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IV. Orthogonal Frequency Division Multiplexing (OFDM)

IV. Orthogonal Frequency Division Multiplexing (OFDM). OFDM a New Idea?. The idea of OFDM has been out there since the 1950s OFDM was first used in military HF radios in late 1950s and early 1960s

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IV. Orthogonal Frequency Division Multiplexing (OFDM)

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  1. IV. Orthogonal Frequency Division Multiplexing (OFDM)

  2. OFDM a New Idea? • The idea of OFDM has been out there since the 1950s • OFDM was first used in military HF radios in late 1950s and early 1960s • Early use of OFDM has been limited in commercial communication systems due to the high costs associated with the requirements for hundreds/thousands of oscillators • The use of OFDM has experienced a breakthrough in the 1990s with advancements in DSP hardware • Currently, OFDM has been adopted in numerous wire-line and wireless communications systems, such as: • Digital audio and video broadcasting • Digital subscriber lines (DSL) • Wireless LAN 802.11 • WiMAX 802.16 • LTE (Long term Evolution), 4G Cellular Networks

  3. OFDM & DFT (Discrete Fourier Transform) OFDM Signal over 4 Sub-carriers Ts -f1 f1 -f2 f2 -f3 f3 -f4 f4 OFDM Signal: Freq. Domain OFDM Signal: Time Domain

  4. OFDM & DFT (Discrete Fourier Transform) OFDM Signal over 4 Sub-carriers OFDM Signal: Time Domain OFDM Signal: Freq. Domain DFT is means to generate samples of the OFDM signal in the frequency and time domain without the use of oscillators At the transmitter OFDM uses IDFT to convert samples of the spectrum of the OFDM signal into a corresponding equal number of samples from the OFDM signal at the time domain At the receiver OFDM uses DFT to restore the signal representation in the frequency domain and proceed with symbols detection

  5. OFDM & DFT (Discrete Fourier Transform) OFDM Signal over 4 Sub-carriers (Separated by 1/2Ts) We need to compute the composite spectrum in the frequency domain to be able to compute the 4 samples used by the IDFT

  6. OFDM & DFT (Discrete Fourier Transform) OFDM Signal over 4 Sub-carriers (Separated by 1/Ts) The separation between carriers guarantee that samples from individual spectrum of sub-carriers correspond to samples from the composite spectrum

  7. Number of Subcarriers in OFDM with DFT • For band-limited FDM if the system bandwidth is B, number of sub-carriers is given by: • For OFDM if the system bandwidth is B, Number of sub-carriers is given by: OFDM with DFT has the potential to at increase the number of sub-carriers compared to FDM for α>0 (remember that α=0 filter is not physically realizable ) DFT implementation of OFDM avoids the needs for oscillators to generate the OFDM signal

  8. Inverse Discrete Fourier Transform Example for Interpretation The IDFT corresponds to samples from the signal generated by the summation of signals generated from the angular motion of particles with frequencies 0, 1, 2, 3 Im-Axis Im-Axis Im-Axis Im-Axis f=3 f=1 f=2 f=0 + + + Re-Axis Re-Axis Re-Axis Re-Axis

  9. Inverse Discrete Fourier Transform Example for Interpretation Im-Axis Im-Axis Im-Axis Im-Axis f=0 f=1 f=2 f=3 Re-Axis Re-Axis Re-Axis Re-Axis

  10. Inverse Discrete Fourier Transform Example for Interpretation Im-Axis Im-Axis Im-Axis Im-Axis f=0 f=1 f=2 f=3 Re-Axis Re-Axis Re-Axis Re-Axis

  11. Inverse Discrete Fourier Transform Example for Interpretation Im-Axis Im-Axis Im-Axis Im-Axis f=0 f=1 f=2 f=3 Re-Axis Re-Axis Re-Axis Re-Axis

  12. Inverse Discrete Fourier Transform Example for Interpretation Im-Axis Im-Axis Im-Axis Im-Axis f=0 f=1 f=2 f=3 Re-Axis Re-Axis Re-Axis Re-Axis

  13. Inverse Discrete Fourier Transform Example for Interpretation Im-Axis Im-Axis Im-Axis Im-Axis f=0 f=1 f=2 f=3 Re-Axis Re-Axis Re-Axis Re-Axis

  14. Inverse Discrete Fourier Transform Example for Interpretation

  15. Inverse Discrete Fourier Transform Generalization: x(0) X[0] IDFT x(1) X[1] x(2) X[2] x(N-1) X[N-1] • Notes: • The IDFT does not specify the time between samples x(0), x(1), x(2), …, x(N-1) • In essence the frequency samples X[i] may be interpreted as the frequency that corresponds to i cycles of a moving particle within the time of interest T • In other words • X[0] corresponds to a DC signal • X[1] corresponds to a particle moving with frequency of 1 cycle every T sec (i.e., f=1/T) • X[2] corresponds to a particle moving with frequency of 2 cycles every T sec (i.e., f=2/T) • …

  16. OFDM with DFT Block Diagram Modulation Binary Encoder Demod. IDFT DFT x Modulation Binary Encoder Demod. Wireless Channel Carrier Frequency • The output from the IDFT process is a baseband signal • In OFDM a carrier frequency is used to carry the output samples from the IDFT process • The output from the IDFT process are complex samples. For each sample: • The real part is modulated by a cosine wave • The imaginary part is modulated by a sine wave Modulation Binary Encoder Demod.

  17. OFDM with DFT: Modeling X[0] IDFT Parallel to Serial X[1] x2 xN-1 x 0 x1 x X[2] X[N-1] Carrier (fc ) Multi-Path Channel Tx Y[0] DFT Serial to Parallel Y[1] Correlator with fc y0 y1 y2 yN-1 Y[2] Y[N-1] Rx Multiplying by the carrier at the transmitter and the correlator at the receiver are inverse operation. The could be abstracted in the system model for OFDM

  18. OFDM with DFT: Modeling X[0] IDFT Parallel to Serial X[1] x2 xN-1 x 0 x1 X[2] X[N-1] Multi-Path Channel h=[h0 h1 … hp] Tx Y[0] DFT Serial to Parallel Y[1] y0 y1 y2 yN-1 Y[2] Y[N-1] Rx Multiplying by the carrier at the transmitter and the correlator at the receiver are inverse operation. The could be abstracted in the system model for OFDM

  19. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  20. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  21. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  22. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  23. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  24. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  25. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  26. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  27. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  28. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel?

  29. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel? Similarly

  30. OFDM with DFT: Modeling x1 x2 x3 x 0 y2 y1 y0 y 3 What is w for the OFDM system to behave as a flat fading channel? This means that setting w=h1x3 makes the OFDM system over each sub-carrier behaves as a flat fading channel Setting w=h1x3 is achieved by inserting x3 as a cyclic prefix after the completion of the IDFT process at the transmitter. Only x3 is cyclically prefixed because the channel delay corresponds to only one time sample of the transmitting signal

  31. OFDM with DFT & Cyclic Prefix: Modeling Cyclic convolution x3 x1 x2 x3 x 0 y 3 y2 y1 y0 y-1 Removed before reception (cyclic prefix removal) Effect of previous OFDM symbol Function of h0 ,h1 and some phase shift that is a function of the subcarrier index

  32. OFDM with DFT Model Parallel to Serial Serial to Parallel IDFT Insert Cyclic Prefix QAM Modulator x Carrier (fc ) Multi-Path Channel Tx DFT Remove Cyclic Prefix Serial to Parallel Parallel to Serial Correlator with fc QAM De Mod. Rx

  33. Mitigation of Fading • Frequency Equalization: • Divide by received signal Y[i] by H[i] for all sub-carriers • Requires channel estimation • For low values of H[i] equalization results in noise amplification • Precoding • Divide transmitted signal X[i] by H[i] for all sub-carriers • Requires channel estimation knowledge at transmitter • Does not result in any noise amplification at the receiver • For low values of H[i] excessively high transmission power might be needed at the transmitter

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