OFDM with MathCAD

Advanced telecommunications
for wireless systems
Investigating 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.
• Topics:
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
• 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
– 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
• 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
• 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
• Objectives: High capacity and variable bit rate
information transmission with high bandwidth
• 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
• 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
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
s t  
 g  t  kT 
k 
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
g k  t    an,k exp  j 2
n  N 2
N 2
 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
In summary, the OFDM TX signal is serially converted IFFT
of complex data symbols an,k
Orthogonality of subcarriers
( k 1)T
s1 (t )s2 (t )dt  0
Orthogonality over the FFT interval:
 TS 2 m  n
cos  2 mt TS  cos  2 nt TS  dt  
Phase shift in any subcarrier - orthogonality over the
FFT interval should still be retained:
cos  2 mt TS    cos  2 nt TS  dt  0
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
– #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
OFDM Spectra
OFDM in frequency domain
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.
Spectra for multiple carrier
Single subchannel
OFDM spectrum
Subcarrier spacing
= 1/TFFT
Spectral nulls at
other subcarrier
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
– #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
– #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
– 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
– 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.
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
Combating multipath channel
• Multipath prop. destroys orthogonality
• Requires adaptive receiver – channel sensing
required (channel sounding by pilot tones or
using cyclic extension)
• Remedies
– Cyclic extension (decreases sensitivity)
– Coding
• 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:
D f  1 2Tm
Tm  maximum delay spread
• Path Loss
• Shadow Fading
• Multipath:
– Flat fading
– Doppler spread
– Delay spread
• Interference
– Inter-symbol interference (ISI) – flat fading, sampling theorem
must be fulfilled
– Inter-carrier interference (ICI) – multipath propagation (guard
Multipath radio channel
*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/
– #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
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
• #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 MathCAD’s
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
Peak-power problem
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
– #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 wave’s 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
Norwood, MA, USA: Artech House, Incorporated, 2004.
p 150.
Selective mapping
• 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
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
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
• Follow the previous script and…
– Explain how it works by own words
– Comment the result with respect of
information theory
System impairments
Exercise ICI
• Create a MathCAD script to investigate
frequency offset produced ICI
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