I tutorial di EOWEB (capitolo 3): Signal Chain Processing. Analog to Digital Converter
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Elettronica Oggi
Taking into consideration the signal flow block diagram introduced in previous sections, the Analog to Digital Converter is presented first; the presence of the lowpass filter in front of it is in fact a consequence of the converter itself
The Analog to Digital Converter performs two operations at the same time on the incoming signal [x(t)]. It is therefore possible and valuable to represent the Analog to Digital Conversion operation, based on such distinction, with the block diagram in figure 1.
The overall consequence of its use is that it “transforms” the input signal from a time continuous signal (that is an analog signal that has a value for each instant in time) into a time discrete signal [x(n)]. This means x(n) only exists at specific instants in time (the sampling instants) and its value is selected among a restricted set of possible values.
A useful consequence is that (n) is an index that identifies a specific sample in the set of samples taken on the continuous input signal waveform.
The first operation in figure 1, performed by the sampler, consists of the following steps:
1. it “samples” the input value; this is a theoretically instantaneous operation, so that the output value only exists at the sampling instants;
2. since a real device is at hand, the instantaneous value is keep constant (by the hold circuit) for an amount of time long enough for a safe and reliable operation of the Analog to Digital Converter on the analog value, but much shorter (usually) compared to the sampling time.
Fig. 1  Analog to Digital Converter: internal functional blocks
In other terms, a signal [x(t)] that has an analog value, for each instant in time, has been transformed into a signal that only exists at the sampling instants where it however assumes the same analog value it had before (at least for now).
A deep investigation of the consequences of sampling is absolutely fundamental to understand Digital Signal Processing.
Fig. 2  Time and amplitude continuous signal in the time domain
Suppose that the time continuous signal is as represented in figure 2 in the time domain and its frequency domain spectrum is as in figure 3. For simplicity, in this figure and in all the following ones, the signal spectrum is represented as a trapezoid, which is, of course, an abstraction. The key point is that, in general, the real shape of the signal spectrum is of no interest in the argumentation. In most cases one only needs to know how big the portion of the frequency axis is, where the spectrum has amplitude greater than zero (on the frequency axis, from 0 to f_{m}). In the following text, this trapezoid will be referred to as “spectrum of the signal of interest” or “spectrum of interest”.
Fig. 3  Time and amplitude continuous signal in the frequency domain
Be f_{s} the sampling frequency. The key concept is that the sampling operation has the effect, in the frequency domain, to generate replica of the original spectrum around integer multiples of the sampling frequency (+/ n_{fs}, with n integer). This is shown in figure 4. At each multiple of f_{s}, the spectrum presents a symmetrical half on the left side. In effect also the “original” spectrum, the one around the origin, has a left side half (shown dashed in the figure). This is somewhat difficult to accept (a portion of the spectrum should be on the negative frequency axis!), but it is a mathematical trick that makes computations easier.
Fig. 4  Signal spectrum repetition along the frequency axis as an effect of sampling in the time domain
An equivalent way to represent the sampling operation is to think of the frequency plane as split into vertical stripes, each (f_{s}) wide. See figure 5. The effect of sampling is to fold back all the stripes (an infinite number of them) onto the “main” stripe, centered on the origin. The figure shows what happens when a far too small sampling frequency is used, compared to fm: the meaning of the fold back operation is clearly shown with the oblique line (the signal spectrum) being repeatedly folded into the region (–f_{s}/2) to (f_{s}/2).
Fig. 5 The effect of sampling a signal with a far too low sampling frequency (fs) compared to the maximum signal frequency (fm). The spectrum is folded back around the vertical axis
The second operation of the Analog to Digital Converter, performed by the quantizer (refer to Fig. 1) is to transform the sampler output from a signal continuous in amplitude (this means that x(t) can take any real value) to a digitalized signal, that can only take a limited number of finite values. There is therefore a mapping between the input signal x(t) and a limited set of possible output values. For instance in an 8 bit ADC, the output can only assume one out of 2^{8} = 256 discrete values.
Thus this operation (quantization) reads, during every sampling period, a real valued sample from the sampler and “transforms” it into a value close enough to it, but that belongs to the finite set of values that makes the converter output set.
Figure 6 explains the process of quantization: the mapping of the discrete time/continuous amplitude signal into a finite set of discrete time/discrete values. The “ladder” represents the input/output relationship of the Analog to Digital Converter; the dashed green line represents the ideal operation of an ADC having an infinite number of levels.
Fig. 6  Analog to Digital Converter inputoutput relation
Focusing on each interval, it is clear that the quantizer introduces an error that equals the difference between the ideal transfer function and the ladder value. The error generated on the whole xaxis is represented in figure 7. This error signal is limited within –D/2 and D/2, where D is the amplitude of each step (called LSB: Least Significant Bit).
Fig. 7  Quantization error
Quantization Error in Analog to Digital Converters
At this point some specific attention has to be deserved to the quantization error introduced in the previous paragraph.
A fundamental theoretical result is that noise has a behavior that cannot be described by traditional mathematical tools, used for instance to describe a sinusoidal waveform. Noise is a random process. This means that:
1. we cannot describe it in a closed form, that is there is no analytical function (like for instance [sin (t)]) that
can define/describe it;
2. to operate on it, we must use special tools made available from Random Processes Theory. This is a (quite complex) mathematical theory used in the analysis of a number of different signals, all having, as noise, the characteristic of being “nondeterministic”;
3. what we can do is to describe the behavior of the random process in terms of a set of specifications (characteristics) like mean value, variance, correlation, probability density function and so on.
In the following section some concepts and results from the Random Processes Theory will be used. The consequences of the application of concepts of this Theory will be shown. However readers willing to have a deeper understanding should refer to specific books; there are plenty of such excellent books. In the following RP means Random Process and RPT means Random Process Theory.
The main focus here is to describe the consequences of the use of the quantizer in terms of the (quantization) error it generates. Because of the randomness of the error process, we will consider the quantization error power.
RPT shows that one can describe how the error power is distributed on the frequency axis. The reason of the interest in the relationship between the quantization error and frequency is that digital systems (sampled signal systems) are being investigated. The behavior of such a system related to sampling frequency is of maximum interest.
The relevant result that is derived in RPT is that the quantization error power is evenly spread on the frequency axis between –f_{s}/2 and f_{s}/2. The area between this curve (its correct name is spectral power density) and the xaxis is the power of the quantization error, in the selected frequency range –f_{s}/2 and f_{s}/2, as defined by the chosen sampling frequency. This is shown in figure 8a. The shaded area is the quantization error power.
In a system where the sampling frequency is kept constant, a higher quantization error power will have as a consequence that the power spread over the frequency axis will also be higher (Fig. 8 b).
Fig. 8 a  Quantization error power spread over the frequency axis from –fs/2 to fs/2. 10 bit ADC
The final step is that the amplitude of this distribution (A_{1} and A_{2 }in Fig. 8a and 8b) depends on the amplitude of the Least Significant Bit of the Analog to Digital Converter, that is, in other terms, of the number of bits of the Analog to Digital Converter. In fact
eq. 1
Fig. 8 b  Quantization error power spread over the frequency axis from –fs/2 to fs/2. 8 bit ADC
This result is then obtained: an 8 bit resolution Analog to Digital Converter will have a higher quantization error power than a 10 bit resolution Analog to Digital Converter. See figure 8c.
Finally, the contribution to noise of a higher resolution (more bits) Analog to Digital Converter is lower compared to a low resolution (less bits) ADC.
Fig. 8 c  Quantization error power spread over the frequency axis from –fs/2 to fs/2. Comparison between 8 and 10 bit ADC quantization error power levels
Appendix 2
Analog to Digital Converter Quantization Noise
This Appendix investigates with additional details the quantization error. Some terms and concepts from Random Signal Theory will be used without demonstrating them. Interested readers are invited to refer to specialized books on Random Variables and Processes.
As shown in figure 6 and figure 7, in an Analog to Digital Converter, the quantization error amplitude e(n) is limited to:
eq. A2.1
where Δ equals one LSB, that is:
eq. A2.2
e(n) is a random variable, that is each new value may assume any possible value within the indicated range. Some special functions can be used to describe in mathematical terms such a “random” behavior.
The RPT (Random Processes Theory) function commonly used to describe the behavior of e(n) is called “probability density function”, f(e).
Figure 9 represents f(e): it is flat; any value of the error e(n) greater than –D/2 and smaller than D/2 is equally probable.
Fig. 9
This specific shape makes it very easy to compute the variance of the random variable e(n). The importance of such computation is connected to the fact that, in this case, the variance also describes the signal power, that is, the variance corresponds to the quantization error power.
The power of the quantization noise can be computed as:
eq. A2.3
noting that E[x] is zero (zero mean value).
The important results in this equation is that the quantization error power is a function of the amplitude of the LSB (D) of the Analog to Digital Converter at hand. As a consequence, an ADC with 8 bit resolution will have a bigger value (see eq. A2.2) of D compared to a 10 bit resolution ADC for instance. The corresponding error power will be bigger also as shown by eq. A2.3. In other terms, the weight of the quantization error is bigger for smaller resolution converters (which is reasonable). See also figure 8.
Another useful function introduced by RPT is the spectral power density S(f); it describes how the power of the quantization error e(n) is spread over the frequency axis. The importance to relate the error power to the frequency axis is directly connected (as should appear clear later) to the fact that we are dealing with sampled signals.
Quantization error belongs to a class of random processes called “white noise”. This is good information, since such processes are somehow easier to deal with from a mathematical point of view.
In the case of a “white noise” random process the power spectral density is again flat, constant in the range –f_{s}/2 to f_{s}/2. As can be argued from its name, the interesting feature of the power spectral density is that the area enclosed between S(f) itself and the xaxis is the power of the signal. The last notion to stress is that this area can be easily computed as the integral of S(f) in the interval (f_{s}/2) to (f_{s}/2).
In
other terms, the power of the quantization error is
eq. A2.4
Fig. 10  Spectral power density
Putting together the results from eq. A2.3 and eq. A2.4, we have:
eq. A2.5
Since D is the LSB amplitude, we see that, considering a fixed sampling frequency f_{s}, the quantization error power is greater for a low resolution ADC (for instance an 8 bit ADC) compared to a 10 bit resolution ADC.
This result is clearly shown by figure 11, where the amplitude of the power spectral density functions for two different converters (8 and 10 bit) are represented, both converters operating at the same sampling frequency.
Fig. 11  Spectral power density for an 8 and a 10 bit Analog to Digital Converter
As an example the table below shows the value of the LSB and the quantization error power for the 8 and 10 bit resolution ADC we have been referring to often in this appendix (supposing that the full scale range is 5V).
The noise power of the 8 bit ADC is about 16 times greater than that of the 10 bit ADC.
Queequeg2004
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