EE 451

Homework Assignment 8
Due Oct. 18, 2000

  1. Problem 5.21
  2. Problem 6.4. Note that the designer also needs some other chips (adders and multipliers) to complete the job. You should just get an equation where X(k) can be written in terms of three eight-point DFTs X1(k), X2(k) and X3(k).
  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.

Bill Rison, < >