 ## 5G Physical Layer Intro

5G (fifth generation) mobile wireless communications standards are published. From the standards companies can start developing applications that adhere to the 5G standard. We will see emerging devices saying they are 5G capable. But what does that mean? Is it faster? More reliable? Do I care? Is it just some business rebranding to generate more money? To understand the answers to these questions we need to first understand the underlying technology for 5G Physical Layer wireless data transfer.

## 5G Physical Layer – Wireless Data Transfer

Orthogonal Frequency Division Multiplexing (OFDM) is a Multiple-Carrier Modulation Scheme used in the 5G physical layer that allocates symbols in the frequency domain. So to start we first define some notation for the frequency domain signals. If a vector is in the frequency domain we will use a tilde [\tilde{(.)}] to denote it. So the vector whatever $latex \tilde{\mathbf{x}}$ is a frequency domain vector. The length of the frequency domain vector is N; the number of subcarriers, hence a Multiple-Carrier Modulation Scheme.

## OFDM Example

Just like with a single-carrier modulation scheme we map the bits we are trying to send to the receiver to symbols. You can pick your favorite constellation. Once we have the symbols we can assign N of the symbols to the N available subcarriers. In the 5G physical layer N can be 2048 or even larger. For our example, we have N=4 and we use BPSK to assign subcarriers then for bits  we have:

\begin{equation}
\label{eq:subcar}
\tilde{\mathbf{x}} = [-1~1~-1~1]^T.
\end{equation}
\noindent
The T is the transpose operator, we define all vectors as column-vectors. Once our [\tilde{\mathbf{x}}] is defined we then can go to the time domain:

\begin{eqnarray}
\mathbf{x} &=& \operatorname{IFFT}\left[\tilde{\mathbf{x}}\right]\
&=& [0~0~-2~0]^T.
\label{eq:ofdmtime}
\end{eqnarray}

x is now in the time domain and is referred to as an OFDM symbol. We now can generate OFDM symbols for all the bits we want to send but if we stack them back to back we will still have ISI induced on the receive signal, but now the symbol interference is referring to the OFDM symbol. To distinguish this difference OFDM symbol interference is referred to as IBI. Where an OFDM symbol is termed a block.

As an aside: the reason for all the terms and what may seem like splitting hairs is that when you are designing a system and working with a lot of people communicating the steps of the processing can get tedious. To alleviate some of the confusion we try to avoid overloading terms like symbol.

## Inter-Block Interference and the Cyclic Prefix

To avoid IBI we will use a CP. A CP is just a copy of the last, [N_{CP}] samples of the OFDM block. The [N_{CP}] are then pre-pended to the OFDM block. For our example we can set our [N_{CP}=2]. For 5G Physical layer the CP is a lot longer. Then we can now define our OFDM block with a CP as:

\begin{equation}
\mathbf{x}_{cp}= [-2~0~0~0~-2~0]^T.
\end{equation}

For our little example [N_{CP}=2] was the only interesting number to choose but in a real system we chose the [N_{CP}] by estimating L. The optimal value for [N_{CP}] is L, but we never know what it is. The danger here is that if we pick to small of a value then we could introduce IBI or if we pick to high of a value we reduce our data rate, since we are sending redundant data. As a general rule [N_{CP}=0.9\hat{L}], where [\hat{L}] is the estimate of L.

The transmitted signal is then [\mathbf{x}_{CP}] and assume that we set [N_{CP}] appropriately we experience no IBI then we can demodulate y which is modeled in the time domain as:

\begin{equation}
y[n] = \sum_{\ell=0}^{L-1}x_{CP}[n-\ell]h[\ell].
\end{equation}
\noindent

After receiving the signal we first remove the CP by taking the last N samples of the received OFDM block,

\begin{equation}
\mathbf{y} = y\left[N_{CP}:N_{CP}+N]\right].
\end{equation}
We can then go to the frequency domain with the FFT operation,

\begin{equation}
\label{eq:rxfft}
\tilde{\mathbf{y}} = \operatorname{FFT}\left[\mathbf{y}\right].
\end{equation}

And finally we can we can map the symbols on each subcarrier to estimate what bits where sent.

## Channel Estimation

We can still improve the performance of the receiver if we equalize the effect of h. To improve the performance we will need to estimate the frequency response of the channel, [\tilde{\mathbf{h}}]. To do this we will need to structure the transmit waveform a little differently to give ourselves the ability to estimate the channel at the receiver. We will place pilot tones on a subset of subcarriers.

A pilot tone is a known value determined a priori and is known at the transmitter and receiver. This way when the receiver has [\tilde{\mathbf{y}}] some are pilot tones and some are data tones. We have N subcarriers to allocate as pilot subcarriers and data subcarriers. We will use [N_{pil} \geq L]. So we then have [N-N_{pil}] data tones to transmit data. In the 5G physical layer pilot tones are dynamically allocated, they are added with more accurate channel estimation is needed and removed with a higher data rate is needed.

### Step 1: Estimate Channel at Pilot Tones

The receiver starts by analyzing all the pilot tones. For a single subcarrier that is designated as a pilot, say index p, the received signal is modeled in the frequency domain as:

\begin{equation}
\label{eq:psubcar}
\tilde{y}_p = \tilde{x}_p\tilde{h}_p.
\end{equation}

The only unknown is [\tilde{h}_p], and we can solve for [\tilde{h}_p] by simple division. Once we have the frequency response estimated at all the pilot tones we can then interpolate the full frequency response.

### Step 2: Interpolate and Estimate Impulse Response

To estimate the full frequency response we will need to use a sub-matrix of the DFT matrix. First we define [\mathbf{\mathcal{F}}_N=\operatorname{DFTMTX}(N)], the \ac{DFT} matrix is a matrix when multiplied by a column vector the result is the same as FFT of the column vector.

Then we remove the columns of [\mathbf{\mathcal{F}}_N] that correspond to the data tones, leaving a matrix [\mathbf{\mathcal{F}}{pil}] that is [(N\times N_{PIL})] where [N_{PIL}] is the number of pilot tones used. Then the estimated impulse response is then calculated as:

\begin{equation}
\hat{\mathbf{h}} = \mathbf{\mathcal{F}}{pil}^{-1}\tilde{\mathbf{h}}{pil}.
\end{equation}

### Step 3: Calculate the Interpolated Frequency Response

Our estimated frequency response is just an \ac{FFT} of the estimated impulse response,

\begin{equation}
\tilde{\mathbf{h}} = \operatorname{FFT}\left[\hat{\mathbf{h}}\right].
\end{equation}

## Step 4 : Equalize the Data Tones

Now that we have our estimated frequency response we can estimate the received symbols. For each data tone, which we will index with k, we can equalize the received symbol as,

\begin{equation}
\hat{\tilde{x}}_k= \frac{\tilde{y}_k}{\hat{\tilde{h}}_k}
\end{equation}

We then can map then [\hat{\tilde{x}}_k] symbols back to bits and we will have better performance.

The channel estimation and equalization is the last concept needed to understand the workings of an OFDM transceiver. Next we will discuss an issue called PAPR that appears in OFDM. Lastly we will discuss the ability for OFDM to be able to handle multiple users.

## Peak to Average Power Ratio

To understand the PAPR issue with OFDM we will need to use the CLT. The CLT states that if we have a set of N random numbers, [{x_1, \dots, x_N}], and we find the mean of that set where we define, S_N as:

\begin{equation}
S_N := \frac{1}{N}\sum_{n=1}^{N}x_n.
\end{equation}

Then no matter the distribution of [x_n], [S_N] converges to a normal distribution. We can apply this to the time domain OFDM block. We can apply the CLT because of the structure of the FFT calculation. In the FFT we have summing and scaling calculations that do fit the definition for the CLT.

The result of understanding that the CLT applies to the time domain signal is that no matter what we are encoding onto the subcarriers we will see the samples in the time domain follow a Gaussian distribution. This finding is problematic since the Gaussian distribution has infinite support, meaning it is possible to get quite large results for a particular sample.

The PAPR is clearly defined as the peak power, which as we have just discussed can be quite large due to the CLT. The average power however stays relatively low since the Gaussian distribution is symmetric. Which results in a large PAPR. To transmit a signal there is a power amplifier in the transmit chain. Power amplifiers need to operate in a linear region for the range of the transmitted signal. With a large PAPR the power amplifier needs to have a large linear region of operation, making the power amplifier more expensive in terms of power consumption and dollars to make the system.

For a cell phone that follows a 5G physical layer standard this problem was needed to be solved this because they needed to make the cell phones as cheaply as possible. Also battery life is a large issue for consumers. Increasing the power required to transmit signal would be very costly for a cell phone. So they came up with a slight variation of OFDM called SC-FDMA is the topic of the next post.

## Mulitple-User Access

Before moving on to SC-FDMA which has the ability to support multiple users. Since 5G supports multiple users we need to understand how physical layer changes to support multiple users. We will discuss the multiple user case of OFDM, namely OFDMA. In OFDMA a user is assigned specific subcarriers and has to blank the subcarriers that are not alloted for the user. So as we are defining our symbols in the frequency domain we allocate our symbols in our negotiated subcarriers and put zeros where we can not transmit,

\begin{equation}
\tilde{\mathbf{x}}=\left[0,\dots,0,s,\dots,s,0,\dots,0\right]^T.
\end{equation}
We then continue with our processing just like before. In this scheme we could support up to N users, each getting one subcarrier. This scheme also allows for easy dynamic subcarrier allocation. The cell phone tower does dynamic carrier allocation where a user (with a cell phone) is inactive, that cell phone is allocated the minimal connection number of subcarriers. Then as the phone becomes more active then as the cell phone requests are answered by the 5G tower the cell phone can be allocated more subcarriers in the physical layer to get the data requested back to the user in a timely manner.