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.
Advanced telecommunications for wireless systemsInvestigating OFDM by MathCAD
Timo Korhonen, Communications Laboratory, TKK
Spectra of a sinus wave.mcd
Fourier transformation of a sinusoidal wave.mcd
Examples of OFDM-systems
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:
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!
ofdm spectra by rect windowed sinc.mcd
burst windowing and ofdm spectra.mcd
http://www.eng.usf.edu/wcsp/OFDM_links.html ~ Aalborg-34-lecture13.pdf
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
*Spike distance depends on impulse response
Pr: Received mean power
Gs: Shadow fading
Log r ~ Fast fading
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
- 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
OFDM in a multipath channel.mcd
Determine by using MathCAD’s linterp-function the maximum rate for delay spread of 70 us!
Rate and delay spread.mcd
Prasad, Ramjee. OFDM for Wireless Communications Systems.
Norwood, MA, USA: Artech House, Incorporated, 2004. p 150.
b) peak cancellation
How would you modify this function to
simulate unipolar system?