Unit 2: Linear time-invariant systems - Subjective Questions

An LTI system is a system that satisfies both the linearity and time-invariance properties.

• Linearity: A system is linear if it obeys the principle of superposition, which combines additivity and homogeneity (scaling):
  

• Time-Invariance: A system is time-invariant if a time shift in the input causes an identical time shift in the output:
  

Significance:

• LTI systems can be completely characterized by their impulse response or .
• The output for any arbitrary input can be found using convolution.
• Powerful analysis tools like Fourier and Laplace transforms apply directly.
• They form the foundation of most practical filter and system designs.

Step 1 — Representing a signal using impulses:
Any discrete-time signal can be written as a weighted sum of shifted unit impulses:

Step 2 — Applying the system operator :

Step 3 — Using linearity:

Step 4 — Using time-invariance:
Let be the impulse response. Then .

Result — The Convolution Sum:

This shows that the output of a discrete-time LTI system is the convolution of the input with the impulse response.

Step 1 — Representing a signal using impulses:
A continuous-time signal can be expressed via the sifting property:

Step 2 — Applying the system operator:

Step 3 — Using linearity (integration is a limiting sum):

Step 4 — Using time-invariance:
Let . Then .

Result — The Convolution Integral:

Thus the output of a continuous-time LTI system is the convolution of the input signal with the system's impulse response.

Convolution obeys three key algebraic properties:

1. Commutative Property:

• The roles of input and impulse response can be interchanged without affecting the output.

2. Associative Property:

• Interpretation: Two LTI systems in cascade (series) are equivalent to a single system whose impulse response is the convolution of the individual impulse responses:
  

3. Distributive Property:

• Interpretation: Two LTI systems in parallel are equivalent to a single system whose impulse response is the sum of the individual impulse responses:
  

These properties greatly simplify the analysis of interconnected systems.

Examples:

• Continuous-time: An RC circuit, an analog amplifier, a mechanical spring-mass system.
• Discrete-time: A digital filter, a bank account with monthly interest, a moving-average computation on sampled data.

Key point: Continuous-time systems process signals defined for all real , while discrete-time systems process signals defined only at integer time indices, often obtained by sampling a continuous signal.

Causality means that the output of a system at any time depends only on present and past values of the input, never on future values.

Condition on impulse response:

For a discrete-time LTI system to be causal:

For a continuous-time LTI system to be causal:

Reasoning: In the convolution sum , if for (i.e. ), then future inputs with do not contribute to .

Example:

• is causal.
• is non-causal (depends on future inputs).

All physically realizable real-time systems must be causal.

BIBO Stability (Bounded-Input Bounded-Output): A system is stable if every bounded input produces a bounded output.

Derivation (discrete-time):
Let for all . The output is:

Taking magnitude:

For to be bounded, we require:

Continuous-time condition:

Conclusion: An LTI system is BIBO stable if and only if its impulse response is absolutely summable (discrete) or absolutely integrable (continuous).

Given:

• for
• for

Using the convolution sum:

Length of output = samples ( to ).

Computing each value:

Result:

This can also be verified using the sliding strip (tabular) method or polynomial multiplication.

Static (Memoryless) System:

• The output at any instant depends only on the present input at that same instant.
• Mathematically: .
• Impulse response: (a scaled impulse).
• Examples: , a resistor .

Dynamic (Memory) System:

• The output depends on past or future values of the input in addition to (or instead of) the present.
• Examples:
  • (depends on past)
  • (integrator)
  • Capacitors and inductors store energy, giving memory.

Key distinction: Memory arises whenever the system involves storage elements or delays. A memoryless LTI system's impulse response is nonzero only at the origin.

Convolution:

• Involves folding (time-reversal), shifting, multiplication, and summation.
• Used to find the output of an LTI system.

Correlation:
Cross-correlation of and :

• No folding of the signal; one signal is only shifted.

Key differences:

Applications:

• Convolution: filtering, LTI system response.
• Correlation: signal detection, pattern matching, radar/sonar range estimation, measuring similarity/time-delay between signals.

Relation: Correlation of and equals convolution of with the folded version of : .

Given: , causal system.

Step 1 — Set input to impulse: Let , so :

Step 2 — Recursively solve (with for ):

• :
• :
• :
• :

Step 3 — Recognize the pattern:

Verification: The system is stable since .

Invertibility: A system is invertible if distinct inputs produce distinct outputs. If invertible, there exists an inverse system that recovers the input from the output.

Condition for LTI systems:
If a system has impulse response and its inverse has impulse response , then their cascade must reproduce the input exactly:

Similarly for discrete time:

Example:

• A pure delay system has impulse response .
• Its inverse is an advance with .
• Check: . ✓

Application: Channel equalization in communication systems uses an inverse filter to undo channel distortion.

The step response is the output when the input is the unit step .

Continuous-time:

Conversely, the impulse response is the derivative of the step response:

Discrete-time:

Conversely:

Significance:

• The step response is often easier to measure experimentally than the impulse response.
• Once is known, (which fully characterizes the LTI system) can be obtained by differentiation/first-difference.

For cascade (series) connection, the overall impulse response is the convolution:

Substituting (, ):

Limits: and , so for :

Result:

This confirms the associative property — cascaded LTI systems combine via convolution.

The convolution can be evaluated graphically in four steps:

1. Folding (Flip):

• Reflect about the vertical axis to obtain .

2. Shifting:

• Shift by to obtain . For shift right, for shift left.

3. Multiplication:

• Multiply the overlapping portions of and .

4. Integration:

• Compute the area under the product to get for that particular .

Repeat steps 2–4 for all values of , identifying distinct intervals where the overlap region changes.

Tips:

• Carefully determine the limits of integration for each range of based on overlap.
• The output support extends from to .

Software simulation implements the convolution sum numerically so that system outputs can be computed for arbitrary inputs without manual calculation. Tools like MATLAB (conv), Python (numpy.convolve), or Scilab are commonly used.

Algorithm for linear convolution of (length ) and (length ):

Input: x[0..N-1], h[0..M-1]
Output length L = N + M - 1
Initialize y[0..L-1] = 0

for n = 0 to L-1:
for k = 0 to N-1:
if (n - k) >= 0 and (n - k) < M:
y[n] = y[n] + x[k] * h[n-k]
return y

Example in Python:
python
import numpy as np
x = [1, 2, 3]
h = [1, 1]
y = np.convolve(x, h)
print(y) # Output: [1 3 5 3]

Advantages:

• Fast, accurate for long sequences.
• Easily extended to correlation and 2-D (image) convolution.
• Enables visualization and verification of theoretical results.

LTI systems described by difference equations fall into two classes:

FIR Systems:

• General form: (no feedback terms).
• Impulse response has finite duration.
• Also called non-recursive systems.
• Always stable.
• Can have exactly linear phase.
• Example: .

IIR Systems:

• General form: (has feedback).
• Impulse response has infinite duration.
• Also called recursive systems.
• May be unstable if poles lie outside the unit circle.
• More efficient (fewer coefficients for sharp filters).
• Example: .

Summary: FIR trades higher order for guaranteed stability and linear phase; IIR offers efficiency but requires careful stability checking.

(a)

Linearity: For inputs with output and :

→ Linear. ✓

Time-Invariance: Shift input by : . But shifting output gives . These are not equal.
→ Time-Variant. ✗

Conclusion (a): Linear but Time-Variant.

(b)

Linearity: , which .
→ Non-linear. ✗

Time-Invariance: .
→ Time-Invariant. ✓

Conclusion (b): Non-linear but Time-Invariant.

The impulse response (or ) is the output of an LTI system when the input is a unit impulse (or ).

Why it completely describes an LTI system:

• Any input signal can be decomposed as a weighted sum/integral of shifted impulses.
• Using linearity and time-invariance, the output is the same weighted combination of shifted impulse responses.
• This directly yields convolution:
  

System properties determined from :

• Causality: for .
• Stability: (absolutely integrable).
• Memory: memoryless iff .
• Invertibility: exists with .

Conclusion: Because the entire input-output behavior for any input can be predicted from , the impulse response is a complete and sufficient characterization of an LTI system.

Given: , causal system.

Step 1 — Set , so :

Step 2 — For (homogeneous solution, since impulse acts only at ):

Step 3 — Determine using the initial condition from the impulse:
Integrating the differential equation across gives a jump: . Since the system is causal, , so , giving .

Result:

Verification of stability: , so the system is BIBO stable.