1. Learning Objectives Static and Dynamic Characteristics of Signals
Signal Decomposition
Data Sampling and Acquisition

2. Signals, Systems, Data
A Signal is the function of one or more independent variables that carries some information to represent a physical phenomenon.
A continuous-time signal, also called an analog signal, is defined along a continuum of time.
Systems process input signals to produce output signals.
Output signals are often converted to digital information with an analog to digital converter.
Transfer Function
How the analog input signal relates to the analog or digital output signal. This can be represented as a graph or a calibration curve.

3. Signal / Sensor Characteristics
Static characteristic: Comparison between output signal and ideal output when the input is constant.
Dynamic characteristics: Comparison between output signal and ideal output when the input changes.

4. Instrument Static Characteristics
Accuracy
Relation of the instrument output to the true value.
Typically shown as percent error relative to true value as determined through calibration.
Precision
The repeatability of an instrument when reading the same input.
High accuracy means that the mean is close to the true value, while high precision means that the standard deviation σ is small.
Systematic error:
High Precision, low accuracy.

5. Static Characteristics
Example : Two pressure gauges (pressure gauge A and B) have a full scale accuracy of ± 5%. Sensor A has a range of 0-1 bar and Sensor B 0-10 bar. Which gauge is more suitable to be used if the reading is 0.9 bar? Answer : Sensor A : Equipment max error = ± 5 x 1 bar = ± 0.05 bar 100 Equipment 0.9 bar ( in %) = ± 0.05 bar x 100 = ± 5.6% 0.9 bar Sensor B : Equipment max error = ± 5 x 10 bar = ± bar ( in %) = ± 0.5 bar x 100 = ± 55% Conclusion : Sensor A is more suitable to use at a reading of 0.9 bar because the error percentage (± 5.6%) is smaller compared to the percentage error of Sensor B (± 55%).
Source: D. Veeman

6. Instrument Static Characteristics
Range
The difference of reading between the minimum value and maximum value for the measurement of an instrument.
Bias
Constant error which occurs during the measurement of an instrument.
This error is usually rectified through calibration.
Linearity
Largest deviation from linear relation between input and output.
Shown as full scale percentage (% fs).
Sensitivity
Ratio of change in output towards the change in input at a steady state condition.
Resolution
The minimum detectable change in signal – (% fs).

7. Instrument Static Characteristics
Most sensitive
Variation of the physical variables
Source: D. Veeman

8. Instrument Static Characteristics
Dead Band - The range of input reading when there is no change in output (unresponsive system). Threshold - Minimum value before a response is observed. Hysteresis - Lag in sensor reading returning to previous value.
Output
Reading - +
Measured
Variables
Dead Band
Source: D. Veeman

9. Dynamic Characteristics
Behaviour of instruments when the input signal is changing.
Characterized by standardized inputs –
Step
Sudden change in input
Transient response
Ramp
Linear change
Ramp response
Sine wave
Harmonic input
Frequency response
Input
Response
Time

10. Dynamic Characteristics
Response from a 2nd order instrument: Rise Time ( tr ) - Time taken for the output to rise from 10% to 90 % of the steady state value. Settling time (ts) - Time taken for output to reach a steady state value.
Source: D. Veeman

11. Classification of Signals
Deterministic & Non Deterministic Signals
Periodic & A periodic Signals
Even & Odd Signals

12. Source: Dr. AJAY KUMAR, BCET Gurdaspur
Elementary Signals
Sinusoidal & Exponential Signals
Sinusoids and exponential signals arise naturally in physical systems and mathematical representations.
x(t) = A sin (2Пfot+ θ)
= A sin (ωot+ θ)
x(t) = Aeat Real Exponential
= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential
θ = Phase of sinusoidal wave
A = amplitude of a sinusoidal or exponential signal
fo = fundamental cyclic frequency of sinusoidal signal
ωo = radian frequency
Sinusoidal signal
Source: Dr. AJAY KUMAR, BCET Gurdaspur

13. Time versus Frequency Domain
Source: Data Communications and Networking:

14. Composite periodic signal
Periodic analog signals can be classified as simple or composite. A simple periodic analog signal, a sine wave, cannot be decomposed into simpler signals. A composite periodic analog signal is composed of multiple sine waves.
According to Fourier analysis, any composite signal is a combination of simple sine waves with different frequencies, amplitudes, and phases.
If the composite signal is periodic, the decomposition gives a series of signals with discrete frequencies;
if the composite signal is nonperiodic, the decomposition gives a combination of sine waves with continuous frequencies.
Source: Data Communications and Networking:

15. Decomposition of a composite periodic signal in the time and frequency domains
Source: Data Communications and Networking:

16. Mathematical Modeling of Continuous Systems
Most continuous time systems represent how continuous signals are transformed via differential equations.
E.g. RC circuit:
System indicating car velocity:
Source: Dr. AJAY KUMAR, BCET Gurdaspur

17. Discrete-Time Signals
Sampling is the acquisition of the values of a continuous-time signal at discrete points in time
x(t) is a continuous-time signal, x[n] is a discrete-time signal
Source: Dr. AJAY KUMAR, BCET Gurdaspur

18. Discrete Time Sinusoidal Signals
Source: Dr. AJAY KUMAR, BCET Gurdaspur

19. Mathematical Modeling of Discrete Time Systems
Most discrete time systems represent how discrete signals are transformed via difference equations
e.g. bank account, discrete car velocity system
Source: Dr. AJAY KUMAR, BCET Gurdaspur

20. Discrete Time Exponential and Sinusoidal Signals
DT signals can be defined in a manner analogous to their continuous-time counter part
x[n] = A sin (2Пn/No+θ)
= A sin (2ПFon+ θ)
x[n] = an
n = the discrete time
A = amplitude
θ = phase shifting radians,
No = Discrete Period of the wave
1/N0 = Fo = Ωo/2 П = Discrete Frequency
Discrete Time Sinusoidal Signal
Discrete Time Exponential Signal
Source: Dr. AJAY KUMAR, BCET Gurdaspur

21. Source: D. Gheith Abandah - http://www.abandah.com/gheith/
Signal Processing
Signal processing involves systems that process input signals to produce output signals.
A system is combination of components that manipulate one or more signals to accomplish a function and produces some output.
system
output signal
input signal
Source: D. Gheith Abandah -

22. Analog to Digital Conversion
Most physical signals are analog.
Analog signals are captured by sensors or transducers.
Examples: temperature, sound, pressure, …
Need to convert to digital signals to facilitate processing by the microcontroller.
The device that does this is analog-to-digital converter (ADC).
Source: D. Gheith Abandah -

23. Analog v. Digital Signals
Source: Data Communications and Networking:

24. Analog vs. Digital Property Analog Digital Representation
Continuous voltage or current
Binary Number
Precision
Infinite range of values
Limited by the number’s length
Resistance to Degradation
Weak
Tolerant to signal degradation
Processing
Limited
Powerful
Storage
Impossible
Possible
Source: D. Gheith Abandah -

25. Elements of a data acquisition system
Source: D. Gheith Abandah -

26. Elements of a data acquisition system
Transducers: physical to electrical
Amplify and offset circuits
The input voltage should traverse as much of its input range as possible
Voltage level shifting may also be required
Filter: get rid of unwanted signal components
Multiplexer: select one of multiple inputs
Sampler: the conversion rate must be at least twice the highest signal frequency (Nyquist sampling criterion)
ADC
Source: D. Gheith Abandah -