 Hello, welcome to this lecture on digital communication using GNU radio. In this lecture, we are going to take a look at the complex baseband representation of signals. In terms of complex baseband equivalent, we will be defining what are baseband and passband signals. We will then distinguish between real and complex baseband signals. We will then see what the complex baseband equivalent signal is, its significance and how to construct it. And finally, up conversion of the complex signal to passband and how you can recover the complex signal from the real passband signal by down conversion. So, what are baseband and passband signals? Roughly speaking, baseband signals are those that occupy frequencies near DC, that is, they generally contain lower frequency components. Baseband signals, on the other hand, occupy a narrow band of frequencies closer to the so-called carrier frequency. The carrier frequency is generally much greater than the bandwidth of the actual signal. As an example, 2.4 GHz may be the carrier frequency that may be carrying data signals of 10 MHz bandwidth. In this situation, the 10 MHz signal is the baseband signal and the carrier frequency is 2.4 GHz. On what basis are these values chosen? So, this is dependent on various parameters and of course, how the standard is designed. So, there are propagation characteristics, licenses available for transmission and several other factors that determine the passband signals characteristics both in terms of the carrier frequency as well as the bandwidth. Visually, if we inspect the spectrum, the baseband and passband signals can be very distinctly characterized. In this example, let us say that S1 is a baseband signal and its spectrum or Fourier transform is S1 of f. S1 of f is available over here and as you can see, it occupies a frequency between minus w and w generally closer to the lower frequency lower frequency range. If you now consider the signal S2 of t that is constructed as S1 of t multiplied by cos 2 pi fct where fc is chosen to be larger generally much larger than w, we obtain the passband signal S2 of f. In this particular situation, if you remember the Fourier transform of cos, it has two impulses one at fc and one at negative fc. Therefore, the convolution of the spectrum of S1 will give two copies one around fc, one around minus fc which is why this passband signal S2 occupies the frequencies between fc minus w and fc plus w and the corresponding negative frequencies. One issue however is in this particular picture is that if S1 of t is a real signal, you will recall that its spectrum has the conjugate even property that is S1 of f is the same as S1 of minus f conjugate. This means that the information contained in the positive frequencies of S1 allows you to directly infer the information contained in the negative frequencies of S1 which means that keeping only one of them is enough. Of course, you may argue that for a real signal the conjugate even property has to be satisfied. However, at passband you can see that there are two copies and having only one of these and the corresponding negative is sufficient for you to obtain a real signal that has all the information needed to reconstruct S1. In other words, this particular approach of multiplying the real baseband signal by a cosine is inefficient because the information is duplicated in the passband. So, for efficient usage of the spectrum how do we remove this redundancy and how do we actually use the bandwidth between or the frequency range between fc minus w and fc plus w to maximize the information transmission. This is what motivates the so called complex baseband representation of signals. The key idea here is to take two real valued baseband signals for reasons that will be clear shortly. We will call these signals sc of t and ss of t remember both are real baseband signals that occupy the same frequency range of minus w to w and we call that a bandwidth of w. As these signals are real both of these signals have conjugate symmetric Fourier transforms that is sc of f is equal to sc conjugate of minus f ss of f is the same as ss conjugate of minus f. We now construct the complex baseband signal s of t as sc of t plus jss of t. In general unless there are some specific properties that sc and ss may have this is a complex signal and it does not have a conjugate symmetric Fourier transform. In other words we will shortly see that this particular construction eliminates the so called redundancy in the frequency domain and allows you to transmit two real signals two real valued signals within the same bandwidth w if you permit the use of complex signals. So, to convince ourselves that s of t is equal to sc of t plus jss of t is not generally real is very simple. We know that the real nature of sc and ss means that sc conjugate of f is the same as sc of minus f ss conjugate of f is the same as ss of minus f. Then by the linearity of the Fourier transform we can directly write s of f as sc of f plus jss of f. Then let us see if s of f satisfies the so called conjugate even property. If you take the conjugate of s of f we get sc of f plus jss of f the whole conjugate expanding this puts the conjugate on the j. So, you get a minus j over here you get sc conjugate of f minus jss of f using the properties of our sc and ss defined above we get sc of minus f minus jss of minus f. If s of f were corresponding to a real signal then the conjugate even property would have meant that s of f conjugate would have been sc of f plus jss of minus f sc of minus f plus jss of minus f which is not the case. Therefore, this so called complex base band construction or representation allows you to very easily just design two real signals and combine them by just using the complex addition operation that is you take the first signal plus j times the second signal and you end up with two real signals that occupy the same bandwidth minus w to w as a complex signal. As a visual queue if we have a sc and ss we are depicting the spectrum being similar as having a hatch over here and a hatch over here this particular information over here and this particular information over here they are the same information except that they have the conjugate even property and similar properties hold over here but in case you construct the so called complex base band equivalent signal you end up with a resulting signal sc of f plus jss of f the spectrum of that signal that is that need not have the property that the real part and the imaginary part are just mirror images of each other that is we occupy the same bandwidth minus w to w and now have a complex signal in the form of two real signals that occupy the same bandwidth range but this is nice but we need to be able to transmit this signal as a real signal because whenever you have any transmission medium you have to be able to send a real signal the question arises as to how we can make this into a real pass band signal that can be transmitted over any media to do this we have the base band to pass band transformation which is very simple we will now start dealing with s of t directly where s of t is now a complex signal that occupies the frequency range between minus w and w in other words it occupies a bandwidth of w but it is a complex signal thus it consists of two real signals we are now going to define sp of t as real part of root 2 times s of t e power j 2 pi fct for a minute we can ignore the root 2 which is just a scaling factor but what we are doing is we are multiplying or modulating s of t with the complex exponential e power j 2 pi fct if you remember your Fourier transform properties multiplication with e power j 2 pi fct convolves the spectrum of s of t and places it around fc therefore this signal s of t e power j 2 pi fct is going to occupy the frequency range between fc minus w to fc plus w the real operation just places a copy at minus fc we will soon see that sp of t is a real pass band signal that has information of sc and ss together the bandwidth footprint as we mentioned is between fc minus w and fc plus w of course real signal so there is a corresponding conjugate at minus fc minus fc plus w minus fc minus w pictorially what happens is that this spectrum baseband spectrum which is unsymmetric and corresponds to a complex signal is brought to around fc and a copy of the same with the conjugate operation is also brought to minus fc therefore the conjugate symmetry property is definitely satisfied but you can clearly make out that all the information about both the positive frequency part and the negative frequency part of the baseband signal are present very much and there should be a way to recover them of course the way we motivated this was by taking a complex signal and then taking real part v power j 2 pi fct and so on but one question that arises naturally is that can we get sp of t directly from sc and ss sc and ss are just a proxy for s of t because s of t essentially has sc as its real part and ss at its imaginary part so can we construct sp of t directly from sc and ss to do this we consider sp of t as the real part of root 2 s e power j 2 pi fct and we can expand the real expand this by writing e power j 2 pi fct as cos 2 pi fct plus j sin 2 pi fct so let us now perform this operation if you now expand e power j 2 pi fct as cos 2 pi fct plus sin j sin 2 pi fct you will find that the real part that remains after this expansion is sc of t cos 2 pi fct minus ss of t sin 2 pi fct therefore the way to construct your pass band signal sp of t is to modulate sc of t with a cosine at fc and to modulate the ss of t with a sign at the same frequency and add or subtract them therefore by performing this operation you are able to construct sp of t directly from sc ss of t and sc of t the c and s subscripts should now become somewhat clear because sc of t rides on cos 2 pi fct in the sense of being modulated by cos 2 pi fct and ss of t rides on 2 pi sin 2 pi fct which is why the subscripts make sense sc is for cos ss is for sin conventionally because of the fact that sc of t is multiplied by cos 2 pi fct we refer to sc of t as the i component or the in phase component and ss of t as the q component or the quadrature component this should be consistent with the nomenclature that is used in the context of circuits and power systems phases and so on. To understand this pass band transition better we will take a small detour that considers combining these signals from the base band to obtain pass band signals. We will now take a detour to make a very simple experiment in glue radio wherein we will construct a complex base band signal using very simple cosine and sine we will take the base band signals cos 2 pi f0t and sine 2 pi f0t implicitly assuming that f0 is small and therefore close to dc thus these can be treated as base band signals since both of these are real signals they have conjugate symmetric Fourier transforms and we are going to use the conventional approach of taking the first signal plus j times the second signal and observing the spectral characteristic and showing that the symmetry is no longer present thereby indicating that you have a base band signal that has two distinct real signals. Let us begin by first adding a signal source we will use the conventional approach control f or command f type signal grab a signal source place it in our flow graph we want a real signal therefore we will double click this and change the type to float since we also want a sign we can pull in another signal source or we can just select the signal source by clicking on it and hitting control c or command c and control v or command v to produce a copy we then double click the signal source and change this to sign and say okay our next course of action is to construct the first signal plus j times the second signal grew radio offers a convenient approach to do this we will pull in the float to complex block control f or command f type float and we have the float to complex next we connect the signal source one to the real part the second one to the imaginary part and we are ready with our complex base band signal but we wish to view the spectra of this signal along with the original signals so we would need a complex qt gi frequency sync we will press control f or command f f req get the qt gi frequency sync let's not forget the throttle control f or command f throttle we can connect our signal to the throttle and we want to visualize three signals in the frequency sync so we will double click it we will say grid yes auto scale yes and we will say three inputs and say okay we will connect the throttle to the third input now to view the signal source on the in the complex gi frequency sync we need to convert this signal source again to complex for that we will again use float to complex but we will keep only the real part an easy way is to just select this float to complex hit control c or command c to copy and control v or command v to paste control v to paste again so that we get two of them connect the output to the real part connect this output to the real part and then we will play a trick to make sure that the signal which comes out has only a real part we will create a constant source that emits zero so control f or command f c o n s d const we will get a constant source the constant source that always outputs the real number zero so double click this change it to float say okay and we connect the output of this complex sorry constant source to the complex by imaginary part over here and the imaginary part over here we can then connect the first source to the input zero second source to the input one and our flow graph is ready let us execute this flow graph now what is the interpretation let us inspect only the first signal by disabling the second and third ones this is our cosine as expected you see two peaks one at one kilohertz and the other at minus one kilohertz if we then see the sign it's essentially overlapping with the cosine because both the sign and cosine have similar magnitude spectra but they differ only in phase in fact to get a finer line we can middle click hit the control panel and change this to a rectangular window in which case you will see the lines distinctly but when you now bring the third signal the third signal only has a peak at exactly one kilohertz and no peaks elsewhere why is this the case this is because cos 2 pi f0 t plus j sin 2 pi f0 t is actually e power j 2 pi f0 t which has a Fourier transform of delta of f minus f0 therefore as expected it has a single distinct peak at one kilohertz but from the perspective of baseband signals here is a baseband signal e power j 2 pi f0 t that embeds cos 2 pi f0 t and sin 2 pi f0 t which are two real signals and does not have a complex conjugate which is equal thereby having a complex baseband signal so this is an example of a very simple complex baseband signal as a final step let us also convert this to a passband signal by performing the operation real part of s of t e power j 2 pi fct to do this let us take a complex signal source say control f for command f signal source this signal source is going to be e power j 2 pi fct so let us choose the carrier frequency as say 6000 hertz we will then multiply this s of t that we obtained by combining the cos and sin with this so control f for command f we get the multiply block we multiply the signal multiply the other signal and all we need to do is to take the real part of this and look at the spectrum so let us now take the real part so we say control f for command f and type real we get the complex to real block connect the output to here and we get a real qt jui frequency sync control f for command f wrap the frequency sync over here we double click it and change it to float connect it over here hit run now as you can see the complex baseband signal was e power j omega 0 t when you now modulate it with a carrier at 6 kilohertz you will get a copy at 7 kilohertz and minus 7 kilohertz the reason is because 6 kilohertz is the center frequency and your signal appears 1 kilohertz to the right and over here 1 kilohertz to the left of minus 6 kilohertz this is a very simple example of obtaining a passband signal from a complex baseband signal we will soon expand this to more sophisticated signals in the next detour