Frequency Response of Linear Systems

Frequency Response of Linear Systems

• Linearity: A system is linear if it satisfies the principles of superposition and homogeneity.
• Frequency Response: Represents how a linear time-invariant (LTI) system responds to sinusoidal inputs of varying frequencies.
  • For a continuous-time system, the frequency response is represented by the transfer function H(s) where s = jω (j is the imaginary unit and ω is angular frequency).
  • The impulse response h(t) is related to the frequency response H(jω) through the Fourier transform.
• Magnitude Response: Describes how the system modifies the amplitude (magnitude) of each frequency component in the input signal. Represented as |H(jω)|.
• Phase Response: Describes the phase shift introduced by the system for each frequency component. Represented as ∠H(jω).
• Bode Plots: A graphical representation of the frequency response. Magnitude and phase responses are plotted against frequency (logarithmic scale) for visualization.
  • Bode Plot Steps:
    • Choose appropriate parameters: (K, τ) to create a range of Bode plot examples.
    • Use signal.bode function from scipy to automatically calculate the Bode magnitude and phase responses for the specific parameters.
• Important System Properties:
  • Bandwidth: The range of frequencies a system can process effectively.
  • Resonance: The frequency at which the system's response to a sinusoidal input is amplified due to its natural frequency characteristics.
  • Stability: A stable system has a frequency response where all poles of H(s) have negative real parts.

Applications and Practical Uses

• Filter Design: Frequency response analysis helps design filters (low-pass, high-pass, band-pass, etc.) to achieve desired frequency characteristics.
• Signal Processing: Understanding frequency response is crucial in audio processing, telecommunications, control systems, and more.

Polar Plots for System Analysis

• Polar plots visualize the frequency response, especially useful when phase response matters.
• Types of Systems Represented in Polar Plots:
  • Minimum-Phase System: Magnitude and phase are causal and stable. In a polar plot:
    • Magnitude decreases gradually with increasing frequency, phase shift occurs concurrently.
    • Example: Minimum-phase low-pass filter.
  • Non-Minimum Phase System: Phase responses include zeros with positive real parts or poles with negative real parts.
    • Phase response has peaks and dips, indicating phase lead or lag.
    • Magnitude response might have ripples.
    • Example: Systems with zeros in the right half-plane or poles in the left half-plane.
  • All-Pass System: Constant magnitude response (equal to 1) for all frequencies, but phase response varies.
    • In a polar plot:
      • Magnitude is constant (a circle of radius 1).
      • Phase changes linearly with frequency.
    • Example: A simple all-pass filter (first-order).
  • Linear Phase System: Phase response is linearly proportional to frequency: ∠H(jω) = ω ⋅ θ (θ is a constant).
    • In a polar plot:
      • Phase response increases linearly with frequency.
      • Magnitude response varies depending on the system.
    • Example: Certain types of FIR (Finite Impulse Response) filters.
  • Unstable System: Poles with positive real parts, leading to unbounded responses.
    • In a polar plot:
      • Magnitude may grow indefinitely as frequency increases.
      • Phase response shows unstable behavior (oscillations).
    • Example: An unstable filter design.
  • Bandpass and Bandstop Systems: Specific frequency ranges where magnitude response is significant or zero, respectively.
    • In polar plots:
      • Bandpass systems show peaks in magnitude at specific frequencies.
      • Bandstop systems show dips (zeros) in magnitude at specific frequencies.
    • Example: A bandpass filter will have a peak in magnitude at the center frequency of its passband.

Practical Uses of Polar Plots

• System Design: Understand how a system behaves across different frequencies.
• Stability Analysis: Determine stability based on pole locations.
• Phase Margin and Gain Margin Analysis: Critical for control system design.

Estimating Transfer Functions from Frequency Response

• Methods: Several methods used depending on available information and system complexity.
• Structured Approach:
  • Step 1: Analyze the frequency response to identify key features.
  • Step 2: Determine potential pole and zero locations based on these features.
  • Step 3: Use approximation techniques to estimate the transfer function.
  • Step 4: Validate the estimated transfer function by comparing its frequency response to the original data.

Frequency Response of Linear Systems

• Linear Systems: A system where the output is proportional to the input, and superposition and homogeneity principles apply.
• Frequency Response: Describes how a system responds to sinusoidal inputs of varying frequencies.
  • For continuous-time systems, the transfer function, H(s), with s = jω (ω = angular frequency), represents the frequency response.
  • The impulse response (h(t)) is related to the frequency response through the Fourier transform.
• Magnitude Response: Indicates how the system changes the amplitude of each frequency component in the input signal, represented by |H(jω)|.
• Phase Response: Shows the phase shift introduced by the system for each frequency component, represented by ∠H(jω).
• Bode Plots: Graphical representations of the frequency response.
  • Magnitude and phase responses are plotted against frequency on a logarithmic scale.
• Bandwidth: The range of frequencies that a system can process effectively without significant attenuation or distortion.
• Resonance: The frequency at which the system's response to a sinusoidal input is amplified due to its natural frequency characteristics.
• Stability: A stable system has a frequency response where all poles (roots of the denominator polynomial of H(s)) have negative real parts.
• Practical Importance: Frequency response analysis is crucial for:
  • Filter Design: Designing filters (low-pass, high-pass, band-pass, etc.) based on desired frequency characteristics.
  • Signal Processing: Applications in audio processing, telecommunications, control systems, etc.
• Mathematical Formulation: The frequency response H(jω) is obtained by evaluating the transfer function H(s) on the imaginary axis (s = jω).

Bode Plots

• Bode Plots: Logarithmic frequency response plots, consisting of two graphs:
  • Magnitude Response: Shows gain (dB) vs. frequency.
  • Phase Response: Shows phase shift (degrees) vs. frequency.

Polar Plots

• Polar Plots: Useful for visualizing frequency response, especially when phase response is critical (control systems, communications).
  • Minimum-Phase System: A system where both magnitude and phase are causal and stable. The polar plot shows a gradual decrease in magnitude with frequency increase, along with a corresponding phase shift.
  • Non-Minimum Phase System: Systems with zeros with positive real parts or poles with negative real parts.
    • Phase response with peaks and dips, indicating phase lead or lag.
    • Magnitude response might have ripples.
  • All-pass System: Magnitude response constant (1) for all frequencies, but phase response varies. The polar plot shows constant magnitude (circle of radius 1) with phase changing linearly with frequency.
  • Linear Phase System: Phase response linearly proportional to frequency. The polar plot shows phase response increasing linearly with frequency.
  • Unstable System: Poles with positive real parts, leading to unbounded responses. The polar plot shows increasing magnitude with frequency and unstable phase response.
  • Bandpass and Bandstop Systems: Specific frequency ranges where magnitude response is significant or zero.
    • Bandpass: Peaks in magnitude at specific frequencies.
    • Bandstop: Dips (zeros) in magnitude at specific frequencies.
• Practical Uses of Polar Plots:
  • System Design
  • Stability Analysis
  • Phase Margin and Gain Margin Analysis

Estimating Transfer Functions from Frequency Response

• Methods involve using frequency response data to approximate a system's transfer function. Considerations:
  • Available information
  • System complexity