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  1. Advanced telecommunications for wireless systemsInvestigating OFDM by MathCAD Timo Korhonen, Communications Laboratory, TKK

  2. Motto • If you tell me – I forget • If you show me – I will remember • If you involve me – I can understand- a Chinese proverb

  3. Topics • The objective of workshop OFDM module is to get familiar with OFDM physical level by using MathCAD for system studies. • Topics: • OFDM Signal in time and frequency domain • Channel model and associated effects to OFDM • Windowing • Cyclic prefix • Peak-to-average power ratio (PAPR) • OFDM transceiver • Water-pouring principle • System modeling: Constellation diagram, error rate • System impairments

  4. References for exercises • • Bahai, Ahmad R. S: Multi-Carrier Digital Communications : Theory and Applications of OFDM • Hara, Shinsuke: Multicarrier Techniques for 4G Mobile Communications • Prasad, Ramjee: OFDM for Wireless Communications Systems • Xiong, Fuqin: Digital Modulation Techniques. Norwood, MA, USA •

  5. Exercise: Using MathCAD • Plot the sinc-function • Create a script to create and draw a rectangle waveform. • Demonstrate usage of FFT by drawing a sin-wave and its spectra. • Determine Fourier-series coefficients of a sinusoidal wave and plot the wave using these coefficients • Prepare a list of problems/solutions encountered in your tasks.

  6. Rect

  7. Spectra of a sinus

  8. Fourier transformation of a sinusoidal

  9. Introduction

  10. Background • Objectives: High capacity and variable bit rate information transmission with high bandwidth efficiency • Limitations of radio environment, also Impulse / narrow band noise • Traditional single carrier mobile communication systems do not perform well if delay spread is large. (Channel coding and adaptive equalization can be still improve system performance)

  11. OFDM • Each sub-carrier is modulated at a very low symbol rate, making the symbols much longer than the channel impulse response. • Discrete Fourier transform (DFT) applied for multi-carrier modulation. • The DFT exhibits the desired orthogonality and can be implemented efficiently through the fast fourier transform (FFT) algorithm.

  12. Basic principles • The orthogonality of the carriers means that each carrier has an integer number of cycles over a symbol period. • Reception by integrate-and-dump-receiver • Compact spectral utilization (with a high number of carriers spectra approaches rectangular-shape) • OFDM systems are attractive for the way they handle ISI and ICI, which is usually introduced by frequency selective multipath fading in a wireless environment. (ICI in FDM)

  13. Drawbacks of OFDM • The large dynamic range of the signal, also known as the peak-to-average-power ratio (PAPR). • Sensitivity to phase noise, timing and frequency offsets (reception) • Efficiency gains reduced by guard interval. Can be compensated by multiuser receiver techniques (increased receiver complexity)

  14. Examples of OFDM-systems • OFDM is used (among others) in the following systems: • IEEE 802.11a&g (WLAN) systems • IEEE 802.16a (WiMAX) systems • ADSL (DMT = Discrete MultiTone) systems • DAB (Digital Audio Broadcasting) • DVB-T (Digital Video Broadcasting) OFDM is spectral efficient, but not power efficient (due to linearity requirements of power amplifier=the PAPR-problem). OFDM is primarily a modulation method; OFDMA is the corresponding multiple access scheme.

  15. OFDM Signal

  16. Multiplexing techniques

  17. OFDM signal in time domain OFDM TX signal = Sequence of OFDM symbols gk(t) consisting of serially converted complex data symbols The k:th OFDM symbol (in complex LPE form) is where N = number of subcarriers, TG + TS= symbol period with the guard interval, and an,k is the complex data symbol modulating the n:th subcarrier during the k:th symbol period. In summary, the OFDM TX signal is serially converted IFFT of complex data symbols an,k

  18. Orthogonality of subcarriers Definition: Orthogonality over the FFT interval: Phase shift in any subcarrier - orthogonality over the FFT interval should still be retained:

  19. Exercise: Orthogonality • Create a MathCAD script to investigate orthogonality of two square waves • #1 Create the rect-function • #2 Create a square wave using #1 • #3 Create a square wave with a time offset • #4 Add the waves and integrate

  20. Exercise: Orthogonality of OFDM signals • Create and plot an OFDM signal in time domain and investigate when your subcarriers are orthogonal • #1 Create a function to generate OFDM symbol with multiple subcarriers • #2 Create a function to plot comparison of two subcarriers orthogonality (parameter is the frequency difference between carriers) • Note: also phase continuity required in OFDM symbol boarders • #3 Inspect the condition for orthogonality and phase continuity


  22. OFDM Spectra

  23. OFDM in frequency domain TG TFFT Square-windowed sinusoid in time domain => "sinc" shaped subchannel spectrum in frequency domain See also A.13 in Xiong, Fuqin. Digital Modulation Techniques. Norwood, MA, USA: Artech House, Incorporated, 2006. p 916.

  24. Spectra for multiple carrier Single subchannel OFDM spectrum Subcarrier spacing = 1/TFFT Spectral nulls at other subcarrier frequencies

  25. Next carrier goes here!

  26. Exercise: Analytical spectra • Draw the spectra of OFDM signal by starting its frequency domain presentation (the sinc-function). Plot the spectra also in log-scale • #1 Plot three delayed sinc(x) functions in the range x = -1…2 such that you can note they phase align correctly to describe the OFDM spectra • #2 Plot in the range from f = -20 to 20 Hz an OFDM spectra consisting of 13 carriers around f=0 in linear and log-scale

  27. Ofdm

  28. Exercise: Spectra modified • Investigate a single OFDM carrier burst and its spectra by using the following script: • How the spectra is changed if the • Carrier frequency is higher • Symbol length is altered

  29. OFDM Spectra by MathCAD for a single carrier ofdm spectra by rect windowed

  30. Spectral shaping by windowing

  31. Exercise: Windowed spectra • The next MathCAD script demonstrates effect of windowing in a single carrier. • How the steepness of the windowing is adjusted? • Why function win(x,q) is delayed by ½? • Comment the script

  32. burst windowing and ofdm

  33. Modeling OFDM Transmission

  34. Transceiver • Some processing is done on the source data, such as coding for correcting errors, interleaving and mapping of bits onto symbols. An example of mapping used is multilevel QAM. • The symbols are modulated onto orthogonal sub-carriers. This is done by using IFFT • Orthogonality is maintained during channel transmission. This is achieved by adding a cyclic prefix to the OFDM frame to be sent. The cyclic prefix consists of the L last samples of the frame, which are copied and placed in the beginning of the frame. It must be longer than the channel impulse response.

  35. OFDM and FFT ~ Aalborg-34-lecture13.pdf

  36. Exercise: Constellation diagram of OFDM system • Steps • #1 create a matrix with complex 4-level QAM constellation points • #2 create a random serial data stream by using outcome of #1. Plot them to a constellation diagram. • #3 create complex AWGN channel noise. Calculate the SNR in the receiver. • #4 form and plot the received complex noisy time domain waveform by IFFT (icfft-function) • #5 detect outcome of #4 by FFT and plot the resulting constellation diagram

  37. Exercise : Constellation diagram of OFDM Ofdm