LTI systems
In general, a Linear system is a system that takes and inputΒ X, and produces an outputΒ Y, that can be defined by a series of linear operators. Recall a linear operator is an operation that satisfies the principles of scalar multiplication and superposition. More mathematically, for vectors x,y, scalarsΒ
the operatorΒ AΒ is linear if:
is satisfied. Examples of linear operators are:
- Differentiation/Integration
- Convolution
- Gradient/Divergence/Curl
- Laplace/Fourier transforms
Note that if there are multiple Linear systems, their cascaded effect can be described by composing each subsequent system with the output of the other. This is equivalent to combing each of the operators the systems are implementing and forming a compound operator.
Time invariance is the property that for a given system, ifΒ y(t) is the output of the system when given an inputΒ x(t), then applying a delayed inputΒ x(tβT)Β will produceΒ y(tβT), a delayed version ofΒ y(t). In the Time domain, this system has an impulse responseΒ β(t)Β and relatesΒ y(t),x(t)Β by the relation:
WhereΒ βΒ is the convolution operation. The act of convolution can be difficult, so it is oftentimes more convenient to operate in another domain through some sort of transformation. A common transform is the Fourier transform, whereby a time domain signal is represented in the frequency domain. A consequence of the transform is the convolution operation is now reduced to multiplication.
WhereΒ H(f)Β is now the transfer function or frequency response of the system. Canonically, the transfer function is written as:
Every LTI system can be described by a given transfer function, with various systems being formed by various coefficientsΒ a,b. In this lab, we will form a small number of systems with real components, and examine their behaviors.
Transfer Function
The transfer function is a fundamental concept in the analysis and design of linear time-invariant (LTI) systems. It represents the relationship between the input and output signals of a system in the frequency domain. By examining the transfer function, we can understand how the system processes different frequency components of the input signal. It captures the systemβs frequency response characteristics, such as amplification, attenuation, and phase shifts.
Mathematically as mentioned above, the transfer function, denoted as H(f), is defined as the ratio of the output signal Y(f) to the input signal X(f):
The transfer function is obtained through the Fourier transform of the systemβs impulse response, simplifying the analysis of the systemβs behavior. It is typically expressed as a rational function of complex variables, with the numerator and denominator represented by polynomials. This allows us to determine the systemβs gain and phase response. Understanding the transfer function enables engineers and researchers to design LTI systems with desired frequency response characteristics, optimizing their performance for specific applications. By leveraging the transfer function, we can accurately analyze and predict how a system amplifies or attenuates specific frequencies and introduces phase shifts. This knowledge is invaluable for tailoring the systemβs behavior to meet the desired requirements.
Types of Filters
Filters are LTI systems used to modify the frequency content of a signal. They can be classified into different types based on their frequency response characteristics. A low-pass filter allows frequencies below a certain cutoff frequency to pass through while attenuating higher frequencies. It is commonly used to remove high-frequency noise and retain the low-frequency components of a signal. A high-pass filter, on the other hand, allows frequencies above a certain cutoff frequency to pass through while attenuating lower frequencies. It is useful for extracting high-frequency information and eliminating low-frequency noise. A band-pass filter allows a specific range of frequencies to pass through while attenuating frequencies outside that range. It finds applications in tasks such as signal isolation and frequency selection. Finally, a band-stop filter, also known as a notch filter, attenuates a specific range of frequencies while allowing other frequencies to pass through. It is effective in removing unwanted frequency components or interference.
Frequency Response
The frequency response of a linear time-invariant (LTI) system provides crucial insights into its behavior across different frequencies. It characterizes how the system processes and modifies signals of varying frequencies. The frequency response consists of two components: the magnitude response and the phase response.
The magnitude response represents the systemβs gain or attenuation for each frequency component. It quantifies how much the system amplifies or attenuates different frequencies in the input signal. Mathematically, the magnitude response is denoted as
where
is the transfer function of the system. By examining the magnitude response, we can understand how the system enhances or diminishes specific frequency components, thereby shaping the overall spectral content of the output signal.
The phase response, on the other hand, reveals the phase shift introduced by the system at each frequency. It indicates the time delay or advance experienced by different frequency components of the input signal. The phase response is denoted as
and is essential for applications where phase synchronization or time relationships between signals are critical. By analyzing the phase response, we can determine the phase characteristics of the system and how it influences the timing of the output signal relative to the input signal at different frequencies.
Both the magnitude and phase responses are typically plotted as functions of frequency to visualize their characteristics. These frequency response plots provide valuable information about the systemβs behavior, such as frequency selectivity, gain variations, and phase distortions. They serve as a powerful tool for analyzing and designing systems in various fields, including signal processing, communication, audio engineering, and control systems.
Impulse Response
In the time domain, the impulse response of an LTI system is a key concept for understanding its behavior. It characterizes the systemβs response when the input signal is an impulse function, often represented ash(t). The impulse response provides valuable insights into how the system processes signals over time and shapes the output signal based on its inherent properties. Mathematically, the output signal y(t) of an LTI system can be obtained by convolving the input signal x(t) with the impulse response h(t) as expressed by the integral equation:
This equation represents the superposition of the weighted contributions of the input signal at different time instances, where the weights are determined by the impulse response. The impulse response reveals the systemβs temporal characteristics and provides information about its response to instantaneous changes in the input signal. By examining the impulse response, we can understand phenomena such as signal distortion, time-domain filtering, and transient behavior of the system.
Stability
Stability is a fundamental property of LTI systems that ensures their reliable and predictable operation. A stable system maintains a bounded output for bounded input signals, providing confidence in its performance and behavior. Stability analysis is crucial in evaluating the robustness and reliability of LTI systems. The stability of an LTI system can be determined by examining its transfer function or impulse response. Various stability criteria, such as the location of poles in the complex plane or the boundedness of the impulse response, are employed to assess stability. A stable system exhibits desirable characteristics, such as controlled response to inputs, absence of oscillations, and suppression of noise and disturbances. In contrast, an unstable system may exhibit erratic behavior, uncontrollable oscillations, or even diverging responses. Ensuring stability is of utmost importance in the design and analysis of LTI systems. It enables reliable signal processing, accurate control systems, and effective communication.
By analyzing the stability of an LTI system, we can gain insights into its performance limitations and make informed decisions in system design and implementation. Stability considerations become particularly crucial in applications where precision, robustness, and error-free operation are essential. Stable LTI systems have widespread applications in various fields, including control systems, telecommunications, audio processing, and image processing. They provide a foundation for designing systems that exhibit desired behaviors, such as accurate signal reproduction, noise suppression, and precise control of dynamic processes. Stability analysis allows engineers and researchers to ensure the reliability and safety of LTI systems in real-world scenarios, where external disturbances, noise, and uncertainties may be present.
Furthermore, stability analysis plays a pivotal role in stability-based control design. Controllers are designed to stabilize unstable systems or improve the stability margins of marginally stable systems. Stability analysis techniques help identify the critical parameters and system characteristics that affect stability, guiding the selection and adjustment of control parameters to achieve desired stability and performance objectives.
In summary, understanding and analyzing the stability of LTI systems is crucial for ensuring their reliable operation, robustness, and optimal performance. Stability considerations guide system design, control design, and decision-making processes in various engineering disciplines. By evaluating the stability characteristics of LTI systems, engineers and researchers can create systems that meet performance requirements, mitigate undesirable effects, and deliver reliable and predictable results in diverse applications.
Conclusion
In conclusion, measuring the frequency response of a circuit is a valuable tool for evaluating its performance and validating theoretical models. By comparing the measured results with the expected behavior, we can identify any discrepancies and refine our understanding of the circuitβs characteristics. Additionally, accounting for uncertainties in component values enhances the accuracy of our analysis. Overall, frequency response measurements provide crucial insights for circuit design, optimization, and ensuring desired functionality in practical applications.
Experiment for this Course:
Single stage Low Pass circuitSingle stage High Pass circuitCascading FiltersCascading Filters with Variable StagesExperiments related to filters
Experiments & Lessons
Gallery view
Modifying a comparator
An OpAmp multivibrator
Transformer-Based Full Wave Rectification
Resistor Voltage Divider
Ping Pong
Moving average on Red Pitaya
Simple Calculator
Frequency Counter
Stopwatch
Knight Rider
Simple LED blinker
LED Counter
Vivado Project Setup
Programming the FPGA
Installation of Vivado 2020.1
Tasks and Functions in Verilog
Statements and Loops in Verilog
Assignments in Verilog
Operators in Verilog
Data Types and Values in Verilog
Introduction to Verilog
Boolean Functions
Karnaugh Map
Other Gates
Boolean Algebra
Mathematical Operations with the Binary Base
Digital Information and Numerical Bases
Common Emitter Transistor Amplifier
Transistors
Driving an LED with Analog PWM
Smoothing a PWM Signal using an RC Filter
Driving an LED with a PWM Signal
OpAmp Triangular Wave Generator
OpAmp Sine Wave Generator
OpAmp Multivibrator
Pulse Width Modulation
Function Generators
R-2R DAC
Binary Weighted DAC
Resistor Voltage Divider DAC
Digital to Analog Converter
Band-Pass Filter
Subtraction with OpAmps
Addition with OpAmps
Inverting and noninverting Schmitt Trigger
OpAmp Inverting Amplifier
OpAmp Non-Inverting Amplifier
Active Filters
Comparator & Schmitt Triggers
Analog Addition and Subtraction
Operational Amplifiers
Transformer-Based Full Wave Rectification
Full Wave Rectifier
OpAmp-Controlled LED Circuit
Diode Experiment
Full Wave Rectifiers and Transformers
Diodes
RC Transfer Function with Bode Analyzer
RC Circuit Response
Transient Response
Analog Filters
Resistors in Superposition
Superposition
Measuring the Oscillating Frequency of RLC Circuits
Measuring Frequency
Measuring Inductance using the Impedance Measurement Method
Measuring Inductance
Measuring Capacitance using the Rise/Fall Time Method
Measuring Capacitance
Measuring Uncertainty in Voltage Measurement
Measuring Uncertainty
Resonant Filter
Running Sum Filter
FIR vs. IIR Filters
Digital Filtering
Digital Filters
Half Bridge Rectifier
A Brief Tour of Linear Systems
Cascading Filters with Variable Stages
Cascading Filters
Single stage High Pass circuit
Single stage Low Pass circuit
Frequency Response of LTI Systems
Signal Modulation Experiment
Signal Modulation
Measuring external waveforms
Measuring External Waveforms with Red Pitaya
Resistor circuits
Measurements in Time Domain
Properties of periodic waveforms
Measuring the OpAmp board Gain using Oscilloscope application
Calculate gain Using SCPI commands in Python
Calculate Gain of OpAmp and Find -3dB point in Jupyter Notebooks
Calculate gain, phase and turn on LEDs in Jupyter Notebooks
Generate, acquire and plot a signal in Jupyter Notebooks