Exercises

1. A signal $x[n]$ is 1 for $n=0$ and $0$ otherwise (an impulse). What is its Fourier transform as a function of $k$?
2. For what integer values of $k$ is the Fourier transform of the $N$-point Hann window function nonzero?
3. Suppose an N-point Fourier transform is done on a complex sinusoid of frequency $2.5\omega$ where $\omega=2\pi/N$ is the fundamental frequency. What percentage of the signal energy lands in the main lobe, channels $k=2$ and $k=3$? If the signal is Hann windowed, what percentage of the energy is now in the main lobe (which is then channels 1 through 4)?