# Analogue Digital Conversion

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```					Analogue to Digital Conversion

Digital Signal Processing
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A digital signal is an approximation of an analog one Levels of signal are sampled and converted to a discrete bit pattern. Resistor networks can be used to convert digital signals into analogue voltages

Step (discrete) approximation
sample “stair-step” approximation of original signal

level

more samples give greater accuracy time hold time for sample

This Lecture
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Methods of analogue to digital conversion
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flash counter ramp successive approximation

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Sample interval and aliasing problems Sample and hold circuits

The Comparator
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Most A-D converters use a comparator as part of the conversion process A comparator compares 2 signals A and B
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if A > B the comparator output is in one logic state (0, say) if B > A then it is in the opposite state (1, say)

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A comparator can be built using an op amp with no feedback
+

analogue input reference voltage

-

7V

4V

1V

input signal

+

-

-

000 001 010 011 100 101 110 111

2V

+

A 0 1 1 1 1 1 1 1

B 0 0 1 1 1 1 1 1

C 0 0 0 1 1 1 1 1

D 0 0 0 0 1 1 1 1

E 0 0 0 0 0 1 1 1

F 0 0 0 0 0 0 1 1

G 0 0 0 0 0 0 0 1

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Converter input range (V) <1 >1-2 >2-3 >3-4 >4-5 >5-6 >6-7 >7

Comparator Outputs

Encoder Output

3V

+

-

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+

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-

5V

+

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-

6V

+

-

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Uses a reference and a comparator for each of the discrete levels represented in the digital output Number of comparators = number of quantisation levels Not practical for more than 10 bit converters generally fast but expensive

+

Flash Converter

G

F

E

encoder
D

digital output

C

B

A

Counter-ramp Converter
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Comprises a D-A converter, a single comparator, a counter, a clock and control logic When a conversion is required
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A signal (conversion request) is sent to the converter and the counter is reset to zero a clock signal increments the counter until the reference voltage generated by the D-A converter is greater than the analogue input At this point in time the output of analogue input the comparator goes to a logic 1, which notifies the control logic the comparitor conversion has finished D-A Converter The value of the counter is output as the digital value
Counter +

-

clock and control logic

Counter-ramp Converter
conversion request

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The time between the start and end of the conversion is known as the conversion time A drawback of the counter-ramp converter is the length of time required to convert large voltages We must assume the worst case when calculating conversion times

comparitor output d.c input voltage

D-A converter output

0

1

2

3

4

5

6

6

6

6

6

6 counter output
clock

Successive Approximation Converter
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Counter replaced by a register Contents of register decided by clock and control logic When a conversion is required:
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Vd = 0

Vin

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if Vc = 0 then Vd < Vin => leave MSB set if Vc =1 then Vd > Vin => clear MSB

D-A Converter

Vd

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Repeat previous step for other bits in MSB to LSB order

4-bit reg b3 b2 b1 b0

-

MSB set to a 1

+

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contents of register cleared

analogue input

Vc
comparitor

clock and control logic

The successive approximation A-D converter
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Example:
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A 4-bit successive approximation A-D converter has a full-scale input of +15V. Show how the A-D converter would convert the analogue voltages 10.9V and 3.1V into their digital equivalents

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Total conversion time = n+1 cycles where n = the number of bits in the code word

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Assume D-A converter output has stepped up to V1. Because Vi > V1, the output has stayed at a logic 0. On the next clock pulse the DA output rises to V2. V2 > Vi, comparator output becomes logic 1 and conversion is completed. Maximum possible error = q.

V2 VI V1

Quantisation
7V 6V 111 110 101
100 011

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Output from an A-D converter can only be one of a limited number of possible codes
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5V
4V 3V

2V 1V 0V

010 001 000

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Hence quantisation errors will arise. Possible to reduce this error to half by adding q/2 to the output of the D-A converter Equivalent of “rounding” decimal numbers.

7V

111 110 101 100 011 010

6V 5V
4V 3V 2V

1V 0V 001 000

Quantisation
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Quantisation errors can be reduced by increasing the number of bits Common for A-D converters to have 16 bit or better resolution However the accuracy of the reference voltage must be of the same precision Example:
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Consider a A-D converter where Vref is only accurate to within 1%

Summary
Device Flash  Counter ramp Sucessive approx
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Comme nt fast and expensive simple but slow w idely used

Conve rsion time Bes t Average Worst 1 1 1 1 2n/2 2n n+1 n+1 n+1

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One way to reduce quantisation errors is to use a larger number of bits in the codeword absolute accuracy of conversion may not be as good as the resolution if the error tolerance for reference voltages gets too large A multiplexer enables one A-D converter to be switched between several signal inputs

Multiplexers
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The A-D converters described above have all been single-input devices It is often necessary to convert several analogue signals to binary code words Integrated circuit multiplexers are available which can select one of its analogue inputs at a time and present it to a single A-D converter 1
Analogue inputs 2 3 4 digital control lines Selected analogue output switch decoding logic

Conversion of a.c. signals
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The A-D converters that we have looked at present no special problems with d.c.
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Example consider reading room temperature and plotting against time Not possible to sample at every instant in time
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rate at which we take samples is known as the sampling rate
temp

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sampling too fast can be inefficient
A3
A2

A1

time

Conversion of a.c. signals
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Sampling too slowly can cause information to be lost
temp

A2

A1

t1

t2

time

Sample Time vs Frequency
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Consider what happens when the signal frequency is higher than the sampling frequency.
voltage

time

sample frequency is number of samples / second

Conversion of a.c. signals
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Effects of under-sampling
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voltage

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possible to interpolate high frequency components as low frequency ones these errors are said to be caused by aliasing important to preceed A-D converter with a low pass filter to remove high frequencies known as an anti-aliasing filter

time

Sample frequency must be at least twice the highest signal frequency (2f is also called the Nyquist Frequency).

Example
What

is the maximum frequency of input signal that can be converted by an A-D convertor with a conversion time of 0.25 mS?
samples

per second = 1000 / 0.25 = 40,000

frequency in input signal has to be half this or 20kHz.

Maximum

Sample-and-hold devices
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Sampling rule tells us at what rate to make conversions, but there is still another problem associated with changing signals

voltage

t1

t2

time

Sample-and-hold devices
voltage



t1

t2

time

To remove the problem a sample and hold device which
samples the input and holds this value until the end of the conversion is often used
switch

storage capacitor

Sample-and-hold devices
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A number of problems exist with the previous sample and hold circuit
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load placed on the input of the circuit by charging the capacitor during the sample phase current flowing from the capacitor used in the conversion will reduce the voltage stored on the capacitor
+ + C

sample/hold control line

What you should be able to do
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Explain the operation of binary weighted resistor and R2R ladder networks. Recall their general layout. Calculate the output voltage given an input 4-bit value. Explain quantisation with reference to D-A conversion. Explain the operation of flash, counter ramp and successive approximation A-D convertors. Recall their general layout. Recall their conversion time relative to number of bits required. Explain quantization with reference to A-D conversion. Explain the aliasing problem and the relationship between sample rate and input signal frequency.

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