Govt. Exams
Entrance Exams
At DC (ω=0): |H(j0)| = 1. At ω = ωc: |H(jωc)| = ωc/√(ωc² + ωc²) = 1/√2. This is the -3dB cutoff frequency, a fundamental property of first-order filters.
Using partial fractions or standard Z-transform tables: X(z) = z/(z-0.5) corresponds to x[n] = 0.5^n × u[n] where u[n] is the unit step function. The ROC |z| > 0.5 confirms a causal right-sided sequence.
Energy = 1² + 2² + (-1)² + 0² = 1 + 4 + 1 + 0 = 6.
For an even signal, the Fourier transform is real and even: X(-f) = X(f) and X*(f) = X(f).
X(z) = Σ(1/2)^n z^(-n) = z/(z - 1/2) = z/(z - 0.5) with ROC |z| > 0.5.
∫₀^∞ e^(-3t) dt = 1/3 < ∞. For stability, BIBO condition requires absolute integrability, which is satisfied.
The modulation theorem (or frequency convolution property) states that x(t)×y(t) ↔ X(f)*Y(f).
The maximum frequency in the signal is 50 Hz (from 100π = 2πf). Nyquist frequency = 2 × 50 = 100 Hz.
Denominator s²+3s+2 = (s+1)(s+2) = 0, giving poles at s = -1 and s = -2.
Linear convolution of signals of length M and N produces output of length M+N-1 = 5+5-1 = 9.