Signals & Systems (Electronics)

Signal Types:

• Analog vs Digital.
• Continuous-time vs Discrete-time.
• Periodic vs Aperiodic.
• Deterministic vs Random.
• Energy vs Power signals.

Basic signals:

• Impulse δ(t), δ[n].
• Step u(t), u[n].
• Ramp r(t) = t·u(t).
• Sinusoidal: sin(ωt + φ).
• Exponential: e^(±at).

System Properties:

• Linearity: superposition holds.
• Time-invariance: delay in/out same.
• Causality: output depends on past + present, not future.
• Stability: bounded input → bounded output.
• Memoryless: output depends only on current input.

LTI (Linear Time-Invariant) Systems:

• Characterized by impulse response h(t) or h[n].
• Output = convolution: y(t) = x(t) * h(t).

Convolution:

• Continuous: y(t) = ∫ x(τ)h(t-τ) dτ.
• Discrete: y[n] = Σ x[k]h[n-k].

Fourier Series (periodic signals):

• x(t) = Σ aₙ·e^(jnω₀t).
• Coefficients give frequency content.

Fourier Transform (aperiodic):

• X(ω) = ∫ x(t)·e^(-jωt) dt.
• Time domain → frequency domain.

Discrete-time Fourier Transform (DTFT):

• For discrete signals.

Discrete Fourier Transform (DFT):

• Finite-length discrete signal.
• N-point: O(N²) direct, O(N log N) FFT.

Z-Transform:

• Discrete equivalent of Laplace.
• X(z) = Σ x[n]·z⁻ⁿ.

Sampling:

• Nyquist theorem: sampling rate ≥ 2× max signal frequency to avoid aliasing.
• Aliasing: above Nyquist → frequency folding.

Digital Signal Processing:

• FIR filters (Finite Impulse Response).
• IIR filters (Infinite IR).
• Filter design: window method, frequency sampling.

Applications:

• Audio/video compression.
• Speech recognition.
• Communications (modems).
• Image processing.

RRB JE focus: Nyquist criterion, basic convolution, signal types, simple Fourier ideas.