Tools

Study of Frequency division multiplexed schemes

Orthogonal Frequency Division Multiplexing (OFDM)

OFDM divides a high-bandwidth signal into multiple narrowband subcarriers to reduce inter-symbol interference (ISI) and simplify equalization in multipath channels.

Mathematical Formulation of OFDM

Let the input data symbols be \(X_k\), for \(k = 0, 1, ..., N-1\), where \(N\) is the number of subcarriers. The OFDM signal in the time domain is obtained using the Inverse Fast Fourier Transform (IFFT):

$$ x_n = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X_k \, e^{j 2 \pi k n / N}, \quad n = 0, 1, ..., N-1 $$

Here:

  • \(X_k\) = frequency-domain symbol on the \(k\)-th subcarrier
  • \(x_n\) = time-domain OFDM sample
  • \(N\) = total number of subcarriers
  • Normalization factor \(1/\sqrt{N}\) ensures constant average power

At the receiver, the Fast Fourier Transform (FFT) recovers the transmitted symbols:

$$ Y_k = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} y_n \, e^{-j 2 \pi k n / N}, \quad k = 0, 1, ..., N-1 $$

Where:

  • \(y_n\) = received time-domain OFDM sample

Cyclic Prefix (CP)

To prevent ISI, a cyclic prefix is added. If \(x_n\) is the original OFDM symbol of length \(N\), the transmitted symbol with CP of length \(N_{CP}\) is:

$$ \tilde{x}_n = \begin{cases} x_{N-N_{CP}+n}, & 0 \le n < N_{CP} \\ x_{n-N_{CP}}, & N_{CP} \le n < N+N_{CP} \end{cases} $$

The sequence x̃ₙ is formed by appending a cyclic prefix of length N_CP to the original signal xₙ. The first N_CP samples of x̃ₙ are copies of the last N_CP samples of xₙ, while the remaining samples correspond to the original signal shifted by N_CP. This results in a new sequence where the end of the signal is repeated at the beginning, a technique commonly used in OFDM systems.

This ensures that linear convolution with the channel becomes circular convolution, preserving orthogonality of subcarriers.

Subcarrier Spacing and Orthogonality

The subcarrier spacing \(\Delta f\) is defined by the sampling frequency \(f_s\) and the number of subcarriers \(N\). It is also the reciprocal of the useful symbol duration \(T_{sym}\):

$$ \Delta f = \frac{f_s}{N} = \frac{1}{T_{sym}} $$

This specific spacing guarantees that the subcarriers are orthogonal (mutually independent):

$$ \sum_{n=0}^{N-1} e^{j 2 \pi (k-m) n / N} = \begin{cases} N, & k=m \\ 0, & k \ne m \end{cases} $$

Where \(k\) and \(m\) are subcarrier indices. Orthogonality ensures no inter-carrier interference (ICI) in ideal conditions.

OFDM Transmitter

ofdm_image
Fig 1: OFDM Transmitter Block Diagram

1. Modulation: The input message bits are mapped into complex symbols using digital modulation schemes such as Quadrature Amplitude Modulation (QAM) or Phase Shift Keying (PSK).
2. Serial-to-Parallel Conversion: The modulated symbols are grouped into blocks and converted from a serial stream into parallel streams, each corresponding to a subcarrier.
3. IFFT (Inverse Fast Fourier Transform): An N-point IFFT is applied to the parallel data to convert the frequency-domain symbols into a time-domain OFDM signal while maintaining orthogonality among subcarriers.
4. Cyclic Prefix Addition: A cyclic prefix is added to each OFDM symbol to reduce inter-symbol interference caused by multipath propagation. It is formed by copying the last portion of the symbol and appending it to the beginning.
5. Parallel-to-Serial Conversion: The parallel OFDM symbols are converted back into a serial stream to produce a continuous signal.
6. Digital-to-Analog Conversion and Filtering: The discrete-time OFDM signal is converted into a continuous-time analog signal using a digital-to-analog converter (DAC). A reconstruction filter is then applied to smooth the signal and remove unwanted spectral components.
7. Transmission: The resulting analog signal is transmitted over the communication channel.

OFDM Receiver

ofdm_image
Fig 2: OFDM Receiver Block Diagram

1. Signal Reception: Receive the incoming OFDM signal from the channel.
2. Filtering & Analog-to-Digital Conversion: Filter the received signal and convert it from analog to digital form for further processing.
3. Serial-to-Parallel Conversion: Convert the incoming serial data stream into parallel data blocks corresponding to OFDM symbols.
4. Cyclic Prefix Removal: Remove the cyclic prefix from each OFDM symbol to eliminate redundancy added during transmission.
5. N-Point FFT: Apply the FFT to transform the time-domain signal into the frequency domain and recover subcarrier data.
6. Parallel-to-Serial Conversion: Convert the parallel frequency-domain data back into a serial stream.
7. De-mapping: Map the received symbols back to their corresponding bit sequences to recover the transmitted bits.