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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 1 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 19V 19 x 10 -6 x100 % x102 % 2.5V 2.5 19 x 10 -4 % Error % 7.6 x 10 4 % .00076% 2.5 2 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. 3 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. 4 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. 5 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 6 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 7 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 8 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 9 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 10 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 11