Actually, this is only the first of the mirrored frequencies.  Signals are mirrored again and again⁵ like in a barbershop mirror.  But each time they get mirrored, their amplitude decreases, so we typically ignore all but the first mirror.

 

Picking a Sample Frequency

It is always safest and easiest to pick[7] a sampling frequency well above the highest anticipated frequency.  A factor of ten to hundred times higher provides a comfortable margin.  But there are times when it is not feasible to pick such a high frequency.  Your converter may be incapable of sampling fast enough, or the desired rate may be technologically infeasible.  Even if a converter with the desired sample rate exists, it may be prohibitively expensive; the cost of ADCs and DACs increase rapidly with sampling speed.  For instance a 250kS/s (kilo Sample/sec) ADC costs just $375 today (2005): a 10MS/s card costs $4000, and a 1GS/s card costs $10,000.   And then, even if you do have an appropriately fast card, you may find that the fast data rate overwhelms your system, and you may be forced to sample slower anyway.

 

Nyquist’s theorem states that you need only sample at twice the desired frequency.  What happens if you are close to the Nyquist limit?  The sampled signal may not appear to resemble the actual signal.  For instance, if you sample a sine wave at 2.35 times its frequency, you get:

Sampling at 2.05 times the frequency yields:

Neither sampled signal resembles the original signal.  Is Nyquist wrong?  Comparing the spectrum of the original signal to the sampled signal illuminates the problem:

The original signal contains one clean peak that the original frequency.  The sampled signal has an extra peak at a frequency a bit higher than the original.[8]  The amplitude modulations in the sampled signal are due to these two peaks in the sampled spectrum beating against each other.  We can largely recover the original frequency by low-pass filtering the sampled signal to destroy the spurious peak:

Unfortunately, filtering is not always easy to accomplish.  It is relatively easy to filter a signal on a computer by taking its Fourier transform and discarding the spectral content about the Nyquist frequency.  But filtering with real-world components in real time is not nearly so easy.  A simple one pole RC filter does not roll off fast enough, so we need to cascade filters to make them roll off faster:

The 6^th order RC filter depicted above rolls off quite fast; its transfer function is down by two orders of magnitude one octave about its design corner at 1Hz.  But it has also significantly reduced the signal amplitude in the passband below 1.   Signal aliasing may not be a problem, but the original signal will be distorted. 

 

Filter design is an exceedingly complicated topic, and there are better filters exist than simple RC designs.  Typical advanced designs are called Butterworth, Chebyshev, and Bessel filters, and their response is plotted below. 

The order of the filters have been adjusted so that they attenuate by a factor of 100 one octave about their design frequency of 1Hz.  The Butterworth and Chebyshev filters look very attractive; they cut off quickly, but leave the pass-band relatively unmodified.  Unfortunately, their transient response is poor.  Feed a step into them, and the will distort the signal:

All the filters delay the onset of the step, and The Butterworth and Chebyshev filters ring significantly.  The Bessel filters has the best step response, but its frequency response is not as good as the Butterworth and Chebyshev filters.  In sum, perfect filters cannot be constructed outside of a computer.  Even in a computer, it is difficult to make a perfect filter.  The spectrum of the signal sampled at 2.05 contains one major spurious peak above the Nyquist frequency, but is also contains smaller spurious peaks below the Nyquist frequency.  These peaks distort the signal.  For instance, if we sample a signal consisting of two equal amplitude sine waves at frequencies of 0.7 and 1, at a sampling frequency of 3, we can do a fair job of reconstructing the original signal:

       

Nonetheless, the spectrum shows many spurious peaks, whose effects are visible on careful comparison with the original signal.  With the ideal, computer filter, however, the spectrum is much closer to the original signal spectrum and differences to the original signal are insignificant.   The effects of imperfect filtering are even more visible for an AM modulated carrier wave:

 

  

 

One should remember that the sampled signal is just a table of values.  When we plot the sampled data, we need to interpolate between the sampled points.  There are several ways to interpolate.  Starting from uninterpolated data:

we can step interpolate (used on most DACs), ramp interpolate (used on some expensive DAQs) or comb interpolate.

Comb interpolation, which uses a sequence of infinite delta functions, each with area equal to the amplitude of sample at successive points, is difficult to use in the real world, because delta functions are unphysical.  Surprisingly though, it is theoretically optimal, as can be seen from the proof of Nyquist’s theorem, http://en.wikipedia.org/wiki/Nyquist_theorem

 

Digital to Analog Converters

There are several ways to build digital to analog converters, but the simplest way is with a scaled resistor network:

This is just an Op-Amp adder circuit, in which each bit’s is added in with weight .  In practice, it is difficult to make a high resolution DAC with a scaled resistor network because it requires precisely scaled resistors over a very wide range.  It is much easier to make precise resistors over a narrow range;[9] a common DAC that takes advantage of this fact uses a ladder network:

Like the scaled resistor ADC, this circuit also adds together appropriately weighted bits. 

 

Practical Filtering and Aliasing Advice

 

ADCs:   Always make sure that you are not being fooled by aliasing artifacts.  The best way to test for the presence of artifacts is to look at the signal with your entire detector, amplifier, and ADC chain in place and adjusted to your contemplated settings, but with your experiment turned off so that you are receiving no real data.  If you observe a significant signal, you then know that you are observing an artifact.

·       Occasionally, the received signal is so free of noise that aliasing of unwanted frequencies is unimportant.  If so, either: (1) sample at twenty or more times the desired signal frequency, or (2) sample closer to twice the desired signal frequency and use digital filtering to remove the artifacts.

·       The maximum frequency of most received signals is naturally limited by the detector that produced the signal, or by the detector amplifiers.  If possible, sample at a frequency that is higher than twice the maximum frequency of the received signal, using digital filtering, as necessary, to remove the artifacts.

·       If the desired signal is centered around a narrow band of frequencies, you may be able to ignore aliasing artifacts.  Set your ADC to sample at several times the band center frequency.  Establish that you can ignore aliasing artifacts by the test described above, concentrating only on signals in the band of interest.  If you do observe artifacts, you may be able to get rid of them by changing your sample frequency so that the spurious signals alias to a different frequency that is out of your band of interest.

·       In necessary, employ a filter to remove the high frequency noise.  Some ADCs come with anti-aliasing filters built-in.  These are filters are likely to be quite good; use them before building your own.  Otherwise, use your own anti-aliasing filter.  If you can, use a simple RC low-pass filter designed to cut off at a much higher than your signal frequency, (so that the pass-band attenuation at the signal frequency is small) and sample at a frequency much higher than the filter’s cut off frequency (so that the filter attenuation is large above the Nyquist frequency.)  If you cannot space your cutoff and sample frequency so generously, obtain a commercial filter, or, if all else fails, build a sophisticated multipole filter optimized for your application.

 

DACs:

·       If your DAC and memory can support it, use a sample frequency 20 to 200 times greater than your signal frequency.

·       The device you are driving may not care about spurious high frequency components.  In this case, you can run quite close to the Nyquist frequency without filtering.

·       If your signal is close to the Nyquist frequency, use a filter.  Some DAC’s come with high quality filters; use them if available.  Otherwise, use a commercial filter, or a simple RC filter.  If you signal is close to a pure sine wave, and you can tolerate variations in its amplitude with frequency, you may be able to generate relatively undistorted signals quite close to the Nyquist frequency.

·       If nothing else works, design a higher order filter optimized for your signal. 

 

Analog to Digital Converters

The simplest way to make an n-bit ADC is to have  separate comparators, each comparing  the input voltage to one of the set of all the voltage levels allowed by the resolution of the ADC:

An encoder determines the highest “on” comparator, and encodes this information into a number.  Such flash, or parallel recorders are very fast, and are used in high frequency applications.  But they are not feasible for high resolution ADCs, as every voltage level requires a separate comparator.

 

Conversion is more commonly done by a successive approximation ADC.  This type of ADC successively refines a guessed value in a loop.  The steps in the loop are:

1.     Convert the guess to a voltage using a DAC.

2.     Compare this guess voltage to the input voltage.

3.     Refine the guess:

a.     If the guess is too low, make a new guess that is higher.  The new guess should be half way between the current guess and the last known guess that was too high.

b.     If the guess is too high, make a new guess that is lower. The new guess should be half way between the current guess and the last known guess that was too low.

4.     Begin the cycle again if the conversion has not exhausted all the DAC bits.  Otherwise, call the final guess the answer and end the loop.

The loop is initialized by setting the last know limits to be the minimum and maximum values accepted by the ADC, and setting the first guess to be half way in between.  This process is best illustrated by a tree diagram, shown here for an ADC whose range is 0 to 16V:

 

Here the input signal is 5.3.  In the first loop iteration the ADC guesses that the answer is 8, and compares this guess with 5.3.  Since 5.3 is less than the guess, the ADC refines the guess to be half as big: namely 4.  This guess is smaller than the input, so the ADC refines its guess on the next loop to be 6, half way between 4 and 8.  Six is too large, so the next guess is 5, half way between 4 and 6.  Finally the best guess is determined to be 5.5, half way between 5 and 6, because 5 is less than 5.3.

Successive approximation methods are as accurate as their internal DAC.  They can be quite fast, as they can chase down an n bit tree in n steps, giving 2^{^n} resolution.  For increased speed, some DACs combine flash converters and successive approximation converters.  The flash converts are used to quickly find the highest order bits, perhaps the first 5, and successive approximate used to find the remaining bits. There are many other techniques as well; manufacturers aggressively develop improved devices.

 

In the lab

 

Problem 10.1 - Aliasing and Sample Rate

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Open the file U:\BSC Share\LAB_10\Lab Stations\Sampling_Simulator.llb, and then run Sampling Simulator.vi from the LLB Manager.  With filtering set to No Filter, explore the effects of the sample rate on Pure Sine and Square waves.  Use both Flat Interpolation and Linear Interpolation.   In your lab notebook, record your answers to the following questions:

1.     Sample a 1Hz, phase 90 Sine wave at 4Hz.  Do you get a reasonable facsimile of the original sine wave?  Now change the phase.  What happens?  Do you still get a reasonable facsimile?  Restore the phase to 90.  What happens when you change the sample rate to 4.01?  Can you account for the difference?

2.     Repeat these measurements with a Square wave, and record your answers.

3.     Using a 1Hz Sine wave, try some inadequate sampling rates, say 1.9, 1, 0.9, 0.5 and 0.47Hz.  Can you account for the features you observe?

4.     Explore the effects of inadequate sampling rages on a Square wave.

5.     What sample rate is necessary to give a reasonable facsimile of a sine wave, and what sample rate gives a reasonable facsimile of a square wave?  Why are these frequencies different?

6.     Now find appropriate sample rates for the Two Tone, AM, and FM signals.  Ironically, you may conclude that these signals can be sampled at a lower rate than the pure sine wave.   This is probably because these signals are so complex that the sampling artifacts so clearly visible with the Sine signal are masked.

7.     Now try the Realistic Low Pass Filter with the Sine signal and various sample rates.  Can you employ lower sample rates?  Look at the amplitude as well as the form of the sampled and filtered signals.  Is  the amplitude sometimes incorrect even when the form is correct?

8.     Look at Square waves with the Realistic Low Pass Filter.  What is the effect of the sample rate?  Using the Expanded Time Axis, cycle between Realistic Low Pass Filter and No Filter.  Note that the filter causes the rise and falls to be delayed.  What is the size of the delay?  Does it depend on the sample rate?  Note that the realistic filter is a 3^rd order Butterworth.  Does this fact explain the delay?

9.     Look at the Two Tone, AM, and FM signals.  Does the filter help?

10. Now go back to a 1Hz Sine wave, and set the sample rate to just above 2Hz, say 2.1Hz.  Does any interpolation method produce a reasonable facsimile of the wave with the Realistic Low Pass Filter? (Hint: Ask what is reasonable?)  Switch to the Ideal Low Pass Filter. Is it better?   What do your signals look like for the different interpolations?  Which one gets the amplitude correct?  With Comb Interpolation, how low a sample frequency can you use? Try sample rates of 2.02 and 2.01Hz.  In principle, you should be able to go down to 2.00Hz, but, in practice, 2.01Hz has problems.  Can you explain why?  (This is hard and is not obvious from the material covered in this writeup or in lecture.)

11. Now go to a sample rate of 1.95Hz.  Is the Ideal Low Pass Filter any better than the Realistic Low Pass Filter?

12. With the Ideal Low Pass Filter and Comb Interpolation, explore sampling ranges above 2.00Hz for the Two Tone, AM, and FM signals.  How good are the sampled signals?  Try a Square Wave?  Why is the sampled signal no good?

Show and discuss your results to the TA’s.

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Problem 10.2 - Digital to Analog Conversion

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Build a scaled resistor DAC:

 

Use the digital outputs^[10] of your DAQ card to drive the outputs; in particular, use the low order bits marked P0.0 (lowest) to P0.4.  These digital outputs switch between approximately 0 and +5V; consequently your converter will output negative voltages.  You will have to scale your resistors up from the values used in the above schematic; determine the base value (the “1”) from the requirement that the current drawn from any of the digital outputs individually should be less than 2.5mA.  We do not have the precise resistor values that you will need.  You will have to synthesize the correct values by combining two or more resistors.

 

Correct your Pre-Lab Question calculation of the full scale voltage of the DAQ by measuring the actual high level output of the DAQ.^[11]  Build a LabVIEW driver for your DAC.^[12]  Your driver’s front panel should look something like: [Note added: The front panel text "DAC Output Voltage" should be "Desired DAC Output Voltage" which is an input number provided by the user, not a measured output voltage from the DAC.]

where the Bit Pattern indicator is included for debugging purposes.  You will have to scale the output voltage to a bit pattern.  LabVIEW already has Vis for many common tasks such as this, try to find the right one for the job before you implement your own (Try searching “Boolean array”).  You should make sure that you cannot drive your DAC with an out-of-range bit pattern.  A convenient way to assure that you don’t is to use LabVIEW’s In Range and Coerce operator,  which can be found amongst the comparison operators menu.   Use your scope or multimeter to test your DAC.  Note that even though you are driving the digital lines with one byte (actually only 5 bits in one byte) of data, the DAQ Assistant expects an array.  Use the Build Array operator  to turn your byte into a one element array of bytes.

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Problem 10.3

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Write a program to test your DAC.[13]  The program’s front panel should look something like

 

The program should:

1.     Loop through all of the possible DAC output levels.

2.     At each level

a.     Display the Desired DAC output with an indicator.

b.     Drive the DAC

c.     Wait twenty milliseconds for the DAC to settle.  Use the Wait operator   found under Time and Dialog Operators menu.

d.    Read the output of the DAC with one of the analog input channels of the DAQ.

Hint: Use a Stacked Sequence Structure for steps b-d

e.     Output the DAC voltage.

                                                             i.     Display the DAC output with an indicator.

                                                           ii.     Store the current Desired Output and Actual Output in an array and use an XY plot to display a graph of the Desired Output vs. the Actual Output.  The best way to do this is with shift registers.  The program Running XY Graph Example demonstrates the technique.

Hint: Also use Build Arrays and the Bundle functions. If you’re having trouble, be sure to review the Array section of the LabVIEW Basics I tutorial.

                                                         iii.     Subtract the Desired Output from the Actual Output to find the DAC Deviation, and display a running graph.

Explain the systematic features in the DAC Deviation graph.  What is the maximum error?  Demonstrate your program and the DAC Driver SubVI to the TA’s and explain their block diagrams.

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Problem 10.4 - Analog to Digital Conversion

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Build a successive approximation ADC by adding a comparator to your DAC:

Write the following programs to complete the ADC implementation:

1.     Compare Trial.vi.  This subroutine should accept a Trial Voltage, write it to the DAC, wait 10ms for the DAC to settle, measure the output voltage of the comparator, and compare this voltage to some fixed voltage to determine if the Trial Voltage is greater than or less than the ADC Input Voltage.  The front panel of the vi should look something like:

Hint: Again, The Stacked Sequence Structure works well for timing control.

 

2.     ADC.vi. This routine should guess successive Trial Voltage’s until the ADC converges.   You may find it helpful to organize your iterations in a for loop, remembering the previous high and low guesses from iteration to iteration with shift registers, Also Case Structures and Expression Nodes are helpful.  For debugging purposes, you might wish to put in an indicator giving the current guess, and a switch controlling a 1s per iteration delay. Your front panel should look something like:

 

To help you get started, working versions of these programs are available on the lab computers.  The block diagrams of these programs are inaccessible.  You may also use the program Set Voltage to be Converted to set precise voltages on the DAQ analog output channel AO-0.

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Problem 10.5

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Write a program to test the ADC.  The program should be similar to your DAC tester except that it should convert an essentially continuous voltage ramp.  The front panel of the program should look something like:

 

Explain the systematic features in the ADC Deviation graph. What is the maximum error?  Demonstrate your program and its SubVIs to the TA’s, and explain their block diagrams.

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Problem 10.6

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The routine Test DAQ ADC and DAC.vi is very similar to the Test ADC routine that you just wrote, but tests the DAC and ADC of the DAQ.  After attaching the AO-0 output to the AI-7 input, run the routine to test the accuracy of the DAQ converters.  The DAQ DAC and ADC are both 16 bit converters.  How big is a 1bit error?  Are any of the errors larger than one bit? 

Note: The output on the front panel of Test DAQ ADC and DAC.vi that outputs the number of errors larger than 1 bit is incorrect - you should figure this out on your own.  We'll fix the software as soon as permission issues are resolved.

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Problem 10.7 - Filtering

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Use the program Sine_DAQ_Drive.vi to output a 5V sine wave on AO-0. The sample rate is 10kHz.  The drive frequency is adjustable; look at the output on a scope for a range of frequencies. (Don’t exceed the Nyquist frequency.) About when is the signal acceptable?  Now use 3.3k resistors to construct simple low pass filters with cut off frequencies near 10kHz and 1kHz.  Drive the filters with the output from the DAQ.  Take pictures of the waveforms, and determine when the signal is acceptable.  Note both amplitude changes and phase delays.

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