Question

How to write an expression for magnitude s(w)?

J.cos -ku] (cos-kt ) + COs CoS Exercise Derivn (4) Exercise 3 Use (9) to write an expression for Su)

Answer 1

* If x(t) is of the form

                                                                                  

. (t) = A cos(wit)

Then the single sided power spectrum is given by

                                                      

0 ,(اند - بی) کا 1 | = (ن) S

* If x(t) is a combination of two sinusoids like

                                         

r(t) = A1 cos(wit+01) + A2 cos(wyt + 02)

Then the single sided power spectrum is given by

                            

Sa(w) = ( 4 )owwi)+( 4 ) sw – ws), w.unto (1)

provided that

                                                                                    

این پ المی

In general it extends for infinite number of sinusoids of arbitrary frequencies and phase provided all frequencies are different.

The given signal is

                               

X0 s(t) = Jo(n) cos wct + Jx()[1 +(-1)] cos (wc – kwm) t k=1

s(t) consists of infinite sinusoids. We will assume that

                                                                 

We + kwm, for any 'k

So all frequencies present in s(t) are different. So using properties of (1) and extending to infinite no of terms, we have single sided power spectrum for s(t) as

                             

s(w) – Hayn) (w – wd + Sign) (1 + (-1)45(w – (we – k'wm)