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Chapter 7
Introduction to signal
processing
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Key Highlights
 Analogue and Digital Signal Processing
 Analogue filtering
 Signal Amplification
 Introduction to DSP
 Analogue-to-Digital-Converters
 Digital-to-Analogue Converters
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Introduction
 Signal processing is concerned with improving the
quality of the reading or signal at the output of a
measurement system, and one particular aim is to
attenuate any noise in the measurement signal that
has not been eliminated by careful design of the
measurement system
 Other
functions are signal filtering, signal
amplification, signal attenuation, signal linearization
and bias removal
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Analogue and Digital Signal
processing
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 In the past it was analogue using various types of
electronic circuit. However, the ready availability of
digital computers in recent years has meant that
signal processing has increasingly been carried out
digitally
 DSP is inherently more accurate than analogue
techniques, but for measurements coming from
analogue sensors and transducers, because an
ADC stage is necessary before the digital
processing can be applied, thereby introducing
conversion errors.
 Also, analogue processing remains the faster.
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Analogue signal filtering
 Signal filtering consists of processing a signal to
remove a certain band of frequencies within it. The
band of frequencies removed can be either at the lowfrequency end of the frequency spectrum, at the highfrequency end, at both ends, or in the middle of the
spectrum. Filters to perform each of these operations
are known respectively as low-pass filters, high-pass
filters, band-pass filters and band-stop filters (also
known as notch filters). All such filtering operations
can be carried out by either analogue or digital
methods.
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Analogue signal filtering
 A filter is a circuit that passes certain frequencies and
attenuates or rejects all other frequencies.
 The passband of a filter is the range of frequencies
that are allowed to pass through the filter with
minimum attenuation (usually defined as less than -3
dB of attenuation).
 The critical frequency, fc
(also called the cutoff
frequency) defines the end of the passband and is
normally specified at the point where the response
drops -3 dB (70.7%) from the passband response.
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Analogue signal filtering
 Following the passband is a region called the
transition region that leads into a region called the
stopband. There is no precise point between the
transition region and the stopband.
 The band of frequencies removed (stop band) can be
either at the low-frequency end of the frequency
spectrum, at the high-frequency end, at both ends, or
in the middle of the spectrum. Filters to perform each
of these operations are known respectively as lowpass filters, high-pass filters, band-pass filters and
band-stop filters (also known as notch filters).
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Passive Filters-Low Pass Filter
Low-pass filter circuit and its frequency
characteristics.
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Passive Filters-Low Pass Filter
 The voltage gain for the network
is :


, the time constant. The amplitude
characteristic is

/
 And the phase characteristic is

 Note that at fc,
the amplitude is
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Passive Filters-High Pass Filter
High-pass filter circuit and its frequency
characteristics.
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Passive Filters-High Pass Filter
 The voltage gain for this network is

 Where once again
. The magnitude of the
function is

/
 And the phase is

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Passive Filters-High Pass Filter
 The half-power frequency is
and the phase at
this frequency is 45°.
 The magnitude and phase curves for this high-pass
filter are shown in figure above.
 At low frequencies the magnitude curve has a slope of
+20 dB/decade due to the term
in the numerator .
 Then at the break frequency the curve begins to
flatten out.
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Passive Filters-Band pass and
Band rejection filters
Band-pass and
rejection
filters
characteristics.
13
bandand
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Passive Filters-Band pass and
Band rejection filters
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 Ideal and typical amplitude characteristics for simple
band-pass and band-rejection filters are shown in
Figs. a and b above, respectively.
 Simple networks that are capable of realizing the
typical characteristics of each filter are shown in
figures c and d above. ω0 is the center frequency of
the pass or rejection band and the frequency at which
the maximum or minimum amplitude occurs.
are the lower and upper break
frequencies or cutoff frequencies, where the amplitude
is of the maximum value.
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Passive Filters-Band pass and
Band rejection filters
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 The width of the pass or rejection band is called
bandwidth, and hence

 Consider the band-pass filter.

/
 And therefore the amplitude characteristic is


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Passive Filters-Band pass and
Band rejection filters
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

.
 Therefore, the frequency characteristic for this filter is
shown in figure e. The center frequency is
 At the lower cutoff frequency

 Or
.
Solving this equation gives
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Passive Filters-Band pass and
Band rejection filters

/
17
/
 At the upper cutoff frequency

 Or

. solving this equation gives
/
/
 And the bandwidth of the filter is given by:

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Passive Filters-Example
 Consider the frequency-dependent network in Fig.
below. Given the following circuit parameter values:
L=159 μH, C=159 μF, and R=10 Ω, let us demonstrate
that this one network can be used to produce a lowpass, high-pass, or band-pass filter.
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Passive Filters-Example
 The voltage gain
is found by voltage division to
be

/
.
.
.
 which is the transfer function for a band-pass filter.
 At resonance,
, and hence
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Passive Filters-Example
 Now consider the voltage gain
/
.
.
 which is a second-order high-pass filter transfer
function.
 Again, at resonance,

 Similarly, the voltage gain
is
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Passive Filters-Example

.
.
.
 which is a second-order low-pass filter transfer
function. At the resonant frequency

 Thus, one circuit produces three different filters
depending on where the output is taken. This can be
seen in the Bode plot for each of the three voltages in
figure below, where VS is set to
.
 KVL is satisfied. i.e.
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Passive Filters-Example
Bode plots for network in figure above
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Passive Filters-Exercise
 Given the filter network shown below, sketch the
magnitude characteristic of the Bode plot for
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Passive Filters-Exercise
 A band-pass filter network is shown below. Sketch the
magnitude characteristic of the Bode plot for
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Active Filters
 Passive filters have some serious drawbacks. One
obvious problem is the inability to generate a network
with a gain>1 since a purely passive network cannot
add energy to a signal.
 Another serious drawback of passive filters is the need
in many topologies for inductive elements. Inductors
are generally expensive and are not usually available
in precise values. In addition, inductors usually come
in odd shapes (toroids, bobbins, E-cores, etc.) and are
not easily handled by existing automated printed
circuit board assembly machines.
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Active Filters
 By applying operational amplifiers in linear feedback
circuits, one can generate all of the primary filter types
using only resistors, capacitors, and the op-amp
integrated circuits themselves.
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Active Filters-Example 1
 Let us determine the filter characteristics of the
network shown in figure below.
Operational amplifier filter circuit.
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Active Filters-Example 1

/

/
 Therefore, the voltage gain of the network is
/

 Note that the transfer function is that of a low-pass
filter.
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Active Filters-Example 2
 We will show that the amplitude characteristic for the
filter network in Fig. a is as shown in Fig. b.
Operational amplifier circuit and its amplitude characteristic
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Active Filters-Example 2


 Where
and
 Since
, the amplitude characteristic is of the
form shown in Fig. b. Note that the low frequencies
have a gain of 1; however, the high frequencies are
amplified. The exact amount of amplification is
determined through selection of the circuit parameters.
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Active Filters-Exercise
 Given the filter network shown below, determine the
transfer function
sketch the magnitude
and identify
characteristic of the Bode plot for
the filter characteristics of the network.
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Second Order Filters
 All the circuits considered so far in this section have
been first-order filters. In other words, they all had no
more than one pole and/or one zero. In many
applications, it is desired to generate a circuit with a
frequency selectivity greater than that afforded by firstorder circuits.
 The next logical step is to consider the class of
second-order filters. For most active filter applications,
if an order greater than two is desired, one usually
takes two or more active filter circuits and places them
in series so that the total response is the desired
higher-order response.
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Second Order Filters
 In general, second-order filters will have a transfer
function with a denominator containing quadratic poles
. For high-pass and low-pass
of the form
circuits,
and
For these
circuits, is the cutoff frequency, and is the
damping ratio.
 For band-pass circuits,
and and
where
is the center frequency and Q is the quality
factor for the circuit. Notice that Q is a measure of the
selectivity of these circuits. The bandwidth is as
discussed previously.
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Second Order Filters
 The transfer function of the second-order low-pass
active filter can generally be written as

 Where
is the dc gain.
 A circuit that exhibits this transfer function is illustrated
in Fig. 6.10 and has the following transfer function:

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Second Order Filters
Figure; Second-order low-pass filter.
Question: Determine the damping ratio, cutoff frequency
and dc gain for the network in figure 6.10 if
0.1
(2000 rad/s, 1.5. -1)
5
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Differential amplification
1
If
, then
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Instrumentation amplifier
1
2
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Signal attenuation
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Exercise
Read about and make notes about
1. Signal linearization
2. Bias (zero drift) removal
3. Signal integration
4. Voltage comparator
5. Phase-sensitive detector
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Introduction to DSP
 A review of the difference between digital and analog
quantities:
 Digital quantities—values can take on one of two
possible values.
 Actual values can be in a specified range, so exact
value is not important.
 Analog quantities—values can take on an infinite
number of values, and the exact value is important.
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Introduction to DSP
 Most physical variables are analog, and can take on any
value within a continuous range of values.
 Normally a nonelectrical quantity.
• A transducer converts the physical variable to
an electrical variable.
– Thermistors, photo-cells, photodiodes, flow meters,
pressure transducers, tachometers, etc.
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Introduction to DSP
 The transducer’s electrical analog output is the
analog input to the analog-to-digital converter.
 The ADC converts analog input to a digital output
 Output consists of a number of bits that represent the
value of the analog input.
 The binary output from the ADC is proportional to
the analog input voltage.
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Introduction to DSP
 The digital representation of the process variable
is transmitted from the ADC to the digital computer
 The digital value is stored & processes according
to a program of instructions that it is executing.
 The program might perform calculations or other
operations to produce output that will eventually be
used to control a physical device.
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Introduction to DSP
 Digital output from the computer is connected to a DAC
 Converted to a proportional analog voltage/current.
• The analog signal is often connected to some device
or circuit that serves as an actuator to control the
physical variable.
– An electrically controlled valve or thermostat, etc.
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DSP
 Digital techniques achieve much greater levels of
accuracy in signal processing than equivalent
analogue methods. However, the time taken to
process a signal digitally is longer than that required
to carry out the same operation by analogue
techniques, and the equipment required is more
expensive.
 Therefore, some care is needed in making the
correct choice between digital and analogue
methods.
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Signal sampling
 The process of AD conversion consists of sampling
the analogue signal at regular intervals of time. Each
sample of the analogue voltage is then converted into
an equivalent digital value. This conversion takes a
certain finite time, during which the analogue signal
can be changing in value. The next sample of the
analogue signal cannot be taken until the conversion
of the last sample to digital form is completed. The
representation within a digital computer of a
continuous analogue signal is therefore a sequence of
samples whose pattern only approximately follows the
shape of the original signal.
 This pattern of samples taken at successive, equal
intervals of time is known as a discrete signal.
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Sampling
…Discrete signal matches the analog signal well
- - -Discrete signal does not match the analog signal
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Aliasing
 If the rate of sampling was very much less than the
frequency of the raw analogue signal, such as 1
sample per second, only the samples marked ‘X’ in
the figure would be obtained. Fitting a line through
these ‘X’s incorrectly estimates a signal whose
frequency is approximately 0.25 cycles per second.
 This phenomenon, whereby the process of
sampling transmutes a high-frequency signal into a
lower frequency one, is known as aliasing.
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Quantisation
 It’s a factor that affects the quality of the signal..
 Quantization describes the procedure whereby the
continuous analogue signal is converted into a number
of discrete levels. At any particular value of the analogue
signal, the digital representation is either the discrete
level immediately above this value or the discrete level
immediately below this value.
 If the difference between two successive discrete levels
is represented by the parameter Q, then the maximum
error in each digital sample of the raw analogue signal is
±Q/2. This is the quantisation error a to the no. of bits
used to represent the samples in a digital format.
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Sample and hold circuit
It holds the input signal at a constant level whilst the AD
conversion process is taking place reducing the
conversion errors that would occur if the input was
changing.
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Sample and hold circuit
 The operational amplifier circuit shown in Figure
above provides this sample and hold function.
 The input signal is applied to the circuit for a very short
time duration with switch S1 closed and S2 open, after
which S1 is opened and the signal level is then held
until, when the next sample is required, the circuit is
reset by closing S2.
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Analogue-to-digital
converters
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 Important factors in the design of an analogue-to-
digital converter are the speed of conversion and the
number of digital bits used to represent the analogue
signal level.
 The minimum number of bits =8. (resolution of 1 part
in 256)
 Several types of analogue-to-digital converter exist.
These differ in the technique used to effect signal
conversion, in operational speed, and in cost.
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The counter ADC
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The counter ADC
 Following reset of the counter at the start of the
conversion cycle, clock pulses are applied continuously
to the counter through the AND gate, and the analogue
signal at the output of the digital-to-analogue converter
gradually increases in magnitude.
 At some point in time, this analogue signal becomes
equal in magnitude to the unknown signal at the input to
the comparator. The output of the comparator changes
state in consequence, closing the AND gate and
stopping further increments of the counter. At this point,
the value held in the counter is a digital representation
of the level of the unknown analogue signal.
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The flash ADC
+VREF
Op-amp
comparators
R
Input from
sampleand-hold
The flash ADC uses a series
high-speed comparators that
compare the input with
reference voltages. Flash
ADCs are fast but require 2n –
1 comparators to convert an
analog input to an n-bit binary
number.
+
–
R
+
–
R
+
–
R
R
R
7
6
5
+
–
4
+
1
0
–
R
Priority
encoder
1
2
3
2
4
D0 Parallel
D1 binary
output
D2
EN
+
–
+
–
Enable
pulses
R
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The flash ADC
 The flash method utilises comparators that compare reference
voltages with the analogue input voltage. When the input
voltage exceeds the reference voltage for a given comparator,
a HIGH is generated. Figure above shows a 3-bit converter
that uses seven comparator circuits; a comparator is not
needed for the all-0s condition.
 The reference voltage for each comparator is set by the
resistive voltage-divider circuit. The output of each
comparator is connected to an an input of the priority encoder.
The encoder is enabled by apulse on the EN input, and a 3-bit
code representing the value of the input appears on the
encoder’s outputs. The binary code is determined by the
highest-order input having a HIGH level.
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The dual-slope ADC
Vin
+
I
–
CLK
C
SW
–
R
≈0 V
+
A1
–
C
A2
+
R
–VREF
Counter
n
Control
logic
Latches
EN
D7 D6 D5 D4 D3 D2 D1 D0
1. The dual-slope ADC integrates the input voltage for a
fixed time while the counter counts to n.
2. Control logic switches to the VREF input.
3. A fixed-slope ramp starts from –V as the counter counts.
When it reaches 0 V, the counter output is latched.
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Digital to Analog Conversion
 Many A/D conversion methods utilize the D/A
conversion process.
 Converting a value represented in digital code to a
voltage or current proportional to the digital value.
DAC output is technically not an
analog quantity because it can
take on only specific values.
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Digital to Analog Conversion
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 For each input number, the D/A converter output
voltage is a unique value—in general:
…where K is the proportionality factor and is a constant value
for a given DAC connected to a fixed reference voltage.
• The quantity of possible output values can be
increased, and the difference between successive
values decreased—by increasing the input bits.
Allowing output more & more like an analog quantity
that varies continuously over a range of values.
A “pseudo-analog” quantity, which approximates
pure analog, referred to as analog for convenience.
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Digital to Analog Conversion
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 Each digital input contributes a different amount
to the analog output—weighted according to their
position in the binary number.
Weights are successively doubled
for each bit, beginning with the LSB.
VOUT can be considered to be the
weighted sum of the digital inputs.
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Digital to Analog Conversion
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 The Resolution of a D/A converter is defined
as the smallest change that can occur in analog
output as a result of a change in digital input.
Always equal to the weight of the LSB,
called the step size, it is the amount
VOUT will change as digital input value
changes from one step to the next.
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Digital to Analog Conversion
 Resolution (step size) is the same as the DAC
input/output proportionality factor:
…where K is the proportionality factor and is a constant value
for a given DAC connected to a fixed reference voltage.
is the analogue full-scale output
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Digital to Analog Conversion
 Percentage Resolution
 Although resolution can be expressed as the amount
of voltage or current per step, it is also useful to
express it as a percentage of the full-scale output. To
illustrate, the DAC of Figure 6.19 has a maximum fullscale output of 15 V (when the digital input is 1111).
The step size is 1 V, which gives a percentage
resolution of

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Digital to Analog Conversion
 Example
 A 10-bit DAC has a step size of 10 mV. Determine the
full-scale output
voltage and the percentage
resolution
 Solution
 With 10 bits, there will be
= 1023 steps of 10
mV each. The full-scale output will therefore be 10
mV X 1023 = 10.23 V, and
.
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Binary Weighted Resistors DAC
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Simple DAC using an op-amp summing amplifier with binary1
1
1
weighted resistors.
2
4
8
Input resistor values are binarily weighted.
Starting with MSB, resistor values
increase by a factor of 2, producing
desired weighting in the voltage output.
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R/2R Ladder
2
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R/2R Ladder
 The R-2R ladder requires only two values of resistors.
By calculating a Thevenin equivalent circuit for each
input, you can show that the output is proportional to the
binary weight of inputs that are HIGH.
 Each input that is HIGH contributes to the output:
.
 where VS = input HIGH level voltage
n = number of bits

i = bit number
 For accuracy, the resistors must be precise ratios, which
is easily done in integrated circuits.

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R/2R Ladder
 An R-2R ladder has a binary input of 1011. If a HIGH
= +5.0 V and a LOW = 0 V, what is Vout?
 Solution
Apply
. to all inputs
that are HIGH, then sum the results.
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