Question

Question We are given the following signal (t), which is also known as "infinite train" of Dirac delta functions. w(t) TITI t(ms) -20 -10 0 10 20 30 Write a mathematical expression for w(t) in the time domain. Find the Fourier coefficients of the signal using the definition and its Fourier transform W(S) [Note that results without correct derivation will get minimal points]. Sketch the signal magnitude spectrum | W (f). Is this signal an energy or a power signal? Please provide detailed explanation.

Answer 1

All the parts has been solved and solution has been attached.

Q: soun: GILLE 20 lo 30 tims) who is a Periochic Impulse Train with Time Period IT=loms time domain Expression, W(4= 2816) + 28( told +28(*90)+28 (4-33) + f 28(6+10)+28 (*+20) +-- wra= 2 § 8(+-KI) Ks-o0 where, t in ms, T=loms (Period) ری -2,-1,0,1,2,3 k is Cir Integer : at solni Fourier coefficient of a signal arts is Tz given as, Cn= Meth Jowot, -T2 where, T= time period of signal. wo=21: angular Freequency here, Cn = 4 5 wlis é inwot at T2 Woch -72 consider one Period of 5 Cn = 1 To Wrth we have at 5 5 - ㅗ 10 dt 2 slt) ē Jowot so te s sime de 5

or cn= 2 up 1 15 set dt = 1 so 25 Area under Impulse signal = L. to sits at =1 also, allt lt1= xlo) 81* Jowot : Do wolo) f(t)=sit e f(thé = e So , = f(t) hence, [ca- it or Im = 2 / 1 trioms where T=loms Ang Fourier Transform weth=2 { flt-kt) ke-

is. Tourier Transform of a Periodic signal is Extension version of of Fourier Fourier wefficient with each coefficient value multiplied by replace by Impulse signal here, Fourier coefficient na į or, Kal=ht lal and 2 T C KHz) -36=2wym wo 0 Wo qwo 3 wo nwo د ولہ 2 ) 2 TT - TT T 10 5x153 1 Cal Elm Kong x103 radian th ㅗ 5 > bebe x103 radian (nwo) -21) — 17 MS 27775 3775

henre is replaced by its Fourier Transform Impulge signal multiplied by an. wifi con be plotted hodc.fo Wo= I xla radian ass 2г ATT 54 2 . Wo (x103 radian) 2wo o دلہا -wo -zwo in frequency axis, W-auf, hence T f= الب 0.21T =0.1=len lo ау 5 2 T 2 multiple f = 0.1 kkz will be hence, frequency aris of 0.1 kuuz? wlf) २॥ 20 at hallh kuz) Uil 0:2 f -02-01

Soini mu nagnitude de spectruma (wlfl 2T1 2T halla 02 0 0.1 f(km2) -0.2 @ its Power signal because its a Periodic signale for a Power signal, Energy = 60; and Argi finite Calculate its Energy: E=90 (NCA) 12 at į aid (t_kT) Power= :h) لمه ka

hence its Energy, E=461 Its at a sositio) at [ Sures? In de *st*• **---- (po zimes] E sa s²ltat a finite = 1 but addition of its oo times beads to o. lang= Average lower of the sign and with period 'Ti Pang= 1 / 2 5l we alle at Parga - 5 m . gal to at cTz w(+7=28lar, - 7 / < 4</ 2 Pong to 's 4.8² (tat = -5 4 x x1 = on krist Pang= 4X1 = 0.4 k-unit = noo unit Ang locms) I Pang (finite) Hence its a power signal.