 So, hi I am Ashwit and I am Pratik and in this video we are going to talk about some basic signals that we will see very frequently in the course and first we look at the sinusoid. You can see in the plot that we have a sinusoidal signal and symbolically it is written as sin of omega t plus phi and in these we see various quantities. So, the first quantity we look at is omega and omega is called the angular frequency it is measured in radiance per second and in this particular case the angular frequency is 125.66 radiance per second. So, you probably not familiar with this some of you the quantity you are probably familiar with is the frequency which you studied in high school physics. The frequency is measured in hertz or cycles per second and it talks about how many cycles of the periodic waveform we have in one second. So, this plot has a frequency of 20 hertz of course, it is also important to talk about the fact that the sinusoid is periodic. So, it repeats with time and at every time period in which in this case is 0.05 seconds we know that the cycle repeats the waveform keeps repeating and it gets back to where it was. So, this is the sinusoid all right of course, the time period is distance between peaks and I am sure you know this from earlier classes. So, the next signal we look at is the growing exponential the growing exponential is written as exponential of e raised to the power of alpha times t where alpha is a constant which tells you how much the exponential grows or decays. In this case we have taken alpha is 0.9 and when alpha is positive as you can see over here the exponential keeps rising and grows over time. Very simple signal and of course, if alpha is made negative we get the decaying exponential and in this case we take alpha is negative 0.4 both signals are actually quite simple, but we will see it several times during the course all right. So, we studied two types of signals we studied an exponential signal we studied a sinusoidal signal we can also combine signals. So, we can take the exponential we can take the sinusoid multiply the two and we can get what is called as the exponentially growing sinusoid and you can see it is represented as e raised to alpha times t into sin of omega t plus 5 and you can see that we have the envelope of the function which is which actually is e raised to the power of alpha t and of course, on the negative side it is negative e raised to the power of alpha t and between that envelope you see the sin which is growing over time and this is the exponentially growing sinusoid all right. So, Pratik you seen the case where alpha is greater than 0 it is 0.8 in this case what if I keep alpha negative what will I see well if you keep alpha negative the exponential will decay and as a result the amplitude of sin will also decay and you will just get the same graph just flipped about the y axis that is perfect mirror image of this that is perfect. So, here you can see what is called as the exponentially decaying sinusoid of course, it is important to mention exponentially because it can decay in other ways and yeah it is quite simple enough in this case we have taken alpha is negative 0.4 and Pratik can you think of some application where we can use this signal to represent something well there are many physical phenomena which will give rise to sinusoidal waves. So, if the sinusoidal wave decays over time say because sound attenuates. So, if the sound is from far away from the source the sound you the amplitude of the sound is less. That is perfectly right. So, speaking of sound we can talk about music that is my favorite application of signals and systems and as you can see in the graph the graph reduces over time just like you pluck a string if you hear the string it will be loud in the beginning and as time progresses the amplitude of or rather the loudness of the string reduces over time. Of course, a string is not just a simple sinusoid, but that is the story we will see later on in the course. So, now that we studied exponentially decaying and growing sinusoid I would like to talk about another notation for complex exponentials and we get that from Euler's formula and those of you studied complex analysis would have seen this before it is e raised to the power of j omega t and according to Euler's formula that is cos of omega t plus j times sin of omega t. Those of you are mathematicians and not electrical engineers would probably wonder what j is in electrical engineering we use j which is the same as i that is the square root of minus 1. We do not use i very often because i is also used for current and we do not want to confuse notation. So, this is the complex exponential this is Euler's formula and very simple formula and of course we can add phase as well. So, you can see in the second case we put e raised to the power of j times omega t plus phi and that looks like cos of omega t plus phi plus j times sin omega t plus phi alright. So, this is the complex exponential we can of course plot each part the real part in the imaginary part. So, the imaginary part is on top the real part is at the bottom right. So, we have the complex exponential we can take it further and I like you the students of this course to tell me what this plot would look like. So, I have e raised to the power of sigma plus j times omega t plus phi and I want you to tell me what this plot would look like what would the real part look like what is the imaginary plot look like plot some graphs show it in the forums and I have some more questions for you. So, we have discussed one application. So, we discussed the application of the decaying exponential I would like you to tell me applications of every other signal we have discussed. So, take a physical phenomenon and tell me how we can apply one of these signals to model those physical phenomena. So, that is another question for you the third question I would like you to solve is you have seen complex exponential and Euler's formula this can be plotted really nicely using a 3D graph and you get a really nice figure. So, I would like you to do that. So, those of you who can plot 3D figures I would like you to plot a 3D figure for e raised to the power of j omega t and show us in the forums what it looks like and from this 3D figure you can also see which part is the cosine part which is the sine part that is the real and imaginary part and if you can do that as well in the forums it will be interesting. So, three questions for you the first question would be what would this signal look like and the second question is give me examples of physical phenomena where you use the other signals to model them. So, the last question would be to plot e raised to the power of j omega t on a 3D graph right. So, I will see you in another video. Thank you.