in this lecture we will study periodic and non periodic signals non periodic signals are also called as a periodic signals non periodic signals are also called as a periodic signals his signal is said to be periodic if it repeats itself after a regular interval of time so if there is a signal which repeats itself after a regular interval of time we call that signal periodic signal by reputation after regular interval of time means this signal is having the same value after this particular time interval let's take one example to understand this let's say the time interval delta T is equal to 5 seconds for periodic signal XT and according to definition of periodic signals XT will remain same if we increment the time by 5 seconds so XT is equal to X T plus 5 and if we again increment the time by 5 seconds XT plus 5 will remain same as XT plus 10 X T plus 5 will remain same as XT plus 10 in the same way X T plus n minus 1 multiplied with 5 is equal to X T plus and 5 this is what we have for this particular case and we can generalize it in place of XT plus + 5 we can have X T plus n capital T not this 5 is the time interval and this time interval is nothing but fundamental period and fundamental period is represented by T not so we can say that in case of periodic signals XT is equal to XT plus small n capital T not this is the condition for periodicity the signal XT must be equal to XT plus and D not were small n small n is an integer and capital T naught is the fundamental period the fundamental period is the smallest positive value of time for which signal is periodic I will write this down this is the definition of fundamental period it is the smallest small lest positive value of time for which signal is periodic is periodic and this is smallest positive value is fixed this is the definition of fundamental period T naught we will understand why we have smallest in this definition but first we have to understand one more thing in the periodic signals delta T is 5 seconds XT will also remain same for X T minus 5 when we decrement the time by 5 seconds the value of signal will remain same so in place of plus we have plus minus the signal XT will have the same value as compared to XT plus and T naught as well as XT minus and t naught so this is the condition of periodicity and now we will take one example in which I will explain what do we mean by fundamental period and why it is smallest this is the plot of sine theta and you can clearly see when theta is equal to 0 sine theta is 0 when theta is equal to PI sine theta is again equal to 0 and you will say the fundamental period T naught is equal to pi because sine theta is having the same value at 0 and pi but pi is not the fundamental period because signal is having the same value at 0 and pi but it is not true for all the values of theta and also signal is not repeating itself you can see before theta equal to zero sine theta was negative but before theta equal to PI sine theta is positive so signal is not repeating itself when T not the fundamental period is equal to pi you can clearly understand this thing when you calculate the value of sine theta at PI by 2 it is equal to 1 and if you calculate sine theta at 3 pi by 2 you will find it is equal to minus 1 so if fundamental period is equal to PI the value of signal is not same at PI by 2 it is 1 at 3 PI by 2 it is minus 1 so the fundamental period in case of sine function is equal to 2 pi sine theta is having the same value when theta is equal to 0 and when theta is equal to 2 pi and also the signal is repeating itself so if we add 2 pi to theta or subtract 2 pi from theta the value of signal will remain same as sine theta and therefore sine theta is equal to sine theta plus minus and 2 pi where 2 pi is the fundamental period and n is an integer when n is equal to 2 we are having sine theta plus minus 4 pi so the signal sine theta is periodic for 4 pi also when n is equal to 3 the signal is periodic for period 6 pi as well so they are simply the period for which this signal is periodic but they are not the fundamental period because they are not smallest the smallest one is 2 pi therefore 2 pi is the fundamental period this is all you need to know about the fundamental period the next thing is fundamental frequency we will talk about fundamental fundamental frequency in questions you have to calculate the time period the fundamental time period as well as the fundamental frequency so it is important to understand what do we mean by fundamental frequency it is denoted by F naught and it is equal to 1 by T naught if we have the value of T naught we can easily calculate F naught the unit is cycles per second or simply Hertz the next thing is fundamental angular frequency fundamental angular frequency sometimes you have to calculate the fundamental angular frequency in the problems it is denoted by Omega naught and it is equal to 2 pi F naught F naught is equal to 1 by T naught so fundamental angular frequency is equal to 2 pi by P naught so the first thing is the calculation of fundamental period and once we have the fundamental period we can easily calculate the fundamental frequency as well as fundamental angular frequency the unit of fundamental angular frequency is radians per second this is the unit of fundamental angular frequency the next thing is the condition of periodicity in case of discrete time signals this is the condition of periodicity for continuous-time signals now we will talk about condition of periodicity in case of discrete time signals the discrete-time signal is represented by small X inside the square bracket small N and if this signal this discrete-time signal is periodic then it must be equal to small X inside the square bracket small n plus minus small M capital n where small M is an integer Capital n is the fundamental period of this discrete-time signal and this capital n must be an integer in this case T naught may or may not be an integer but in case of discrete time signals the fundamental period must be an integer this is very important point the next thing is the frequency or we can say nominal frequency denoted by capital F and it is equal to 1 by capital n the unit will remain same now I have one question for you this question is a true/false question and the statement is DC value DC value is periodic in nature you need to tell me whether this statement is true or false you are having five seconds and your time starts now this is very interesting problem and I will explain why DC value is a periodic signal this statement is true and let's take a random DC value XT is equal to two I will plot this and the plot will look something like this XT T the amplitude is equal to 2 XT is 2 for all the values of time T so if we add or subtract any period to this time T the value of signal will remain same which is equal to 2 so X T XT is equal to XT plus minus and T naught where T naught may be anything if it is 1 then also the signal is having the value equal to 2 if T naught is 2 then also signal is having the value equal to 2 in the same way you can take any value of the fundamental time period this signal is going to be 2 therefore the fundamental time period is undefined and defined in case of DC value this is very important point the fundamental period is undefined in case of DC value and if we talk about fundamental frequency F naught it is equal to 0 now why fundamental frequency is equal to 0 the fundamental frequency is defined as cycles per second and here you can see the cycles are equal to 0 therefore fundamental frequency is also equal to zero the relation F naught equal to one by T naught is not valid for the DC value because T naught is undefined and as it is undefined we cannot use this relation so this is all you need to know about the periodicity of DC value DC value is a periodic signal it is not a periodic this is one common mistake so write this point somewhere so that you may revise it before your exam this is all for this lecture in the next lecture I will explain how to calculate the fundamental time period