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291 8.1. We sample a periodic continuous-time signal with a sampling rate: (a) The sampled signal is given by: Expressed as a Discrete Fourier Series: 2 a{n} = > aye a=ne We note that, in accordance with the discussion of Section 8.1, the sampled signal is represented by the summation of harmonically-related complex exponentials. The fundamental frequency of this set of exponentials is 2x/N, where N = 6. Therefore, the sequence z[n] is periodic with period 6. For any bandlimited continuous-time signal, the Nyquist Criterion may be stated from Eq. (4.14b) as: (o F, 2 2Fy, where F, is the sampling rate (Hz), and Fy corresponds to the highest frequency component in the signal (also Hz). As evident by the finite Fourier series representation of z,(¢), this continuous-time signal is, indeed, bandlimited with a maximum frequency of Fn = qx Hz. Therfore, by sampling at a rate of F, = pts Hz, the Nyquist Criterion is violated, and aliasing results. (c) We use the analysis equation of Eq. (8.11): e Nei = Zk = Vo ge . n=0 From part (a), £[n] is periodic with N = 6. Substitution yields: tt it iM- Xi (= 9 > yn 2124/8) Eh ma-9 satin) te “, ‘We reverse the order of the summations, and use the orthogonality relationship from Example 8.1: eo X[k] = 6m YS slm-k+rnj me ‘Taking the infinite summation to the outside, we recognize the convolution between am and shifted impulses (Recall an, = 0 for |mj > 9). Thus, X[k = 6 ys Ch 6r roo Note that from X[k], the aliasing is apparent. © 
