Transcript Conversion New

Data Representation
• Computers use base 2,

n
0
1
2
3
a
2

a
2

a
2

a
2

a
2

n
0
1
2
3
n 0
instead of base 10:

n
0
1
2
3
a
10

a
10

a
10

a
10

a
10

n
0
1
2
3
n 0
• Internally, information is represented by binary digits; “switches”
that are either on or off.
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111 BSC Data Acquisition and Control
Data Converters
• Signals must be converted to their digital
representation:
Computer
– ADC: Analog to Digital Converter.
• Digital information must be converted to analog
Computer
signals:
– DAC: Digital to Analog Converter.
• ADCs and DAQs are imperfect. Important parameters
include:
– Sample rate.
– Resolution.
– Linearity.
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Conversion: Sampling
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Conversion: Sample Rate
• High Sample Rates can better represent high
frequency waveforms.
f  1 f s  10.4
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f  1 f s  2.35
111 BSC Data Acquisition and Control
Conversion: Nyquist Theorem
• What is the lowest
sample rate that can
represent a signal?
f 1
fs  2
• The Nyquist Theorem states that a wave can be correctly
represented when sampled at a rate equal to twice the
highest frequency of the wave.
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Conversion: Aliasing
• Sampling below the Nyquist frequency leads to aliasing:
f  1 f s  0.53
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f 1
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f s  1.7
Conversion: What rate?
• Preferably, one should operate far above the Nyquist limit.
f  1 f s  10.4
f  1 f s  2.35
• Sampling 10 to 100 times higher than the signal frequency
generally works very well.
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111 BSC Data Acquisition and Control
Conversion: What rate?
• Unfortunately, it is often impossible to sample this fast.
– The employed device may not be capable of sampling at the
desired rate.
– The desired rate may be technologically impossible.
– Even if it is possible, you may not be able to afford the
required device.
• ADC’s
–
–
–
–
250kS/s --- $375 for a computer card.
10MS/s --- $4000.
200MS/s --- $6000.
1GS/s ---$10,000.
• DAC’s
– Static --- 8 Channels for $700.
– 1MS/s --- 4 Channels for $800.
– 200M/s --- 1 Channel for $6000.
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111 BSC Data Acquisition and Control
Conversion: What rate?
• Unfortunately, it is often impossible to sample this fast.
– The employed device may not be capable of sampling at the
desired rate.
– The desired rate may be technologically impossible.
– Even if it is possible, you may not be able to afford the
required device.
– Fast sampling may produce or require too much data.
• Limited buffer sizes
• Limited computational speeds.
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111 BSC Data Acquisition and Control
Conversion: Near Nyquist Sampling
• Using the “Sampling Simulator,” explore the effects of various
sampling rates on different waveforms with interpolation off.
• Real world signals are continuous. Sampling is
discontinuous. Interpolation is used to turn the
discontinuous samples into a continuous signal.
No Interpolation
“Flat” Interpolation
“Ramp” Interpolation
Comb Interpolation
Commonly used by DACs
Option on expensive DACs
Theoretically optimal
• Using interpolation, explore the effects of various sampling
rates on different waveforms. Note: that well above the Nyquist
frequency, ramp interpolation represents the signal better than flat
interpolation.
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111 BSC Data Acquisition and Control
Conversion: Near Nyquist Sampling
• Below the Nyquist Frequency, aliasing can produce
deceptively pretty waveforms. Be careful.
• Just above the Nyquist Frequency, the sampled
waveforms look nothing like the original waveform.
Is the Nyquist Theorem wrong?
f  1 f s  2.05
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111 BSC Data Acquisition and Control
Conversion: Near Nyquist Sampling
• The sampled spectrum has two peaks;
– One at the original signal frequency.
– One above the Nyquist frequency.
f  1 f s  2.05
• We observe an apparent beat between these frequencies.
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111 BSC Data Acquisition and Control
Conversion: Near Nyquist Sampling
• The sampled spectrum has two peaks;
– One at the original signal frequency.
– One above the Nyquist frequency.
f  1 f s  2.05
• We observe an apparent beat between these frequencies.
• The higher frequency can be filtered away to recover the
original signal from the sampled signal.
• Filtering must be done carefully.
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111 BSC Data Acquisition and Control
Conversion: Near Nyquist Sampling
Two Tone Signal
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f  0.7,1
fs  3
111 BSC Data Acquisition and Control
Conversion: Near Nyquist Sampling
AM Modulated Signal f  0.05 f c  0.93
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111 BSC Data Acquisition and Control
fs  2
Conversion: RC Filtering
• We need to kill frequencies higher than the Nyquist Frequency.
• Could use an RC filter:
f Low  1
• First Order RC filters are not sharp enough.
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111 BSC Data Acquisition and Control
Conversion: RC Filtering
• Try a 2nd order filter:
f Low  1
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111 BSC Data Acquisition and Control
Conversion: RC Filtering
• We need even higher order.
f Low  1
• A 6th order RC filter kills the amplitude by a factor of 100 one
octave above its cutoff.
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Conversion: RC Filtering
• But the signal is significantly reduced in the passband as well!
f Low  1
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Conversion: Sharper Filters
• Filter designs using inductors (or gyrator synthesized
inductors) are much sharper.
• Using, as a figure of merit, a reduction by a factor of 100 one
octave above the cutoff:
f Low  1
• Chebyshev has the best frequency response.
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111 BSC Data Acquisition and Control
Conversion: Temporal Response
• Unfortunately, good frequency response generally yields poor
temporal response.
f Low  1
• Bessel filters have the best temporal response.
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111 BSC Data Acquisition and Control
Conversion: When is Filtering Required?
• Both DACs and ADC usually require filters.
– DACs:
• Filtering turns the discontinuous output from your DAC into a
continuous signal.
• Occasionally, the device being driven by the DAC is insensitive to
the high frequency components in the unfiltered DAC output. If so,
filtering is unnecessary.
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111 BSC Data Acquisition and Control
Conversion: When is Filtering Required?
• Both DACs and ADC usually require filters.
– ADCs:
• Filtering prevents aliasing.
– Input signals are often noisy, and this noise may extend above the
Nyquist frequency.
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111 BSC Data Acquisition and Control
Conversion: When is Filtering Required?
• Both DACs and ADC usually require filters.
– ADCs:
• Filtering prevents aliasing.
– Input signals are often noisy, and this noise may extend above the
Nyquist frequency.
On sampling:
• Frequencies above the Nyquist Frequency mirror:
f Observed  f Nyquist   f Actual  f Nyquist 
 fSample  f Actual
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111 BSC Data Acquisition and Control
Conversion: When is Filtering Required?
• Both DACs and ADC usually require filters.
– ADCs:
• Filtering prevents aliasing.
– Aliasing artifacts confuse the spectrum and distort the waveforms.
– Unless the spectrum is very quiet above the Nyquist frequency, the
signal must be filtered before it is converted by the ADC.
– But filtering itself introduces artifacts:
» Spectral amplitude errors in the passband.
» Distortions to the temporal waveform.
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111 BSC Data Acquisition and Control
Conversion: When is Filtering Required?
• Both DACs and ADC usually require filters.
– ADCs:
• Filtering prevents aliasing.
• Very occasionally aliased signals can still be used.
– Spectrum is predictable, but reversed.
– The DAC’s analog bandwidth may make
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111 BSC Data Acquisition and Control
Conversion: When is Filtering Required?
• Both DACs and ADC usually require filters.
– ADCs:
• Filtering prevents aliasing.
• Filtering turns the discontinuous measurements from your ADC
into a continuous signal.
• “Ideal” filters for static signal reconstruction can be developed using
Fourier Transforms.
• Filtering is unnecessary if you are only interested in the spectral
content of your signal.
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111 BSC Data Acquisition and Control
Conversion: Resolution
• Resolution specified in number of bits.
• n-bit converter can represent 2n levels.
22  4
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24  16
111 BSC Data Acquisition and Control
26  64
BSC Data Acquisition Card
Function
Number
Rate
Resolution
ADC
8
200kS/s*
12 bits
DAC
2
1MS/s
12 bits
Digital In/Out
8 bits
*Though not spec’d to do this the card will digitize faster than 1.6MS/s when acquiring a single channel.
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111 BSC Data Acquisition and Control
DAC Circuits: Scaled Resistor
• bn is either 0 (off) or 1 (on.) Then:
Vout  b0  2b1  4b2  8b3  16b4
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111 BSC Data Acquisition and Control
DAC Circuits: Scaled Resistor DAC Errors
• Focus on a low and high order bit:
Vout  b0 
 4096b12 
• What happens if the high order bit
resistor is off by 1%?
Say: Vout  b0   4055b12 
or: Vout  b0   4137b12 
• Instead of 0111111111111  1000000000000 changing the
output from 4095 to 4096, it would change to 4055 or 4137.
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111 BSC Data Acquisition and Control
DAC Circuits: Scaled Resistor DAC Errors
• A 16bit DAC requires resistors accurate to 0.002% over
a 1:65536 resistance range.
• Such accurate resistors cannot be fabricated.
• Accurate resistors can be fabricated over a narrow
resistance range.
•Laser trimming.
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DAC Circuits: R-2R
Virtual Ground
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DAC Circuits: R-2R Ladder
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DAC Circuits: R-2R Ladder
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DAC Circuits: R-2R Ladder
Vout 
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1
1
1
1
b0  b1  b2  b3  b4
16
8
4
2
111 BSC Data Acquisition and Control
ADC Circuits: Flash (Parallel) Converters
• Very fast.
• Low Resolution
• Expensive
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111 BSC Data Acquisition and Control
1. ADC Circuits: Successive Approximation
1.
2.
3.
4.
Make a guess.
Convert the guess to a voltage with a DAC.
Compare the guess voltage to the actual voltage.
Refine the guess.
.
.
.
5. Stop when satisfied with the accuracy of the
answer.
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