Questions & Answers of Continuous-time and discrete-time Fourier series

The magnitude and phase of the complex Fourier series coefficients ak of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation C is the set of complex numbers, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

Let x~n=1+cosπn8 be a periodic signal with period 16. Its DFS coefficients are defined by $a_k=\frac1{16}\sum\limits_{n=0}^{15}\widetilde x\left[n\right]exp\left(-j\frac{\mathrm\pi}8kn\right)$ for all k. The value of the coefficients a31 is_____.

Let g ( t ) = e π t 2 , and h(t) is a filter matched to g(t) . If g(t) is applied as input to h(t) , then the Fourier transform of the output is

The trigonometric Fourier series of an even function does not have the

The Fourier series of a real periodic function has only

P. Cosine terms if it is even
Q. Sine terms if it is even
R. Cosine terms if it is odd
S. Sine terms if it is odd

Which of the above statements are correct?