LTI System Analysis and Signal Processing Principles
Fundamental System Properties
System Properties define how a system behaves across all inputs and time:
- Linearity (Additivity and Homogeneity): A system is linear if it follows the principle of superposition. Mathematically, if x₁(t) → y₁(t) and x₂(t) → y₂(t), then ax₁(t) + bx₂(t) → ay₁(t) + by₂(t).
- Shift-Invariance (Time-Invariance): A system is shift-invariant if a delay in the input results in an identical delay in the output. If x(t) → y(t), then x(t – t₀) → y(t – t₀).
- Causality: A system is causal if the output at any time t depends only on the present and past values of the input, not on future values.
- Stability (BIBO): A system is Bounded-Input Bounded-Output (BIBO) stable if every bounded input results in a bounded output.
Aliasing
If the sampling rate is lower than the Nyquist rate (f_s < 2f_m), high-frequency components “overlap” into lower frequencies during reconstruction, causing distortion called aliasing.
LTI System Characterization
Linear Time-Invariant (LTI) systems are uniquely characterized by their Impulse Response h(t):
- Causality in LTI: For a continuous-time LTI system to be causal, h(t) = 0 for all t < 0.
- Stability in LTI: An LTI system is BIBO stable if its impulse response is absolutely integrable.
State Transition Matrix (STM)
In state-space analysis, the STM (often denoted as Φ(t)) describes how the internal state of the system evolves over time from an initial state when the external input is zero.
Sampling Theorem and Signal Reconstruction
Sampling Theorem: To accurately reconstruct a signal from its samples, the sampling frequency (f_s) must be at least twice the maximum frequency component (f_m) present in the signal (f_s ≥ 2f_m). This minimum rate is called the Nyquist rate.
Signal Reconstruction Methods
After sampling, the signal must be reconstructed using an interpolator:
- Ideal Interpolator: Uses a low-pass filter (sinc function) to perfectly reconstruct the signal, assuming the sampling theorem was followed.
- Zero-Order Hold (ZOH): Holds the value of a sample constant until the next sample arrives, creating a “staircase” waveform.
- First-Order Hold (FOH): Connects consecutive samples with straight lines (linear interpolation), providing a smoother approximation than ZOH.
ROC and System Stability
The Region of Convergence (ROC) is the set of values in the complex plane for which the Laplace or Z-transform integral/sum converges.
- Laplace Transform Stability: A system is stable if the ROC of its system function H(s) includes the jω (imaginary) axis. For causal systems, all poles must be in the Left Half Plane (LHP).
- Z-Transform Stability: A discrete system is stable if the ROC of H(z) includes the unit circle (|z| = 1). For causal systems, all poles must be inside the unit circle.
Mathematical Principles and Convolution
Parseval’s Theorem
States that the total energy of a signal computed in the time domain is equal to the total energy computed in the frequency domain.
Fourier Duality
This property suggests a symmetry between time and frequency; if a signal has a specific shape in the time domain, its transform will have a similar shape in the frequency domain (e.g., a rectangular pulse in time becomes a sinc function in frequency, and vice versa).
Convolution in LTI Systems
Convolution is a mathematical operation used to determine the output of an LTI system.
- Continuous-time: y(t) = ∫ x(τ)h(t – τ) dτ
- Discrete-time: y[n] = Σ x[k]h[n – k]
Steps of Convolution
- Flip one signal
- Shift it
- Multiply
- Integrate or sum
Significance
It provides the output of any LTI system and completely defines system behavior using the impulse response.
System Realization and Value Theorems
Direct Form (I & II)
These forms represent a discrete-time system and directly implement the difference equation. Direct Form II is “canonic,” meaning it uses the minimum number of delay elements (z⁻¹) by sharing them between the poles and zeros.
- Cascade Form: The system function H(z) is factored into smaller first or second-order subsystems and connected in series. The output of one is the input to the next.
- Parallel Form: H(z) is broken down into a sum of simpler fractions using partial fraction expansion. The subsystems work independently in parallel, and their outputs are summed together.
STM Key Properties
- Identity at Origin: Φ(0) = I (the identity matrix).
- Inverse Property: Φ⁻¹(t) = Φ(-t).
- Semi-group Property: Φ(t₁ + t₂) = Φ(t₁)Φ(t₂). This means moving from time 0 to t₂, then t₂ to t₁ + t₂, is the same as moving directly from 0 to t₁ + t₂.
Initial and Final Value Theorems
- Initial Value Theorem: If x[n] is causal, then x[0] = lim (z → ∞) X(z).
- Final Value Theorem: If the system is stable, the steady-state value is x[∞] = lim (z → 1) (z – 1)X(z).