I tutorial di EO-WEB (capitolo 5): Signal Chain Processing. Digital Signal Processor

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The only thing that should be kept in mind is that when talking of a digital signal processor, a system capable to perform the needed operations in real time should always be considered. In most cases this means that each time the system receives a new input value, it must be able to generate the corresponding output value, before the next input value is acquired.

Digital to Analog Converter
The Digital to Analog Converter is a device that from a digital word generates the corresponding analog value. Somehow it can be considered to perform the inverse operation of the Analog to Digital Converter. The input to the DAC is a set of samples, the result of the digital signal processing. As already explained, the spectrum of such a signal is composed by the periodic repetition (at +/- nf_s, n being any integer) of the spectrum of the signal of interest.
The Digital to Analog Converter operation is to keep the value of the samples constant during each period T_s=1/f_s. The signal at the output of the Digital to Analog Converter is then as in figure 1.

Fig. 1 – DAC output (zeroth-order hold)

From a mathematical point of view, such a signal can be considered as the convolution in the time domain of the impulse train that represents the signal (Fig. 2a) and a rectangular pulse (Fig. 2b).

Fig. 2a – DAC input
Fig. 2b – Rectangular pulse

In the frequency domain the convolution operation corresponds to the product of the spectra of the two signals. The signal spectrum, as already seen, is made of the periodic repetition of the spectrum at the origin. In the frequency domain, the transform of the rectangular pulse is called sinc(f) and is represented in figure 3a and figure 3b (modulus).

Fig. 3a – Sinc waveform

Fig. 3b – Sinc waveform. Modulus

The expression of the sinc(f) function is as follows:

[eq. 1]

With this background we can now consider what happens: it is represented in fig 4a.

Fig. 4a – Signal spectrum repetition and transfer function of the Digital to Analog Converter

In the figure, in the frequency domain, the following items are represented:

1. Spectrum of interest and its repetitions due to the fact that a digital system is under consideration
2. Spectrum of the sinc(f) (black)
3. Shaded area (blue) that represents the area of interest on the frequency axis.

As stated above, the overall result is the product of the spectrum repetition times the sinc(f) spectrum.

It should then be clear that the overall effect of the introduction of the DAC in the signal path is:

1. The reduction in amplitude of the repetition of the spectra (the sinc(f) function is zero at all multiples of f_s), which is good
2. The main spectrum (the one of interest) at the origin is distorted by the sinc(f) function, which is bad (shaded area in Fig. 4a).

This effect is probably very often disregarded, but it is always present. Of course the importance of the distortion of the spectrum at the origin is lower and lower as the sampling frequency increases.
There are applications where the distortion introduced by the Digital to Analog Converter is not relevant; in other cases it is not possible to disregard the presence and the effects of such distortion. A typical example is the spectrum shaping digital filter used in telecommunication equipments.
In line of principle, it is easy however to correct the distortion. It is in fact sufficient to introduce a filter whose transfer function is:

[eq. 2]

This function is plotted in figure 4b. Note that the function is only defined in the range (0 to f_s/2).

Fig. 4b – Signal spectrum of [1/sinc(f)]

A first idea for implementing this correction is to implement the filter as part of the reconstruction filter that follows the Digital to Analog Converter.
However, implementing this correction in the reconstruction filter may be challenging since it is an analog low pass filter. Some good design tools are needed to enable performing the synthesis of such a filter. Another approach that can be sometimes used is to add this pre-distortion in the digital filter that is implemented (if any) by the digital processing that precedes the Digital to Analog Converter itself. In this case the operation is sometimes easier: the desired filter transfer function must be multiplied by the 1/sinc(f) function, in the frequency range (0 to f_s/2).

Output low pass filter (Reconstruction filter)
It is also called “reconstruction” filter, and its main use is to enable the system to recover the spectrum of the signal of interest, that is the spectrum close to the origin, removing all the periodic repetitions. The choice of the reconstruction filter is strictly connected to the Sampling Theorem. This fundamental theorem reads as follows: “in a digital system the minimum sampling frequency must be at least twice the maximum frequency of the signal of interest”.

Pretty easy, but its simplicity hides some very fundamental concepts of digital signal processing that must be thoroughly understood to be able to design a working digital system. The reason for this should be already clear from the previous discussion. Figure 5 represents the spectrum of the sampled signal; the blue spectrum at the origin is the one of interest.
In order to be able to recover the original spectrum of interest removing all the periodic repetitions, we must have the main spectrum and the first repetition clearly distinct: no overlap of the two spectra is allowed.

Fig. 5 – Spectrum repetition due to sampling in the time domain

A situation like the one in figure 6 (see also previous sections) would not allow recovering the original spectrum of interest without distortion. Distortion is represented by the shaded area.

Fig. 6 – Overlapping of spectra of a signal sampled at a too low frequency. Portion of the first repetition of the spectrum overlapping the spectrum of interest

It is then clear that the minimum sampling frequency should be twice the bandwidth of the spectrum of interest. This brings to the situation of figure 7.

Fig. 7 – Minimum sampling frequency to avoid spectra overlap

The spectra are close to each other, they just touch each other, but there is no overlap, no distortion. In this case:

[eq. 3]

This is in fact the (minimum) condition that ensures the two spectra (the one of interest, in the lower end of the frequency axis, and the first repetition) are distinct and not-overlapping.
Should we have such a system, we would be perfectly satisfying the Sampling Theorem. However to recover the spectrum of interest one should implement, as the reconstruction low pass filter, a filter having the transfer function shown in figure 8 (black line). The filter is referred to as “ideal
brick-wall” filter. Remembering what was described in a previous section, the order of the filter increases with the sharpness of the transition band. In the case of the brick-wall filter the transition band is infinitely small (it is a vertical line). This means that a filter of order infinite should be implemented: this is obviously impossible.

Fig. 8 – Brick-wall low-pass filter used to recover the spectrum of interest

So in the real world designers have to select a sampling frequency higher than strictly necessary to enable the design of the reconstruction filter (Fig. 9).

Fig. 9 – Higher sampling frequency positions the spectra repetition far away from each other. This allows to use a less selective low-pass filter to extract the spectrum of interest

In other terms, a trade-off has to be accepted. Designers increase the system sampling frequency the amount that is required to allow the implementation of a physically realizable low-pass reconstruction filter. In doing so, one obviously pays the fact that the digital system will have to operate at a higher than theoretically minimum frequency. In general this means higher performances are needed from the processor, that is higher costs are normally involved.

Two steps beyond
The development of high resolution low cost Analog to Digital Converters and Digital to Analog Converters allows to design new architectures with a high degree of accuracy, simplicity and versatility. One key feature of these components is the fact that the reference voltage, which must be supplied externally, may have a very wide range: from a few tenth of mV up to Vcc (the parts usually operate from single supply, for instance from 2.7V to 5.25V). The reference voltage of an A/D converter has a direct impact on the input voltage range: in fact, the former directly sets the latter. Depending on which component we are considering, the input range can be 0 ÷ V_REF or 0 ÷2V_REF or – V_REF ÷ +V_REF. In the first case, since the LSB is given by the reference voltage divided by 2n, where n is the resolution in number of bits, the following Table 1 shows the wide range of values it can assume in two extreme cases (n = 12):

Vref       LSB = Vref / 4096
50 mV 12.2 mV
5 V       1.22 mV

Table 1

The crucial point is that the number of bits of resolution is not affected by this scale changing: in any case the LSB is obtained as 1/4096 of the input voltage range.

The idea for a different approach to the problem of extending the dynamic range in Analog to Digital Converters is to adapt the reference voltage to the incoming signal, increasing or decreasing it as the input voltage increases or decreases. The overall effect is to obtain a gain, that is the same effect discussed previously with an amplifier in front of the Analog to Digital Converter. Now the input signal needs not be amplified any more to reach the Analog to Digital Converter full scale, but, one can say, it is the converter which adapts itself to the incoming signal (Fig. 10).

Fig. 10 – Traditional Analog to Digital Conversion Approach

A second small step forward: a Digital to Analog Converter is used to generate the reference voltage through a direct connection of its output to the reference input of the Analog to Digital Converter (Fig 11). A closed loop system is created, where an intelligent controller (a microcontroller) can set the Analog to Digital Converter reference voltage and input range in such a way to always exploit it at best.

Fig. 11 – Alternative Analog to Digital Conversion approach where a DAC , controlled by a micro, is used to set the (variable) Analog to Digital Converter input dynamic range

Two advantages of this solution are clear at once:

1. the number of “gain values” are much more than in the previously considered solutions (equal to 2n, where n is the Digital to Analog Converter resolution, versus typically 4)
2. they can be changed almost continuously in small steps (down to the Digital to Analog Converter Least Significant Bit amplitude).

Of course this solution has a drawback too: as we reduce the reference voltage, the relative weight of errors inherent to the converter becomes more and more important. In an A/D converter the noise sources are many: external (noise incoming with the signal, noise from the power supply rails,), and internal (quantization noise, comparator noise, resistors Johnson noise, switching noise, …).
Random Process Theory gives us the mathematical tools to add all this different noise contributions. The result is that the performances of an Analog to Digital Converter are slightly inferior to what may be expected from the bare number of bits.
The definition of Effective Number of Bits (ENOB) allows taking into consideration all these noise sources and determine which is the “real” resolution of the Analog to Digital Converter.

All these sources (E_n1, E_n2, …) are statistically independent and the total effect is obtained as sum of the rms values:

[eq. 4]

The combined effect of all these errors is represented at best by the parameter ENOB (effective number of bits) whose value is in its first approximation given by:

[eq. 5]

Typical examples of Resolution of an Analog to Digital Converter and its ENOB are:

Theoretical resolution: 10 bits   ENOB: 9.7 bits
Theoretical resolution: 12 bits   ENOB: 11.6 bits

Something can be done as far as some of the noise sources are concerned. As already analyzed in previous sections, the noise generated internally in the Analog to Digital Converter is essentially white (Gaussian). This means it is evenly spread along the frequency axis in the range (-f_s/s to f_s/2).
Improvements from this point of view can be obtained making some sort of averaging or filtering. In order to keep the computational burden low (so that even small microcontrollers can be used) we will consider only two possibilities: moving average filters and FIR filters.

Queequeg2004

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