘wirey’ or ‘sandy’ coarse quality. The sandiness is the quantisation
noise which can be considered in effect as a distortion signal added to
the tone which reduces its purity or quality.
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Speech:
The following examples are samples of speech where the phrase
spoken is:
‘The possibility of a Mann Act conviction, resulting in disbarment
proceedings and total loss of his livelihood, was a key factor in his
decision‘.
Speech quantised with 8 bits resolution:
Speech-Mann Act 8 bits.wav
Speech quantised with 3 bits resolution:
Speech-Mann Act 3 bits.wav
Speech quantised with 2 bits resolution:
Speech-Mann Act 2 bits.wav
The following examples are samples of speech where the phrase
spoken is:
‘Rimmer I’m bored…... Bored?, This is essential routine maintenance.
It’s absolutely vital for the well-being of this crew, this mission and
this ship……. Dispenser 172 – Chicken Soup nozzle clogged!’.
Speech quantised with 8 bits resolution: Speech-Red Dwarf 8 bits.wav
Speech quantised with 4 bits resolution: Speech-Red Dwarf 4 bits.wav
Speech quantised with 2 bits resolution: Speech-Red Dwarf 2 bits.wav
Speech quantised with 1 bits resolution: Speech-Red Dwarf 1 bits.wav
As can be heard the quantisation process gives the speech a ‘gravelly’
or ‘gritty’ coarse character. This becomes more pronounced as the
quantisation resolution is lowered. It appears to be more perceptible
here at higher resolution because it interferes with the natural
distinctive quality of the human voice which is more elaborate than
merely containing varying tones. However, if the quality is not made
the focus of the listening exercise, but rather attention is directed
towards the intelligibility of the speech, then it can be observed that
what is being said by the speaker can still be understood at quite low
bit resolutions. That is to say the majority of the ‘information content’
is discernible even at low resolution. This illustrates the fact that
human speech is remarkably resilient to interference and distortion
when it comes to simply transferring the information contained in it.
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Music:
When music signals are digitised, they are much more sensitive
to the effects of quantisation noise. This is because the energy in a
music signal is spread across a much greater part of the audio
frequency spectrum. There is much more energy at higher frequencies
than is the case in a speech signal. The higher frequency energy very
often defines subtle minor changes in the signal which affect tone,
timbre, pitch or harmonic quality of the musical content of what is
heard. This is appealing to the ear and mind in a musical context and is
a significant part of what makes the music ‘enjoyable’. This means that
the information content is entirely different than is the case with
speech. It is not merely a factual message that is being conveyed but a
spectrum of musical audio colouration and psychoacoustic emotional
experience.
Music quantised at 16 bits resolution: Eleanor Rigby 16 bits mono.wav
Music quantised at 8 bits resolution: Eleanor Rigby 8 bits mono.wav
Music quantised at 4 bits resolution: Eleanor Rigby 4 bits mono.wav
Music quantised at 2 bits resolution: Eleanor Rigby 2 bits mono.wav
It can be observed that the quantisation resolution has again the same
gravelly effect on quality. However, because of the different nature of
the information content, the quantisation error has a more noticeable
degrading effect at higher quantisation resolution. In effect it
interferes in a more unacceptable way with the features of the music
which are central to its enjoyment, which must be considered its main
purpose, rather than its mere intelligibility.
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Images:
An image is commonly thought of as a two dimensional object.
However, when digitised it essentially becomes a three dimensional
object since the brightness of the image must be quantised also to
convert it into digital form. Initially, let us ignore what is commonly
referred to as the digital resolution which is the number of pixels (or
picture elements) forming the image. This is evaluated as the number
of horizontal pixels multiplied by the number of vertical pixels so that
we obtain a measure of the digital area of the image in pixels.
Instead, let us consider a simple black-and-white image where
any element of the image has a value of brightness from zero
representing total blackness to some maximum level representing
total whiteness as indicated in Fig. 5.2 below. When the image is
converted into digital form the degree of brightness is quantized into a
finite number of levels which are then encoded into binary form as for
the speech and sound signals examined previously.
Fig. 5.2 Quantized Levels of Brightness in Black & White
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The number of bits determines the degree of resolution of the
brightness as shown in the examples in Fig. 5.3. This affects the
degree of detail which can be made out in the image. The basic content
of the image can be discerned even at low resolution, as for example
in the top-left image with only single-bit resolution. As the resolution
is increased, more detail can be made out in the image.
Top Left–1 Bit, Top Right–2 Bits, Bottom Left–3 Bits, Bottom Right–4
Bits
Fig. 5.3 Varying Resolution in Quantization of Image Brightness
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The image shown in Fig. 5.4 is a famous image known as ‘Lena’ which
is used as a worldwide benchmark in image processing research. The
same effect can be seen in the images here having different degrees of
brightness quantisation where N represents the number of levels
rather than the number of bits in the quantisation process. At high
resolution any difference between images is difficult to discern and
only affects very minor detail. At medium degrees of resolution
‘banding’, ‘shading’ and ‘blotching’ effects begin to become apparent.
At low resolution significant differences are clearly visible. Note that
the image is still recognisable at 1-bit resolution. At 2-bit resolution a
considerable amount of detail can be made out.
Fig. 5.4 Varying Resolution in Image Brightness Quantization
N = No of Levels
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However, more commonly the digital resolution of images is
concerned with the number of picture elements or ‘pixels’ which make
up the image. This is normally expressed as the ‘number of pixels per
row’ multiplied by the ‘number of pixels per column’, i.e. as row x
column. Examples of varying degrees of pixel resolution are shown in
Fig. 5.5. In this case it can be seen that little or nothing of the image
can be discerned at low degrees of resolution. Even at the highest
number of pixels of 72 x 72 in this example the image is very badly
defined.
An example of an image having higher pixel resolutions is shown
in Fig. 5.6. It can again be seen that detail in the image can only be
made out at the highest degrees of resolution.
The effect of limited pixel resolution is even more noticeable in
colour images as can be seen from the example shown in Fig. 5.7.
Fig. 5.5
Examples of the Image ‘Lena’ at Different Pixel Resolutions
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Fig. 5.6
Fig. 5.7.
Examples of an Image of a Machine Part at Varying
Resolutions
A Colour Image with Two Different Pixel Resolutions
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In practice a much higher resolution is used in obtainig the image than
can be appreciated with the naked eye. This principally to allow for
digital ‘resizing’ or ‘cropping’ of the image or ‘zooming’ in on sections
of the picture. It can also help with printing quality. This can be
appreciated from the examples of Fig. 5.8. and is the main reason why
the image resolution of modern digital cameras runs to megapixels.
Fig. 5.8
Resizing, Cropping and Zooming of Digital Images
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