1. Timo Korhonen, Communications Laboratory, TKK
Advanced telecommunications for wireless systems Investigating OFDM by MathCAD
Timo Korhonen, Communications Laboratory, TKK

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
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

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.

7. Rect waveform.mcd

8. Spectra of a sinus wave.mcd

9. Fourier transformation of a sinusoidal wave.mcd

10. Introduction

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 a&g (WLAN) systems
IEEE a (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.

17. OFDM Signal

18. Multiplexing techniques

20. 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

21. Orthogonality of subcarriers
Definition:
Orthogonality over the FFT interval:
Phase shift in any subcarrier - orthogonality over the FFT interval should still be retained:

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

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

27. Orthogonality.mcd

29. OFDM Spectra

30. OFDM in frequency domainTG
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, p 916.

31. Spectra for multiple carrier
Single subchannel
OFDM spectrum
Subcarrier spacing = 1/TFFT
Spectral nulls at other subcarrier frequencies

32. Next carrier goes here!

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

35. Ofdm spectra.mcd

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

39. Spectral shaping by windowing

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

41. burst windowing and ofdm spectra.mcd

43. Modeling OFDM Transmission

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.

48. http://www.eng.usf.edu/wcsp/OFDM_links.html ~ Aalborg-34-lecture13.pdf
OFDM and FFT
~ Aalborg-34-lecture13.pdf

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

50. Exercise : Constellation diagram of OFDM
Ofdm system.mcd

52. Channel

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:
Pilot subcarriers at some, selected frequencies
Time
Between pilot subcarriers, some form of interpolation is necessary!
Frequency
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:
maximum delay spread
Time
Frequency

56. Channel Path Loss Shadow Fading Multipath: Interference OFDM:
Flat fading
Doppler spread
Delay spread
Interference
OFDM:
Inter-symbol interference (ISI) – flat fading, sampling theorem must be fulfilled
Inter-carrier interference (ICI) – multipath propagation (guard interval)

57. Multipath radio channel

58. *
*Spike distance depends on impulse response

59. Multipath channel model

60. Received signal Pr: Received mean power Gs: Shadow fading
Log r ~ Fast fading

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
Comment how realistic this simulation is? Rayleigh or Rice fading channel?
impulse response radio ch.mcd

63. Frequency response
Frequency response shown by swapping left-hand side of the fft

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!

66. Path loss

67. Shadow fading
N(0,s2): Log normal distribution

68. Doppler spread

69. DELAY SPREAD IN TIME DOMAIN
Small delay spread
Large delay spread

70. Delay spread in frequency domain

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’!

73. The channel
OFDM in a multipath channel.mcd

75. Single carrier transmission and rms delay spread
Determine by using MathCAD’s linterp-function the maximum rate for delay spread of 70 us!

76. Rate and delay spread.mcd

77. Cyclic prefix

80. Example: a two-path channel

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!

82. cyclic prefix.mcd

84. Detection

85. Peak-power problem

86. Envelope Power Statistics
The OFDM signal represents the vector additions of a large number of relatively uncorrelated
carriers. For this reason the power envelope of an OFDM signal is not constant. Often a
single metric, peak-to-average ratio (PAR) is used to describe the amount of headroom
required in an amplifier. For OFDM signals, this metric is not very useful as the “real peak”,
whatever that is, may not occur very often.
It’s usually more meaningful for OFDM signals to associate a percentage probability with a
power level. For example, in the plot shown, the signal exceeds the average power (red line)
40% of the time. (It would be 50% only if the mean and median were identical). It exceeds a
level that is 4 dB above the average, 5% of the time. In other words, if we ran this particular
signal through a PA with 4 dB of headroom, or back-off, the signal would clip 5 % of the time.
This is more useful than knowing that the peak-to-average for this signal is 8dB.

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)

88. OFDM papr.mcd

91. Transfer characteristics of power amplifier

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

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, p 150.

95. Selective mapping

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

99. Peak cancellation

101. Comparing clipping and peak cancellation
a) undistorted
b) peak cancellation
c) clipping

102. Effect of peak cancellation on packet error rate (PER)

103. OFDM transceiver

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

105. DESIGN
EXAMPLE

106. Exercise: BPSK error rate

108. How would you modify this function to
simulate unipolar system?

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

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

116. System impairments

120. Exercise ICI
Create a MathCAD script to investigate frequency offset produced ICI

121. Summary