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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.

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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.

  • 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


    • 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.


Spectra of a sinus

Fourier transformation of a sinusoidal



  • 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.

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)

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 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:

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

OFDM Spectra

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

Single subchannel

OFDM spectrum

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


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

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

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 ~ 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



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:

maximum delay spread




  • Path Loss

  • Shadow Fading

  • Multipath:

    • 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)

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/ Rice

    • #4 Plot #3 in frequency domain

Comment how realistic this simulation is? Rayleigh or Rice fading channel?

impulse response radio

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 histogram-function)

  • #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 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’!

The channel

OFDM in a multipath

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

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!



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 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)


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.

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 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



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

System impairments

Exercise ICI

  • Create a MathCAD script to investigate frequency offset produced ICI


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