# Exercise 4.3: Subcarrier Mapping

From LNTwww

The diagram shows two transmission schemes that play a role in connection with Long Term Evolution $\rm (LTE)$. These block diagrams are referred here neutrally as "arrangement $\rm A$" or "arrangement $\rm B$".

- The light grey blocks represent the transition from the time to the frequency domain.
- The dark grey blocks represent the transition from the frequency to the time domain.

We refer here to the following links:

- Discrete Fourier Transform ⇒ $\rm DFT$,

- Inverse Discrete Fourier Transform ⇒ $\rm IDFT$.

For the number of interpolation points of DFT and IDFT, realistic numerical values of $K = 12$ and $N = 1024$ are assumed.

- The value $K = 12$ results from the fact that the symbols are "mapped" to a certain bandwidth by the "subcarrier mapping". The smallest addressable block for LTE is $180 \ \rm kHz$. With the subcarrier spacing of $15 \ \rm kHz$ the value $K = 12$ results.
- With the number $N$ of interpolation points of the IDFT $($with arrangement $\rm A)$ , up to $J = N/K$ users can thus be served simultaneously. For subcarrier mapping, there are three different approaches with DFDMA, IFDMA and LFDMA.
- The first two users are shown in green and turquoise in the diagram. In subtask
**(5)**you are to decide whether the sketch applies to DFDMA, IFDMA or LFDMA.

*Note:*

- The task belongs to the chapter The Application of OFDMA and SC-FDMA in LTE.

### Questions

### Solution

**(1)** __Proposed solution 2__ is correct:

- Both arrangements show "Single Carrier Frequency Division Multiple Access" $\text{(SC–FDMA)}$, recognisable by the DFT and IDFT blocks.
- The advantage over "Orthogonal Frequency Division Multiple–Access" $\text{(OFDMA)}$ is the more favourable Peak–to–Average Power–Ratio $\text{(PAPR)}$.
- A large PAPR means that the amplifiers must be operated below the saturation limit and thus at poorer efficiency in order to prevent excessive signal distortion.
- A lower PAPR also means longer battery life, an extremely important criterion for smartphones.
- This is why SC-FDMA is used in the LTE uplink. For the downlink, the aspect mentioned here is less significant.

**(2)** __Proposed solutions 1 and 2 __are correct:

- While in OFDMA the data symbols to be transmitted directly generate the various subcarriers, in SC-FDMA a block of data symbols is first transformed into the frequency domain using DFT.
- To be able to transmit multiple users, $N > K$ must apply. An input block of a user thus consists of $K$ bits. It is thus obvious that arrangement $\rm A$ applies to the transmitter.
- Arrangement $\rm B$, on the other hand, describes the receiver of the LTE uplink and not the transmitter.

**(3)** __Both statements__ are correct:

- The measures are necessary to be able to process a continuous bit stream at the transmitter,
- or to ensure a continuous bit stream at the receiver as well.

**(4)** The DFT also generates $K$ spectral values from $K$ input values.

- The subcarrier mapping does not change anything.
- Further users also occupy $K$ (bits) of the total of $N$ (bits).
- Thus $J = N/K = 1024/12 = 85.333$ ⇒ $J \ \underline{= 85}$ users can be supplied.

**(5)** __Proposed solution 3__ is correct:

- The graph conforms to the current 3gpp specification, which provides for "Localized Mapping".
- Here, the $K$ modulation symbols are assigned to adjacent subcarriers.

**(6)** __Solutions 2 and 3__ are correct:

- The realisation of DFT or IDFT as an (inverse) "Fast Fourier Transform" is only possible if the number of interpolation points is a power of two.
- For example, for $N = 1024$, but not for $K = 12$.