Advanced telecommunications for wireless systemsInvestigating OFDM by MathCAD
Timo Korhonen, Communications Laboratory, TKK
- If you tell me – I forget
- If you show me – I will remember
- If you involve me – I can understand- a Chinese proverb
- The objective of workshop OFDM module is to get familiar with OFDM physical level by using MathCAD for system studies.
- OFDM Signal in time and frequency domain
- Channel model and associated effects to OFDM
- Cyclic prefix
- Peak-to-average power ratio (PAPR)
- OFDM transceiver
- Water-pouring principle
- System modeling: Constellation diagram, error rate
- System impairments
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
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.
Slide 7 Slide 8
Spectra of a sinus wave.mcd
Fourier transformation of a sinusoidal wave.mcd
- 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)
- 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.
- 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
Orthogonality over the FFT interval:
Phase shift in any subcarrier - orthogonality over the FFT interval should still be retained:
- 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
Slide 27 Slide 29
OFDM in frequency domain
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.
Spectra for multiple carrier
Subcarrier spacing = 1/TFFT
Spectral nulls at other subcarrier frequencies
Next carrier goes here!
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
Slide 35 Slide 37
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 ½?
- Comment the script
burst windowing and ofdm spectra.mcd
Modeling OFDM Transmission
- 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
- #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
Combating multipath channel
- Multipath prop. destroys orthogonality
- Requires adaptive receiver – channel sensing required (channel sounding by pilot tones or using cyclic extension)
- Cyclic extension (decreases sensitivity)
- One can deal also without cyclic extension (multiuser detection, equalizer techniques)
- More sensitive receiver in general
- More complex receiver - more power consumed
Pilot allocation example
To be able to equalize the frequency response of a frequency selective channel, pilot subcarriers must be inserted at certain frequencies:
Pilot subcarriers at some, selected frequencies
Between pilot subcarriers, some form of interpolation is necessary!
Subcarrier of an OFDM symbol
Pilot allocation example cont.- A set of pilot frequencies
The Shannon sampling theorem must be satisfied, otherwise error-free interpolation is not possible:
maximum delay spread
- Path Loss
- Shadow Fading
- Flat fading
- Doppler spread
- Delay spread
- Inter-symbol interference (ISI) – flat fading, sampling theorem must be fulfilled
- Inter-carrier interference (ICI) – multipath propagation (guard interval)
Multipath radio channel
*Spike distance depends on impulse response
Multipath channel model
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 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 histogram-function)
- #3 add the theoretical pdf to #2
- Note that true pdf area equals unity, how could you adjust the above for this?
- Add comparison to the theoretical Rayleigh distribution!
N(0,s2): Log normal distribution
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 in
- 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’!
OFDM in a multipath channel.mcd
Single carrier transmission and rms delay spread
Determine by using MathCAD’s linterp-function the maximum rate for delay spread of 70 us!
Rate and delay spread.mcd
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!
Slide 82 Slide 84
Envelope Power Statistics
- 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)
Slide 88 Slide 91
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 wave’s amplitudes
- #3 Apply #2 to #1 and plot the result
- #4 compare #1 to #3 in frequency domain log-scale with different levels of clipping
- 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.
- Cancel the peaks by simply limiting the amplitude to a desired level
- Out-of band radiation
- Side effects be reduced by applying different clipping windows
Comparing clipping and peak cancellation
b) peak cancellation
Effect of peak cancellation on packet error rate (PER)
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
Slide 105 Slide 106
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
- 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
- Create a MathCAD script to investigate frequency offset produced ICI