we have already completed trigonometric Fourier series expansion and now we will move to the next type of Fourier series which is complex exponential Fourier series this particular type is very important and therefore we will discuss more questions on this type I will first write down the expansion and then we will understand what are different points related to it whenever you have a periodic signal XT and you want the complex exponential Fourier series expansion then you need to perform the summation starting from n equal to minus infinity to infinity and the summation of C n multiplied to e raise to power J n Omega naught T where CN is the complex exponential Fourier coefficient I will write this point CN is the complex exponential exponential Fourier coefficient we have seen Fourier coefficients in case of trigonometric Fourier series expansion and like there here also we have one Fourier coefficient and CN is equal to 1 divided by T not integration over T not signal XT multiplied to e raise to power minus J and Omega naught T DT so the first step is to calculate the Fourier coefficient this is the first step and once you have the Fourier coefficient CN you can substitute its value here and you will have the Fourier series expansion for signal XT which is a periodic signal you can convert complex exponential Fourier series to trigonometric Fourier series and you can also convert trigonometric Fourier series to complex exponential Fourier series and therefore there must be some relation between coefficients n BN and C n where a and BN who are the coefficients which we saw in trigonometric Fourier series expansion and C n is the complex exponential Fourier coefficient so there must be some relation between a and B n n CN and using that we can convert complex exponential Fourier series to trigonometric Fourier series and vice versa in the coming presentation we will see the relation between a n BN and C n now we will move to another important point which is actually the property of CN if you perform the reversal operation reversal operation this means in place of n if you have minus n you will have C minus n equal to 1 over T naught integration over T naught X T multiplied to e raise to power J n Omega naught T in place of n when you put minus n you will have positive J n Omega naught T and let's say this is the first equation after this we will perform the conjugate operation we already know what is conjugate operation it is very simple operation in which we simply reverse the sign of the imaginary part CN is a complex coefficient it is having real as well as imaginary part and after performing the reversal operation we are performing the conjugate operation this will give us C minus n its conjugate equal to 1 divided by T naught integration over T naught conjugate of signal XT multiplied to e raise to power minus J and Omega naught T DT due to conjugate operation we will have negative of G n Omega naught T and let's call this equation number 2 now if C n is conjugate symmetric if C n is conjugate symmetric this means C n is equal to C minus n its conjugate and if you see equation 1 and equation 2 you will find the left-hand side is same and therefore right-hand side will also be same and on comparing you will find XT will be equal to XT conjugate so this implies XT is equal to conjugate of XT and this implies XT is real in nature because whenever you find the conjugate and you have the same signal this clearly means there is no imaginary part and therefore the signal is purely real signal so this is one important result we are having now we will move to the next point and after that we will have the conclusion to understand this point I will write C n has magnitude of C n and e raise to power J angle of CN cn is a complex coefficient therefore it is having two different parts the first part is the magnitude which is represented by mod CN and the second part is the angle which is angle C n now we will repeat the process we will perform the reversal this means in place of n we will have minus n so we have C minus n equal to magnitude of C minus n e raised to power J C minus n now we will perform the conjugate operation this will give us C minus n conjugate on the left hand side and on the right hand side we will have C minus ends magnitude he raised to power minus J angle C minus n we have negative sign here because of conjugate operation let's call this equation number 3 and let's call this equation number 4 and as you can see here we are discussing the case when C n is conjugate symmetric this means C n is equal to C minus n conjugate so left hand side of third and fourth equations are same therefore right hand side will also be same so magnitude C n e raise to power J angle of CN is equal to magnitude of C minus n magnitude of c- n e raise to power minus J angle of C minus n minus J angle of C minus n on comparing you can clearly see the two magnitudes must be same this means magnitude of CN is same as magnitude of C minus n magnitude of CN his same as magnitude of C minus n and angle of CN is same as negative half angle of C minus n angle of CN is same as negative half angle of C minus N or we can write negative of angle CN is equal to angle C minus n now if you see the two magnitudes you will find after performing the reversal we have the same magnitude so if we have signal XT and CN is the complex exponential Fourier coefficient of the given signal XT and when you calculate the magnitude after performing the reversal you have the same magnitude this implies it is having the purely even nature and if you see the angle you will find after performing the reversal we have negative of the angle this implies it is having the odd nature now we will move to the conclusion who when XT is a real signal C n which is the complex exponential Fourier coefficient his conjugate symmetric and when CN his conjugate symmetric the magnitude of C n is even and the angle of CN or phase of CN his odd so I will summarize this important conclusion when XT is real this implies CN is conjugate symmetric and this implies the magnitude of C n is even and the angle of CN or phase of CN is odd now we will solve one example based on this important conclude in this example in this example signal XT is there which is a periodic signal and it is found that the exponential coefficient CN is equal to n square e raised to power J and cube the solution is very easy you can see the magnitude part is n square this means it is even and the angle part is n cube this means it is odd so magnitude is even and phase or angle is odd this implies CN is conjugate symmetric conjugate symmetric and as it is conjugate symmetric the corresponding signal which is XT is real in nature so this is all for the introduction of complex exponential Fourier series expansion in the coming presentations we will solve more questions based on the exponential Fourier series so if you have any doubt you may ask in the comment section I will end this lecture here see you in the next one [Applause] [Music] you [Music]