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.