A continuous-time signal x(t) = 5cos(2πt + π/4) has a fundamental frequency of:
From x(t) = 5cos(2πt + π/4), ω = 2π, so f = ω/2π = 1 Hz
Which of the following is a causal system?
A causal system does not depend on future values. Only option B satisfies this condition.
The Fourier transform of a rectangular pulse rect(t/T) is:
The Fourier transform of rect(t/T) is T·sinc(πfT) where sinc(x) = sin(x)/x
A system is BIBO stable if:
Bounded Input Bounded Output stability requires the impulse response to be absolutely summable: ∑|h[n]| < ∞
The energy of signal x(t) = e^(-2t)u(t) is:
E = ∫₀^∞ e^(-4t)dt = [-e^(-4t)/4]₀^∞ = 41 = 0.25 J
Advertisement
Which property does NOT apply to the Laplace transform?
The Laplace transform does not preserve periodicity of signals in the frequency domain
A signal x[n] is time-reversed to get x[-n]. This operation is:
Time reversal is linear (satisfies superposition) but time-variant (violates shift invariance)
The Z-transform of x[n] = aⁿu[n] is:
Z-transform: X(z) = ∑aⁿu[n]z⁻ⁿ = z/(z-a) for |z| > |a|
Nyquist sampling rate for a signal with maximum frequency 5 kHz is:
Nyquist sampling rate = 2 × (maximum frequency) = 2 × 5 = 10 kHz
For a discrete-time LTI system with H(z) = 1/(1-0.5z⁻¹), the ROC should be:
For a causal system, ROC is outside the outermost pole. Pole at z=0.5, so |z| > 0.5
The convolution of two sequences is commutative, meaning:
Convolution is commutative: x[n] * h[n] = h[n] * x[n]
A signal is called energy signal if:
Energy signal has 0 < E < ∞ and P = 0. Power signal has P > 0 and E = ∞
The inverse Fourier transform of δ(f-f₀) is:
IFT of δ(f-f₀) = e^(j2πf₀t) using sifting property of delta function
For a real signal x(t), its Fourier transform X(f) satisfies:
For real signals, Fourier transform exhibits conjugate symmetry: X(-f) = X*(f)
A system with H(z) = (z-0.5)/(z-0.8) has:
Numerator gives zeros: z=0.5. Denominator gives poles: z=0.8
Which window function has the BEST frequency resolution but WORST spectral leakage?
Rectangular window has narrowest main lobe (best resolution) but highest sidelobe (worst leakage)
Parseval's theorem states that energy in time domain equals energy in frequency domain. For discrete signals:
Parseval's theorem for discrete-time: ∑|x[n]|² = (21π)∫|X(e^jω)|²dω over 2π
For a stable IIR filter with difference equation y[n] = 0.5y[n-1] + x[n], the DC gain is:
DC gain = H(z=1) = 1/(1-0.5) = 01.5 = 2
The concept of 'group delay' in filters refers to:
Group delay τ_g(ω) = -dθ(ω)/dω represents dispersion in filter's phase response
A continuous-time signal x(t) = e^(-2t)u(t) is applied to a system. What is the Laplace transform X(s)?
For x(t) = e^(-at)u(t), the Laplace transform is 1/(s+a) with ROC Re(s) > -a. Here a=2, so X(s) = 1/(s+2).
