8. OFDM transmission is used over a

Table of Contents

Ideas for Solving the Problem

For Question 8:

OFDM (Orthogonal Frequency Division Multiplexing) performance in dispersive channels
Relationship between FFT size and subcarrier bandwidth
Effect of FFT size on channel flatness per subcarrier
Trade-off between FFT size and channel characteristics

For Question 9:

Linear Minimum Mean Square Error (LMMSE) estimator properties
Optimality conditions for MMSE estimators
Statistical properties of different distributions
Relationship between linear estimators and Gaussian distributions

For Question 10:

OFDM bit rate calculation formula
Accounting for different modulation schemes
Impact of cyclic prefix on effective data rate
Conversion from symbol rate to bit rate

Calculation Steps

Question 8: OFDM FFT Size for Flat Fading

In OFDM systems, the total bandwidth is divided into N subcarriers (where N is the FFT size). When the FFT size increases:

Each subcarrier occupies a narrower bandwidth (BW/N)
Narrower bandwidth per subcarrier means the channel is more likely to be flat within each subcarrier
The coherence bandwidth of the channel remains constant

For a dispersive channel, we want each subcarrier’s bandwidth to be much smaller than the coherence bandwidth to approximate flat fading. Among the given options, the largest FFT size (512) will create the narrowest subcarriers, making the channel closest to flat for each subcarrier.

Question 9: LMMSE Estimator Optimality

The Linear Minimum Mean Square Error (LMMSE) estimator is given by:

x̂ = E[x] + Cxy(Cyy)^(-1)(y – E[y])

For this estimator to be the optimal MMSE estimator (not just the best linear estimator):

The joint distribution of the parameter and observation must be Gaussian
This is because the conditional expectation E[x|y] is linear only when x and y are jointly Gaussian
For other distributions, the optimal MMSE estimator may be non-linear

Question 10: OFDM Effective Bit Rate Calculation

Step 1: Calculate total bits per OFDM symbol

512 subcarriers with QPSK (2 bits/symbol) = 512 × 2 = 1,024 bits
512 subcarriers with QAM-16 (4 bits/symbol) = 512 × 4 = 2,048 bits
Total bits per OFDM symbol = 1,024 + 2,048 = 3,072 bits

Step 2: Calculate total symbol duration including cyclic prefix

OFDM symbol length = 1,024 points
Cyclic prefix length = 10 points
Total length = 1,024 + 10 = 1,034 points

Step 3: Calculate effective bit rate

Symbol rate = 1,000 symbols per second
Effective bit rate = (Total bits per symbol / Total symbol length) × Symbol rate
Effective bit rate = (3,072 / 1,034) × 1,000 = 2,970.99 bits per second

Final Answer

Question 8: (d) 512
The largest FFT size (512) divides the bandwidth into the most subcarriers, making each subcarrier’s bandwidth narrower and thus closer to experiencing flat fading in a dispersive channel.

Question 9: (c) Gaussian
The LMMSE estimator is the optimal MMSE estimator when the parameters and noise follow a Gaussian distribution, as this is the only case where the conditional expectation is guaranteed to be linear.

Question 10: 2,970.99 bits per second
The effective bit rate falls within the acceptable range of [2960, 2980] bits per second.

Highlights

In OFDM systems, increasing the FFT size narrows the bandwidth per subcarrier, which helps combat frequency-selective fading by making the channel appear more flat within each subcarrier.
The LMMSE estimator becomes the optimal MMSE estimator specifically for Gaussian distributions because the conditional expectation is linear only in this case.
When calculating OFDM bit rates with mixed modulation schemes, it’s important to account for the different bits per symbol for each modulation type and the overhead introduced by the cyclic prefix.
The cyclic prefix reduces the effective bit rate as it carries no new information but consumes transmission time.