Consider a continuous-time finite-energy signal $f(t)$ whose Fourier transform vanishes outside the frequency interval $-\omegac, \omegac$, where $\omegac$ is in rad/sec. The signal $f(t)$ is uniformly sampled to obtain $y(t) = f(t) p(t)$. Here, $$p(t) = \sum{n=-\infty}^{\infty} \delta(t - \tau - nTs),$$ with $\delta(t)$ being the Dirac impulse, $Ts > 0$, and $\tau > 0$. The sampled signal $y(t)$ is passed through an ideal lowpass filter $h(t) = \omegac Ts \frac{\sin(\omegac t)}{\pi \omegac t}$ with cutoff frequency $\omegac$ and passband gain $Ts$. The output of the filter is given by .

Consider a continuous-time, real-valued signal $f(t)$ whose Fourier transform $F(\omega) = \int{-\infty}^{\infty} f(t) \exp(-j \omega t) dt$ exists. Which one of the following statements is always TRUE?

Let $f(t)$ be a periodic signal with fundamental period $T0 > 0$. Consider the signal $y(t) = f(\alpha t)$, where $\alpha > 1$. The Fourier series expansions of $f(t)$ and $y(t)$ are given by $f(t) = \sum{k=-\infty}^{\infty} ck e^{j\frac{2\pi}{T0}kt}$ and $y(t) = \sum{k=-\infty}^{\infty} dk e^{j\frac{2\pi}{T0}\alpha kt}$. Which of the following statements is/are TRUE?

Let $xn$ be a discrete-time signal whose $z$-transform is $X(z)$. Which of the following statements is/are TRUE?

The radian frequency value(s) for which the discrete time sinusoidal signal $xn = A \cos(\Omega n + \pi/3)$ has a period of 40 is/are .

The relationship between any $N$-length sequence $xn$ and its corresponding $N$-point discrete Fourier transform $Xk$ is defined as $$Xk = \mathcal{F}\{xn\}$$ Another sequence $yn$ is formed as below $$yn = \mathcal{F}\{\mathcal{F}\{\mathcal{F}\{\mathcal{F}\{xn\}\}\}\}$$ For the sequence $xn = \{1, 2, 1, 3\}$, the value of $Y0$ is .

A causal and stable LTI system with impulse response $h(t)$ produces an output $y(t)$ for an input signal $x(t)$. A signal $x(0.5t)$ is applied to another causal and stable LTI system with impulse response $h(0.5t)$. The resulting output is .

For a causal discrete-time LTI system with transfer function $H(z) = \frac{2z^2+3}{(z+\frac{1}{3})(z-\frac{1}{3})}$, which of the following statements is/are true?