1. How many complex numbers z are there such that z3= 1?
2. Translate x(t) = -cos(πt + π/3) into standard form A⋅sin(2πft+ φ) (There are multiple correct answers)
3. If fs = 100 Hz, what are three aliasingfrequencies for f = 80 Hz?
4. A signal x is delayed by one sample and scaled by −1/2,producing a new signal y[n] = −1 2 x[n − 1]. (a) How does Y[m]relate to X[m]? (b) What about |Y[m]| and |X[m]|?
5. If a signal x’s DFT has entirely real coefficients X[m] (hasno imaginary component), what can you deduce about x? Consider thephase for each analysis frequency.
6. Let x[n] be a signal with N = 5000 samples. (a) If youcompute an STFT with a frame length of K = 1000 and hop length h =250, what will be the shape of the resulting spectrogram? (b) Howwould it change if you set K = 2000? (c) What if you set h =100?
