A continuous-time signal x(t) = 5cos(2πt + π/4) has a fundamental frequency of:
A 1 Hz B 2 Hz C 0.5 Hz D 4 Hz
From x(t) = 5cos(2πt + π/4), ω = 2π, so f = ω/2π = 1 Hz
Which of the following is a causal system?
A y[n] = x[n+1] + x[n] B y[n] = x[n] + x[n-1] C y[n] = x[-n] D y[n] = x[n] + x[n+2]
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:
A T·sinc(πfT) B sinc(fT) C T·sinc²(πfT) D πT·sinc(fT)
The Fourier transform of rect(t/T) is T·sinc(πfT) where sinc(x) = sin(x)/x
A system is BIBO stable if:
A ∑|h[n]| = ∞ B ∑|h[n]| < ∞ C All poles are in right half-plane D Impulse response is periodic
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:
A 0.5 J B 0.25 J C 2 J D 1 J
E = ∫₀^∞ e^(-4t)dt = [-e^(-4t)/4]₀^∞ = 4 1
Which property does NOT apply to the Laplace transform?
A Linearity B Time shifting C Frequency shifting D Periodicity preservation
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:
A Linear and time-invariant B Nonlinear but time-invariant C Linear but time-variant D Neither linear nor time-invariant
Time reversal is linear (satisfies superposition) but time-variant (violates shift invariance)
The Z-transform of x[n] = aⁿu[n] is:
A z/(z-a) B 1/(z-a) C z/(z+a) D a/(z-1)
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:
A 2.5 kHz B 5 kHz C 10 kHz D 20 kHz
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:
A |z| < 0.5 B |z| > 0.5 C |z| = 0.5 D All z except z=0
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:
A x[n] * h[n] ≠ h[n] * x[n] B x[n] * h[n] = h[n] * x[n] C Convolution is only associative D Convolution is only distributive
Convolution is commutative: x[n] * h[n] = h[n] * x[n]
A signal is called energy signal if:
A Its energy is infinite and power is zero B Its energy is finite and power is zero C Both energy and power are finite D Its power is infinite
Energy signal has 0 < E < ∞ and P = 0. Power signal has P > 0 and E = ∞
The inverse Fourier transform of δ(f-f₀) is:
A e^(j2πf₀t) B cos(2πf₀t) C sin(2πf₀t) D e^(-j2πf₀t)
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:
A X(-f) = X(f) B X(-f) = X*(f) C X(-f) = -X(f) D X(f) = X(f)*
For real signals, Fourier transform exhibits conjugate symmetry: X(-f) = X*(f)
A system with H(z) = (z-0.5)/(z-0.8) has:
A One zero at z=0.5 and one pole at z=0.8 B One pole at z=0.5 and one zero at z=0.8 C Two poles at z=0.5 and z=0.8 D Pole-zero cancellation
Numerator gives zeros: z=0.5. Denominator gives poles: z=0.8
Which window function has the BEST frequency resolution but WORST spectral leakage?
A Hamming window B Hann window C Rectangular window D Blackman window
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:
A ∑|x[n]|² = (1/N)∑|X[k]|² B ∑|x[n]|² = (2 1 C ∑|x[n]|² = ∑|X[k]|² D ∑|x[n]| = ∫|X(f)|df
Parseval's theorem for discrete-time: ∑|x[n]|² = (2 1
For a stable IIR filter with difference equation y[n] = 0.5y[n-1] + x[n], the DC gain is:
A 0.5 B 1 C 2 D 1.5
DC gain = H(z=1) = 1/(1-0.5) = 0 1
The concept of 'group delay' in filters refers to:
A The delay of magnitude spectrum B The negative derivative of phase with respect to frequency C The delay between input and output signals D Filter settling time
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)?
A 1/(s+2), Re(s) > -2 B 1/(s-2), Re(s) > 2 C 2/(s+2), Re(s) > -2 D 1/(s+2)², Re(s) > -2
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).
