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
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)
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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
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)
An LTI system has impulse response h(t) = e^(-3t)u(t). Is this system stable?
For stability, ∫|h(t)|dt must be finite. Here, ∫₀^∞ e^(-3t)dt = 31, which is finite. System is stable.
The output of a system is y[n] = 0.5y[n-1] + x[n]. What is the transfer function H(z)?
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).
A sinusoidal signal x(t) = 5sin(2πf₀t + π/4) is sampled at fs = 10f₀. What is the Nyquist frequency?
Nyquist frequency = fs/2 = 10f₀/2 = 5f₀. The Nyquist rate is 2×(highest frequency) = 2f₀.
Which window function provides the narrowest main lobe but highest side lobes in frequency domain?
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.
For a sequence x[n] = {1, 2, 1, -1}, what is the circular convolution with itself using 4-point DFT?
Circular convolution in time domain equals multiplication in frequency domain. y[n] = IDFT{X[k]×X[k]} = IDFT{X[k]²}.
A signal has autocorrelation R_x(τ) = Ae^(-2|τ|). What is its bandwidth (3dB) approximately?
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.
A filter has poles at z = 0.7 and z = 0.9. What can we infer about its stability and 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.
An analog filter has magnitude response |H(jω)| = 1/√(1+(ω/ωc)⁴). What is the order of the filter?
The denominator power indicates filter order: (ω/ωc)⁴ represents 4th-order filter response.
A real discrete signal x[n] has DTFT X(e^jω). What is the relationship between X(e^jω) and X(e^-jω)?
For real signals, the DTFT exhibits Hermitian symmetry: X(e^-jω) = X*(e^jω), meaning magnitude is even and phase is odd.
A system is described by y[n] = 0.8y[n-1] + x[n]. The steady-state gain is:
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.
An IIR filter has poles at z = 0.6e^j0.4π and z = 0.6e^-j0.4π. The filter order is:
Two distinct poles indicate a second-order filter. Complex conjugate pairs count as one pole pair.
