[REFERENCE](#page-14-0)

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REFERENCE

1. Mitra, S. K. Digital Signal Processing: A Computer-Based Approach, 2nd ed. McGraw-Hill, New York, 2001.

PROBLEMS

9.1-1 Find the discrete-time Fourier series (DTFS) and sketch their spectra |Dr| and D^r for 0≤r≤N0−1 for the following periodic signal:

x[n] = 4 cos 2.4πn+2 sin 3.2πn

• 9.1-2 Repeat Prob. 9.1-1 for x[n] =cos 2.2πncos 3.3πn.
• 9.1-3 Repeat Prob. 9.1-1 for x[n]=2 cos 3.2π(n−3).
• 9.1-4 Determine and sketch the DTFS spectrum D^r of a 7-periodic signal x[n] that over 0 ≤ n ≤ 6 is given by

$$[0, 1, -2, 3, -4, 5, -6]$$

How does the spectrum D^r change if x[n] is time reversed?

• 9.1-5 Find the discrete-time Fourier series and the corresponding amplitude and phase spectra for the x[n] shown in Fig. P9.1-5.
• 9.1-6 Repeat Prob. 9.1-5 for the x[n] depicted in Fig. P9.1-6.
• 9.1-7 Repeat Prob. 9.1-5 for the x[n] illustrated in Fig. P9.1-7.
• 9.1-8 An N0-periodic signal x[n] is represented by its DTFS, as in Eq. (9.3). Prove Parseval’s theorem

Figure P9.1-5

Figure P9.1-6

Figure P9.1-7

2

(for the DTFS), which states that

$$\frac{1}{N_0}\sum_{n=\langle N_0\rangle} |x[n]|^2 = \sum_{r=\langle N_0\rangle} |\mathcal{D}_r|$$

In the text [Eq. (9.36)], we obtain Parseval’s theorem for the DTFT. [Hint: If w is complex, then |w| ² = ww^∗ and use Eq. (8.15).]

• 9.1-9 Answer yes or no, and justify your answers with an appropriate example or proof.
  • (a) Is a sum of aperiodic discrete-time sequences ever periodic?
  • (b) Is a sum of periodic discrete-time sequences ever aperiodic?
• 9.2-1 Show that for a real x[n], Eq. (9.18) can be expressed as

$$x[n] = \frac{1}{\pi} \int_0^{\pi} |X(\Omega)| \cos(\Omega n + \angle X(\Omega)) d\Omega$$