Exercises

1. A signal $x[n]$ is 1 for $n=0$ and $0$ otherwise (an impulse). What is its ($N$-point) Fourier transform as a function of $k$?
2. Assuming further that $N$ is an even number, what does the Fourier transform become if $x[n]$ is 1 at $n=N/2$ instead of at $n=0$?
3. For what integer values of $k$ is the Fourier transform of the $N$-point Hann window function nonzero?
4. In order to Fourier analyze a 100-Hertz periodic tone (at a sample rate of 44100 Hertz), using a Hann window, what value of $N$ would be needed to completely resolve all the partials of the tone (in the sense of having non-overlapping peaks in the spectrum)?
5. 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)?