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**Learning Objectives Static and Dynamic Characteristics of Signals**

Signal Decomposition Data Sampling and Acquisition

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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.

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**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.

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**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.

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**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

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**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).

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**Instrument Static Characteristics**

Most sensitive Variation of the physical variables Source: D. Veeman

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**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

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**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

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**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

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**Classification of Signals**

Deterministic & Non Deterministic Signals Periodic & A periodic Signals Even & Odd Signals

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**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

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**Time versus Frequency Domain**

Source: Data Communications and Networking:

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**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:

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**Decomposition of a composite periodic signal in the time and frequency domains**

Source: Data Communications and Networking:

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**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

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**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

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**Discrete Time Sinusoidal Signals**

Source: Dr. AJAY KUMAR, BCET Gurdaspur

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**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

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**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

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**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 -

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**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 -

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**Analog v. Digital Signals**

Source: Data Communications and Networking:

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**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 -

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**Elements of a data acquisition system**

Source: D. Gheith Abandah -

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**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 -

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