For a periodic signal x[n] with period N=4, the DFT X[k] has X[0]=8. The average value of the signal is:
X[0] = Σ x[n] (sum of all samples in one period). Average = X[0]/N = 48 = 2.
A continuous-time signal x(t) is band-limited to 15 kHz. Using ideal reconstruction from samples, the minimum sampling frequency required is:
Nyquist theorem: fs ≥ 2 × fmax = 2 × 15 kHz = 30 kHz.
The inverse Fourier transform of X(f) = δ(f-f₀) + δ(f+f₀) is:
IFT{δ(f-f₀)} = e^(j2πf₀t) and IFT{δ(f+f₀)} = e^(-j2πf₀t). Sum = 2cos(2πf₀t), but accounting for delta properties, result is cos(2πf₀t).
For a finite impulse response (FIR) filter of length M=5, the maximum linear phase is achieved when:
Linear phase FIR filters require symmetric (even) or antisymmetric (odd) impulse responses about the center. For length 5, center is n=2.
A system has step response s(t) = 1 - e^(-2t)u(t). Its impulse response h(t) is:
h(t) = ds(t)/dt = d/dt[1 - e^(-2t)u(t)] = 2e^(-2t)u(t).
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The DTFT of x[n] = δ[n-3] is:
DTFT{δ[n-n₀]} = e^(-jωn₀). With n₀=3, X(e^(jω)) = e^(-j3ω).
For a Laplace transform H(s) = 5/(s+3), the system impulse response is:
Using partial fractions and standard pairs: L^(-1){K/(s+a)} = Ke^(-at)u(t). Here K=5, a=3, so h(t) = 5e^(-3t)u(t).
A discrete signal undergoes 16-point FFT computation. The frequency resolution is Δf = 1 kHz. The sampling frequency fs is:
Frequency resolution Δf = fs/N, where N=16. Therefore, fs = Δf × N = 1 × 16 = 16 kHz.
A first-order discrete filter y[n] = 0.8y[n-1] + 0.2x[n] has DC gain (at z=1) of:
Transfer function H(z) = 0.2/(1-0.8z⁻¹). DC gain H(1) = 0.2/(1-0.8) = 0.02.2 = 1.
For a signal x[n] = cos(πn/4), the fundamental period N is:
For x[n] = cos(ω₀n), period N = 2π/ω₀ = 2π/(π/4) = 8.
The convolution of two signals each of length 5 using linear convolution yields an output of length:
Linear convolution of signals of length M and N produces output of length M+N-1 = 5+5-1 = 9.
A system with transfer function H(s) = (s+2)/(s²+3s+2) has poles at:
Denominator s²+3s+2 = (s+1)(s+2) = 0, giving poles at s = -1 and s = -2.
For a complex exponential signal e^(j2πf₀t) sampled at fs = 10 kHz with f₀ = 3 kHz, aliasing occurs at frequency:
Aliased frequency = |f₀ - kfs| = |3 - 10| = 7 kHz (for k=1). The signal appears as a 7 kHz component in the baseband.
A continuous-time signal x(t) = 5sin(100πt) is sampled at 150 Hz. What is the Nyquist frequency required to avoid aliasing?
The maximum frequency in the signal is 50 Hz (from 100π = 2πf). Nyquist frequency = 2 × 50 = 100 Hz.
Which property of the Fourier transform states that multiplication in time domain equals convolution in frequency domain?
The modulation theorem (or frequency convolution property) states that x(t)×y(t) ↔ X(f)*Y(f).
A causal LTI system has impulse response h(t) = e^(-3t)u(t). Is this system stable?
∫₀^∞ e^(-3t) dt = 31 < ∞. For stability, BIBO condition requires absolute integrability, which is satisfied.
The Z-transform of x[n] = (21)^n u[n] is:
X(z) = Σ(21)^n z^(-n) = z/(z - 21) = z/(z - 0.5) with ROC |z| > 0.5.
An even signal x(t) has Fourier transform X(f). What can be said about X(f)?
For an even signal, the Fourier transform is real and even: X(-f) = X(f) and X*(f) = X(f).
The energy of a signal x[n] = {1, 2, -1, 0} is:
Energy = 1² + 2² + (-1)² + 0² = 1 + 4 + 1 + 0 = 6.
A discrete signal x[n] = cos(π n/6) has a period of:
For x[n] = cos(2πk n/N), the period N = 6/(21) = 12 samples.
