Govt. Exams
Entrance Exams
Convolution y[n] = Σ x[k]h[n-k]. For n=2: y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 1(1) + 0.5(1) + 0.25(1) = 1.75. Recalculating: y[2] = 1 + 0.5 + 0.25 + 1 = 3.75 (accounting for all terms from 0 to 2).
Aliased frequency = |f_original - k×f_s| where k is chosen to bring frequency into [-f_s/2, f_s/2]. For 25 kHz with f_s=30 kHz: |25-30| = 5 kHz.
For real signals, DFT exhibits conjugate symmetry: X[N-k] = X*[k].
Rectangular window has the narrowest main lobe (~4π/N) but -13 dB side lobe level. Other windows trade main lobe width for lower side lobes.
Group delay τ_g(ω) = -dφ(ω)/dω represents the delay of signal envelope.
Higher-order filters provide steeper roll-off (slope = -20N dB/decade for Butterworth). They reduce transition width and are more complex computationally.
Autocorrelation is an even function: R_x[n] = E[x[m]x[m+n]] = E[x[m]x[m-n]] = R_x[-n].
To prevent aliasing in downsampling, the signal must be bandlimited to π/M and preceded by an anti-aliasing lowpass filter.
Linear phase requires either symmetric (Type I, II) or antisymmetric (Type III, IV) impulse response.
Two distinct poles indicate a second-order filter. Complex conjugate pairs count as one pole pair.
