Advanced telecommunications for wireless systems Investigating 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 sinwave 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.
Rect waveform.mcd Spectra of a sinus wave.mcd Fourier transformation of a sinusoidal wave.mcd
Introduction 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 rectangularshape) 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 Multiplexing techniques
OFDM signal in time domain OFDM TX signal = Sequence s t g k t kTS
k of OFDM symbols gk(t) consisting of serially converted complex data symbols
The k:th OFDM symbol (in complex LPE form) is N 2 n g k t an ,k exp j 2 t
TS n N 2 n 0 k 1 TS t kTS
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 ( k 1)T
s (t )s (t )dt 0 Definition: 1
2 kT Orthogonality over the FFT interval: TFFT
TS 2 m n cos 2 mt TS cos 2 nt TS dt m n 0
0 Phase shift in any subcarrier - orthogonality over the FFT interval should still be retained: TFFT
cos 2 mt T S 0
cos 2 nt TS dt 0 m n 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 Orthogonality.mcd
OFDM Spectra OFDM in frequency domain TG TFFT
Square-windowed sinusoid in time domain => "sinc" shaped subchannel spectrum in frequency domain sinc fTFFT sin fTFFT
fTFFT 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 = -12 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 Ofdm spectra.mcd 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 Spectral shaping by windowing
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 ?
Comment the script burst windowing and ofdm spectra.mcd Modeling OFDM Transmission
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
*Spike distance depends on impulse response Multipath channel model Received signal
Pr: Received mean power Gs: Shadow fading Log r ~ Fast fading Exercise: Modeling channel
Create a MathCAD script to create artificial impulse and frequency response of a multipath channel (fast fading) #1 Create an array of complex AWGN #2 Filter output of #1 by exp(-5k/M) where M is the number of data points
#3 Plot the time domain magnitude of #2 Is this a Rayleigh or Rice fading channel? How to make it the other one than Rayleigh/ Rice #4 Plot #3 in frequency domain
Comment how realistic this simulation is? Rayleigh or Rice fading channel? impulse response radio ch.mcd Frequency response
Frequency response shown by swapping left-hand side of the fft Exercise: Rayleigh distribution #1 create a Rayleigh distributed set of random numbers (envelope of complex Gaussian rv.)
#2 plot the pdf of #1 (use the histogramfunction) #3 add the theoretical pdf to #2 Rayleigh distribution - Note that true pdf area equals unity, how could you adjust the above for this?
- Add comparison to the theoretical Rayleigh distribution! Path loss Shadow fading
N(0,s2): Log normal distribution Doppler spread DELAY SPREAD IN TIME DOMAIN
Small delay spread Large delay spread Delay spread in frequency domain
Exercise: Variable channel Discuss a model of a channel with flat/ frequency selective characteristics and report its effect to the received modulated wave Amplitude and phase spectra What happens to the received frequency components
Flat fading Frequency selective fading Time invariant / time variant channel Doppler effected channel Exercise: ODFM in a multipath
channel #1 Create an impulse response of 256 samples with nonzero values at h2=16, h10=4+9j, and h25= 10+3j and plot its magnitude spectra #2 Create OFDM symbol for three subcarriers with 1,2 and 3 cycles carrying bits 1,-1 and 1
#3 Launch the signal of #2 to the channel of #1 and plot the OFDM signal before and after the channel to same picture. #4 Detect (Integrate an dump) the bits after and before the channel and compare. See the generated ICI also by detecting the 4:n carrier!
The channel OFDM in a multipath channel.mcd Single carrier transmission and
rms delay spread Determine by using MathCADs linterp-function the maximum rate for delay spread of 70 us!
Rate and delay spread.mcd Cyclic prefix Example: a two-path channel
Home exercise: Orthogonality and multipath channel Demonstrate by MathCAD that the orthogonality of OFDM signal can be maintained in a multipath channel when guard interval is applied #1 modify syms2-signal to include a cyclic prefix
#2 introduce multipath delays not exceeding the duration of cyclic prefix (apply the rot-function) #3 determine integrate and dump detected bits for #2 and especially for carriers that are not used to find that no signal is leaking into other subcarriers -> ICI is avoided!
cyclic prefix.mcd Detection Peak-power problem
Envelope Power Statistics Exercise Create an OFDM signal in time domain and determine experimentally its PAPR
Experiment with different bit-patterns to show that the PAPR is a function of bit pattern of the symbol #1 create 64 pcs BPSK LPE bits #2 define a function to create OFDM symbol with the specified number of carriers (with 256 samples)
carrying bits of #1 #3 check that the carriers are generated correctly by a plot #4 determine PAPR for a set of 64 OFDM subcarriers. Compare with different bit patterns (eg. evaluate #1 again by pressing F9)
OFDM papr.mcd Transfer characteristics of power amplifier
Exercise: Non-linear distortion Demonstrate build-up of harmonics for a sinusoidal wave due to non-linearity of a power amplifier #1 Create a sinusoidal wave, 256 samples and 8 cycles
#2 Create a clipping function that cuts a defined section of waves amplitudes #3 Apply #2 to #1 and plot the result #4 compare #1 to #3 in frequency domain logscale with different levels of clipping PAPR suppression
Selective mapping (coding) Cons: Table look-up required at the receiver Signal distortion techniques Clipping, peak windowing, peak cancellation Cons: Symbols with a higher PAPR suffer a
higher symbol error probability Prasad, Ramjee. OFDM for Wireless Communications Systems. Norwood, MA, USA: Artech House, Incorporated, 2004. p 150. http://site.ebrary.com/lib/otaniemi/Doc?
id=10081973&ppg=166 Selective mapping Clipping Cancel the peaks by simply limiting the
amplitude to a desired level Self-interference Out-of band radiation Side effects be reduced by applying different clipping windows
Peak cancellation Comparing clipping and peak cancellation
a) undistorted b) peak cancellation c) clipping Effect of peak cancellation on packet error rate (PER)
OFDM transceiver System error rate In AWGN channel OFDM system performance same as for single-carrier
In fading multipath a better performance can be achieved Adjusts to delay spread Allocates justified number of bits/subcarrier DESIGN
EXAMPLE Exercise: BPSK error rate How would you modify this function to simulate unipolar system?
Bit allocation for carriers Each carrier is sensed (channel estimation) to find out the respective subchannel SNR at the point of reception (or channel response) Based on information theory, only a certain,
maximum amount of data be allocated for a channel with the specified BT and SNR OFDM bit allocation policies strive to determine optimum number of levels for each subcarrier to (i) maximize rate or (ii) minimize power for the specified error rate
Water-pouring principle Assume we know the received energy for each subchannel (symbol), noise power/Hz and the required BER Assume that the required BER/subchannel is the
same for each subchannel (applies when relatively high channel SNR) Water-pouring principle strives to determine the applicable number of levels (or bit rate) for subcarriers to obtain the desired transmission
Home exercise 2: Water pouring principle Follow the previous script and Explain how it works by own words Comment the result with respect of information theory
System impairments http://www.eng.usf.edu/wcsp/OFDM_links.html Exercise ICI
Create a MathCAD script to investigate frequency offset produced ICI Summary
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