 The next task is for us to come back from the pass band to the base band. Consider sp of t multiplied by root 2 cos 2 pi fct. To obtain sp of t root 2 cos 2 pi fct we can just write the expansion of sp of t which we had from the previous slide as sp of t being sc of t cos 2 pi fct minus s s of t sin 2 pi fct of course, there are root 2's and that yields 2 times sc of t cos 2 pi fc square of t minus 2 times s s of t sin 2 pi fct cos 2 pi fct by performing trigonometric simplifications we get this result that is sp of t times root 2 cos 2 pi fct is actually sc of t plus sc of t times cos 4 pi fct minus s s of t sin 4 pi fct. What is important here is to notice that this component sc of t cos 4 pi fct and s s of t sin 4 pi fct are actually signals that are near the frequency 2 times fc while sc of t is a signal which is near base band because we began this with the assumption that sc of t was a real base band signal that occupied the frequency range between minus w and w. So, clearly from this particular result if we somehow eliminate the frequency components near 2 fc we will be able to recover the original sc of t. This is possible because in general 2 fc is a far away frequency and using an appropriately designed low pass filter can be used to cut these signals off and get only the component of signals that are close to dc which is what you want you want sc. In a very similar fashion multiplying sp of t by sin 2 pi fct instead of the cos will give you a similar expression except that sc ss of t will now come out and the other components related to sc and ss will once again be close to 2 fc. Therefore, this technique allows you to recover sc and ss from the pass band signal without any loss of information. To look at this up and down conversion we can now pictorially depict how we can construct sp of t from sc of t and ss of t that is multiply by root 2 cos 2 pi fct in the case of sc of t multiply by root minus root 2 sin 2 pi fct in the case of ss of t and add these two to get sp of t. Now to get back your ss and sc you then take sp of t multiply it by cos 2 pi fct root 2 times add a low pass filter that cuts off the 2 fc frequencies you get sc of t. In a similar manner you multiply by minus 2 minus root 2 sin 2 pi fct add a low pass filter that cuts off your 2 fc frequencies and you end up with ss of t. We will now take a detour and demonstrate how you can perform this up and down conversion for a pair of real base band signals using GNU radio. As you would recall the simplified structure shown here can be used directly in order to produce our up converted pass band signal and then down converted to obtain our original base band signals as well. We will now construct a very simple example that takes two base band real valued signals up converts them to pass band and then down converts them back to base band and obtains the two real signals once again. For our example we will produce two signals namely a sink and a sink square. Let us begin by first importing num pi so that we can use the num pi python functions conveniently. So control f or command f and type import we grab the import block double click it and we type import num pi. Next we create an array consisting of the time values at which the sink has to be evaluated. We press control f or command f and type variable. We grab the variable block put it here double click it the name will be t underscore vales and the value will be num pi a range from minus 512 to 512 upon 32. This will allow us to produce a sink that is 32 milliseconds long and has spacing between its successive zero crossings and you know the peak value and the successive zero crossings of exactly 1 millisecond. Let us add our common elements namely a vector source our time sink frequency sink control f or command f vector source and we have our vector source over here. We double click the vector source change the type to float and the vector that we want is num pi dot sink of t vales. We add our throttle command f or control f and throttle we double click the throttle to convert the type to float we then add a time sink so control f or command f and time sink double click convert it to float we add a grid we also auto scale and we add a frequency sink so control f or command f f req grab the frequency sink double click it convert it to float we want a rectangular window scroll down a bit we want a grid we want auto scale we say okay we then connect the vector source to the throttle the throttle to the sink as well as the frequency sink and we then run our flow graph so we get a nice sink and its spectrum this is familiar let us now construct our second signal that is sink square to do that we will just take the vector source that we already have hit control c to copy hit control v to paste if you are using a mac please use command c and command v we double click the vector source and we replace or we add a star star 2 that corresponds to a square of every element in the array so this will yield us sink square for the same t-vals let us now increase the number of signals in the time sink and frequency sink we double click the time sink and say two two inputs we double click the frequency sink and say two inputs connect the second input to the second second signal to the second inputs and executing this will yield a sink squared which arguably is narrower and it has a slightly wider spectrum because its spectrum can be obtained by convolving the rect with the rect because it will be triangular in the log domain it appears in this fashion great our next task is to upconvert this by multiplying the first signal by cos 2 pi fct and the second signal by sin 2 pi fct let us first add a range slider for fc so we press ctrl f for command f type range grab this range and double click it we will change the id to fc the type will be float let us keep the default value as 3000 and let us keep 3000 as the start value and let us say 8000 as the end value step one our next task will be to produce cos 2 pi fct and sin 2 pi fct let us do that a little bit below on the flow graph we press ctrl f or command f and say signal source we grab the signal source and place it here we double click the signal source the frequency is going to be fc and it is going to be float we can then create a copy of this by hitting ctrl c or command c and ctrl v or command v dragging it below and changing this to sin we now have two signal sources let us move this a little bit we now have our two signal sources that will produce a sin and cos we will multiply the first vector source with the cos and the second vector source with the sin we will temporarily ignore the root 2 factor but you can always add it right by right over here by changing the amplitude of the signal source to root 2 let us grab multiply blocks by hitting ctrl f or command f and typing multiply we drag this multiply block we double click it make it float we can copy this multiply block and paste it again the first signal gets multiplied by the cos the second signal gets multiplied by the sin negative sin so let us make a small change we change the amplitude of the sin to minus 1 and these are then added up ctrl f for command f and add we get the add block we double click this we change it to float we then connect these two and we then observe the spectrum of the resulting signal by adding a separate frequency sync so ctrl f for command f and f req grab a qt gi frequency sync double click change it to float change it to a rectangular window we'll add a grid we'll add auto scale we'll say okay connect the out to the in run the flow graph you can see that the combination of the two spectra appear at fc which we have chosen as 3000 hertz let us change it to 4000 hertz you will see that the spectra have translated to be around 4000 hertz but the values are still between 4.5 and minus 4.5 for the blue curve which is faithful because the original baseband signal was between minus half kilo hertz and half kilo hertz the second one is a little wider because of the convolution but it will largely be between minus one kilo hertz and one kilo hertz which is what you see roughly over here it is between three kilo hertz and five kilo hertz this spectrum is a magnitude spectrum and you can clearly see that it has similar characteristic on the positive and negative frequencies and it is the spectrum of a real signal whose baseband equivalent is a complex signal that embeds these two signals namely the sync and the sync square let us now work on getting back our original signal from this complex baseband rather this passband signal that contains the complex baseband signals we close this what we will now do is to multiply this particular signal by cos multiply it by minus sign and again grabbing the resulting signals and putting them in a sync however before that we must remember that we need the low pass filter to cancel the components along 2 fc let us now build this particular system so let us actually you we can use the same signal source so we can just take a multiply block i am going to hit ctrl c and ctrl v to copy this multiply block take this signal add it to the input one take the cosine add it over here of course you could use separate signal sources as well but here to reduce the number of blocks i am keeping it simple and you reusing the same signal source we then take the second this multiply block and create a second multiply block by hitting ctrl c and ctrl v or command c command v we connect the same output signal to the input and connect the minus sign over here we then have to add two low pass filters that cut off the frequencies around 2 fc to do this we use the inbuilt low pass filter in GNU radio we hit ctrl f or command f and say low and we take the low pass filter place it over here we double click the low pass filter and say it is float to float interpolating it does not matter because we are just setting the interpolation to 1 we will now set the cut off frequency to fc it is safe to set it to fc because any frequencies above fc are cut off and our original signal is actually just at baseband and it is now it is actually the components that we obtain near 2 fc that we want to remove so we press ok we then copy this and paste it again to get a second filter let us make some space for these filters we connect the first output to one filter second output to another filter and we will grab another QT GUI frequency sync we already have this one over here so we can say ctrl c and ctrl v to get a two input GUI frequency sync connect these outputs over here and we need to set the transition width apologies we will set the transition width to a thousand hertz in both the filters let us now run this flow graph if you now observe the QT GUI frequency sync both the syncs give you roughly the same type of signal of course there are some artifacts over here because of the filtering but this is well below about minus 90 dB indicating that it is very very minimal and you largely get the characteristic that you wanted in order to ensure that this is indeed the case let us add a time sync as well we will just copy this time sync place it over here connect this output here connect this output over here in fact we will temporarily remove this time sync or we will disable it by right clicking and saying disable right clicking and saying disable let us run the flow graph now you can see that the only time sync that we have is this one and that displays the signal and the other spectra are as we expect let us actually view all the four signals at once let us stay remove this QT GUI time sync and let us just add four signals to this QT GUI time sync which are the third being this one and the fourth being this one therefore we would expect the signal one and three to be similar and two and four to be sorry zero and two to be similar one and three to be similar that is the first and the third and second and the fourth should be similar if we now look at what we have over here the signal one and the signal three do look similar like a sync the signal two and signal four two look similar like a sync square if you now compare the amplitude it's one over here and about point five over here the reason for this is the root two that we chose to ignore let us address that by changing the amplitude of our cosine and sine to root two or maybe one point four one four the approximate root two minus one point four one four and run our flow graph you can see that the signal one and signal three are very similar signal two and signal four are very similar meaning that barring the delay that is produced by the low pass filter you are able to recover the signals with reasonable accuracy and that confirms that this approach of converting the baseband signals to passband and converting them back to baseband is able to preserve all the characteristics of your signal the band limited baseband signals and you are not wasting any spectral redundancy because you are sending two real signals that occupy the same bandwidth within fc plus w and fc minus w so to summarize for complex baseband signals in particular for digital communications we will henceforth directly design complex signals because there is no real constraint for us to use pairs of real signals it is very natural for us to just design and construct complex baseband signals in the case of digital communication so we will be moving to designing complex baseband signals that convey information effectively while taking into account constraints such as bandwidth power usage and so on and thus to emphasize we will be considering only complex baseband signals as a summary real signals there is a redundancy in the spectrum this as you have seen is a natural consequence of the fact that sc of f for example is the same as sc star of minus f that is there is a conjugate even symmetry in the spectrum the complex baseband signals utilize the full bandwidth effectively because they break this redundancy by allowing you to transmit two real signals concurrently over the same bandwidth by then up converting this complex baseband signal we then obtain real passband signals that capture all this information and are ready for transmission during reception these very real passband signals can then be used to recover the complex baseband signals thank you