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
• 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
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.
11. 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)
12. 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.
13. 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)
14. 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)
15. 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.
20. OFDM signal in time domain
2
,
2
0
exp 2
N
k n k
n N S
n
n
g t a j t
T
1 S S
k T t kT
k S
k
s t g t kT
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
21. Orthogonality of subcarriers
0
2
cos 2 cos 2
0
FFT
T
S
S S
T m n
mt T nt T dt
m n
Orthogonality over the FFT interval:
Phase shift in any subcarrier - orthogonality over the
FFT interval should still be retained:
0
cos 2 cos 2 0
FFT
T
S S
mt T nt T dt m n
( 1)
1 2
( ) ( ) 0
k T
kT
s t s t dt
Definition:
22.
23.
24. 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
25.
26. 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
30. OFDM in frequency domain
TG TFFT
Square-windowed sinusoid in time domain
=>
"sinc" shaped subchannel spectrum in frequency domain
sinc sin
FFT FFT FFT
fT fT fT
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=1
0160973&ppg=932
31. Spectra for multiple carrier
Single subchannel OFDM spectrum
Spectral nulls at
other subcarrier
frequencies
Subcarrier spacing
= 1/TFFT
34. 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
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
38. OFDM Spectra by MathCAD for a single carrier
ofdm spectra by rect windowed sinc.mcd
40. 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
45. 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.
49. 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
53. 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
54. Pilot allocation example
To be able to equalize the frequency response of a
frequency selective channel, pilot subcarriers must be
inserted at certain frequencies:
Between pilot
subcarriers, some
form of interpolation
is necessary!
Frequency
Time
Pilot subcarriers at some,
selected frequencies
Subcarrier of an OFDM symbol
55. Pilot allocation example cont.
- A set of pilot frequencies
The Shannon sampling theorem must be satisfied,
otherwise error-free interpolation is not possible:
Frequency
Time
m
1 2
f m
D T T
f
D
maximum delay spread
61. 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
62. Comment how realistic this simulation is? Rayleigh or Rice fading channel?
impulse response radio ch.mcd
64. 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
65. Rayleigh distribution
- Note that true pdf area equals unity, how could you adjust the above for this?
- Add comparison to the theoretical Rayleigh distribution!
71. 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
72. 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’!
81. 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!
87. 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)
92. 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
93.
94. 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&p
pg=166
96. 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
104. 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
108. How would you modify this function to
simulate unipolar system?
109.
110. 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
111. 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
112.
113.
114.
115. 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
