Govt. Exams
Entrance Exams
DC gain H(z)|_{z=1} = 1/(1-0.8) = 5. Alternatively, setting y[n] = y[n-1] = y_ss: y_ss = 0.8y_ss + x_ss gives gain = 5.
For real signals, the DTFT exhibits Hermitian symmetry: X(e^-jω) = X*(e^jω), meaning magnitude is even and phase is odd.
The denominator power indicates filter order: (ω/ωc)⁴ represents 4th-order filter response.
For discrete-time systems, stability requires all poles inside the unit circle (|z| < 1). Both 0.7 and 0.9 satisfy this, so the filter is stable.
Power spectral density S_x(ω) = Fourier transform of R_x(τ) = 4A/(4+ω²). At 3dB point: 4+ω² = 8, so ω ≈ 2 rad/s.
Circular convolution in time domain equals multiplication in frequency domain. y[n] = IDFT{X[k]×X[k]} = IDFT{X[k]²}.
Rectangular window has the narrowest main lobe (4π/N) but the highest side lobes (-13 dB). Other windows trade main lobe width for lower side lobes.
Nyquist frequency = fs/2 = 10f₀/2 = 5f₀. The Nyquist rate is 2×(highest frequency) = 2f₀.
From y[n] = 0.5y[n-1] + x[n], taking Z-transform: Y(z) = 0.5z⁻¹Y(z) + X(z). So H(z) = Y(z)/X(z) = 1/(1-0.5z⁻¹).
For stability, ∫|h(t)|dt must be finite. Here, ∫₀^∞ e^(-3t)dt = 1/3, which is finite. System is stable.
