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}(Ω) is zero for |Ω| ≥ 2π(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}(Ω) for as many values of Ω as you can in the region |Ω| < 2π(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^{jω}. You wish to be able to resolve sinusoidal signals that are separated by as little as π/100 in ω. In addition, your window length L is constrained to be a power of 2. What is the minimum length L = 2^{ν} 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 π/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}(Ω) = 0 for |Ω| ≥ 2π(100). We wish to estimate the continuous-time spectrum X_{c}(Ω) 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}(Ω) 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.