Lecture on Real and Complex Exponential Signals

hi again this is mr pinter your instructor for signals and systems welcome to part two of our lecture on chapter one in this part of the video we'll be looking at real and complex exponential signals before we start talking about these exponential signals i just want to mention that exponential signals are an extremely important class of signals for the study of signals and systems because exponential signals are the solutions to differential equations which describe a lot of systems so we'll be seeing this throughout the course we'll be encountering these exponential signals so let's start out by talking about continuous time real exponential signal so mathematically our our signal here can be represented by the function x of t which is equal to c some constant we'll let that be a positive number times e raised to the a t now um the exponent here is some coefficient a multiplied by time and a has there's two options for a it can be positive or negative so when a is greater than 0 this is a growing exponential function and when we plot it it looks something like this of course when we put in a zero for t remember that e to the zero is one so our function gives us c which uh is this number out in front here um and so that's plotted here and then of course this thing grows exponentially on the other hand if a is less than zero in other words if we have a minus sign up here like a minus two t or something like that this is a decaying exponential function and it's going to look something like this it gets smaller and decays exponentially with time so here's a simple activity just sort of a like a warm up for this part of the video go ahead and make a sketch of the following signal x of t is equal to 5 times e to the minus t so you don't have to do anything fancy here just make a quick sketch uh you know on a piece of paper or something like that it doesn't even have to be graph paper go ahead and pause the video and sketch this and we'll check your results so here's the solution if x of t is equal to 5 times e to the minus t then the function is going to look something like this notice it passes through the value of five at t equals zero and because there's a minus sign up here this is a decaying exponential function so it's going to eventually go to zero for large values of t the next important class of signals we're going to look at are exponential signals or functions that have a complex exponent okay so before we dig into those uh let's do a quick complex number review and again this is very fundamental uh information for this course we're going to be dealing with complex numbers over and over in just about every chapter so um this is you know should be review for you this is stuff you learned in your calculus class and so forth but the good news is that pretty much everything you need to know about complex numbers is shown on this one page but make sure that you understand this uh well in order to you know be able to do this course and be successful in this course so let's just do the review here so we can represent a complex number by any variable we want but let's use the letter z here to represent our complex number there are two forms that we can put the complex number in there's the so-called polar form and the rectangular form so the polar form has the magnitude out in front i called it r here so r is the magnitude of z okay so i've got the you can see the absolute value bars around z there and then we have e raised to the j theta where theta represents the phase angle so the polar form um basically is magnitude and phase of the complex number now you may be wondering what is this j here well j is what um your math textbook probably called i it's the uh imaginary unit basically it's the square root of minus one now why do we use j and not i well electrical engineering um textbooks tend to use j so that it's not to be confused with i because current is represented by the letter i so in this course we will be using j and you'll get used to that very quickly now the other form that we can put this same number in is the rectangular form which looks like a plus j times b where a is the real part of the complex number and b is the so-called imaginary part of the complex number notice it it has the uh j um in front of it here so one way to sort of uh reconcile these two different forms you know they're obviously very different forms but they represent the exact same thing so how is this well it's useful to plot z either as a point or even more uh it's actually more illustrative just to plot it as a vector in the so-called complex plane so the complex plane remember this is the real axis down here and the vertical axis is the so-called imaginary axis so this real axis represents the real part of z the vertical axis represents the imaginary part of z so here is z z can be plotted as a point right here okay let's say this is our value z it has both real parts and imaging parts or we can plot it as a vector so let's take a look at these two forms so in the polar form remember r is the magnitude out front well that's just the magnitude of this vector in other words how long is this arrow right here and theta of course is the phase angle which represents the angle of this vector with respect to the real axis so here our r and theta this is the polar way to represent our complex number z now of course z can also be represented in the rectangular form where a here is if we project z down to the real axis a is the real part of this z and b is projected across to the imaginary axis b is the imaginary part of z so this this picture here you know as they say is worth a thousand words because this tells you everything you could possibly want to know about this single complex number it tells you its real part its imaginary part its magnitude and its phase and going further we can relate the polar and rectangular forms by using basic trig so trigonometry you know notice this is a right triangle here allows us to express r theta a and b in terms of each other so for example this angle here you remember from you know trig or math class is the inverse tangent of the opposite side which is b right this is this height here is b and this length down here is a so it's inverse tangent of b over a so this is a relationship between theta and a and b we can use the pythagorean theorem to find out the value of r with respect to a and b all right so r is the square root of a squared plus b squared again because this is a right triangle furthermore we can relate a down here is equal to r times the cosine of theta so here's a nice relationship of how to find the value a if you know r and theta so in other words if you're given a a complex number in polar form it's very straightforward to find out the rectangular form by using a equals r cosine theta and likewise b is equal to r sine theta now we can do something that's actually quite profound here if we notice that z remember our complex number in polar form is given by r times e to the j theta and we said that's equal to a plus b where a is our cosine theta and b is our sine theta well notice our rectangular form here a plus j b if we eliminate r in other words divide get rid of this r here this r here and this r here just by dividing then what we're left with is the famous euler's formula which says that e to the j theta is equal to cosine theta plus j sine theta and again euler's formula is something that we will be using fairly often in this course before we leave our review of complex numbers there's one more uh slide here just on this is sort of some special cases and these are very convenient forms uh to use so first of all remember if we take a complex number z and represent it in polar form as r times e to the j theta there's a special case where we let r equal one and this is r equal one refers to what we call is the unit circle so this is a circle with a radius of one this r here you know basically represents the radius of that circle so when we set r equal to one then there are some special values here um going around the circle that will come up now and then so for example if we have e to the j times 2 pi all right or e to the j 4 pi well remember we start here if we go around 2 pi we end up right back here if we go around another two pi so we've gone around four pi radians now we still end up back here and this is the same as e to the zero all right zero radians is the same as two pi radians because zero radians you know that means theta is zero we're right here at plus one and if we go around we're back at plus one so these forms up here if ever we see those forms obviously they are just equal to one now look at this point up here well this has an angle of 90 degrees which is pi over 2. so e to the j pi over 2 is up here and that's equal to j that's equal to positive j right it's a purely imaginary number um likewise instead of going up by pi over 2 if we go down and around by 3 pi over 2 which is 270 degrees we also end up at j so these two expressions here even though they look very different they are identical to each other and they're equal to the value of j likewise when we start here and we go by a full pi radians over here 180 degrees we go from plus one all the way to minus one so e to the plus or minus j pi is equal to e plus or minus j three pi you know five pi so so on and so forth and they're all equal to -1 and our last special case is down here if we go down by pi over 2 in other words negative pi over 2 radians we end up at minus j or we can go in the positive direction 3 pi over 2 okay so all these forms here can be simplified to very basic fundamental numbers either 1 you know plus or minus 1 or plus or minus j so now that we've completed our review of complex numbers we're ready to dig into complex um exponential or imaginary exponential signals and we're gonna we're doing this for continuous time first we'll deal with discrete time later so the general form is uh this looks familiar this is our exponential signal again x of t equals c e to the a t the only difference now and it's a big difference is that a is not a real number like it was earlier now a is going to be purely imaginary in other words it has no real part it's just got the j times some number and that number we're going to call omega and you'll see why shortly so substituting this into here we see that our imaginary exponential signal is x of t is equal to some constant c times e to the j omega t and what we can do is invoke euler's formula remember euler's formula says if we have e to the j something that's equal to cosine something plus j sine something so by plugging this into euler's formula x of t can be written as c times cosine omega t plus j times c sine of omega t so let's look at the the real part of this function you know there's a real part here and this is the imaginary part because it's got that j in front of it but the real part of this function is just a cosine function it's oscillating it's an oscillating function so if we were to plot the the real part all right of x of t that's what this means the r e for real part of x of t it's going to look something like this it's a cosine function and likewise the imaginary part oscillates also and notice it oscillates at the frequency given by omega where omega was um up here in between the j and the t so that is our radian frequency so this um provides us an easy way to find the radian frequency of a periodic um signal just by inspection so if we ever see you know some form like a times cosine omega t or a times sine omega t or a times e to the j omega t in all cases here the number in front of the t and in this case you know it's in between the j and the t that number gives us the radian frequency so the radian frequency omega is of course in radians per second we can convert that to the so sometimes they call this the physical frequency f which is equal to omega divided by two pi because remember there are two pi radians per person you know circling around the circle there are two pi radians per cycle so the physical frequency is given not in radians per second but in cycles per second and um this is also known as a hertz the unit hertz or hz is when we're talking about the physical frequency the period of this this signal here t is equal to one over the frequency you'll remember that from you know your physics uh class when you learn about waves and that is equal to two pi over omega if you're going to use omega instead of f and the period of course has units of seconds and finally the amplitude of this signal here is given by whatever this number is out in front of the sine or the cosine or the complex exponential notice in all these cases here the amplitude is designated by a and if it's shown like this in this form you would say the amplitude is equal to c so here's an example or an activity for you to practice what we just learned so let's say you're given the following signal x of t is equal to 5 cosine of 6 pi t so your task is to find the amplitude the radiant frequency the physical frequency and the period of this signal right here and just to make this a little bit easier so you don't have to go back to the previous slide i put all the equations here to figure out these various parameters down here so again go ahead and pause the video and see if you can figure out what these are what a omega f and t are for this signal here all right so the solutions here the amplitude is found just by inspection that's 5 the radian frequency is also found by inspection remember omega is the number that is multiplying in front of t so omega is equal to six pi and the units of course are radians per second the physical frequency we get from just taking omega which is six pi and dividing it by two pi so we get three cycles per second and finally the period t is just one over f so we have one over three gives us one third or point 0.333 seconds let's now take a look at a more general type of continuous time complex exponential function this time we're going to start you know pretty much with the same form x of t is equal to c times e to the a t but now instead of a being purely imaginary a is going to be complex in other words a is going to have a real part and a imaginary part and also just to make things completely general here let's let this number out in front c also be a complex number it's not just a real number anymore so the trick to to making some sense of this is um to let this c here be in polar form so we're going to let c equal to the magnitude of c because remember c is complex times e to the j theta where theta is the phase and this is the magnitude of c and then um we're going to let a which is up here in the exponent we're going to let that one be in rectangular form so we're going to let a equals sigma plus j omega where sigma is the real part and omega is the imaginary part so when we substitute these expressions here for a and c into this up here we get the following x of t is equal to magnitude c times e to the j theta times e to the quantity sigma plus j omega and the quantity times time now what we need to do we can lump together all the real parts so the real part here is obviously the magnitude of c is real and also remember sigma is a real number so e to the sigma times t is also a real uh quantity all right this is just an exponential function there's no j or anything in it okay so we're going to lump these two real terms together and they constitute what is called the envelope of this function now what's left are the imaginary parts we have e to the j theta and we have e to the j omega t so i'm going to factor out the j and in parentheses i have omega t plus theta so what these mean now is that omega is still the frequency it's the radiant frequency because it's the number that is in front of the t the theta here is what's called a phase constant okay it's not related to the frequency but it determines sort of where this function starts at time zero and furthermore this sigma going back to our our real part here the envelope the sigma is related to the time constant of how fast this signal either grows or decays in fact sigma is one over the time constant so let's look at a couple different cases here if sigma is a positive number right sigma greater than zero then this envelope here is a growing function in other words the amplitude of this signal is getting bigger with time so that's represented by these dotted lines notice the amplitude is growing and an example of this is in resonance when we have resonance we have a signal that grows with time an oscillating signal that grows in time now notice the actual signal though is oscillating up and down now why is that well because of this portion the um this complex exponential here remember this is e to the j omega t you know and then there's a phase constant but the e e to the j omega t remember is oscillating it has a cosine and a sine uh involved with it so this is what um this signal would look like for a positive value of sigma it is a growing uh you know an oscillating function that is growing with time the other um case of course is when sigma here is a negative number then our envelope is decaying with time it's getting smaller and smaller and we have an oscillating signal that is um you know decaying with time an example of this would be damped harmonic motion or damped oscillations due to say friction so let's take a look at a couple of examples and uh before we do in fact uh this is uh these are activities uh for you to figure out here you can do these um on your own and then we'll look at the solutions so um i put the you know the expression from the last page here uh to help us kind of remember what's what so for this first example y of t is equal to 12 times e to the j five plus two all in parentheses times t so the question is what is omega what is the phase constant and then is the amplitude constant decreasing or increasing with time the next example x of t is equal to 7 times e to the 3j times e to the j2t so again what is omega what is the phase constant and what's happening with the amplitude is it constant is it getting smaller or is it getting bigger with time so um go ahead and again pause the video and see if you can figure these out on your own and we'll check it soon so here are solutions to the example so for the first one what is omega well the trick to doing these um i think is is so you don't you'll be less likely to make an error is to take our original signal here and let's factor in the t so we have 12 times e to the 2t plus i'm sorry times e to the j 5t okay now um one thing we can do here is we can immediately see that the 5 is multiplying the t so that gives us our omega now what is the phase constant well there is none in this form notice like you know if we compare this to up here you can think of this five t here is the same as five t plus zero so now it looks exactly like this form up here and you can see that theta is indeed zero now what about the amplitude well what we have to look at is the real stuff out in front so we have the 12 and then what's important is we have the e to the 2t this is an increasing function with time so our amplitude is increasing for the second one down here what i'm going to do is sort of the opposite i'm going to take these two forms here and i'm going to combine them all right by using our rules for exponents remember when we have two numbers like this multiplied together we simply add the exponents together so when we add these together i can factor out a j and i get two t plus three and now it's pretty um simple to see that omega is the number in front of t it's two and our phase constant theta is the three here all right again i'm looking at this up here for reference theta is three radians and then the amplitude notice is just seven there's a seven out here all by itself so that is a constant amplitude that's not changing with time let's now turn to real and complex discrete time exponential signals so we'll start with discrete time real exponential sequence and we can write it mathematically as x of n remember sequences have the index n inside of those square brackets there is equal to c some we're going to let c be a real number times alpha which is sort of the base here and we'll let alpha in this case be a real number also and alpha is raised to the n remember n is kind of like our time variable here so there's a bunch of different cases here depending on the value of alpha here so let's take the first case where alpha is greater than one okay so alpha could be two three four something like that or it could even be a fraction like 1.5 or something like that well if this number is greater than 1 and we're raising it to the nth power then we have a sequence that is growing exponentially like this so this would be x of n um on the other hand um if alpha is between zero and one okay like alpha is one half or 0.7 or something like that then as we raise this number alpha to a higher power it's going to get smaller and smaller because alpha is less than 1. and so this would represent the case where we have exponential decay when alpha is between zero and one now what about alpha uh for negative values well here things get um a little more crazy because if alpha is a negative number then remember n is an integer so n you know goes one two three four five but when we square a negative number it's positive but if we raise a negative number to an odd power it's negative so in other words even powers of alpha or when n is even are positive and when n is uh an odd number designating an odd power our sequence is going to be negative so what we see is the sequence actually alternates it goes positive negative positive negative and in this case alpha is between negative one and zero so it's an alternating exponential decay and if alpha is less than negative one for example if alpha is negative two negative 3 whatever then we have an alternating exponential growth and again notice that our sequence x of n alternates between positive and negative values as it grows so here is an activity for you to practice um go ahead and make a sketch of x of n which is equal to one half to the n from the values of n equal negative two to n equal to so go ahead and pause the video and do this activity so here's our solution remember what we can do here is if we're going to start let's start at n equal negative two so we have one half to the negative two power so remember a negative exponent means you take the reciprocal of whatever you're you know uh raising it uh whatever you're raising so the reciprocal of two i'm sorry the reciprocal of one-half is two so one-half to the minus two is the same as two squared which is four all right and so that's how you do the negative one so this sequence then when n is negative two it has a value of four and then it goes two one one half and one fourth so if you didn't get that uh make sure you know you go back right now and make sure that you can redo this and understand this let's now move on to discrete time imaginary exponential sequences and again this is something that we're going to um encounter in this course later on so this is why we're learning about these now so in this case we have a sequence x of n that is uh described by the function e to the a n where a is a purely imaginary number like j omega and again we're using omega because this is going to turn out to be the frequency of this signal so let's substitute j omega for a put it in there and we have e to the j omega n and now we can use euler's formula to write this out in terms of cosines and sines so this is equal to cosine omega n plus j sine omega n and here again we see that you know this this complex exponential sequence here is going to be oscillating all right this thing is represented the real part is represented by a cosine function cosine omega n and you know also remember that n here is our index all right like our time index but it really doesn't have it's not time because it's a dimensionless integer it doesn't carry any units likewise omega even though we can we sort of call it our frequency here we have to be careful it really represents an angle in radians all right now there's something else that um is is sort of uh maybe a little bit surprising about this function cosine omega n now cosine omega n it clearly oscillates just like cosine of omega t would however this may not be a periodic function so you know if you're wondering well if it oscillates then doesn't it have to be periodic well the answer there is no and we'll we'll investigate this next so let's take a look at um let's go back to continuous time if i have a function x of t is equal to cosine times t over six and that's plotted here all right remember this is a continuous time function here well then by inspection i can see that uh my frequency is the number in front of the t right so it's 1 6 and that's my omega my radian frequency the physical frequency is then omega divided by 2 pi which is 1 over 12 pi because remember omega is 1 6. so my frequency is 1 over 12 pi the period of this function here is equal to 1 over the frequency so we take the reciprocal of 1 over 12 pi and we get a period of 12 pi and of course this function is periodic because it repeats itself over and over every 12 pi seconds however let's take a look at x of n is equal to cosine of n over six so notice this is sort of like the discrete time version of this we're just replacing t over here with n right the t right here becomes an n here well if we plot it okay it looks like a cosine function it goes up and down and up and down however it's not periodic it doesn't exactly repeat itself okay and the one way to show this is that notice by inspection here again omega is 1 6 because omega remembers the number in front of the n the frequency then is omega over 2 pi which is 1 over 12 pi all right nothing different that's exactly what we did here and the period which by the way in uh when we talk about periods in discrete time we're going to use the uppercase n instead of the uppercase t well the other the the period is always one over the frequency and that gives us 12 pi but here's a problem 12 pi for a period is not an integer and remember discrete time sequence has for its index it has to be an integer in other words n equals 0 1 2 3 4 5 6 and so forth n is never equal to 12 pi right because it's not an integer so this is one way of showing that this thing is not periodic it doesn't exactly repeat itself here here's another uh maybe a little more better more clear way to see that not all oscillating sequences are periodic so this is an example of a non-periodic sequence so our sequence that i plotted here in excel is very simple it's x of n is equal to cosine of n over two so you know here's n i went from zero out to forty and here's the values of x of n and you can see they're plotted here so you know if we kind of think of connecting the dots we can obviously see this is a nice cosine function okay everything looks fine and dandy however this thing never repeats itself notice up here at the peak notice the placement of these two um you know parts of the sequence well on the next cycle over here they're in different places they've shifted over and over here you know they looks kind of close to that but it's not they're they're always shifted slightly from from each other and it'll never ever repeat itself you could look at you know 100 million cycles of this thing and some of the values may get very very close but it will never actually repeat itself so um this is an example where we have a sinusoidal signal or graph but it is not periodic so the question now becomes is there such thing as a periodic discrete time sequence and the answer is yes however there is a special uh periodicity requirement that has to be met in other word uh in order for a discrete time sequence to be periodic and this is the way you can uh prove and derive this you're welcome to go through this yourself it's you know it's nothing that difficult here we're just basically what we're doing is imposing the periodicity requirement that a sequence is periodic if x of n plus the period n is equal to x of n in other words it has to have the same exact value as x of n when we add the period to it okay and and you know again is our discrete time period it has to be remember a um an integer value okay so when you go through all the math here you'll end up with this requirement this is the important thing this is like the test the period of our signal or sequence n has to be equal to 2 pi times m over omega now we know what omega is omega is the uh the frequency of our signal which we can get you know by just inspection by looking at it m is the smallest non-zero integer needed to make n an integer so in other words if we can find a value of m and remember m has to be one two three four five six seven one one of the integers all right not zero if we can find a value of m that makes n an integer then we have a periodic sequence so let's look at some examples so um here's we're going to show how to find the frequency and period of a discrete time sequence so finding the frequency omega is pretty straightforward by inspection so in other words if we have cosine of omega n well the frequency is just this whatever that number is omega if we have sine of omega n again we use whatever is in front of the n or if we have it in complex exponential form like e to the j omega and again the omega is the number right in front of the n there so for example if x of n is equal to cosine of seven pi n over three then just by inspection here i can immediately see that omega is seven pi over three because that's what's multiplying n right here now to find the period remember what i have to do is i have to i have to say set n equal 2 pi m over omega all right and i have to try to find a value of m that makes n an integer so let's see if we can do it so what is omega down here well we said omega was seven pi over three so i'm going to plug that in here so here's the seven pi and i put the three up here so notice this three up here and the two multiply together to get six notice the other thing and this is kind of the key thing this pi up here which you know we started with here cancels the pi down there so when those cancel out we're left with six m over seven now can we turn this fraction into an integer yes just sort of by trial and error you can see that if you let m equal to seven it cancels the seven down here and we're left with n our period is equal to six right here okay so this signal right here this sequence has a frequency of seven pi over three and it has a period of 6 meaning it repeats itself every 6 values so finding the period is you know not quite as obvious you can't do it just by inspection it's not nothing obvious there you have to apply this test here here's another example x of n is equal to sine of 2 pi n over 12 plus 1. so let's do this again so first of all let's find the omega well remember the omega is whatever is multiplying n so that's the 2 pi over 12 which we can simplify to pi over 6. now to find the period n we have to set it equal to 2 pi m over omega where remember omega is pi over 6. so i'm going to plug that in here's the pi i put the 6 up here this 6 and that 2 multiplied together to get 12. this pi on top cancels this pi on the bottom so i get 12 m over one which is of course just equal to 12 m and obviously i can find that m the lowest you know possible number to make this an integer is when m is equal to one and so therefore our period is n equal 12. and this is the function plotted here okay so in other words this this sequence up here is plotted right here and you can see that this thing repeats itself every 12 values if you count you know so let's say you start here you go 1 2 3 4 5 6 7 8 9 10 11 and at 12 we're back to where we started from so here's a chance for you to practice these techniques go ahead and do these activities for each of these you're asked to calculate the amplitude the frequency and the period for the following discrete time sequences there's two examples here and uh you know you may need to go back to the previous uh you know go back in the the video to the last slide um especially to find the period you have to apply that technique there so go ahead and give these a try all right so here are the activity solutions for the first one here by inspection we can see that omega is just pi over eight and then to find the period we say n is equal to two pi m over omega where omega is pi over eight so here's the pi here's the eight eight times two is sixteen the pi's cancel so i get 16 m and the lowest value of m to make this an integer is of course m equal one which gives us our period of n is equal to 16. all right and that is plotted right here the next one here x of n equals cosine three pi n over two plus five um again by inspection i can see that omega is three pi over two that's just the stuff in front of the n there we don't need to worry about this five and to figure out the period we we set n equal two pi m over omega we plug in omega which is three pi over two um simplify right multiply two times two is four the pi's cancel we have four m over three the smallest value of m that makes this an integer is m equal three which gives us our period of four and by the way this um uh this sequence is plotted right here and you can see this thing repeat itself every four values because that's the fundamental period of this sequence in this section we're going to look at the concept of discrete time frequency in a little more detail as there are some subtle and important differences in frequency between discrete time and continuous time signals so this whole concept of discrete time frequency um is kind of interesting because um if if we consider you know let's say our sequence x of n is equal to cosine of omega n um this is you know a a cosine function that's obviously oscillating with a frequency of omega but in discrete time omega is actually an angle in radians it's not really a frequency because the reason for this um you know is again it's kind of subtle this n here is sort of like t like time except n is a dimensionless integer it's just an index it doesn't carry the units of time so remember that you know the argument of a cosine function has to be an angle so omega times n has to be in radians and therefore omega has to be radiant so omega technically is just an angle it's not really a frequency now what what's makes this what's significant about this is that remember angles only vary from 0 to 2 pi before they repeat themselves in other words theta and theta plus 2 pi are the exact same angle so you know we can see this on the on the circle here if this represents the angle theta right here if we add 2 pi to that angle we go all the way around the circle and we end up right back to where we started from so what's what's kind of weird about this is that this when we talk about you know cosine of omega n the um first the first of all you know this frequency omega can only go from zero to two pi and if we go above two pi we can always subtract the two pi to get back to the same frequency so you know there's some significant differences here in continuous time the frequencies range from zero to infinity and they can have any value all right from zero to infinity but in discrete time frequencies range from zero to two pi only there is no other frequency um beyond you know this range of just from zero to two pi um if you do have a frequency higher than that you can always basically simplify it back to one of the angles around on the circle which is you know from zero to two pi um often times the frequencies are said to range from negative pi to positive pi and you know notice though this is still a total frequency range of only two pi um this is sometimes a more convenient way to think about it so let's dig into this discrete time frequency a little bit more and you know notice i kind of put it in quotes here because although you know we call it a discrete time frequency remember it really represents an angle that repeats itself now the thing is about this um let's plot um our cosine function cosine of omega n let's plot it for a bunch of different values of omega so if we let omega equal zero well cosine of zero times n is just the cosine of zero which is one so here's the sequence this is x of n which has no frequency now this makes sense if it has no frequency then it never changes and it's just a constant value so this is what you know really a really low frequency signal would look like basically it's not changing at all so that all makes sense this is cosine of pi n over eight so we have a frequency of one eighth all right and so now you can see it oscillating up and down here's uh cosine of pi n over four so here the frequency was pi over eight here it's pi over four so this has twice the frequency of this one you can see it's wiggling up and down twice as often all right twice as fast and then we have cosine of pi n over 2 so now it's oscillating you know even faster than this one twice as fast again and then finally look at this one here's um cosine of pi n so the frequency here omega if you can do it by inspection is pi but look what's happening this thing is oscillating up and down as fast as it possibly can it's going from max value to min value back to max you can't go any faster than that right that's the fastest it can possibly oscillate so um what this tells us oh and then by the way if we keep going notice it starts slowing down again slowing down and when we get back to two pi we're back to our signal that doesn't oscillate at all because remember 2 pi is the same as 0. so what this tells us is that with discrete time a frequency of 0 or two pi or four pi or any multiple of two pi represents a low frequency with slow variation and a frequency of pi or three pi or five pi is the highest possible frequency that represents rapid variation and that was the uh image shown in here so this is um you know again a kind of a a little different from continuous time it's actually very different frequencies can only vary from 0 to 2 pi there's nothing else that's possible for a discrete time sequence there's one more really meaningful concept about discrete time frequency and this is actually not really shown in your book it may be mentioned much later in your book in chapter 7 on sampling but your book doesn't really stress this concept but i i like to include it here because i think it gives some some good uh insight into uh visualizing and thinking about what a discrete time frequency is and that is um we can think of it as radians per sample because really that in a sense that's what it that's what it means so let's start with the sinusoidal continuous time signal which is this red line here let's call it x of t is equal to sine of omega t and what we're going to do is we're going to take samples of this signal every you know sampling period okay so we're going to sample x of t with a sample of period of t sub s to generate the sequence x of n so our sequence now are these black dots here this represents our sampled version or now we have a discrete time sequence so um if we plot x of t and x of n again here you can see them again but instead of plotting them versus time here or versus n we're going to plot them versus their phase remember one full cycle is 2 pi radians so in other words this is phase here so we have zero here and when we've gone around one full cycle we're back to where we started that would be two pi radians all right so halfway through is pi radians and then a quarter way is pi over two one eighth way through is pi over four so this is just you know the same graph up here but we're plotting it versus the phase of the cycle here so the discrete time frequency omega uh of the little d under there for discrete uh it tells us how many radians per sample there are and we can see that each one of these samples is pi over 4 radians all right and so the the way to get from one to the other is we notice that radians per sample is equal to radians per second times seconds per sample right because these seconds cancel right here so radians per sample that is our discrete time frequency omega sub d radians per second that is our continuous time frequency omega and finally seconds per sample well that's just the sample period t sub s right that's how many uh that's how many seconds it took for each of these samples to take place so what we end up with is a way to sort of uh there's a couple things here first of all it tells us that discrete time frequency carries the units really of radians per sample even though technically it's just radians and also we can get the discrete time frequency by multiplying the continuous time frequency by the sample period that was used to create the sequence so here's an example in fact this is the last thing we'll do in this part of a lecture here we have a two megahertz continuous time signal that's the red line here and we're told that it's sampled at 10 million samples per second what is the frequency of the resulting discrete time sequence okay so um i put all the equations that you should need to solve this activity to do this activity right here so go ahead and see if you can figure out you know basically we're taking this 2 megahertz signal here the red line and we're turning into this sequence which are the black dots but we want to know what is the frequency of this sequence when we're all said and done so again go ahead and pause your video and see if you can do this activity all right so here's the solution to our activity um so again we've got this 2 megahertz continuous time signal it's sampled at 10 million samples per second we want to figure out what is the frequency of the resulting discrete time sequence so um we know that our sampling frequency f sub s we're doing 10 million samples per second so that's 10 to the seventh samples per second so our sample period t sub s is one over the sampling frequency so one over ten to the seventh just gives us 10 to the minus seven and you can see that's 10 to the minus seven seconds per sample so in other words we have a this is a tenth of a microsecond per sample all right we can figure out what omega is well remember this is continuous time now omega is just two pi f so we have two pi radians per cycle times 2 times 10 to the 6 cycles per second where did i get this number well that was given to us right 2 megahertz so that's equal to 1.26 times 10 to the 7th radians per second finally we use our equation that were you you know we were told the discrete frequency omega sub d is equal to the continuous frequency omega times the sample period so i have 1.26 times 10 to the seventh radians per second times 10 to the minus seven seconds per sample we multiply these together actually do this in your head you can see these exponents cancel each other and we end up with 1.26 radians per sample so notice that this number here 1.26 is less than 2 pi right 2pi is around 6. so this original signal which had a high frequency of 2 million hertz when we sample it the sample frequency um and um in the uh the sequence frequency the discrete time frequency of the resulting sequence is only 1.26 radians per sample okay so that is it for uh part two of this lecture uh we will continue on and i'll see you in part three goodbye

hi again this is mr pinter your instructor for signals and systems welcome to part two of our lecture on chapter one in this part of the video we&#39;ll be looking at real and complex exponential signals before we start talking about these exponential signals i just want to mention that exponential signals are an extremely important class of signals for the study of signals and systems because exponential signals are the solutions to differential equations which describe a lot of systems so we&#39;ll be seeing this throughout the course we&#39;ll be encountering these exponential signals so let&#39;s start out by talking about continuous time real exponential signal so mathematically our our signal here can be represented by the function x of t which is equal to c some constant we&#39;ll let that be a positive number times e raised to the a t now um the exponent here is some coefficient a multiplied by time and a has there&#39;s two options for a it can be positive or negative so when a is greater than 0 this is a growing exponential function and when we plot it it looks something like this of course when we put in a zero for t remember that e to the zero is one so our function gives us c which uh is this number out in front here um and so that&#39;s plotted here and then of course this thing grows exponentially on the other hand if a is less than zero in other words if we have a minus sign up here like a minus two t or something like that this is a decaying exponential function and it&#39;s going to look something like this it gets smaller and decays exponentially with time so here&#39;s a simple activity just sort of a like a warm up for this part of the video go ahead and make a sketch of the following signal x of t is equal to 5 times e to the minus t so you don&#39;t have to do anything fancy here just make a quick sketch uh you know on a piece of paper or something like that it doesn&#39;t even have to be graph paper go ahead and pause the video and sketch this and we&#39;ll check your results so here&#39;s the solution if x of t is equal to 5 times e to the minus t then the function is going to look something like this notice it passes through the value of five at t equals zero and because there&#39;s a minus sign up here this is a decaying exponential function so it&#39;s going to eventually go to zero for large values of t the next important class of signals we&#39;re going to look at are exponential signals or functions that have a complex exponent okay so before we dig into those uh let&#39;s do a quick complex number review and again this is very fundamental uh information for this course we&#39;re going to be dealing with complex numbers over and over in just about every chapter so um this is you know should be review for you this is stuff you learned in your calculus class and so forth but the good news is that pretty much everything you need to know about complex numbers is shown on this one page but make sure that you understand this uh well in order to you know be able to do this course and be successful in this course so let&#39;s just do the review here so we can represent a complex number by any variable we want but let&#39;s use the letter z here to represent our complex number there are two forms that we can put the complex number in there&#39;s the so-called polar form and the rectangular form so the polar form has the magnitude out in front i called it r here so r is the magnitude of z okay so i&#39;ve got the you can see the absolute value bars around z there and then we have e raised to the j theta where theta represents the phase angle so the polar form um basically is magnitude and phase of the complex number now you may be wondering what is this j here well j is what um your math textbook probably called i it&#39;s the uh imaginary unit basically it&#39;s the square root of minus one now why do we use j and not i well electrical engineering um textbooks tend to use j so that it&#39;s not to be confused with i because current is represented by the letter i so in this course we will be using j and you&#39;ll get used to that very quickly now the other form that we can put this same number in is the rectangular form which looks like a plus j times b where a is the real part of the complex number and b is the so-called imaginary part of the complex number notice it it has the uh j um in front of it here so one way to sort of uh reconcile these two different forms you know they&#39;re obviously very different forms but they represent the exact same thing so how is this well it&#39;s useful to plot z either as a point or even more uh it&#39;s actually more illustrative just to plot it as a vector in the so-called complex plane so the complex plane remember this is the real axis down here and the vertical axis is the so-called imaginary axis so this real axis represents the real part of z the vertical axis represents the imaginary part of z so here is z z can be plotted as a point right here okay let&#39;s say this is our value z it has both real parts and imaging parts or we can plot it as a vector so let&#39;s take a look at these two forms so in the polar form remember r is the magnitude out front well that&#39;s just the magnitude of this vector in other words how long is this arrow right here and theta of course is the phase angle which represents the angle of this vector with respect to the real axis so here our r and theta this is the polar way to represent our complex number z now of course z can also be represented in the rectangular form where a here is if we project z down to the real axis a is the real part of this z and b is projected across to the imaginary axis b is the imaginary part of z so this this picture here you know as they say is worth a thousand words because this tells you everything you could possibly want to know about this single complex number it tells you its real part its imaginary part its magnitude and its phase and going further we can relate the polar and rectangular forms by using basic trig so trigonometry you know notice this is a right triangle here allows us to express r theta a and b in terms of each other so for example this angle here you remember from you know trig or math class is the inverse tangent of the opposite side which is b right this is this height here is b and this length down here is a so it&#39;s inverse tangent of b over a so this is a relationship between theta and a and b we can use the pythagorean theorem to find out the value of r with respect to a and b all right so r is the square root of a squared plus b squared again because this is a right triangle furthermore we can relate a down here is equal to r times the cosine of theta so here&#39;s a nice relationship of how to find the value a if you know r and theta so in other words if you&#39;re given a a complex number in polar form it&#39;s very straightforward to find out the rectangular form by using a equals r cosine theta and likewise b is equal to r sine theta now we can do something that&#39;s actually quite profound here if we notice that z remember our complex number in polar form is given by r times e to the j theta and we said that&#39;s equal to a plus b where a is our cosine theta and b is our sine theta well notice our rectangular form here a plus j b if we eliminate r in other words divide get rid of this r here this r here and this r here just by dividing then what we&#39;re left with is the famous euler&#39;s formula which says that e to the j theta is equal to cosine theta plus j sine theta and again euler&#39;s formula is something that we will be using fairly often in this course before we leave our review of complex numbers there&#39;s one more uh slide here just on this is sort of some special cases and these are very convenient forms uh to use so first of all remember if we take a complex number z and represent it in polar form as r times e to the j theta there&#39;s a special case where we let r equal one and this is r equal one refers to what we call is the unit circle so this is a circle with a radius of one this r here you know basically represents the radius of that circle so when we set r equal to one then there are some special values here um going around the circle that will come up now and then so for example if we have e to the j times 2 pi all right or e to the j 4 pi well remember we start here if we go around 2 pi we end up right back here if we go around another two pi so we&#39;ve gone around four pi radians now we still end up back here and this is the same as e to the zero all right zero radians is the same as two pi radians because zero radians you know that means theta is zero we&#39;re right here at plus one and if we go around we&#39;re back at plus one so these forms up here if ever we see those forms obviously they are just equal to one now look at this point up here well this has an angle of 90 degrees which is pi over 2. so e to the j pi over 2 is up here and that&#39;s equal to j that&#39;s equal to positive j right it&#39;s a purely imaginary number um likewise instead of going up by pi over 2 if we go down and around by 3 pi over 2 which is 270 degrees we also end up at j so these two expressions here even though they look very different they are identical to each other and they&#39;re equal to the value of j likewise when we start here and we go by a full pi radians over here 180 degrees we go from plus one all the way to minus one so e to the plus or minus j pi is equal to e plus or minus j three pi you know five pi so so on and so forth and they&#39;re all equal to -1 and our last special case is down here if we go down by pi over 2 in other words negative pi over 2 radians we end up at minus j or we can go in the positive direction 3 pi over 2 okay so all these forms here can be simplified to very basic fundamental numbers either 1 you know plus or minus 1 or plus or minus j so now that we&#39;ve completed our review of complex numbers we&#39;re ready to dig into complex um exponential or imaginary exponential signals and we&#39;re gonna we&#39;re doing this for continuous time first we&#39;ll deal with discrete time later so the general form is uh this looks familiar this is our exponential signal again x of t equals c e to the a t the only difference now and it&#39;s a big difference is that a is not a real number like it was earlier now a is going to be purely imaginary in other words it has no real part it&#39;s just got the j times some number and that number we&#39;re going to call omega and you&#39;ll see why shortly so substituting this into here we see that our imaginary exponential signal is x of t is equal to some constant c times e to the j omega t and what we can do is invoke euler&#39;s formula remember euler&#39;s formula says if we have e to the j something that&#39;s equal to cosine something plus j sine something so by plugging this into euler&#39;s formula x of t can be written as c times cosine omega t plus j times c sine of omega t so let&#39;s look at the the real part of this function you know there&#39;s a real part here and this is the imaginary part because it&#39;s got that j in front of it but the real part of this function is just a cosine function it&#39;s oscillating it&#39;s an oscillating function so if we were to plot the the real part all right of x of t that&#39;s what this means the r e for real part of x of t it&#39;s going to look something like this it&#39;s a cosine function and likewise the imaginary part oscillates also and notice it oscillates at the frequency given by omega where omega was um up here in between the j and the t so that is our radian frequency so this um provides us an easy way to find the radian frequency of a periodic um signal just by inspection so if we ever see you know some form like a times cosine omega t or a times sine omega t or a times e to the j omega t in all cases here the number in front of the t and in this case you know it&#39;s in between the j and the t that number gives us the radian frequency so the radian frequency omega is of course in radians per second we can convert that to the so sometimes they call this the physical frequency f which is equal to omega divided by two pi because remember there are two pi radians per person you know circling around the circle there are two pi radians per cycle so the physical frequency is given not in radians per second but in cycles per second and um this is also known as a hertz the unit hertz or hz is when we&#39;re talking about the physical frequency the period of this this signal here t is equal to one over the frequency you&#39;ll remember that from you know your physics uh class when you learn about waves and that is equal to two pi over omega if you&#39;re going to use omega instead of f and the period of course has units of seconds and finally the amplitude of this signal here is given by whatever this number is out in front of the sine or the cosine or the complex exponential notice in all these cases here the amplitude is designated by a and if it&#39;s shown like this in this form you would say the amplitude is equal to c so here&#39;s an example or an activity for you to practice what we just learned so let&#39;s say you&#39;re given the following signal x of t is equal to 5 cosine of 6 pi t so your task is to find the amplitude the radiant frequency the physical frequency and the period of this signal right here and just to make this a little bit easier so you don&#39;t have to go back to the previous slide i put all the equations here to figure out these various parameters down here so again go ahead and pause the video and see if you can figure out what these are what a omega f and t are for this signal here all right so the solutions here the amplitude is found just by inspection that&#39;s 5 the radian frequency is also found by inspection remember omega is the number that is multiplying in front of t so omega is equal to six pi and the units of course are radians per second the physical frequency we get from just taking omega which is six pi and dividing it by two pi so we get three cycles per second and finally the period t is just one over f so we have one over three gives us one third or point 0.333 seconds let&#39;s now take a look at a more general type of continuous time complex exponential function this time we&#39;re going to start you know pretty much with the same form x of t is equal to c times e to the a t but now instead of a being purely imaginary a is going to be complex in other words a is going to have a real part and a imaginary part and also just to make things completely general here let&#39;s let this number out in front c also be a complex number it&#39;s not just a real number anymore so the trick to to making some sense of this is um to let this c here be in polar form so we&#39;re going to let c equal to the magnitude of c because remember c is complex times e to the j theta where theta is the phase and this is the magnitude of c and then um we&#39;re going to let a which is up here in the exponent we&#39;re going to let that one be in rectangular form so we&#39;re going to let a equals sigma plus j omega where sigma is the real part and omega is the imaginary part so when we substitute these expressions here for a and c into this up here we get the following x of t is equal to magnitude c times e to the j theta times e to the quantity sigma plus j omega and the quantity times time now what we need to do we can lump together all the real parts so the real part here is obviously the magnitude of c is real and also remember sigma is a real number so e to the sigma times t is also a real uh quantity all right this is just an exponential function there&#39;s no j or anything in it okay so we&#39;re going to lump these two real terms together and they constitute what is called the envelope of this function now what&#39;s left are the imaginary parts we have e to the j theta and we have e to the j omega t so i&#39;m going to factor out the j and in parentheses i have omega t plus theta so what these mean now is that omega is still the frequency it&#39;s the radiant frequency because it&#39;s the number that is in front of the t the theta here is what&#39;s called a phase constant okay it&#39;s not related to the frequency but it determines sort of where this function starts at time zero and furthermore this sigma going back to our our real part here the envelope the sigma is related to the time constant of how fast this signal either grows or decays in fact sigma is one over the time constant so let&#39;s look at a couple different cases here if sigma is a positive number right sigma greater than zero then this envelope here is a growing function in other words the amplitude of this signal is getting bigger with time so that&#39;s represented by these dotted lines notice the amplitude is growing and an example of this is in resonance when we have resonance we have a signal that grows with time an oscillating signal that grows in time now notice the actual signal though is oscillating up and down now why is that well because of this portion the um this complex exponential here remember this is e to the j omega t you know and then there&#39;s a phase constant but the e e to the j omega t remember is oscillating it has a cosine and a sine uh involved with it so this is what um this signal would look like for a positive value of sigma it is a growing uh you know an oscillating function that is growing with time the other um case of course is when sigma here is a negative number then our envelope is decaying with time it&#39;s getting smaller and smaller and we have an oscillating signal that is um you know decaying with time an example of this would be damped harmonic motion or damped oscillations due to say friction so let&#39;s take a look at a couple of examples and uh before we do in fact uh this is uh these are activities uh for you to figure out here you can do these um on your own and then we&#39;ll look at the solutions so um i put the you know the expression from the last page here uh to help us kind of remember what&#39;s what so for this first example y of t is equal to 12 times e to the j five plus two all in parentheses times t so the question is what is omega what is the phase constant and then is the amplitude constant decreasing or increasing with time the next example x of t is equal to 7 times e to the 3j times e to the j2t so again what is omega what is the phase constant and what&#39;s happening with the amplitude is it constant is it getting smaller or is it getting bigger with time so um go ahead and again pause the video and see if you can figure these out on your own and we&#39;ll check it soon so here are solutions to the example so for the first one what is omega well the trick to doing these um i think is is so you don&#39;t you&#39;ll be less likely to make an error is to take our original signal here and let&#39;s factor in the t so we have 12 times e to the 2t plus i&#39;m sorry times e to the j 5t okay now um one thing we can do here is we can immediately see that the 5 is multiplying the t so that gives us our omega now what is the phase constant well there is none in this form notice like you know if we compare this to up here you can think of this five t here is the same as five t plus zero so now it looks exactly like this form up here and you can see that theta is indeed zero now what about the amplitude well what we have to look at is the real stuff out in front so we have the 12 and then what&#39;s important is we have the e to the 2t this is an increasing function with time so our amplitude is increasing for the second one down here what i&#39;m going to do is sort of the opposite i&#39;m going to take these two forms here and i&#39;m going to combine them all right by using our rules for exponents remember when we have two numbers like this multiplied together we simply add the exponents together so when we add these together i can factor out a j and i get two t plus three and now it&#39;s pretty um simple to see that omega is the number in front of t it&#39;s two and our phase constant theta is the three here all right again i&#39;m looking at this up here for reference theta is three radians and then the amplitude notice is just seven there&#39;s a seven out here all by itself so that is a constant amplitude that&#39;s not changing with time let&#39;s now turn to real and complex discrete time exponential signals so we&#39;ll start with discrete time real exponential sequence and we can write it mathematically as x of n remember sequences have the index n inside of those square brackets there is equal to c some we&#39;re going to let c be a real number times alpha which is sort of the base here and we&#39;ll let alpha in this case be a real number also and alpha is raised to the n remember n is kind of like our time variable here so there&#39;s a bunch of different cases here depending on the value of alpha here so let&#39;s take the first case where alpha is greater than one okay so alpha could be two three four something like that or it could even be a fraction like 1.5 or something like that well if this number is greater than 1 and we&#39;re raising it to the nth power then we have a sequence that is growing exponentially like this so this would be x of n um on the other hand um if alpha is between zero and one okay like alpha is one half or 0.7 or something like that then as we raise this number alpha to a higher power it&#39;s going to get smaller and smaller because alpha is less than 1. and so this would represent the case where we have exponential decay when alpha is between zero and one now what about alpha uh for negative values well here things get um a little more crazy because if alpha is a negative number then remember n is an integer so n you know goes one two three four five but when we square a negative number it&#39;s positive but if we raise a negative number to an odd power it&#39;s negative so in other words even powers of alpha or when n is even are positive and when n is uh an odd number designating an odd power our sequence is going to be negative so what we see is the sequence actually alternates it goes positive negative positive negative and in this case alpha is between negative one and zero so it&#39;s an alternating exponential decay and if alpha is less than negative one for example if alpha is negative two negative 3 whatever then we have an alternating exponential growth and again notice that our sequence x of n alternates between positive and negative values as it grows so here is an activity for you to practice um go ahead and make a sketch of x of n which is equal to one half to the n from the values of n equal negative two to n equal to so go ahead and pause the video and do this activity so here&#39;s our solution remember what we can do here is if we&#39;re going to start let&#39;s start at n equal negative two so we have one half to the negative two power so remember a negative exponent means you take the reciprocal of whatever you&#39;re you know uh raising it uh whatever you&#39;re raising so the reciprocal of two i&#39;m sorry the reciprocal of one-half is two so one-half to the minus two is the same as two squared which is four all right and so that&#39;s how you do the negative one so this sequence then when n is negative two it has a value of four and then it goes two one one half and one fourth so if you didn&#39;t get that uh make sure you know you go back right now and make sure that you can redo this and understand this let&#39;s now move on to discrete time imaginary exponential sequences and again this is something that we&#39;re going to um encounter in this course later on so this is why we&#39;re learning about these now so in this case we have a sequence x of n that is uh described by the function e to the a n where a is a purely imaginary number like j omega and again we&#39;re using omega because this is going to turn out to be the frequency of this signal so let&#39;s substitute j omega for a put it in there and we have e to the j omega n and now we can use euler&#39;s formula to write this out in terms of cosines and sines so this is equal to cosine omega n plus j sine omega n and here again we see that you know this this complex exponential sequence here is going to be oscillating all right this thing is represented the real part is represented by a cosine function cosine omega n and you know also remember that n here is our index all right like our time index but it really doesn&#39;t have it&#39;s not time because it&#39;s a dimensionless integer it doesn&#39;t carry any units likewise omega even though we can we sort of call it our frequency here we have to be careful it really represents an angle in radians all right now there&#39;s something else that um is is sort of uh maybe a little bit surprising about this function cosine omega n now cosine omega n it clearly oscillates just like cosine of omega t would however this may not be a periodic function so you know if you&#39;re wondering well if it oscillates then doesn&#39;t it have to be periodic well the answer there is no and we&#39;ll we&#39;ll investigate this next so let&#39;s take a look at um let&#39;s go back to continuous time if i have a function x of t is equal to cosine times t over six and that&#39;s plotted here all right remember this is a continuous time function here well then by inspection i can see that uh my frequency is the number in front of the t right so it&#39;s 1 6 and that&#39;s my omega my radian frequency the physical frequency is then omega divided by 2 pi which is 1 over 12 pi because remember omega is 1 6. so my frequency is 1 over 12 pi the period of this function here is equal to 1 over the frequency so we take the reciprocal of 1 over 12 pi and we get a period of 12 pi and of course this function is periodic because it repeats itself over and over every 12 pi seconds however let&#39;s take a look at x of n is equal to cosine of n over six so notice this is sort of like the discrete time version of this we&#39;re just replacing t over here with n right the t right here becomes an n here well if we plot it okay it looks like a cosine function it goes up and down and up and down however it&#39;s not periodic it doesn&#39;t exactly repeat itself okay and the one way to show this is that notice by inspection here again omega is 1 6 because omega remembers the number in front of the n the frequency then is omega over 2 pi which is 1 over 12 pi all right nothing different that&#39;s exactly what we did here and the period which by the way in uh when we talk about periods in discrete time we&#39;re going to use the uppercase n instead of the uppercase t well the other the the period is always one over the frequency and that gives us 12 pi but here&#39;s a problem 12 pi for a period is not an integer and remember discrete time sequence has for its index it has to be an integer in other words n equals 0 1 2 3 4 5 6 and so forth n is never equal to 12 pi right because it&#39;s not an integer so this is one way of showing that this thing is not periodic it doesn&#39;t exactly repeat itself here here&#39;s another uh maybe a little more better more clear way to see that not all oscillating sequences are periodic so this is an example of a non-periodic sequence so our sequence that i plotted here in excel is very simple it&#39;s x of n is equal to cosine of n over two so you know here&#39;s n i went from zero out to forty and here&#39;s the values of x of n and you can see they&#39;re plotted here so you know if we kind of think of connecting the dots we can obviously see this is a nice cosine function okay everything looks fine and dandy however this thing never repeats itself notice up here at the peak notice the placement of these two um you know parts of the sequence well on the next cycle over here they&#39;re in different places they&#39;ve shifted over and over here you know they looks kind of close to that but it&#39;s not they&#39;re they&#39;re always shifted slightly from from each other and it&#39;ll never ever repeat itself you could look at you know 100 million cycles of this thing and some of the values may get very very close but it will never actually repeat itself so um this is an example where we have a sinusoidal signal or graph but it is not periodic so the question now becomes is there such thing as a periodic discrete time sequence and the answer is yes however there is a special uh periodicity requirement that has to be met in other word uh in order for a discrete time sequence to be periodic and this is the way you can uh prove and derive this you&#39;re welcome to go through this yourself it&#39;s you know it&#39;s nothing that difficult here we&#39;re just basically what we&#39;re doing is imposing the periodicity requirement that a sequence is periodic if x of n plus the period n is equal to x of n in other words it has to have the same exact value as x of n when we add the period to it okay and and you know again is our discrete time period it has to be remember a um an integer value okay so when you go through all the math here you&#39;ll end up with this requirement this is the important thing this is like the test the period of our signal or sequence n has to be equal to 2 pi times m over omega now we know what omega is omega is the uh the frequency of our signal which we can get you know by just inspection by looking at it m is the smallest non-zero integer needed to make n an integer so in other words if we can find a value of m and remember m has to be one two three four five six seven one one of the integers all right not zero if we can find a value of m that makes n an integer then we have a periodic sequence so let&#39;s look at some examples so um here&#39;s we&#39;re going to show how to find the frequency and period of a discrete time sequence so finding the frequency omega is pretty straightforward by inspection so in other words if we have cosine of omega n well the frequency is just this whatever that number is omega if we have sine of omega n again we use whatever is in front of the n or if we have it in complex exponential form like e to the j omega and again the omega is the number right in front of the n there so for example if x of n is equal to cosine of seven pi n over three then just by inspection here i can immediately see that omega is seven pi over three because that&#39;s what&#39;s multiplying n right here now to find the period remember what i have to do is i have to i have to say set n equal 2 pi m over omega all right and i have to try to find a value of m that makes n an integer so let&#39;s see if we can do it so what is omega down here well we said omega was seven pi over three so i&#39;m going to plug that in here so here&#39;s the seven pi and i put the three up here so notice this three up here and the two multiply together to get six notice the other thing and this is kind of the key thing this pi up here which you know we started with here cancels the pi down there so when those cancel out we&#39;re left with six m over seven now can we turn this fraction into an integer yes just sort of by trial and error you can see that if you let m equal to seven it cancels the seven down here and we&#39;re left with n our period is equal to six right here okay so this signal right here this sequence has a frequency of seven pi over three and it has a period of 6 meaning it repeats itself every 6 values so finding the period is you know not quite as obvious you can&#39;t do it just by inspection it&#39;s not nothing obvious there you have to apply this test here here&#39;s another example x of n is equal to sine of 2 pi n over 12 plus 1. so let&#39;s do this again so first of all let&#39;s find the omega well remember the omega is whatever is multiplying n so that&#39;s the 2 pi over 12 which we can simplify to pi over 6. now to find the period n we have to set it equal to 2 pi m over omega where remember omega is pi over 6. so i&#39;m going to plug that in here&#39;s the pi i put the 6 up here this 6 and that 2 multiplied together to get 12. this pi on top cancels this pi on the bottom so i get 12 m over one which is of course just equal to 12 m and obviously i can find that m the lowest you know possible number to make this an integer is when m is equal to one and so therefore our period is n equal 12. and this is the function plotted here okay so in other words this this sequence up here is plotted right here and you can see that this thing repeats itself every 12 values if you count you know so let&#39;s say you start here you go 1 2 3 4 5 6 7 8 9 10 11 and at 12 we&#39;re back to where we started from so here&#39;s a chance for you to practice these techniques go ahead and do these activities for each of these you&#39;re asked to calculate the amplitude the frequency and the period for the following discrete time sequences there&#39;s two examples here and uh you know you may need to go back to the previous uh you know go back in the the video to the last slide um especially to find the period you have to apply that technique there so go ahead and give these a try all right so here are the activity solutions for the first one here by inspection we can see that omega is just pi over eight and then to find the period we say n is equal to two pi m over omega where omega is pi over eight so here&#39;s the pi here&#39;s the eight eight times two is sixteen the pi&#39;s cancel so i get 16 m and the lowest value of m to make this an integer is of course m equal one which gives us our period of n is equal to 16. all right and that is plotted right here the next one here x of n equals cosine three pi n over two plus five um again by inspection i can see that omega is three pi over two that&#39;s just the stuff in front of the n there we don&#39;t need to worry about this five and to figure out the period we we set n equal two pi m over omega we plug in omega which is three pi over two um simplify right multiply two times two is four the pi&#39;s cancel we have four m over three the smallest value of m that makes this an integer is m equal three which gives us our period of four and by the way this um uh this sequence is plotted right here and you can see this thing repeat itself every four values because that&#39;s the fundamental period of this sequence in this section we&#39;re going to look at the concept of discrete time frequency in a little more detail as there are some subtle and important differences in frequency between discrete time and continuous time signals so this whole concept of discrete time frequency um is kind of interesting because um if if we consider you know let&#39;s say our sequence x of n is equal to cosine of omega n um this is you know a a cosine function that&#39;s obviously oscillating with a frequency of omega but in discrete time omega is actually an angle in radians it&#39;s not really a frequency because the reason for this um you know is again it&#39;s kind of subtle this n here is sort of like t like time except n is a dimensionless integer it&#39;s just an index it doesn&#39;t carry the units of time so remember that you know the argument of a cosine function has to be an angle so omega times n has to be in radians and therefore omega has to be radiant so omega technically is just an angle it&#39;s not really a frequency now what what&#39;s makes this what&#39;s significant about this is that remember angles only vary from 0 to 2 pi before they repeat themselves in other words theta and theta plus 2 pi are the exact same angle so you know we can see this on the on the circle here if this represents the angle theta right here if we add 2 pi to that angle we go all the way around the circle and we end up right back to where we started from so what&#39;s what&#39;s kind of weird about this is that this when we talk about you know cosine of omega n the um first the first of all you know this frequency omega can only go from zero to two pi and if we go above two pi we can always subtract the two pi to get back to the same frequency so you know there&#39;s some significant differences here in continuous time the frequencies range from zero to infinity and they can have any value all right from zero to infinity but in discrete time frequencies range from zero to two pi only there is no other frequency um beyond you know this range of just from zero to two pi um if you do have a frequency higher than that you can always basically simplify it back to one of the angles around on the circle which is you know from zero to two pi um often times the frequencies are said to range from negative pi to positive pi and you know notice though this is still a total frequency range of only two pi um this is sometimes a more convenient way to think about it so let&#39;s dig into this discrete time frequency a little bit more and you know notice i kind of put it in quotes here because although you know we call it a discrete time frequency remember it really represents an angle that repeats itself now the thing is about this um let&#39;s plot um our cosine function cosine of omega n let&#39;s plot it for a bunch of different values of omega so if we let omega equal zero well cosine of zero times n is just the cosine of zero which is one so here&#39;s the sequence this is x of n which has no frequency now this makes sense if it has no frequency then it never changes and it&#39;s just a constant value so this is what you know really a really low frequency signal would look like basically it&#39;s not changing at all so that all makes sense this is cosine of pi n over eight so we have a frequency of one eighth all right and so now you can see it oscillating up and down here&#39;s uh cosine of pi n over four so here the frequency was pi over eight here it&#39;s pi over four so this has twice the frequency of this one you can see it&#39;s wiggling up and down twice as often all right twice as fast and then we have cosine of pi n over 2 so now it&#39;s oscillating you know even faster than this one twice as fast again and then finally look at this one here&#39;s um cosine of pi n so the frequency here omega if you can do it by inspection is pi but look what&#39;s happening this thing is oscillating up and down as fast as it possibly can it&#39;s going from max value to min value back to max you can&#39;t go any faster than that right that&#39;s the fastest it can possibly oscillate so um what this tells us oh and then by the way if we keep going notice it starts slowing down again slowing down and when we get back to two pi we&#39;re back to our signal that doesn&#39;t oscillate at all because remember 2 pi is the same as 0. so what this tells us is that with discrete time a frequency of 0 or two pi or four pi or any multiple of two pi represents a low frequency with slow variation and a frequency of pi or three pi or five pi is the highest possible frequency that represents rapid variation and that was the uh image shown in here so this is um you know again a kind of a a little different from continuous time it&#39;s actually very different frequencies can only vary from 0 to 2 pi there&#39;s nothing else that&#39;s possible for a discrete time sequence there&#39;s one more really meaningful concept about discrete time frequency and this is actually not really shown in your book it may be mentioned much later in your book in chapter 7 on sampling but your book doesn&#39;t really stress this concept but i i like to include it here because i think it gives some some good uh insight into uh visualizing and thinking about what a discrete time frequency is and that is um we can think of it as radians per sample because really that in a sense that&#39;s what it that&#39;s what it means so let&#39;s start with the sinusoidal continuous time signal which is this red line here let&#39;s call it x of t is equal to sine of omega t and what we&#39;re going to do is we&#39;re going to take samples of this signal every you know sampling period okay so we&#39;re going to sample x of t with a sample of period of t sub s to generate the sequence x of n so our sequence now are these black dots here this represents our sampled version or now we have a discrete time sequence so um if we plot x of t and x of n again here you can see them again but instead of plotting them versus time here or versus n we&#39;re going to plot them versus their phase remember one full cycle is 2 pi radians so in other words this is phase here so we have zero here and when we&#39;ve gone around one full cycle we&#39;re back to where we started that would be two pi radians all right so halfway through is pi radians and then a quarter way is pi over two one eighth way through is pi over four so this is just you know the same graph up here but we&#39;re plotting it versus the phase of the cycle here so the discrete time frequency omega uh of the little d under there for discrete uh it tells us how many radians per sample there are and we can see that each one of these samples is pi over 4 radians all right and so the the way to get from one to the other is we notice that radians per sample is equal to radians per second times seconds per sample right because these seconds cancel right here so radians per sample that is our discrete time frequency omega sub d radians per second that is our continuous time frequency omega and finally seconds per sample well that&#39;s just the sample period t sub s right that&#39;s how many uh that&#39;s how many seconds it took for each of these samples to take place so what we end up with is a way to sort of uh there&#39;s a couple things here first of all it tells us that discrete time frequency carries the units really of radians per sample even though technically it&#39;s just radians and also we can get the discrete time frequency by multiplying the continuous time frequency by the sample period that was used to create the sequence so here&#39;s an example in fact this is the last thing we&#39;ll do in this part of a lecture here we have a two megahertz continuous time signal that&#39;s the red line here and we&#39;re told that it&#39;s sampled at 10 million samples per second what is the frequency of the resulting discrete time sequence okay so um i put all the equations that you should need to solve this activity to do this activity right here so go ahead and see if you can figure out you know basically we&#39;re taking this 2 megahertz signal here the red line and we&#39;re turning into this sequence which are the black dots but we want to know what is the frequency of this sequence when we&#39;re all said and done so again go ahead and pause your video and see if you can do this activity all right so here&#39;s the solution to our activity um so again we&#39;ve got this 2 megahertz continuous time signal it&#39;s sampled at 10 million samples per second we want to figure out what is the frequency of the resulting discrete time sequence so um we know that our sampling frequency f sub s we&#39;re doing 10 million samples per second so that&#39;s 10 to the seventh samples per second so our sample period t sub s is one over the sampling frequency so one over ten to the seventh just gives us 10 to the minus seven and you can see that&#39;s 10 to the minus seven seconds per sample so in other words we have a this is a tenth of a microsecond per sample all right we can figure out what omega is well remember this is continuous time now omega is just two pi f so we have two pi radians per cycle times 2 times 10 to the 6 cycles per second where did i get this number well that was given to us right 2 megahertz so that&#39;s equal to 1.26 times 10 to the 7th radians per second finally we use our equation that were you you know we were told the discrete frequency omega sub d is equal to the continuous frequency omega times the sample period so i have 1.26 times 10 to the seventh radians per second times 10 to the minus seven seconds per sample we multiply these together actually do this in your head you can see these exponents cancel each other and we end up with 1.26 radians per sample so notice that this number here 1.26 is less than 2 pi right 2pi is around 6. so this original signal which had a high frequency of 2 million hertz when we sample it the sample frequency um and um in the uh the sequence frequency the discrete time frequency of the resulting sequence is only 1.26 radians per sample okay so that is it for uh part two of this lecture uh we will continue on and i&#39;ll see you in part three goodbye

Transcript for:Lecture on Real and Complex Exponential Signals

Transcript for:
Lecture on Real and Complex Exponential Signals