Sampling, Digital Devices Data Acquisition by xvi11400

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									                ME 310
       Chapter 7 - Digital Methods




Sampling, Digital Devices
  & Data Acquisition

     ME 310 - Instrumentation




                                     Mechanical Engineering
                                      University of Kentucky




                ME 310
       Chapter 7 - Digital Methods




 Digital Sampling and Conversion




                                     Mechanical Engineering
                                      University of Kentucky




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                              ME 310
                     Chapter 7 - Digital Methods

Introduction to the Digital World
• What is a digital signal and what differentiates it from
  other signals?
   – All electronic signals (and thus circuits) are either analog or
     digital. Both are voltage (or current) signals. (See Chap. 1 & 2
     slides for examples.)
• An analog signal is continuous; i.e., infinitely divisible
  into smaller and smaller parts.
• A digital signal is quantized; i.e., the information is
  divided into discrete quantities of a finite size.
• Up to c. 1950, prior to the advent of the transistor (and
  thereafter the computer) all signals were essentially analog.
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                              ME 310
                     Chapter 7 - Digital Methods

Analog versus Digital Numbers
• Since analog numbers are continuous, there are an infinite
  number of values between any two readings.
   – Between 12.7 and 12.8 there exists 12.71, 12.75, 12.762572, etc.
• In a digital system, the values are limited to the smallest
  digital increment.
   – If 0.1 is the smallest increment, then there are no values between
     12.7 and 12.8.
   – Thus, any digital system can be represented by integer values
     alone by proper scaling; 12.7 and 12.8 become 127 and 128 (×10),
     respectively. Since the system is represented by integers alone, no
     real values exist between 127 and 128.

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                                                        University of Kentucky




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                   Chapter 7 - Digital Methods

Binary Representation
• Digital information is transmitted using binary; each
  number is made up of a pre-determined quantity of bits
  and 8 bits make a byte.
• Every bit has a value of 0 or 1; either off or on.
• Examples of a 4-bit number are 0000, 1111, and 1101.
                                              Logic 1 (5 V), high


                                              Logic 0 (0 V), low
               1     1       0       1

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                            ME 310
                   Chapter 7 - Digital Methods

Binary Ranges
• The total number of possible values in a digital system is
  given by 2N where N is the number of bits in each number.
          N          1        4          8       16            32


       Total         2       16      256     65,536        4x109

       Min           0        0          0       0              0

       Max           1       15      255     65,533         Big!


• Alternatively, one can use the range -total/2 to total/2-1.
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                        Chapter 7 - Digital Methods

Counting in Binary
• What is the decimal (base 10) equivalent of 1101?
      – (1 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 8 + 4 + 0 + 1 = 13


  •      0 =        0                              •   8 =     1000
                         1-bit
  •      1 =        1                              •   9 =     1001
                                 2-bit
  •      2 =       10                              • 10 =      1010
  •      3 =       11                              • 11 =      1011
                                           3-bit                             4-bit
  •      4 =     100                               • 12 =      1100
  •      5 =     101                               • 13 =      1101
  •      6 =     110                               • 14 =      1110
  •      7 =     111                               • 15 =      1111


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                                 ME 310
                        Chapter 7 - Digital Methods

Digital Measurements
• Why use digital signals?
      – Most measurement inputs are analog in nature - why bother
        converting to digital?
• Computers are digital - signal measurement can be easily
  automated for recording, analysis and process control.
• A single system can handle different types of input; e.g.,
  temperature, pressure, force, etc.
• Digital signals are noise-resistant; increased accuracy.
• Digital readouts are typically easier to read and less prone
  to error than analog gages.

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                                                              University of Kentucky




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                                   ME 310
                          Chapter 7 - Digital Methods

   Digital Conversion
   • The analog signal of the measuring input must be
     converted to a digital signal for the digital system
     (computer) to read it.
         – Analog-to-digital (A/D) converter.
   • Likewise, digital output can be converted into an
     equivalent analog signal.
         – Digital-to-analog (D/A) converter.
                                      A/D
     V                                                                       V
                                      D/A

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                                   ME 310
                          Chapter 7 - Digital Methods

   Digital Conversion
   • To convert an analog signal from a device, one needs
             Correct cabling          A/D converter (internal or external)
Input
                                  +                                  +
Signal



                   Computer                           Software

                                            +



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                           ME 310
                  Chapter 7 - Digital Methods

Resolution
• The resolution, Q, of a A/D converter is determined by the
  input range (for example, -10 V to +10 V) and the number
  of bits.
                      Q = ΔVFS
                                  2N
• Thus, a -10 to +10 V range has a voltage resolution of 78
  mV, 4.8 mV, or 0.3 mV for a 8-bit (28=256), 12-bit
          €
  (212=4096), or 16-bit (216=65,536) converter, respectively.
  (These are typical bit depths of standard A/D converters.)

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                                                 University of Kentucky




                           ME 310
                  Chapter 7 - Digital Methods

Quantization Error
• The finite resolution of an A/D converter results in a fixed
  error in the measurement since the actual analog signal will
  almost always fall somewhere between two bit values.
• This is called quantization error and can be estimated by

                        uq = εV 2

• The error can only be reduced by using more bits.


            €
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                                  ME 310
                         Chapter 7 - Digital Methods

    Problem 7.14
    • A 12-bit A/D has full scale range of E=5 V and an
       accuracy of 0.03% full scale. Estimate (i) quantization
       error; (ii) total error; (iii) relative uncertainty.
     The quantization error is determined by FSR and depth:
                          Q = 5V /212 = 1.2 mV
                          eQ = ±Q /2 = ±0.6 mV
        The maximum possible error is the sum of all errors:
               €
               emax = eQ + eaccuracy = 0.6 + 5(0.0003) = 2.1 mV
        The probable error is given by the relative uncertainty:
               €
                   2    2
           uE = ± eQ + eaccuracy = ± 0.6 + 5(0.0003) = ±1.6 mV
    €
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€



                                  ME 310
                         Chapter 7 - Digital Methods

    Saturation
    • The upper and lower bounds of A/D converters limit the
      range of input values that can be converted.
    • If the signal falls outside this range, it is saturated.

                                                         saturation


    V                                    V


                                    t                                          t
    • If a signal becomes saturated, it must be appropriately
      conditioned before conversion from analog to digital.
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                              ME 310
                     Chapter 7 - Digital Methods

Sampling
• The sampling rate is the rate at which the analog signal is
  converted to a digital record.
• Each converter has a hardware determined maximum
  sampling rate.
• Typical values are in the 1 kHz (1000 samples/second) to
  100 MHz (100,000,000 samples/second) range.
• Any analog signal which varies with a frequency greater
  (or even equal) than the sampling cannot be accurately
  measured. The signal is undersampled and aliasing may
  result.

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                              ME 310
                     Chapter 7 - Digital Methods

Aliasing Example (cf. 7.1, 7.2)
• Aliasing occurs not only in
  acquiring data in an experiment,
  but can also occur in creating
  data in a simulation
• Say we wish to simulate a sine
  wave; in MATLAB, type
      »x=[0:6:100]
      »y=sin(x)
      »plot(x,y,'bs-')

• We get something that looks
  like a sine wave, right?

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                               ME 310
                      Chapter 7 - Digital Methods

Aliasing Example (cont.)
• Since we are already familiar
  with the behavior of a sine
  wave, we can immediately tell
  that this is wrong
• Try plotting using finer step
  sizes, increasing the resolution
        »x=[0:5:100]
        »y=sin(x)
        »plot(x,y,’ro-')

         »x=[0:1:100]
         »y=sin(x)
         »plot(x,y,’gd-')
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                               ME 310
                      Chapter 7 - Digital Methods

Aliasing (cont.)
• In each of these cases, we have
  undersampled the signal
• The frequency is determined by
  the period, f=1/T=1/2π
• As a first guess, determine step
  size by dividing the period by 10,
  or ~0.6
         »x=[0:0.1:100]
         »y=sin(x)
         »plot(x,y,'bs-')
• Here we used a finer resolution
  to get more detail

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                            ME 310
                   Chapter 7 - Digital Methods

Sampling Rate & Nyquist Frequency
• The sampling rate is determined by the time interval
  between samples, δt
                         fs = 1
                                  δt
• The sampling theorem states that to accurately represent
  the measured signal, the sampling rate should be at least
            of
  twice that€ the highest frequency in the signal
                     f s = 2 f N > 2 f signal
• fN is known as the Nyquist frequency


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                                                  University of Kentucky




                            ME 310
                   Chapter 7 - Digital Methods

Determining Alias Frequencies
• As mentioned, if fs is too low, aliasing occurs -
  misinterpretation of these frequencies as “real” signals or
  frequencies are called alias frequencies
• Using a Fourier series representation of the signal, one can
  show that the frequencies f and f+m/δt are
  indistinguishable, where m is an integer
• The phase of the alias signal will vary from in-phase to
  out-of-phase as one goes from one alias frequency to the
  next (see the folding diagram, Fig. 7.3)

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                                                  University of Kentucky




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                              ME 310
                     Chapter 7 - Digital Methods

Fold-Back Diagram
• The observed alias frequency and the phase relation can be
  determined from the fold-back diagram
                                 Out-
                                     of-p
                                         hase
                                                 3fN
                                     ase
                                In-ph
                      2fN      Out-
                                   of-p
                                       hase
                        0                       fN
• Frequency content that is at frequencies above fN will
  appear as alias frequencies less than fN; Ii.e., superimposed
  (folded back) and appear as lower frequencies
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                              ME 310
                     Chapter 7 - Digital Methods

Problem 7.4
• Find the alias frequency that results from sampling the
  signal f1 at sampling frequency fs                     3fN

• f1=60 Hz, fs=90 Hz                            2fN
    – Nyquist frequency is fN=fs/2=45 Hz
                                                 0                    fN
    – The ratio f/fN=1.33
    – From Figure 7.3, a ratio of 1.33 corresponds to a “measured”
      frequency of 0.67fN
    – Thus, fmeasured=0.67fN=30 Hz and is out-of-phase with the original
      signal



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                   Chapter 7 - Digital Methods

Further Aliasing Issues
• Aliasing also crops up when examining frequencies in
  phase space (spectra)
• Cutting of a periodic signal either with too few periods or
  truncating it out-of-phase will produce (or “leak”) multiple
  spikes in the spectrum plot (see Fig. 7.4)
• This can be solved by taking many periods (at least 10)
  and starting and stopping the data set at the same point in
  the period
• One can also use anti-aliasing filters or windows to help
  alleviate these problems

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                            ME 310
                   Chapter 7 - Digital Methods

Calibration
• Since the analog signal is measured in volts and we really
  want to look at the signal in some real units (K, Pa, N, lb,
  etc.), we need to calibrate the signal during the conversion
  process.
• This requires using some known inputs on the device –
  another instrument is usually required if a calibration
  signal is not available.
                                                  Calibration curve
   V                    A/D     V

                                                                quantity
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                                                  University of Kentucky




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                         ME 310
                Chapter 7 - Digital Methods

In Class Demo
• DAQ System
                                               Computer
    Force             External
 transducer         A/D converter




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                         ME 310
                Chapter 7 - Digital Methods




                   Digital Circuits




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                               ME 310
                      Chapter 7 - Digital Methods

Digital Circuits
• Digital circuits can be used for a number of applications,
  including control and signal conditioning.
• A digital circuit uses high and low voltages to indicate if
  the control is on or off.
                                      on                             5V


           off                                      off               0V


• Typical values are 0 to 5 V. For example, if 0 < V < 2.5 V,
  the control is off. If 2.5 V < V < 5 V, the control is on.
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                               ME 310
                      Chapter 7 - Digital Methods

Mechanical Digital Circuits
• Digital circuits don’t have to be
  electronic (though almost all
  are) they can be mechanical as
  well (see Babbage’s difference
  engine to the right, c. 1833, it
  could keep 32 decimal places)
• The last mechanical computer
  was Harvard-IBM Mark I,
  completed in 1944. It kept 23
  decimal places, had a memory
  of 72 counters, and could do a
  multiply in six seconds.
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                               ME 310
                      Chapter 7 - Digital Methods

Electronic Digital Circuits
• Electronic digital circuits                Vacuum tube
  include
    – Vacuum tubes
    – Relays
    – Magnetic cores or amplifiers
    – Superconducting junctions
    – Transistors




                                             Superconducting
                                            Josephson junction
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                               ME 310
                      Chapter 7 - Digital Methods

Transistors
• A transistor is a simple circuit that can produce high or low
  output from high or low input.
• The exact relationship between the input and output (or
  multiple inputs and outputs) depends on the transistor
  design.
• Transistors are the most basic element of CPU design -
  they determine the overall processing power.




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                               ME 310
                      Chapter 7 - Digital Methods

The First Transistor
• First transistor developed in
  1947 by Bardeen, Brattain, and
  Shockley at Bell Labs (later
  awarded Nobel prize for
  discovery)
• Associates later developed IC
  chip and formed Intel
• Less than a decade later, all
  computers used transistors
  (instead of vacuum tubes or
  relays, for example) for logic,
  and later for memory

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                               ME 310
                      Chapter 7 - Digital Methods

Transistor Operation




                                                      Arrow indicates
                                                    direction of current


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                               ME 310
                      Chapter 7 - Digital Methods

Moore’s Law
• 8086 (c. 1974) computed in 8-
  bits and had 6,000 transistors
• Pentium (1993) computes in 32-
  bits and has 3,000,000
  transistors
• P4 (2001) computers in 32-bits
  and has 42,000,000 transistors




        TI’s First Transistor
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                               ME 310
                      Chapter 7 - Digital Methods

IC Chips
• Transistors have been combined
  with other electronic elements
  such as resistors and diodes to
  make more complex digital
  circuits - these have been
  combined into single devices
  called integrated circuit or IC
  chips.
• Each IC chip consists of several
  logic units along with input
  power (+5 V) and ground.


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                   Chapter 7 - Digital Methods

Logic Units
• Logic units are devices which produce high or low output
  for given input conditions.
• The basic logic units are the inverter (which flips the
  signal) and the AND, OR, NAND, and NOR gates.
• These can all be combined in a single IC chip to perform a
  one function or serve as a single platform to perform a
  single type of operation for different parts of a circuit.
• The input-output relation of a logic unit or gate can be
  determined from a logic or truth table.

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                            ME 310
                   Chapter 7 - Digital Methods

The Inverter
• Inverts signal; low (0) ⇒ high (1) or high (1) ⇒ low (0).
• It has one input and one output.



                                                 A           Q

       A                           Q             0           1

                                                 1           0




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                  Chapter 7 - Digital Methods

The AND Gate
• Both inputs must be high to yield a high output.

                                           A    B        Q

                                           0    0        0
 A                             Q
                                           1    0        0
 B
                                           0    1        0

                                           1    1        1

• AND gates can have more than 2 inputs as well.

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                           ME 310
                  Chapter 7 - Digital Methods

The NAND Gate
• The opposite of the AND gate.
                                           A    B        Q

                                           0    0        1
 A                             Q
                                           1    0        1
 B
                                           0    1        1

                                           1    1        0

• A single NAND gate contains 4 transistors, 3 resistors, a
  diode, and is powered by a separate input voltage (+5 V).
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                   Chapter 7 - Digital Methods

OR and XOR Gates
• OR gate is high if either input is high.
                                                 A   B        Q

                                                 0   0        0
 A                                 Q
                                                 1   0        1
 B
                                                 0   1        1

                                                 1   1        1


• Exclusive OR (XOR) is high only if either output is high,
  but not both.
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                            ME 310
                   Chapter 7 - Digital Methods

IC Families
• Various logic units can be combined so that a IC can
  perform a single function.
• Families of IC chips have been created that have special
  characteristics; the most common of these is the TTL
  family (transistor-transistor logic).
• Some sample TTL chips include
     – 7400    quad 2-input NAND gate
     – 7408    quad 2-input AND gate
     – 7430    8-input NAND gate
     – 7474    dual D-edge-triggered flip-flop
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                  Chapter 7 - Digital Methods

 7408 TTL
 • Quad 2-input AND Gate


TTL Voltage
  Power




        4 2-input AND gates               TTL Ground

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                           ME 310
                  Chapter 7 - Digital Methods

 7430 TTL
 • 8-input NAND gate; VCC max is 5.25 V, min is 4.75 V




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                           Chapter 7 - Digital Methods

7474 TTL
• The flip-flop; used to store 1 bit of information for later
  use; very important for CPUs.          A   B    Q Q’
 A                             Q
                                                        1       1      1      0

                                                        1       1      0      1

                                                        0       1      1      0
                                             Q’
                                                        1       0      0      1
 B
     – If A and B are opposite of one another, then Q follows A and Q' is the inverse of Q.
     – If both A and B are switched to 1 simultaneously, then the circuit remembers what
       was previously presented on A and B.

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                                    ME 310
                           Chapter 7 - Digital Methods

7474 D-Type Flip-Flop




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                             ME 310
                    Chapter 7 - Digital Methods

Sample IC Circuits
• Counter
• Frequency meter
• Timer
   – Most commonly used is the 555.
• Multiplexer




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                             ME 310
                    Chapter 7 - Digital Methods

Frequency Meter
• Counts how many times a signal pulses per second (Hz).
                                                  1/2 Hz square wave
   5 MHz crystal-         Cascade of seven
     controlled             divide by 10
     oscillator               IC chips




           Event
          counter                                  Input signal


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                           ME 310
                  Chapter 7 - Digital Methods

TTL Frequency Divider




                                   A 7474 flip-flop and a 7400
                                            NAND Gate



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                           ME 310
                  Chapter 7 - Digital Methods

CMOS Logic Families
• The complementary metal oxide semiconductor family
  (CMOS) has equivalents to most of the TTL chips.
• CMOS chips are much lower in power requirements
  (drawing about 1 mA) and operate with a wide range of
  supply voltages (typically 3 to 18 volts).
• The CMOS model number will have a C in the middle of
  it, e.g., the 74C04 is the CMOS equivalent to the TTL
  7404. A big drawback is extreme sensitivity to static
  electricity - they must be carefully protected from static
  discharges.

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                                                                           24
                              ME 310
                     Chapter 7 - Digital Methods

Final Comments
• Test
• Expect to see
   – Normal and student-t tables
   – Propagation of error
   – Simple filters (passive, active)
   – Low-pass, high-pass, band-pass filters
   – DAQ systems (issues related to A/D, D/A conversion)




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                              ME 310
                     Chapter 7 - Digital Methods




               Additional Digital Examples




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                                                                              25
                            ME 310
                   Chapter 7 - Digital Methods

Analog versus Digital
• Any signal is either a digital or analog signal – the
  difference is obvious in their appearance.
      Analog (continuous)              Digital (discrete)




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                            ME 310
                   Chapter 7 - Digital Methods

Analog versus Digital Examples
• Most any type of measurement can be digitized – let’s
  look at time and temperature, for example.
                         – While the change in the height of the
                           mercury is gradual, the readings occur in
                           discrete units - we can guess at a more
                           accurate reading but are limited to the tick-
                           marks for accurate measurements.


                                            Mercury thermometer



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                             ME 310
                    Chapter 7 - Digital Methods

Analog versus Digital Examples
• Thermocouple
   – The thermocouple provides a continuous (analog) signal into the
     meter, which is digitized and displayed on a LCD panel.




                                                                       12.8

               Analog signal            Digitized signal             Digital
                                                                     Readout


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                             ME 310
                    Chapter 7 - Digital Methods

Analog versus Digital Examples
• Timekeeping is another example.
   – Time can be measured and displayed in either analog or digital
     fashion.
   – Time displays usually (but not always!) indicate if the system is
     keeping track of time digitally or analog, but beware - an analog
     display may sweep seconds using discretized ticks!.




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