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Let x[n]=cos( 3/2π n)+2sin( 3/7π n),y[n]=cos( 3/π n)−sin( 3/5π n). Determine the Fourier series representation of z[n]=e j5( 3/2π )n x[n]+3y[n−2].

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Final answer:

The question pertains to finding the Fourier series representation of a signal z[n], which is a composed signal of x[n] and y[n]. The Fourier series will involve identifying the fundamental frequencies, harmonics, amplitudes, and phases of the composite signal.

Step-by-step explanation:

The student is asking for the Fourier series representation of a composite discrete-time signal z[n]. We have signals x[n] and y[n], defined as x[n]=cos(3/2πn)+2sin(3/7πn) and y[n]=cos(3/πn)−sin(3/5πn), respectively. To get z[n], we apply the given operations, which include multiplication by a complex exponential and a time shift for y[n].

We are not asked to perform a full Fourier analysis here; rather, we are looking at a conceptually similar process involving trigonometric functions. It's important to note that multiplying by a complex exponential corresponds to modulating the frequency in Fourier analysis and the time-shifting results in a phase shift in the Fourier coefficients.

The discrete-time Fourier series for these types of functions typically involves identifying the fundamental frequencies and their harmonics, as well as their corresponding amplitudes and phases. In this case, since x[n] and y[n] are sums of cosines and sines, their Fourier series are immediate by recognizing each cosine and sine term as a component at a specific frequency.

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