DSP Mock Exam 1 Mock Exam 1

Consider a function $f(t)$ defined over the interval $[0, T]$. The Fourier coefficient $C_n$ in the complex exponential form is given by $C_n = \frac{1}{T} \int_{0}^{T} f(t)e^{-jn\omega_0 t} dt$. If $f(t) = 1$ for $0 < t < T$, what is the value of $C_0$?

Which of the following statements best describes the relationship between a continuous-time signal and its discrete-time representation?

What is the primary purpose of using the complex exponential Fourier series representation instead of the trigonometric Fourier series?

A communication system samples a voice signal with a maximum frequency of $4 \text{ kHz}$ at its Nyquist rate. If the voice signal's bandwidth suddenly expands, increasing its maximum frequency to $6 \text{ kHz}$, what is the minimum integer factor by which the original sampling frequency must be increased to avoid aliasing with the new signal?

What is the key distinction between the trigonometric Fourier series and the complex exponential Fourier series when representing a periodic signal?

What is the primary consequence of violating the Nyquist sampling theorem?

The Fourier series representation of a periodic signal involves summing an infinite number of harmonic components. What determines the contribution of each harmonic component to the overall signal?

If $C_n$ represents the coefficient for the $n^{th}$ harmonic in the complex exponential Fourier series, and $C_n = 0$ for all $|n| > 3$, what does this imply about the signal $f(t)$?

What is the primary purpose of the Fourier series representation in the context of the sampling theorem?

A periodic analog signal x(t) is sampled to produce x*(t). If the sampling frequency $f_s$ is less than twice the maximum frequency component $f_m$ of x(t), what phenomenon is most likely to occur?