Signals and Systems Theory

If a system's impulse response is $h(t) = e^{-at}u(t)$, where $a > 0$ and $u(t)$ is the unit step function, what is the system's transfer function $H(s)$?

Consider a discrete-time system with input $x[n]$ and output $y[n]$ described by the difference equation $y[n] - 0.5y[n-1] = x[n]$. What is the system's response to a unit impulse input $x[n] = \delta[n]$?

A continuous-time signal $x(t) = \cos(2\pi f_0 t)$ is sampled at a rate of $f_s$ samples per second. According to the Nyquist-Shannon sampling theorem, what is the minimum sampling rate $f_s$ required to perfectly reconstruct $x(t)$ from its samples?

In the context of error correction codes, what is the Hamming distance between the codewords 10110 and 11001?

Which statement accurately describes the fundamental difference between frequency modulation (FM) and amplitude modulation (AM)?

What is the primary purpose of applying windowing functions, such as Hamming or Blackman windows, to a signal before computing its Discrete Fourier Transform (DFT)?

A linear time-invariant (LTI) system has an impulse response $h[n] = (0.5)^n u[n]$. Determine the system's output $y[n]$ when the input is $x[n] = \delta[n] - 0.5\delta[n-1]$, where $\delta[n]$ is the discrete-time unit impulse function.

Consider a flat fading wireless channel. How does the coherence bandwidth relate to the channel's frequency selectivity?

A signal $x(t)$ has a Fourier Transform $X(f)$. If $x(t)$ is real and even, which of the following statements is true about $X(f)$?

In digital communication systems, what is the primary advantage of using Orthogonal Frequency Division Multiplexing (OFDM) over single-carrier modulation in frequency-selective fading channels?

A causal LTI system is described by the differential equation $\frac{d^2y(t)}{dt^2} + 5\frac{dy(t)}{dt} + 6y(t) = x(t)$. What are the poles of the system's transfer function?

Consider a signal $x[n]$ with a Z-transform $X(z)$. If $X(z) = \frac{z}{z-0.5}$ and the region of convergence (ROC) is $|z| > 0.5$, what is the signal $x[n]$?

In antenna theory, what is the effective aperture ($A_e$) of an antenna related to its directivity (D) and the wavelength ($\lambda$)?

A binary symmetric channel (BSC) has a crossover probability of $p$. If a codeword of length $n$ is transmitted, what is the probability that exactly $k$ bits are flipped during transmission?

What is the primary advantage of using convolutional codes with Viterbi decoding compared to block codes with algebraic decoding in digital communication systems?

A discrete-time signal $x[n]$ is defined as $x[n] = \begin{cases} 1, & 0 \leq n \leq N-1 \ 0, & \text{otherwise} \end{cases}$. What is the magnitude of its Discrete-Time Fourier Transform (DTFT) at frequency $\omega = 0$?

In the context of wireless communication, what is the Clarke's model primarily used for?

Consider a memoryless source with alphabet {A, B, C} and probabilities P(A) = 0.5, P(B) = 0.25, and P(C) = 0.25. What is the Huffman code average codeword length for this source?

A signal $x(t)$ is bandlimited to 5 kHz. If this signal is ideally sampled at a rate of 12 kHz, what is the highest frequency component present in the reconstructed signal using ideal reconstruction?