- 4 Given That A Deterministic Signal X 1 Can Be Decomposed Into A Sum Of Even And Odd Functions That Is X T Xe T 1 (54.89 KiB) Viewed 34 times
4. Given that a deterministic signal x(1) can be decomposed into a sum of even and odd functions, that is, x(t) = xe(t)+x(t), and given also that the Fourier series of x(t) in trigonometric form is x(t) = ao + En-1{an cos(nwot) + bnsin (nwot)}. (b) If x(t)=1+2cos (200nt) + 3sin(400nt), find the exact expressions of xe(t) and x.(1). What are the values of an and bn for all n? Find the general expressions of xe(t) and xo(t) in trigonometric form. State the condition for the expressions of xe(t) and xo(t) to be valid. (c) If x(t) in part (b) is expressed in its complex Fourier series, what are the values of Cn in polar form for all n? (d) (e) Based on the results obtained from part (c), sketch the two-sided magnitude and phase spectra of x(t). What is the minimum sampling frequency required to covert x(t) in part (b) to discrete-time domain to obtain x[n] without information loss? If the sampling frequency used is 500 Hz, sketch the two-sided magnitude spectrum of x[n] from -1 kHz to +1 kHz.