Understanding Fourier Series Expansion Concepts

From this lecture we will start a new chapter and the name of the chapter is Fourier series expansion. I will introduce Fourier series in this lecture. We will see what is Fourier series, why we use Fourier series and what are different types of Fourier series expansion available. Fourier series, Fourier transform. their applications was given by Joseph Fourier who was French mathematician and physicist. Now we will move on to the main discussion. We will try to understand the use of Fourier series and what is Fourier. Fourier series expansion is used for periodic signals to expand them in terms of their harmonics which are sinusoidal and orthogonal to one another. We will try to understand the meaning of this statement. The first important thing which we can see is the use of Fourier series expansion only in case of periodic signals. We already know what are periodic signals. There are two types of periodic signals. The first one is continuous time periodic signals and the second one is discrete time periodic signals. Now based on continuous time and discrete time we have two types of Fourier series expansions. For continuous time the Fourier series expansion is known as continuous time Fourier series and for discrete time The series is known as discrete time Fourier series. In this particular chapter, we will only discuss continuous time Fourier series expansion. And once we complete discrete time signals, we will move on to discrete time Fourier series expansion. Now we have the Fourier series expansion. Now, what is the use of obtained expansion? We use Fourier series expansion for the analysis purpose and now you will ask when we have Fourier series Then why we moved on? to Fourier transform. To understand this, you have to focus on the first important point mentioned in the above statement. Fourier series expansion is used for periodic signals only. Now what about non periodic signals? This is for Periodic signals. What about non-periodic signals? If we talk about real-life signals, then we don't have the periodic signals. Generally, we have the non-periodic signals. We will see one example of this after few minutes. But we know the real-life signals are non-periodic in nature. Now, for analysis of non-periodic signals, we needed some tool and the tool was given by Joseph Fourier and the tool is known as Fourier transform. Now there are two types of non periodic signals. The first one is continuous time non periodic signals. And for this we have the tool known as continuous time Fourier transform. In the same way we have discrete time Fourier transform and they are for non periodic signals. So I hope you now understand what is the difference between Fourier series and Fourier transform and why we have continuous time Fourier series discrete time Fourier series continuous time Fourier transform and discrete time Fourier transform but again you will ask me what is the use of Laplace transform what is the use of Laplace transform Laplace transform is used for designing purpose Fourier series and Fourier transform are used for analysis purpose whereas Laplace transform is used for designing purpose. We obtain the transfer function using the Laplace transform and by using the different methods available we can easily check the stability of the system and by using the results obtained we can design our system. Now you will ask me what is Z transform or Z transform. Laplace transform is for continuous time and Z transform for discrete time so I hope you now understand the difference between Laplace transform and Z transform and they are all important chapters of this particular course now we will move back to our discussion we have already completed periodic Signals. If you have not seen the lectures on periodic signals, you may watch the lectures by following the playlist. There I have explained every important point related to periodic signals. We have also solved various numerical problems based on periodic signals. But for the explanation purpose, I will quickly explain what do we mean by periodic signals. Periodic signals are those signals in which there is repetition of a particular structure from minus infinity to infinity. Or you can say the periodic signal is a signal in which the signal repeats itself after a particular interval of time. And this particular interval of time is known as time period of the signal. So if we have a periodic signal Xt, it is a periodic signal then we can say that Xt plus minus t is equal to Xt. This is the condition for a signal to be periodic. Now let's try to understand the meaning of this. It says T is a time period. It is a time period and it is equal to n times the fundamental time period. The fundamental time period T0 is the smallest time interval for which the given signal is periodic. And here we are performing the time shifting. We can perform either the left shift or the right shift but the shifting interval should be the time period. So whenever we perform the shifting towards the left or towards the right by the time period of the signal we will get the same signal. And we are getting the same signal because I already said the signal is having a particular structure repeating itself from minus infinity to infinity. This point is very important. If the repetition is not from minus infinity to infinity this condition will not hold true. If the repetition is from minus infinity to infinity small shifting will not affect the whole signal So I hope you now understand or you now remember what we have already discussed in case of periodic signals Now we will move to the next important point And the point is existence of Fourier series. In the statement it is written that Fourier series expansion is used for periodic signals. But it does not mean whenever we have periodic signal the Fourier series expansion will exist. The existence of Fourier series expansion depends on three conditions and the conditions are known as Dirichlet conditions. The conditions are known as Dirichlet conditions. We will discuss Dirichlet conditions in the next lecture. There are three conditions for the existence of Fourier series expansion. Even if the signal is periodic, the three conditions must be satisfied. In case of non-periodic signals, we can straight away say the Fourier series expansion will not exist. But in case of periodic signals, we have to focus on Dirichlet conditions. And the next lecture is going to be based on the Dirichlet conditions. Now we will read the statement further. It says the Fourier series expansion is used for periodic signals to expand them in terms of their harmonics. Now we have a new term. Harmonic and to understand the harmonic you should understand the frequency first. We know the basic definition of frequency. Frequency is defined as the number of cycles per second. It is number of cycles per second. And we can easily calculate the frequency if we have the waveform, I will quickly plot a sinusoidal waveform. And I hope you understand what do we mean by sinusoidal waveform either we can have them. sine waveform or cosine waveform. Here I have taken the waveform of sine function and you can see one cycle. This is the one cycle is completed from 0 to t. 0 to t is the interval after which the signal repeats itself. So this t here is the fundamental time period. So this is the fundamental time period of the sine function and we can see that the signal is repeated. can easily calculate the frequency it is equal to number of cycles per second the signal is taking t seconds for one cycle. So in one seconds, in one seconds, we will have one over T cycles. Therefore, the frequency is going to be one over T cycles per second or simply hertz. So this is the definition of frequency and we already know this. But I told you real life signals are not periodic in nature. For example, I am speaking right now and my voice is converted to the electrical signal by the help of the mic I am using. So the mic is having the transducer which is converting. my voice to the electrical signal and definitely the signal will not look like this. It is not a periodic signal. It depends on the word I am speaking and based on different words we have different spikes in the waveform. For example the waveform. will look like this. Now you cannot use this formula to calculate the frequency in this case because you cannot find out the cycles here. Therefore we need to define the frequency in more general way. So let's define our frequency in more general way. The frequency is defined as the rate of change. So, in this waveform for example when I say letter C you can see the variation is not very sharp I say C. So, this portion here in the waveform is when I say letter C and when I say letter T you can see the variation is very sharp T. So, this portion of the waveform is when I say T. So here the rate of change is very high. So this is having the high frequency. Here the rate of change is not very high. So we have the low Frequency. In this case we had only one frequency but here we have multiple frequencies. There is not only one frequency available. We have different frequencies and strictly speaking this is a signal. This is not a signal. We call it signal but what do we mean by signal is that it carries some information. We can say that it is carrying some information because it is carrying something which we don't know. It is non-deterministic, it is random. But here we already know everything about the sinusoidal waveform either cosine or sine. So it is not carrying some information. Therefore it is practically not a signal. Signal should carry some information and information should not be known to us earlier. So here we have a signal. practical signal in which there are multiple frequencies available. And when we have multiple frequencies, we can have the signal expressed in terms of fundamental component plus the harmonics. Now we will understand what are harmonics. We now know the meaning of frequency and now we will understand the meaning of harmonics. For this, I will take one example. In this example, Signal Xt is a periodic signal and it is expressed as the sum of original signal and the harmonics. It is equal to twice of sin omega t plus sin 2 omega t plus 7 sin 3 omega t. plus other harmonics. So signal Xt is a periodic signal and it is expressed as sum of original signal and the harmonics. 2 sine omega t is the original signal and it is also known as the first harmonic because you can see the frequency here frequency is omega omega is the fundamental frequency or more specifically it is fundamental angular frequency angular frequency is equal to 2 pi f and compared to this if you see the second term you will find the frequency is twice of omega so here the frequency is integral multiple of the fundamental frequency. So whenever you have the frequency as integral multiple of the fundamental frequency we call the term harmonics. They are not desired in the system generally but they are present. In the same way if you see the third term you will find the frequency is is three times omega. So again the frequency is integral multiple of the fundamental frequency. So this one here is also the harmonic. In the same way we have other harmonics present in this signal. And depending on the integer here we have even and odd harmonics. Here you can see two omega is there. Two is even. Therefore this particular harmonic is even harmonic. Here 3 omega is there. 3 is an odd number therefore this harmonic is odd harmonic. Now if you focus on the coefficient you will find here the coefficient is 1 but here the coefficient is 7. 7 is greater than 1 so we can say that the effect of third harmonic is more as compared to the effect of second harmonic. And if the same pattern is followed for the other harmonics also, we can say that odd harmonics are more dominant in the signal. And even harmonics are less dominant. It depends on signal to signal. In some signals odd harmonics are more dominant. In some signals even harmonics are more dominant. And in some signals the dominance depend on the integer here. If the integer is larger, the dominance is smaller and vice versa. So this is all you should know about the harmonic. Whenever you have different frequency components along with the fundamental frequency component we say harmonics are present. present in the signal. And this is what we want to perform the analysis. And we can have the same result from the waveform of the signal using the Fourier series. So we want to analyze this signal and for that we want to analyze the harmonics present. For example, here we got the information that the third harmonic is more dominant as compared to the second harmonic. So we want this form or this expansion of this signal. signal and we can use the Fourier series expansion for the same. But Fourier series expansion is only for the periodic signals. So in this chapter we will only deal with periodic signals and when we have non-periodic signals like this we will use the Fourier transform. By using the Fourier transform we can easily obtain The response known as the frequency response represented by H Omega. It is complex in nature. So it is having two parts. The first one is the magnitude known as the magnitude. response and the second one is the phase of the response known as phase response. We will talk more about Fourier transform in the next chapter but for now we will focus on the Fourier series and we now know why we need to obtain the Fourier series. We want this form to analyze this signal and we can obtain this form. from the waveform using the Fourier series. Now I will show you one example on your screen. You can see the square waveform and in the square waveform you can see different harmonics present. There is fundamental signal and if you try to visualize the waveform from minus infinity to infinity you will find it is like sine waveform. So when you obtain the Fourier series expansion of this particular square wave you will find all sine terms in the expansion and all the different harmonics will be present in the expansion. The harmonics you can see in the waveform will be obtained in mathematical form using the Fourier series. There are other harmonics also but they are not very dominant so there is no need to write down all the harmonics. So I hope you now have the clear understanding of the use of Fourier series. Now we will understand the last point written in this statement. The harmonics are sinusoidal and orthogonal to one another. The harmonics are sinusoidal. This means whenever you have the expansion of the given signal by the help of Fourier series, you will have a DC value plus sine terms plus cosine terms. So we have DC plus sinusoidal terms in the expansion. Sometimes you will have DC and cosine. Sometimes you will have sine. plus dc sometimes you will have dc sine and cosine we will talk more about this in the coming presentations the next important point is the harmonics are orthogonal to one another we already know the meaning of orthogonal this means perpendicular to each other Now let's move to the next point. The next point is Fourier series is also used to calculate the power and phase content of a particular harmonic present in the expansion. Like in this case you can see the harmonics present in the expansion and we can calculate the phase content and the power easily. Therefore Fourier series is also used to calculate the power and phase content of a particular harmonic present in the expansion. The next point is power is equally distributed to the harmonics. Now we will talk about different types of Fourier series expansion. The first one is known as trigonometric Fourier series expansion. Trigonometric Fourier series expansion. The second one is known as complex exponential Fourier series expansion. complex exponential or simply the exponential Fourier series expansion. The third one is known as polar or harmonic Fourier series expansion. polar or harmonic Fourier series expansion. Out of three, we will first study trigonometric Fourier series expansion. It is important for your college examinations. After this, I will explain complex exponential Fourier series expansion this is very important and we will focus more on complex exponential Fourier series expansion the third one is polar or harmonic Fourier series it is not very important and if we have time at the end of this course then maybe I will upload one or two lectures based on polar harmonic Fourier series for now we will focus on the first two types and in the next lecture as I already told you we will start with the conditions for existence of Fourier series that is Dirichlet conditions so this is all for this lecture if you have any doubt you may ask in the comment section this was just an introductory lecture about the Fourier series we will dedicate a lot of lectures on Fourier series and we will try to cover each and everything related to Fourier series. If you are preparing for competitive examinations then it will help you a lot because we will discuss various tricks and tips to solve the questions. So this is all. See you in the next one.

Fourier series, Fourier transform. their applications was given by Joseph Fourier who was French mathematician and physicist. Now we will move on to the main discussion.

We will try to understand the use of Fourier series and what is Fourier. Fourier series expansion is used for periodic signals to expand them in terms of their harmonics which are sinusoidal and orthogonal to one another. We will try to understand the meaning of this statement.

The first important thing which we can see is the use of Fourier series expansion only in case of periodic signals. We already know what are periodic signals. There are two types of periodic signals.

The first one is continuous time periodic signals and the second one is discrete time periodic signals. Now based on continuous time and discrete time we have two types of Fourier series expansions. For continuous time the Fourier series expansion is known as continuous time Fourier series and for discrete time The series is known as discrete time Fourier series. In this particular chapter, we will only discuss continuous time Fourier series expansion. And once we complete discrete time signals, we will move on to discrete time Fourier series expansion.

Now we have the Fourier series expansion. Now, what is the use of obtained expansion? We use Fourier series expansion for the analysis purpose and now you will ask when we have Fourier series Then why we moved on? to Fourier transform. To understand this, you have to focus on the first important point mentioned in the above statement.

Fourier series expansion is used for periodic signals only. Now what about non periodic signals? This is for Periodic signals. What about non-periodic signals? If we talk about real-life signals, then we don't have the periodic signals.

Generally, we have the non-periodic signals. We will see one example of this after few minutes. But we know the real-life signals are non-periodic in nature. Now, for analysis of non-periodic signals, we needed some tool and the tool was given by Joseph Fourier and the tool is known as Fourier transform. Now there are two types of non periodic signals.

The first one is continuous time non periodic signals. And for this we have the tool known as continuous time Fourier transform. In the same way we have discrete time Fourier transform and they are for non periodic signals.

So I hope you now understand what is the difference between Fourier series and Fourier transform and why we have continuous time Fourier series discrete time Fourier series continuous time Fourier transform and discrete time Fourier transform but again you will ask me what is the use of Laplace transform what is the use of Laplace transform Laplace transform is used for designing purpose Fourier series and Fourier transform are used for analysis purpose whereas Laplace transform is used for designing purpose. We obtain the transfer function using the Laplace transform and by using the different methods available we can easily check the stability of the system and by using the results obtained we can design our system. Now you will ask me what is Z transform or Z transform. Laplace transform is for continuous time and Z transform for discrete time so I hope you now understand the difference between Laplace transform and Z transform and they are all important chapters of this particular course now we will move back to our discussion we have already completed periodic Signals. If you have not seen the lectures on periodic signals, you may watch the lectures by following the playlist.

There I have explained every important point related to periodic signals. We have also solved various numerical problems based on periodic signals. But for the explanation purpose, I will quickly explain what do we mean by periodic signals. Periodic signals are those signals in which there is repetition of a particular structure from minus infinity to infinity.

Or you can say the periodic signal is a signal in which the signal repeats itself after a particular interval of time. And this particular interval of time is known as time period of the signal. So if we have a periodic signal Xt, it is a periodic signal then we can say that Xt plus minus t is equal to Xt.

This is the condition for a signal to be periodic. Now let's try to understand the meaning of this. It says T is a time period.

It is a time period and it is equal to n times the fundamental time period. The fundamental time period T0 is the smallest time interval for which the given signal is periodic. And here we are performing the time shifting. We can perform either the left shift or the right shift but the shifting interval should be the time period.

So whenever we perform the shifting towards the left or towards the right by the time period of the signal we will get the same signal. And we are getting the same signal because I already said the signal is having a particular structure repeating itself from minus infinity to infinity. This point is very important.

If the repetition is not from minus infinity to infinity this condition will not hold true. If the repetition is from minus infinity to infinity small shifting will not affect the whole signal So I hope you now understand or you now remember what we have already discussed in case of periodic signals Now we will move to the next important point And the point is existence of Fourier series. In the statement it is written that Fourier series expansion is used for periodic signals. But it does not mean whenever we have periodic signal the Fourier series expansion will exist. The existence of Fourier series expansion depends on three conditions and the conditions are known as Dirichlet conditions.

The conditions are known as Dirichlet conditions. We will discuss Dirichlet conditions in the next lecture. There are three conditions for the existence of Fourier series expansion. Even if the signal is periodic, the three conditions must be satisfied.

In case of non-periodic signals, we can straight away say the Fourier series expansion will not exist. But in case of periodic signals, we have to focus on Dirichlet conditions. And the next lecture is going to be based on the Dirichlet conditions.

Now we will read the statement further. It says the Fourier series expansion is used for periodic signals to expand them in terms of their harmonics. Now we have a new term.

Harmonic and to understand the harmonic you should understand the frequency first. We know the basic definition of frequency. Frequency is defined as the number of cycles per second. It is number of cycles per second.

And we can easily calculate the frequency if we have the waveform, I will quickly plot a sinusoidal waveform. And I hope you understand what do we mean by sinusoidal waveform either we can have them. sine waveform or cosine waveform.

Here I have taken the waveform of sine function and you can see one cycle. This is the one cycle is completed from 0 to t. 0 to t is the interval after which the signal repeats itself. So this t here is the fundamental time period.

So this is the fundamental time period of the sine function and we can see that the signal is repeated. can easily calculate the frequency it is equal to number of cycles per second the signal is taking t seconds for one cycle. So in one seconds, in one seconds, we will have one over T cycles. Therefore, the frequency is going to be one over T cycles per second or simply hertz.

So this is the definition of frequency and we already know this. But I told you real life signals are not periodic in nature. For example, I am speaking right now and my voice is converted to the electrical signal by the help of the mic I am using. So the mic is having the transducer which is converting. my voice to the electrical signal and definitely the signal will not look like this.

It is not a periodic signal. It depends on the word I am speaking and based on different words we have different spikes in the waveform. For example the waveform. will look like this. Now you cannot use this formula to calculate the frequency in this case because you cannot find out the cycles here.

Therefore we need to define the frequency in more general way. So let's define our frequency in more general way. The frequency is defined as the rate of change.

So, in this waveform for example when I say letter C you can see the variation is not very sharp I say C. So, this portion here in the waveform is when I say letter C and when I say letter T you can see the variation is very sharp T. So, this portion of the waveform is when I say T. So here the rate of change is very high.

So this is having the high frequency. Here the rate of change is not very high. So we have the low Frequency. In this case we had only one frequency but here we have multiple frequencies. There is not only one frequency available.

We have different frequencies and strictly speaking this is a signal. This is not a signal. We call it signal but what do we mean by signal is that it carries some information. We can say that it is carrying some information because it is carrying something which we don't know. It is non-deterministic, it is random.

But here we already know everything about the sinusoidal waveform either cosine or sine. So it is not carrying some information. Therefore it is practically not a signal.

Signal should carry some information and information should not be known to us earlier. So here we have a signal. practical signal in which there are multiple frequencies available.

And when we have multiple frequencies, we can have the signal expressed in terms of fundamental component plus the harmonics. Now we will understand what are harmonics. We now know the meaning of frequency and now we will understand the meaning of harmonics.

For this, I will take one example. In this example, Signal Xt is a periodic signal and it is expressed as the sum of original signal and the harmonics. It is equal to twice of sin omega t plus sin 2 omega t plus 7 sin 3 omega t.

plus other harmonics. So signal Xt is a periodic signal and it is expressed as sum of original signal and the harmonics. 2 sine omega t is the original signal and it is also known as the first harmonic because you can see the frequency here frequency is omega omega is the fundamental frequency or more specifically it is fundamental angular frequency angular frequency is equal to 2 pi f and compared to this if you see the second term you will find the frequency is twice of omega so here the frequency is integral multiple of the fundamental frequency. So whenever you have the frequency as integral multiple of the fundamental frequency we call the term harmonics.

They are not desired in the system generally but they are present. In the same way if you see the third term you will find the frequency is is three times omega. So again the frequency is integral multiple of the fundamental frequency.

So this one here is also the harmonic. In the same way we have other harmonics present in this signal. And depending on the integer here we have even and odd harmonics.

Here you can see two omega is there. Two is even. Therefore this particular harmonic is even harmonic. Here 3 omega is there.

3 is an odd number therefore this harmonic is odd harmonic. Now if you focus on the coefficient you will find here the coefficient is 1 but here the coefficient is 7. 7 is greater than 1 so we can say that the effect of third harmonic is more as compared to the effect of second harmonic. And if the same pattern is followed for the other harmonics also, we can say that odd harmonics are more dominant in the signal.

And even harmonics are less dominant. It depends on signal to signal. In some signals odd harmonics are more dominant. In some signals even harmonics are more dominant. And in some signals the dominance depend on the integer here.

If the integer is larger, the dominance is smaller and vice versa. So this is all you should know about the harmonic. Whenever you have different frequency components along with the fundamental frequency component we say harmonics are present.

present in the signal. And this is what we want to perform the analysis. And we can have the same result from the waveform of the signal using the Fourier series. So we want to analyze this signal and for that we want to analyze the harmonics present. For example, here we got the information that the third harmonic is more dominant as compared to the second harmonic.

So we want this form or this expansion of this signal. signal and we can use the Fourier series expansion for the same. But Fourier series expansion is only for the periodic signals. So in this chapter we will only deal with periodic signals and when we have non-periodic signals like this we will use the Fourier transform.

By using the Fourier transform we can easily obtain The response known as the frequency response represented by H Omega. It is complex in nature. So it is having two parts. The first one is the magnitude known as the magnitude. response and the second one is the phase of the response known as phase response.

We will talk more about Fourier transform in the next chapter but for now we will focus on the Fourier series and we now know why we need to obtain the Fourier series. We want this form to analyze this signal and we can obtain this form. from the waveform using the Fourier series.

Now I will show you one example on your screen. You can see the square waveform and in the square waveform you can see different harmonics present. There is fundamental signal and if you try to visualize the waveform from minus infinity to infinity you will find it is like sine waveform.

So when you obtain the Fourier series expansion of this particular square wave you will find all sine terms in the expansion and all the different harmonics will be present in the expansion. The harmonics you can see in the waveform will be obtained in mathematical form using the Fourier series. There are other harmonics also but they are not very dominant so there is no need to write down all the harmonics.

So I hope you now have the clear understanding of the use of Fourier series. Now we will understand the last point written in this statement. The harmonics are sinusoidal and orthogonal to one another.

The harmonics are sinusoidal. This means whenever you have the expansion of the given signal by the help of Fourier series, you will have a DC value plus sine terms plus cosine terms. So we have DC plus sinusoidal terms in the expansion. Sometimes you will have DC and cosine. Sometimes you will have sine.

plus dc sometimes you will have dc sine and cosine we will talk more about this in the coming presentations the next important point is the harmonics are orthogonal to one another we already know the meaning of orthogonal this means perpendicular to each other Now let's move to the next point. The next point is Fourier series is also used to calculate the power and phase content of a particular harmonic present in the expansion. Like in this case you can see the harmonics present in the expansion and we can calculate the phase content and the power easily. Therefore Fourier series is also used to calculate the power and phase content of a particular harmonic present in the expansion. The next point is power is equally distributed to the harmonics.

Now we will talk about different types of Fourier series expansion. The first one is known as trigonometric Fourier series expansion. Trigonometric Fourier series expansion. The second one is known as complex exponential Fourier series expansion.

complex exponential or simply the exponential Fourier series expansion. The third one is known as polar or harmonic Fourier series expansion. polar or harmonic Fourier series expansion.

Out of three, we will first study trigonometric Fourier series expansion. It is important for your college examinations. After this, I will explain complex exponential Fourier series expansion this is very important and we will focus more on complex exponential Fourier series expansion the third one is polar or harmonic Fourier series it is not very important and if we have time at the end of this course then maybe I will upload one or two lectures based on polar harmonic Fourier series for now we will focus on the first two types and in the next lecture as I already told you we will start with the conditions for existence of Fourier series that is Dirichlet conditions so this is all for this lecture if you have any doubt you may ask in the comment section this was just an introductory lecture about the Fourier series we will dedicate a lot of lectures on Fourier series and we will try to cover each and everything related to Fourier series. If you are preparing for competitive examinations then it will help you a lot because we will discuss various tricks and tips to solve the questions. So this is all.

See you in the next one.

Transcript for:Understanding Fourier Series Expansion Concepts

Transcript for:
Understanding Fourier Series Expansion Concepts