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now there's another way to remove the image instead of filtering and what we do is we notice that if the desired signal is down converted to Omega if' its image is converted to minus Omega I F what we notice here is this signal is down converted here and the image is converted here but as we saw before the problem is that there is an image that's converted and dumps on top of our signal so we'd like to somehow remove this image if we could and last time we talked about doing that with filtering well what if we took an idea similar to the Hartley single sideband remember that where we used a 90 degree phase shift and a sine and cosine down conversion in order to basically generate spectrum that has one part as negative one parts positive and if you sum those up they would cancel whereas the other parts would superpose and give you twice the signal and this is exactly how an image rejection works we look at how the signal images are down converted and what we do is we note that if we multiply the signals by cosine so what I have on here is two signals I have the signal we want and I have its image in red and we multiply it by cosine Omega T and this is what we get on the output likewise I can take the signal its image multiply it by sine Omega T and this is what I get on the output and what we notice here is that the signals that are down converted by a cosine have the same sign but when we mix with the sine the signals have opposite signs and now what we do is the same trick we did in the Hartley we're going to add a 90 degree phase shift and what that does is it creates the two original signals to have the same size but the two images to have opposite sides and the reason this is is because the image that comes down on top of our signal is coming from the negative frequency so remember if we're down converting and here is our desired signal it's the image up here that's getting down converted so it's at the negative frequencies and because of that we're able to use this 90 degree shift and the sine to give us a negative answer and you can see right away that these two are just going to cancel one another and we're going to end up with a superposition of the two signals and so that's just like the Hartley did for single sideband but in that case we were trying to eliminate a sideband here we're just trying to eliminate an image and so the complete image reject receiver looks almost just like a Hartley single sideband but reversed where we've got an L oh creating a sine and a cosine we've got a 90 degree phase shift and you've noticed we've removed our image reject filter and we can choose a very low iif to use that low absolute bandwidth of the signal so it seems like if this works perfectly we solved all of our problems but there's a few problems namely that the amplitude of the sine and the phase of the sine generally is going to have some air and even if we made a 90 degree phase shift perfectly for one frequency typically we can't make a perfect 90 degree phase shift for a range of frequencies so they're always going to be some error here so in practice the two signals will have a different amplitude and phase error the two loz the phase shifts not perfectly 90 and the two mixers filters and the combiner may also have mismatches here so we've got a highly complex system that is going to have errors in several different places so if we show all of the amplitude and phase mismatches with the parameters epsilon and delta theta respectively we can actually write the image rejection ratio as the following and this is just a derivation if you went through the math that you could use our previous equation for the IRR before and if we assume that the amplitude mismatch and the phase mismatch are much smaller we can simplify this into a nice handy equation where we can see that the ratio gets reduced by an increase in either of these parameters squared so let's just look an example if an image reject receiver is a 10% amplitude error and a 10 degrees phase error if there's a signal at the image frequency with 10 DB lower power than the desired channel how much will the signal to interference ratio be at if' sounds like a lot of words but we have all the pieces we need so we can use our trusty equation and I'm just going to note here that this is in radians right so this was in degrees and now we're converting it into radians and so what we find with this error is that we end up with a rejection ratio of 20 DB because we've got our 10% error and our 10 degrees of facia