 | by Kenneth Chase | 2 comments

## Properties of Continuous Time Fourier Series – Fourier Series – Signals and Systems | Ekeeda.com

Hi Students in this video we are going to see some properties of the continuous time Fourier series to define a certain properties of a Fourier series let’s make some assumptions so what are the assumptions we are going to make suppose X of T is a periodic signal with time period capital T so that it’s fundamental frequency will become which is omega zero nothing but 2 pie by T then in the Fourier series coefficients denoted by K we will use a notation X of T if I use a Fourier series it will give Fourier series coefficient a K so this notation will tell the relationship between X of T and a K through Fourier series so with respect to this let’s define its property one by one so let’s start with the first property is the linearity it’s quite simple we are having two periodic signals X of T and Y of T we the same fundamental time period t which can be represented like this X of T if you find out a Fourier series it will give up a Fourier series coefficient a K and y of T through a Fourier series they get coefficient b K then if we are having the third signal which is Z of T a linear combination of this two X of T and Y of T as K times X of T plus B times y of T its Fourier series coefficient will be c K and that is nothing but capital A times a K plus capital B times b K so it’s quite simple that this is obeying the property of superposition and homogeneity so A is getting multiplied with X of T similarly on the output side Fourier series coefficient will also get multiplied with the same constant capital A similar exercise will be there for y of T so we’ll get this particular Fourier series coefficients for Z which is nothing but a linear combination of X and Y what let’s go to the next property which is time shifting so what we are having over here we’re having a Y of T as the time shifted version of X of T which is X of t minus t 0 so I repeat this is nothing but time shifted version of X of T so what we’ll get at the Fourier series which we denoted as a b K so what will say Y of T through a Fourier series you will give a coefficient b K and that b K can be calculated like this 1 upon T integral over time period t signal is y of T which is nothing but X of t minus t 0 e raised to minus J K Omega 0 T into DT so we need to solve this integral so what we will do we will put tau as t minus t 0 so that t become tau plus t 0 and DT will be D tau and there won’t be any effect on these limits so what I can say b K will be 1 by T integral over the capital T X of tau e raise to minus JK Omega 0 T will get replaced by tau plus t 0 into D tau so let’s simplify 1 by T integral over capital T X of tau e raise to minus JK Omega 0 tau into e raised to minus J K Omega 0 T0 D tau now tau is the variable with respect to which we will solve this integral for that this term will be a constant so let’s take this term outside the integral so b K will be e raised to minus J K Omega 0 T 0 1 by T integral over capital T X of tau e raise to minus JK Omega 0 tau D tau now it is quite obvious that this particular term is nothing but a K because that we obtained by taking original signal we are just changing the variable from D to tau but in a definite integral that is not going to matter so what I can say in the end I will get b K as e raised to minus J K Omega 0 T0 into a K so in the end I can say X of t minus T 0 through Fourier series what we will say whatever the coefficient earlier was therefore X of T that will just get multiplied with this e raise to minus J K Omega 0 T 0 what we can say over here still this is just a term which is getting multiplied with a K that is not going to effect the magnitude it is just a term coming as a angle angular term rather a complex exponential so what I can say over here b K magnitude is same as a K magnitude so what we can say time shifting does not affect the magnitude of Fourier series coefficients so I can say Fourier series coefficients magnitude are constant or not change though there is a shifting in the signal let’s go to the next property that is time reversal so what we are going to find out over here time reversal if a new signal Y of T is there which is nothing but a time reverse version of X of T we need to find out we need to check what will be the effect on Fourier series coefficient so I can say now X of minus T will become summation K from minus infinity to infinity a K e raise to minus J K 2 pie by T I just substitute Omega 0 is 2 pie by T into T so what I’ve done over here T I replaced by minus T you know original expression of a Fourier series so that I will get this now what I’m going to do I’m going to put K as minus m so that we obtain Y of T which is nothing but X of minus T summation see there wouldn’t be any change over here because just the submission and I’m having a submission from minus infinity infinity only the major effect will happen to this here I say A minus m e raise to J M 2 pie by capital T into small T now this is nothing but a Fourier series representation of any signal where Fourier series coefficients will be this so now what I can say what effect it will have the new coefficient that we obtain by doing the time reversal of a signal will be denoted as b K and that b K is nothing but a minus K so whenever we do a time reverse of a signal what will happen whatever the values you obtain for a raise to repeat a minus K that will be same for a b K so let’s club this two into one line or one statement like this if X of T through Fourier series we are getting a K as Fourier series coefficient then X of minus T through a Fourier series will get a minus K this is nothing but a property which we say a time reversal let’s go to the next property that is time scaling so what we do in a Time scaling X of T is a original signal with a fundamental time period t through Fourier series it is obtaining a K as a Fourier series coefficient now we need to find out if I have a signal X of alpha T I am considering alpha greater than 1 and we need to find out what will be the effect on the Fourier series coefficients so X of alpha T Fourier series representation will be K from minus infinity to infinity a K e raised to J K instead of a T I have to put alpha T into Omega 0 same I can represent in different manner like this so if we see properly over here what I’m having whatever the multiplied we are having with T in the signal that will affect only a fundamental frequency so only a fundamental frequencies multiplier is changing but there is no effect on the Fourier series coefficients so what I can say whenever we have a time scaling in a signal there won’t be no effect on Fourier series coefficients go to the next properties the other we should say its a relationships and we call that as a parseval’s relationship what is showing 1 by T integral over a capital T X of T mod square DT is nothing but summation K from minus infinity to plus infinity mod a K square so this is called as a parseval relationship or a parseval identity of any signal now we know for here a K is a fourier series coefficient so left hand side is nothing but average power that is energy per unit time in one period of the periodic signal which is X of T what I will do X of T I will write in a Fourier series representation so the left-hand side of this particular relation will be 1 by T integral over capital T X of T can be written as mod a K into e raised to J K Omega 0 T Square DT so obviously this is a complex number because mod will be a K only so what I can say now it is 1 by T integral or a capital T more a K square which is nothing but mod a K square so in the end I will get this expression and what is this so mod a K square is the average power in the Kth harmonic component of X of T so parseval’s relation can give a certain relationship let’s discuss what that relationship is so parseval relation states is that the total average power in a periodic signal equals the sum of average powers in all of its harmonic components that’s what it is steady this is nothing but a total average power and this is nothing but average power for individual harmonic component and that component we fill it by selecting the value of K from minus infinity to infinity so it’s a very important relationship which is giving a relationship of total average power of the periodic signal which is average power of harmonic components so here we end properties of a condensed time Fourier series thank you #### Ekeeda

May 5, 2019, 11:42 am Reply

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