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**IERG 4100 Wireless Communications**

Part X: OFDM

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**Introduction OFDM: Orthogonal Frequency Division Multiplexing**

Converts a wideband frequency selective fading channel into a parallel collection of narrow band frequency flat sub-channels Reduces the computational complexity associated with high data-rate transmission over frequency-selective channels

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History of OFDM The basic principles of OFDM was proposed in several publications in the 1960’s. Since 1966 FDM systems with overlapping spectra were proposed The next step is a proposal to realize an FDM system with DFT Finally, in 1971 Weinstein and Ebert proposed a complete OFDM system, which included generating the signal with an FFT and adding a guard interval in the case of multipath channels

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**OFDM Applications Broadcasting WLAN (Wireless local area network)**

DAB (Digital Audio Broadcasting) DVB (Digital Video Broadcasting) WLAN (Wireless local area network) IEEE a HiperLan/2 WMAN (Wireless metropolitan area network) IEEE (WiMax) 4G LTE (Long Term Evolution) 5G ?

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Motivation Inter-symbol interference in high-data-rate wireless communications To avoid ISI, data rate is limited the radio environment – delay spread Otherwise, equalizer is needed at the receiver to overcome ISI OFDM can overcome and take advantage of multipath fading and thus eliminate inherent data rate limitations

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**Time and Frequency Domain Description of Multipath**

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**Inter-symbol interference**

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**Single-Carrier Transmission vs. OFDM**

OFDM (Multi carrier transmission): frequency frequency …… …… time time Each symbol sees a frequency selective fading channel Each symbol on a subcarrier sees a frequency flat fading channel

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**Single Carrier System Sequential Transmission of Waveforms**

Waveforms are of short Duration T Waveforms occupy full system bandwidth 1/T

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**Multi-Carrier System Parallel Transmission of waveforms**

Waveforms are of long duration MT Waveforms occupy 1/Mth of system bandwidth 1/T

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**Subcarriers in the Time Domain**

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**Subcarrier Orthogonality**

In conventional FDMA The whole bandwidth is divided into many narrow sub-channels which are spaced apart and not overlapped. ⇒ Low spectral efficiency In OFDM By using orthogonal carriers with nulls at the center of the other carriers, the subchannels are overlapped. ⇒ Increase spectral efficiency frequency In the frequency domain, the orthogonality is seen by zeros All other subcarriers are zero when one subcarrier peaks

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**OFDM Transmitter and Receiver**

Add Cyclic Prefix & Pulse Shaping Serial to Parallel Parallel to Serial Mixer IFFT fc Time Domain Samples Frequency Domain Samples channel Parallel to Serial FFT Serial to Parallel Matched Filter and Remove Cyclic Prefix Mixer & Filter fc

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**DFT implementation Equivalent baseband notation**

At a sample rate of Ts/N Since (I)DFT can be much more efficiently implemented by (I)FFT

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**DFT implementation Matrix representation s=FHd F: FFT matrix**

Each dn, n=0, 1, …, N-1 is a modulated frequency domain sample Each sn, n=0, 1, …, N-1 is a sample of the OFDM symbol, i.e., time domain sample

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**OFDM Signal in the Time Domain**

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**Guard Interval OFDM deals with ISI within one OFDM symbol (OFDM block)**

Inter-block interference still exists Solution: Insert a guard interval that is longer than the delay spread Guard interval can consist of no signal. In this case, however the problem inter-carrier interference (ICI) would arise, since sub- carriers are no longer orthogonal By cyclic prefix in OFDM symbol, ISI and ICI can be eliminated completely

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Cyclic Prefix When the length of the cyclic prefix is larger than the delay spread, there is no inter-block interference after the cyclic prefix is removed

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**Matrix representation of the ISI channel**

Assume channel impulse response length is P Matrix representation

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Circulant Matrix A Circulant matrix is an n-by-n matrix whose rows are composed of cyclically shifted versions of a length-n list. For example, the circulant matrix on the list l={1, 2, 3, 4} is given by One important property: a circulant matrix can be diagonalized by the Fourier transformation matrix

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Cyclic Prefix In order to form a circulant matrix, instead of transmitting s, we transmit Assume P=1, then

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Cyclic Prefix An effective circulant matrix is created using cyclic prefix Efficiency: with ,since a vector of length will be transmitted for a length-N data vector When N increases, efficiency increases

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**Diagonalization of Circulant Matrix**

Circulant matrix can be diagonalized as where N parallel flat fading subchannels are created Note, the transmitter can diagonalize without knowing any information about Gain of a sub-channel

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Advantages of OFDM With cyclic prefix, intra and inter OFDM symbol ISI can be eliminated completely An effective circulant matrix can be created using cyclic prefix, as a result, ICI can be eliminated completely Implementation complexity is significantly lower than that of a single carrier system with an equalizer Provide frequency diversity Forward error correcting code such as convolutional code with interleaver is needed as some sub-carriers will be in deep fade

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**Fading Across Subcarriers**

Example:

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**Different BERs Across Subcarriers**

Compensation technique Coding across subcarriers Adaptive loading (power and rate)

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**Variable-Rate Variable-Power MQAM**

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**Adaptive Techniques Variable-rate variable-power techniques**

Fixed BER, maximize average data rate Fixed data rate, minimize average BER Fixed BER and data rate, minimize average power

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Formulation BER in non-fading AWGN channel with MQAM (M>=4) modulation and coherent detection: Adaptive MQAM for fixed BER

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**Rate Maximization in Single-Carrier Systems**

Optimal solution: Water filling

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**Power Minimization in Single-Carrier Systems**

Practical (suboptimal) solution: Fix M. Transmit at the minimum power that meets the BER performance Optimal solution: water filling with a carefully chosen water level

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**Constellation Restriction**

M is restricted to {0, …, MN} Carefully design region boundaries Power control maintains target BER

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**Adaptive Loading in Multi-Carrier Systems**

Pros: Smaller rate and power fluctuation Requires smaller buffer size Channel gains are known

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**Rate Maximization Concave maximization**

Transmit power per OFDM symbol is fixed Constellation constraint can be imposed

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**Power Minimization Linear programming**

Data rate per OFDM symbol is fixed Constellation constraint can be imposed

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