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SYED AZEEM AHMED SYED AZEEM AHMED BE(TELECOM) 6th SIGNAL SYSTEM’S BE TELECOM [email protected] (See the last page to download in e-version) page#1 SYED AZEEM AHMED Quantization Error The fundamental difference between digital audio and analog audio is one of resolution. Analog representations of analog signals have a theoretically infinite resolution in both level and time. Digital representations of an analog sound wave are discretized into quantifiable levels in slices of time. We've already talked about discrete time and sampling rates a little in the previous section and we'll elaborate more on it later, but for now, let's concentrate on quantization of the signal level. As we've already seen, a PCM-based digital audio system has a finite number of levels that can be used to specifiy the signal level for a particular sample on a given channel. For example, a compact disc uses a 16-bit binary word for each sample, therefore there are a total of 65,536 (or ) quantization levels available. However, we have to always keep in mind that we only use all of these levels if the signal has an amplitude equal to the maximum possible level in the system. If we reduce the level by a factor of 2 (in other words, a gain of -6.02 dB) we are using one fewer bits worth of quantization levels to measure the signal. The lower the amplitude of the signal, the fewer quantization levels that we can use until, if we keep attenuating the signal, we arrive at a situation where the amplitude of the signal is the level of 1 Least Significant Bit (or LSB). Let's look at an example. Figure 10.1 shows a single cycle of a sine wave plotted with a pretty high degree of resolution (well... high enough for the purposes of this discussion). Figure 8.13: A single cycle of a sine wave. We'll consider this to be the analog input signal to our digital converter. Let's say that this signal is converted into a PCM digital representation using a converter that has 3 bits of resolution - therefore there are a total of 8 different levels that can be used to describe the level of the signal. In a two's complement system, this means we have the zero line with 3 levels above it and 4 below. If the signal in Figure 10.1 is aligned in level so that its positive peak is the same as the maximum possible level in the BE TELECOM [email protected] (See the last page to download in e-version) page#2 SYED AZEEM AHMED PCM digital representation, then the resulting digital signal will look like the one shown in Figure 8.14. Figure 8.14: A single cycle of a sine wave after conversion to digital using 4-bit, PCM, two's complement where the signal level is rounded to the nearest quantization level at each sample. The blue plot is the original waveform, the red is the digital representation. Not surprisingly, the digital representation isn't exactly the same as the original sine wave. As we've already seen in the previous section, the cost of quantization is the introduction of errors in the measurement. However, let's look at exactly how much error is introduced and what it looks like. This error is the difference between what we put into the system and what comes out of it, so we can see this difference by subtracting the red waveform in Figure 8.14 from the blue waveform. Figure 8.15: A plot of the quantization error generated by the conversion shown in Figure 8.14. BE TELECOM [email protected] (See the last page to download in e-version) page#3 SYED AZEEM AHMED There are a couple of characteristics of this error that we should discuss. Firstly, because the sine wave repeats itself, the error signal will be periodic. Also, the period of this complex waveform made up of the will be identical to the original sine wave - therefore it will be comprised of harmonics of the original signal. Secondly, notice that the maximum quantization error that we introduce is one half of 1 LSB. The significant thing to note about this is its relationship to the signal amplitude. The quantization error will never be greater than one half of an LSB, so the more quantization levels we have, the louder we can make the signal we want to hear relative to the error that we don't want to hear. See Figures 8.16 through 8.18 for a graphic illustration of this concept. Figure 8.16: A combined plot of the original signal, the quantized signal and the resulting quantization error in a 3-bit system. Figure 8.17: A combined plot of the original signal, the quantized signal and the resulting quantization error in a 5-bit system. BE TELECOM [email protected] (See the last page to download in e-version) page#4 SYED AZEEM AHMED Figure 8.18: A combined plot of the original signal, the quantized signal and the resulting quantization error in a 9-bit system. As is evident from Figures 8.16, 8.17 and 8.18, the greater the number of bits that we have available to describe the instantaneous signal level, the lower the apparent level of the quantization error. I use the word ``apparent'' here in a strange way - no matter how many bits you have, the quantization error will be a signal that has a peak amplitude of one half of an LSB in the worst case. So, if we're thinking in terms of LSB's - then the amplitude of the quantization error is the same no matter what your resolution. However, that's not the way we normally think - typically we think in terms of our signal level, so, relative to that signal, the higher the number of available quantization levels, the lower the amplitude of the quantization error. Given that a CD has 65,536 quantization levels available to us, do we really care about this error? The answer is ``yes'' - for two reasons: 1. We have to always remember that the only time all of the bits in a digital system are being used is when the signal is at its maximum possible level. If you go lower than this - and we usually do - then you're using a subset of the number of quantization levels. Since the quantization error stays constant at +/- 0.5 LSB and since the signal level is typically lower, then the relative level of the quantization error to the signal is typically higher. The lower the signal, the more audible the error. This is particularly true at the end of the decay of a note on an instrument or the reverberation in a large room. As the sound decays from maximum to nothing, it uses fewer and fewer quantization levels and the perceived quality drops because the error becomes more and more evident because it is less and less masked. 2. Since the quantization error is periodic, it is a distortion of the signal and is therefore directly related to the signal itself. Our brains are quite good at ignoring unimportant things. For example, you walk into someone's house and you smell a new smell - the way that house smells. After 5 minutes you don't smell it anymore. The smell hasn't gone away - your brain just ignores it when it realizes that it's a constant. The same is true of analog tape noise. If you're like most BE TELECOM [email protected] (See the last page to download in e-version) page#5 SYED AZEEM AHMED people you pay attention to the music, you stop hearing the noise after a couple of minutes. Your brain is able to do this all by itself because the noise is unrelated to the signal. It's a constant, unrelated sound that never changes with the music and is therefore unrelated - the brain decides that it doesn't change so it's not worth tracking. Distortion is something different. Distortion, like noise, is typically comprised entirely of unwanted material (I'm not talking about guitar distortion effects or the distortion of a vintage microphone here...). Unlike noise, however, distortion products modulate with the signal. Consequently the brain thinks that this is important material because it's trackable, and therefore you're always paying attention. This is why it's much more difficult to ignore distortion than noise. Unfortunately, quantization error produces distortion - not noise. Download online at: http://www.scribd.com/doc/20710744/SYED -AZEEM-AHMED-Signal-SystemQuantitazation-Error For comments: [email protected] BE TELECOM [email protected] (See the last page to download in e-version) page#6