In this video, we'll discuss about the capacity of single input multiple output, and multiple input single output, AWN channel. Followed by the capacity of fixed single input single output wireless channel. Then we'll calculate the capacity of fixed single input multiple output wireless channel. To start, let us consider a wired channel with single transmit antenna and multiple receive antenna. Here we are considering two receive antennas. The received signal will be given by y_1 is equal to x + n_1. Remember, here we are considering wired channel that's why we don't have any h here. So y_1 is equal to x + n_1, y_2 is equal to x + n_2. The noise is i.i.d, it is independent and identically distributed. Therefore, given by n_1, n_2, complex normal 0, variance n_0. After receive combining, in the case of wired channel simple receive combining where we're just adding up both the signals performs better. We have a combiner, which is just the sum of the received signal of both the antennas, which we get as 2x + n_1 + n_2. Now, if we calculate the signal power, what we're going to get? 4 the power and 2p, only what will be the noise power, which is equal to [inaudible] N_0 + N_0. This we can calculate because we know n_1 and n_2 are independent. Therefore, we will get expectation of this as 0, and we're just getting N_0 + N_0 as 2N_0. Therefore, finally, we have the capacity of the channel given by C equal to log base 2, 1 + SNR. SNR means signal power, which is 4P divided by noise power, which is 2N_0. After substituting these values, what we're getting? Capacity is log base 2, 1 + 2P upon N_0. This is the capacity of a SIMO channel. When compared to SISO channel what do we get? These two factors increase, otherwise in case of SISO channel we have to get log of 1 + P N_0, remember. In case of SIMO with 2 antenna, we're getting 2P by N_0. Now, let us consider a capacity of AWGN MISO channel. First case was AWGN SIMO channel, now we are considering MISO channel. Where we are assuming we are transmitting signal x upon root 2 from first antenna and x upon root 2 from second antenna. Remember why we are doing this, to ensure that the overall transmit power is P. If we don't multiply it by 1 by root 2 and 1 by root 2, the overall power will be P + P, which will become 2P, and that is not the allowed. The power constant should be maintained, and the overall transmit power from the transmitters should be capital P only, irrespective of how many antennas we are transmitting. In this case, like I said, transmit signal from each antenna is equal to x upon root 2. Therefore, the total power efficiency, if we calculate is coming out to be P only. Now, the received signal is given by the sum of both x upon root 2 + x upon root 2 + noise, which has been added at the receive antenna here, which is one receive antenna + n. Therefore, after substituting this value, simplifying this, what we are getting y is equal to root 2 of x + n. Therefore, the SNR is given by 2 upon 2 into P root 2 square into P upon N_0, which is equal to P N_0. Once again, we can see the SNR is coming out to be same, and the capacity C is also same, which is given by log base 2, 1 + 2P by N_0. Therefore, the capacity of SIMO and MISO system is same. Now, let us move towards one interesting, the capacity of fixed wireless channel. Till now we have discussed the capacity of wired channel. Wired single input, multiple output, wired multiple input, single output. Now, we are talking about capacity of wireless SISO channel. Remember I've used one new terminology, fixed. When I'm saying fixed means okay, I'm assuming the channel is known as the transmitter. Transmitter as well as receiver, the channel is known. We'll discuss about other cases, that's why I'm starting with a fixed wireless channel. In that case, we'll have a channel in between the transmitter and receiver, given by h, therefore, the received signal will be given by y, which is equal to h of x + n. Here we assume h to be fixed complex channel gain. When I'm saying fixed, I'm assuming h is known as the receiver. Therefore, at the receiver the SNR, if I calculate what I'll get, SNR or the instantaneous SNR I'm seeing. If I'm assuming h to be known, I'm saying fixed. In that case, P into h squared upon N_0. If I calculate the capacity with channel h known, what I'm going to get, substitute the value of SNR log base 2, 1 + SNR, then SNR is this quantity. We can see and compared to earlier wired channel, we're getting an additional term which is coming from this channel. Otherwise, this used to be unity for the wired channel and we used to get log base 2, 1 + P upon N_0. The unit, if you remember, this was nothing but bits per second per Hertz, because we're calculating the capacity without bandwidth. There's no bandwidth sitting here, that's why it is bits per second per Hertz. Now, we are discussing the capacity of fixed wireless SIMO channel. Now, in case of SIMO channel, what happens? We have multiple output antennas or receive antennas and single transmit antennas. We have two channels, h_1 and h_2. Both of which the received signal at the first antenna will be at h_1x + n_1, received signal at the second antenna will be h_2x + n_2. Now, I'm assuming the channel h_1 and h_2 are known at the receiver, that's why we have taken this fixed. Now, we know in the case of SIMO, we need to use some kind of combining. We know the optimal combining is nothing but the MRC or we also call it matched filtering. [Unclear] combining matched filtering, and there are many names. After using this receive combining as MRC, what we're going to get is the received signal r after doing a MRC is given by h_1 conjugate upon norm h, y_1 + h_2 conjugate upon norm h, y_2, where h is nothing but h_1, h_2. We know with MRC, the idea is to give more importance to receive antenna with better channel gain. Now, if further simplifying the expression of what we have this y_1 and y_2 from this expression, what we're going to get is I'm just simplifying this expression one by one. It is nothing, just equation of the formula that we obtained in the previous slide. What we are going to finally get is, this is my signal term, this is my noise term. Based on this, I can write the SNR expression. But for that I need to know how to calculate the noise power. For that, I've just given a simple simplification here, when we have the expectation of the naught square of h_1 conjugate n_1, we can simplify it as follows, h_1 conjugate h_1 into n_1 conjugate h_1, just doing the conjugate x into x conjugate. This is X, let's say. This is x, this is x conjugate. After simplifying I know I'm assuming h to be fixed. If h is fixed, I can just take it outside the expectation, because it is a constant for me, it is fixed. This is taken outside from left, this is taken constant outside from right side. What we are left with, h_1 conjugate expectation of n_1, n_1 conjugate. We know what is expected from n_1, n_1 conjugate, N_0. Therefore, finally, we are getting all our scalar mod h_1 square into N_0. This is the expression for this. Therefore, using this, we can simplify the noise power term and this signal power term and we're going to get this. We can try out after cancelling the terms in the numerator and denominator, what we're going to get is simple norm of x squared into P by N_0. Therefore, the capacity C is now given as log base 2, 1 + norm h square d into P over N_0. Now see what exactly is happening here. Now, my capacity is coming out to be 1 + norm h square into P by N_0, where x is my, in this case, I'm assuming it to be fixed. The scenario will increase as we are increasing the number of antennas, because the length of the vector will increase, as the length of the vector will increase its norm will also increase. Now, just to show, let us take an example. If h_1 is equal to - 0.2, h_2 is 1, what is the SNR we are obtaining? In this case, we are obtaining an SNR of 1.04P upon N_0. In this video, we discussed the capacity of single input, multiple output and multiple input, single output AWGN channel, and we saw that the capacity of both are always same. We discussed an important point that the total transmit power constraint should always be maintained. Finally, we discussed about the capacity of multi antenna system, which always comes out to be greater than the capacity of single antenna system.
