1. Problem 5.21
2. Problem 6.4
3. Problem 6.11
4. Let $x_c(t)$ be a real-valued, bandlimited signal whose Fourier transform $X_c(\Omega)$ is zero for $\vert\Omega\vert \ge 2 \pi (5000)$. The sequence $x(n)$ is obtained by sampling $x_c(t)$ at 10 kHz. Assume that the sequence $x(n)$ is zero for $n < 0$ and $n > 999$.

   Let $X(k)$ denote the 1000-point DFT of $x(n)$. It is known that $X(900) = 1$ and $X(420)$ = 5. Determine $X_c(\Omega)$ for as many values of $\Omega$ as you can in the region $\vert\Omega\vert < 2 \pi (5000)$.

5. Consider estimating the spectrum of a discrete-time signal $x(n)$ using the DFT with a Hamming window for $w(n)$. A conservative rule of thumb for the frequency resolution of windowed DFT analysis is that the frequency resolution is equal to the width of the main lobe of $W(e^{j\omega}$. You wish to be able to resolve sinusoidal signals that are separated by as little as $\pi/100$ in $\omega$. In addition, your window length $L$ is constrained to be a power of 2. What is the minimum length $L = 2^{\nu}$ that will meet your requirement?

6. Let $x(n)$ be a discrete-time signal whose spectrum you wish to estimate using a windowed DFT. You are required to obtain a frequency resolution of at least $\pi/25$ and are also required to use a window length $N = 256$. A safe estimate of the frequency resolution of a spectral estimate is the main-lobe width of the window used. Which of the windows in Table 8.2 will satisfy the criteria given for the desired frequency resolution?

7. Let $x(n)$ be a discrete-time signal obtained by sampling a continuous-time signal $x_c(t)$ with some sampling period $T$ so the $x(n) = x_c(nT)$. Assume $x_c(t)$ is bandlimited to 100 Hz, i.e., $X_c(\Omega) = 0$ for $\vert\Omega\vert \ge 2\pi(100)$. We wish to estimate the continuous-time spectrum $X_c(\Omega)$ by computing a 1024-point DFT of $x(n)$, $X(k)$. What is the smallest value of $T$ such that the equivalent frequency spacing between consecutive DFT samples $X(k)$ corresponds to 1 Hz or less in continous time?

8. Assume that $x(n)$ is a 1000-point sequence obtained by sampling a continous-time signal, $x_c(t)$ at 8 kHz and the $X_c(\Omega)$ is sufficiently bandlimited to avoid aliasing. What is the minimum DFT length $N$ such that adjacent samples of $X(k)$ correspond to a frequency spacing of 5 Hz or less in the original continuous-time signal?

Note: Problems 4-8 are from Discrete-Time Signal Processing, 2nd Ed. by Oppenheim and Schafer.