Advanced telecommunications for wireless systemsInvestigating OFDM by MathCAD Timo Korhonen, Communications Laboratory, TKK
Motto • If you tell me – I forget • If you show me – I will remember • If you involve me – I can understand- a Chinese proverb
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
References for exercises • http://site.ebrary.com/lib/otaniemi • 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 • www.wikipedia.com
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.
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)
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.
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)
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)
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.
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
Orthogonality of subcarriers Definition: Orthogonality over the FFT interval: Phase shift in any subcarrier - orthogonality over the FFT interval should still be retained:
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
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
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. http://site.ebrary.com/lib/otaniemi/Doc?id=10160973&ppg=932
Spectra for multiple carrier Single subchannel OFDM spectrum Subcarrier spacing = 1/TFFT Spectral nulls at other subcarrier frequencies
Next carrier goes here! http://www.eng.usf.edu/wcsp/OFDM_links.html
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
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
OFDM Spectra by MathCAD for a single carrier ofdm spectra by rect windowed sinc.mcd
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
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.
OFDM and FFT http://www.eng.usf.edu/wcsp/OFDM_links.html ~ Aalborg-34-lecture13.pdf
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
Exercise : Constellation diagram of OFDM Ofdm system.mcd