Original subtitles

So how are the matched filter, the zero forcing receiver and the minimum mean squared error receiver related?

And we're going to look at this digital communications example where x is a vector,

h is a matrix. And this applies for example to an uplink CDMA scenario where the

elements of the x vector are the symbols from the different users.

It also applies to a MIMO scenario where you have a single user but there's multiple antennas.

And this is a standard signal model. For more information on this model and the scalar version where we look at the

channel inversion and the matched filter in that case, there are two other videos on this channel

and you'll find the information in the link below to find the links to those videos.

Okay, so let's now look at what we're going to be doing now. So now we've got our estimate x equals and I'm going to use

the letter w for the filter that we're applying. So over here in the scalar case,

we just had a scalar. We were multiplying either by h to the minus one in the channel inverse case or h

complete conjugate in the matched filter case. Now we have a vector which we're receiving.

So now we're in the vector space as these examples I've discussed down here. And so you've got either the possibility of

multi-user interference, which was in the CDMA case. If the codes were not orthogonal, you would have multi-user

interference. And in the MIMO case, you'd have multi-stream interference. So now we've got not just noise,

we've got interference when we look at the vector case.

Okay, so what are the choices of w that we can make?

Let's now look, I'll put a box around this. This is the filter that we're going to be applying.

We're going to look at different values for this filter. Okay, so let's first of all look at zero forcing.

Okay, so zero forcing is analogous to the channel inverse over here. Okay, so in zero forcing we choose

the, we call the x hat zero forcing equals, we choose w to be

exactly analogous to this. We do take the inverse of the matrix h. Okay, and in this case, as we had in the

scalar case, we get x plus h inverse times n. And exactly the same that we had before in the scalar case,

we've got a problem now of potentially, it's good news that we've got x back to being, we've separated that if it's the

the two data streams in the MIMO case or the two users in the CDMA case, we have now separated those

streams out separately, which is fantastic. We've got them without interference from each other,

because we've zipped and that's what zero forcing means. It's forcing the interference to zero.

But we've got the possibility of noise enhancement. And we've got a real problem if the channel

matrix h is not full rank. So if you can't invert the channel matrix h, then you've got

real problems with the zero forcing approach. Okay, what about the matched filter approach then?

So in match filtering, the approach is so x hat match filter. So in this case, I'll just include

the scaling that we had for this, just so we got the scaling right on here.

But the filter is h complex conjugate. So we use the dagger for complex conjugate of a matrix.

Okay, so in this case, we've got this times hx plus n. Now I'm just going to expand this out

to get a little more insight into this. So in this case, we've got the square root of nt on p.

I'm going to do the expansion that I did down here with the h1 and h2. So I'm going to write

h dagger as h in terms of its components. So now we've got h1 dagger on top of h2 dagger.

That's what this this matrix here is in terms of its components. And this matrix is, of course,

h1 is the first column and h2 is the second column times x plus we've got the h dagger times

n. I'm just going to leave that on its own because that's just the noise. We're not seeing

anything about the interference here. So I want to look at the interference here. So let's look at

this matrix here. So what is this matrix? And don't forget x is it's got its components x1 and x2,

that's a vector. Okay, so I'm just going to take this component here and write out that component

as it's going to be a vector, of course, because this is this is going to be a matrix

times a vector. So we're going to end up with a vector. So the first element of the vector

is going to be h1 dagger times h1 times the first element of x plus h1 dagger times h2

times the second element of x. So that's the first element of the overall vector.

And then the second element is h2 dagger times h1 times the first element of x plus h2 dagger

h2 times the second element of x. So that's a vector that we're getting by multiplying by

the match filter. And let's just observe this vector here for a moment. Let's say for example,

let's say just take the case where the two different channels are orthogonal. So if it was

the CDMA case, and h1 was orthogonal to h2, as we saw before down here, then this term here would be zero,

and this term here would be zero. And what you would get is you would get h1 dagger h1 x1 as the

first element, and h2 dagger h2 as the second element, and there would be no interference.

And that would be exactly what we had over here in the scalar case. So if there is no

interference at all, then a match filter is a good thing to do. It maximizes the signal to

noise ratio. And if there is no interference, then that's a good thing to do. But you can see now,

if there is interference, if these codes were not orthogonal, or in the MIMO case where they are

given to you by the physics of the environment, so they're unlikely to be orthogonal,

then you will have these terms in the matched filter equation. And these terms can be significant.

So even though you are maximizing the signal to noise ratio for the signal of interest, you are

not taking into account the interference, and the interference could cause major problems. And

you can see that it's that's going to be the case here. So if h1 dagger times h2 is not close

to zero, then you won't be able to separate out the two data streams and recover the original

signals. So the matched filter would not be a good thing to do if you have a lot of

interference coming from these two not being orthogonal in your channel. Okay, so just to

recap, so zero forcing is a good thing to do. But it doesn't do much good for your signal

to noise ratio if your channel has an inversion problem. So in this case, you get a good easy way

to detect, but and it's great in terms of you've zeroed out all of the interference, but you've

got a signal to noise ratio problem. In the second case, you've got a good signal to noise

ratio, but if you've got interference, that's going to cause you a problem. So these are two

extremes, you can view these as two extremes of the scenario and how do you go about picking

between the two. And that's where minimum mean squared error comes in. So in minimum mean squared

error decoding, then he will write this one down here. So in this case, w, the minimum mean squared

error answer equals the argument of the when you minimize over w of the mean, that's the mean,

so this is the minimum, this is the mean of the square of the error. So and this is x minus w y

squared. So this is the minimum mean squared error. So you're minimizing over w, you're picking a w,

so you're picking the filter that you're going to apply, which minimizes the mean of the squared

error term between x and w y. So this one, I'm going to write out the solution to this,

not going to derive it all here, but this equals the square root of nt divided by p of h dagger,

or h h dagger plus nt on s nr times i nr to the minus one. And this is where s nr equals p divided

by n. And so this is the equation here that we have where the s nr equals p on n naught. Also you

can use the what's called the matrix inversion lemma. And you can see that it's also equal to nt

divided by p times h dagger h. There's two different versions of it. I'm just going to write it out

here for because both of them are interesting to give insights to the minus one times h dagger.

Okay, so these are two different versions, two different ways of writing the minimum mean squared

error filter. And now let's finally just look at the relationship between the match between this

one and the match filter and the zero forcing. And we can see that this filter gives us a trade

off between those two filters. So let's consider the two extremes cases. So first of all, just for

example, let's consider the case when the s nr is really big. So in the case when the s nr is really

big, then this let's look at this first term here. So in the case when s nr is really big,

this term here essentially goes to zero disappears if s nr goes to infinity, this goes to zero.

And then apart from the scaling term, let's write down what we left with. So in that case,

we are left with so s nr goes really big, we are left with h dagger times in the inverse here,

we have h h dagger in the inverse. And this equals you can put the inverse in here. So you've got

h dagger, and then you've got h dagger inverse times h inverse. And this equals, of course, one. So you

just left with or unity the identity matrix. So you left with h inverse. So in this case,

you have the zero forcing solution. Okay, so if you if you have the sort of an infinite

amount of s nr, then you don't need to worry about the noise enhancement. And you can have

then you can do the zero forcing filter and the Mac and the MMSC filter actually becomes

that filter. So as s nr goes very big, you can see that the MMSC filter approaches the zero

forcing filter. So that's that's a very important observation about how this minimum mean squared

error filter relates to the zero forcing filter. Okay, and what about the another case, the opposite

sort of case, let's say low interference. So in the low interference case, in that case, we've got,

let's look at the second term here, the second way of looking at it in that case.

Well, when there's low interference, then we're going to be close to this situation that we had

up here in the CDMA case where they were actually zero interference. And in the matrix case,

that means that h dagger h is going to be approaching a diagonal matrix. And if h dagger h approaches

a diagonal matrix, then that matrix here approaching a diagonal matrix, well, this is a

diagonal matrix. So this entire term here will be a diagonal matrix, which really means you're

just essentially scaling the elements of what you get from doing h dagger times the received.

So in this case, this just becomes a scalar, a scaling factor. And the only rotation that's

happening is from h dagger. So in this case, you are you are giving the the solution comes to be

h dagger. So that implies h dagger. Okay, so in this case, when the s nr goes high from MMSC,

we've seen that it approaches the zero forcing. And when the interference goes low approaches zero,

then the MMSC filter approaches the matched filter. Okay, because h dagger was the matched filter times

a scalar or a scaling factor. But it's just a scaling factor, it doesn't rotate things around

because it's diagonal. Okay, so hopefully this has given you a lot of insight into the relationship

between the MMSC filter, the zero forcing filter and the matched filter, and how the MMSC gives you

a trade off between the two. And it means you don't have the noise enhancement, so much problem of

zero forcing, or the the problem for of interference when you have simply a matched filter.

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