Introduction to OFDM Characteristics o OFDM (Orthogonal Frequency Division Multiplexing Parallel data transmission with very long symbol duration - Robust under multi-path channels Transormation o a requency-selective channel into N requency-lat channels Cyclic preix (CP is used to eiciently eliminate inter-symbol intererence (ISI. High spectral eiciency o the orthogonal subcarriers Eicient implementation o the transmitter-receiver pair using Inverse Fast Fourier transorm (IFFT and Fast Fourier transorm (FFT High pea-to-average power ratio (PAPR and sensitivity to carrier requency oset (CFO D/A Conversion - 6 / 54
Introduction to OFDM (cont. General system structure o OFDM transmitter Data Sequence Modulation and Coding S/P Inverse DFT Add Cyclic Preix/ Postix Transmit signal to channel RF Device Pulse Shaping D/A P/S Windowing D/A Conversion - 7 / 54
Review o Non-IDFT-Based Signal Model and its PSD Non-IDFT-based signal model or OFDM system Common continuous-time signal model in many literatures or a single OFDM symbol in complex baseband would be Y( t = = X e jπt 0 t < where X is the data symbol transmitted by the th subcarrier whose carrier requency is. The length o a single OFDM symbol and the subcarrier requencies are related as = which leads to the orthogonality o the signals e observation interval with time duration T 0. T 0 or =,,, in the For simplicity, assume that X s are uncorrelated, proper complex, zero-mean random variables with unit variance., T 0 j π t D/A Conversion - 8 / 54
Review o Non-IDFT-Based Signal Model and its PSD (cont. Non-IDFT-based signal model or OFDM system (cont. For the transmission o more than one OFDM symbol, the signal model is modiied to which X [m] is the th subcarrier data symbol or time-index m, /T b is the OFDM bloc transmission rate and w C (t is the windowing unction. T b is the OFDM bloc transmission time and selected as larger than or equal to T + 0 τ max where Yˆ ( t = τ max = X m= [ m] w C ( t mt is the maximum delay spread o the channel. The shape o the roll-o o the windowing unction w C (t is usually designed to reduce the inter-carrier intererence (ICI under carrier requency oset (CFO. e jπt I no CFO, not only IBI but also ICI can be completely removed. b D/A Conversion - 9 / 54
Review o Non-IDFT-Based Signal Model and its PSD (cont. Example o non-idft based continuous-time signal structure with windowing Yˆ ( t = = X m= [ m] w C ( t mt b e jπt Windowing unction (overlapping with adjacent OFDM bloc Guard interval τ max One bloc period T b One OFDM symbol period T 0 t D/A Conversion - 30 / 54
Review o Non-IDFT-Based Signal Model and its PSD (cont. Power spectral density (PSD o non-idft-based signal model The PSD o this signal model is given by S = Yˆ Yˆ ( WC ( Tb = where W C ( is the continuous-time Fourier transorm (CTFT o the windowing unction w C (t. Note that X [m] s are uncorrelated, proper complex, zero-mean random variables with unit variance. I a wideband analog ilter is additionally used, i some subcarriers are null-out, and i dierent transmit power is allocated or each subcarrier, signal model can be rewritten to a more general orm shown in the next page. D/A Conversion - 3 / 54
D/A Conversion - / 54 3 Review o Non-IDFT-Based Signal Model and its PSD (cont. Power spectral density (PSD o non-idft-based signal model (cont. General orm o non-idft-based signal model - p(t is the complex envelope o the bandpass transmit ilter. - is the complex weighting actor or the th subcarrier. ( is zero. The th subcarrier is called a null subcarrier. - X [m] is the th subcarrier data symbol or time-index m. - w C (t is the continuous-time windowing unction. - /T b is the OFDM bloc transmission rate. The PSD o this signal model is given by = = = t j m b C e mt t w m X t p t Z ( ] [ ( ( π α. ( ( ( = = C b ZZ W T P S α
Review o Non-IDFT-Based Signal Model and its PSD (cont. Power spectral density (PSD o non-idft-based signal model (cont. Parameter α =, = 3 Rectangular windowing (a : (b : T b =T 0 T b =.5T 0 Window duration= T b (a : Fourier transorm o the rectangular window unction is a sinc unction. PSD is almost lat except at the edges. (At the edge, the PSD decays approximately at the rate /. = (b : WC ( is a periodic unction with period / T. PSD is not lat in the mid-bands. 0 D/A Conversion - 33 / 54
Review o Non-IDFT-Based Signal Model and its PSD (cont. OFDM signal model in IEEE 80..a Baseband signal rame : r - t SIGNAL = 6µs, t Data = 0µ s Subrame : r PACET ( t = w ( t = r PREAMBLE - w T, SUBFRAME (t : Continuous-time windowing unction with duration t TR - C : Data, pilots, or training symbols ( t + r N ST / - =/ : Subcarrier requency spacing (=/T 0 in our models T FFT SUBFRAME T,SUBFRAME - T GUARD : Guard time to create the cyclic preix SIGNAL ( t t SIGNAL ( t t For the short training seq. (=0 s, long training seq. (=T GI, data OFDM symbols (=T GI Windowing unction may have the time duration extended over than one OFDM period, T FFT, and the transition time overlapped via an adjacent OFDM symbol windowing. (Also in our signal model ( t C = NST / e jπ ( t + r TGUARD DATA DATA D/A Conversion - 34 / 54
Review o Non-IDFT-Based Signal Model and its PSD (cont. OFDM signal model in IEEE 80..a (cont. Timing-related parameter Parameter Value N SD : Number o data subcarriers N SP : Number o pilots subcarriers N ST : Number o total subcarriers : Subcarrier requency spacing T FFT : IFFT/FFT period T PREAMBLE : PLCP Preamble duration T SIGNAL : Duration o the SIGNAL BPS-OFDM symbol T GI : GI duration T GI : Training symbol GI duration T SYM : Symbol interval T SHORT : Short training sequence duration T LONG : Long training sequence duraion 48 4 5 (N SD +N SP 0.35 MHz (=0 MHz/64 3. s ( / 6 s (T SHORT + T LONG 4.0 s (T GI + T FFT 0.8 s (T FFT /4.6 s (T FFT / 4 s (T GI + T FFT 8 s (0T FFT /4 8 s (T GI + T FFT D/A Conversion - 35 / 54
Review o Non-IDFT-Based Signal Model and its PSD (cont. OFDM signal model in IEEE 80.6.e and WiBro OFDM transmit signal NST / s( t = Re C = NST 0 / j π,0 t T - C : Complex number o data symbol (QAM - = F / N : Subcarrier requency spacing s FFT - N FFT : Smallest power by greater than number o used subcarriers - F s : Smallest power by greater than number o used subcarriers - T g : Guard time to create the cyclic preix e ( t t g s D/A Conversion - 36 / 54
IDFT-Based Signal Model and its PSD. IDFT-based discrete-time signal model Problem o the non-idft signal model : Its straightorward implementation requires local oscillators that are tuned to dierent subcarrier requencies { } where κ = : α 0, i.e., is the number o non-null subcarriers. j π t Idea : Using the times sampling o the signal e in the interval at the rate /T 0 or =,,, Orthogonal vectors that are proportional to the column vectors o the - point IDFT matrix The most common discrete-time signal model or a single vector o OFDM symbols is κ 0 t < T 0 κ Y = S X j where S is the -point IDFT matrix whose (, lth entry is given by s [ l] = e, X is the vector consisting o data symbols { X } and Y is the vector o OFDM = symbol. κ πl D/A Conversion - 37 / 54
D/A Conversion - / 54 38 IDFT-Based Signal Model and its PSD (cont.. IDFT-based discrete-time signal model (cont. This procedure can be rewritten as Operation characteristics - The data symbols are orthogonally multiplexed. - The energy o dierent data symbol is centered at dierent subcarrier requencies. + + + = = j j j j j j j j j e e e X e e e X e e e X S X Y π π π π π π π π π Μ Λ Μ Μ { } X =
IDFT-Based Signal Model and its PSD (cont.. IDFT-based discrete-time signal model (cont. For a transmission over a continuous-time channel, the OFDM symbol is D/Aconverted then up-converted using a local oscillator whose requency is tuned to the center requency o the transmitted bandpass signal. perormed eiciently without multiple local oscillators In many case, IDFT multiplication can be implemented by a computationally eicient inverse Fast Fourier Transorm (IFFT instead o IDFT, then multiplication requires only O ( NlogN operations reduced rom O( N Not or a single OFDM symbol but or a sequence o OFDM symbols, what is the signal model considering all o analog wideband ilter, D/A converter, digital interpolation ilter, cyclic preix (or postix, windowing sequence? Needs some modiications to IDFT-based continuous-time signal model D/A Conversion - 39 / 54
IDFT-Based Signal Model and its PSD (cont.. IDFT-based continuous-time signal model Revised system structure or IDFT-based transmitter bloc diagram or OFDM Data Sequence Modulation and Coding S/P Subcarrier weighting IDFT Add Cyclic Preix/ Postix Null subcarrier insertion Transmit signal to channel RF Device (Upconversion Complex baseband signal Z(t Analog iltering D/A P/S Windowing P/S - IDFT matrix point : Digital interpolation - Length o cyclic preix : L+ M, cyclic postix : M L ( channel length, M : # o overlapped entries o the vectors ater P/S - Length o windowing sequence : + L+ M D/A Conversion - 40 / 54
IDFT-Based Signal Model and its PSD (cont.. IDFT-based continuous-time signal model (cont. D/A converter : would be unctions such as the sample-and-hold - I the D/A input is cloced in every T c [sec], then D/A converter can be viewed as a linear modulator employing a rectangular pulse with duration T c. The D/A converter combined with the analog ilter can be viewed as a linear modulator employing a pulse, say p(t with the pulse transmission rate /T c (also when D/A uses a digital interpolation that operates oversampling. Windowing sequence : Usually designed to properly weight the irst M and the last M symbols, The proper shape o the roll-o o the windowing sequence reduces the inter-channel intererence (ICI. Null subcarriers : Not to use an analog ilter with a sharp transition rom passband to stopband, some carriers whose the index s are around being nulled out. - The useul subcarriers will be less than, but still -point IDFT is perormed with data symbols. D/A Conversion - 4 / 54
IDFT-Based Signal Model and its PSD (cont.. IDFT-based continuous-time signal model (cont. General orm o the complex baseband equivalent or IDFT-based continuoustime signal model is, α Z( t = = m= [ m] s - is the subcarrier weighting actor. - X [m] is the mth symbol or the th subcarrier. - T b is the OFDM symol bloc period given by T = ( + L + M. - T c is the transmit pulse time duration. (OFDM subcarrier spacing is /T 0 [Hz], so T c =T 0 - s (t is the th subcarrier waveorm. α X ( t mt b. b T c D/A Conversion - 4 / 54
IDFT-Based Signal Model and its PSD (cont.. IDFT-based continuous-time signal model (cont. s (t is deined by considering the windowing sequence w D [l], IDFT matrix element s [l], and transmit pulse p(t. So, s (t is given by s + L + M ( t = α l= [ l] p( t lt where α [ l] = wd[ l] s[ l] or =,,3,...,, and l =,,3,..., + L + M. πl j (s [l] is rom IDFT matrix (, lth element and is deined as s [ l] = e, but now up to l = + L + M p(t is the equivalent transmit pulse including all the eects o the D/A converter, digital interpolation ilter, and analog wideband ilter. This modiied signal model loos similar to a symbol synchronous DS-CDMA signal model where s (t serves as the signature waveorm and a [l] serves as the signature sequence, both the th users. c D/A Conversion - 43 / 54
D/A Conversion - / 54 44 IDFT-Based Signal Model and its PSD (cont. Power spectral density (PSD o IDFT-based continuous signal model Fourier transorm o the th subcarrier waveorm s (t Using above result, PSD o this signal model is obtained as where W D ( is the discrete-time Fourier transorm o the windowing sequence deined as 0 0 ( ( ( ( = + + = D ZZ T W T M L P S α { } M L l w D l ] [ + + =. ] [ ( + + = = M L l l T j D D c e l w W π = = + + = 0 0 exp T W P( T j ] [l w P( ( S D M L D π
IDFT-Based Signal Model and its PSD (cont. Power spectral density (PSD o IDFT-based continuous signal model (cont. Non-IDFT-based : S ZZ P( ( = α T b = W C ( IDFT-based : S ZZ P( ( = ( + L + M T α WD( 0 = T0 Signiicant dierence rom the non-idft-based model is that the Fourier transorm + L+ M o the windowing sequence { w D [ l] } l= is used instead o that o the continuous windowing unction w c (t [Note] Two properties o W D ( ( W D ( is periodic with a period /T c ( W D ( T0 is periodic in with a period under the assumption T c =T0 D/A Conversion - 45 / 54
IDFT-Based Signal Model and its PSD (cont. Example o PSD o IDFT-based continuous signal model Parameter Rectangular windowing sequence or (a : =0, L+ M =0 (b : =0, L+ M = [Note] W D ( sin( π ( + L+ = sin( πt c M T c Increase in (+L+M Decrease in the width o the main lobe and increase in the amount o in-band energy contained in the interval T0 T0 Periodic with period /T c D/A Conversion - 46 / 54
IDFT-Based Signal Model and its PSD (cont. New insight into PSD o OFDM signals Deinition I the aggregate transmit pulse p(t has a positive excess bandwidth, then, due to the periodicity o W D (, a logical subcarrier can have the signal energy + l concentrated at = or some integers l, that is, a logical subcarrier can have more than one physical subcarriers. T 0 Nominal subcarrier : Physical subcarrier with the greatest energy, and Ghost subcarriers : Other physical subcarriers except the nominal subcarrier For some OFDM logical subcarriers, we may observe two physical subcarriers, the nominal subcarriers and ghost carrier, using IDFT-based continuous signal model. D/A Conversion - 47 / 54
IDFT-Based Signal Model and its PSD (cont. Example : Energy spectral density o the subcarrier waveorms p(t is the SRRC pulse with =0.5, =8, L=0, M=0, no null subcarrier The subcarriers with indexes =3,4, and 5 have the signiicant portion o the energy split into two physical subcarriers located /T c [Hz] apart. D/A Conversion - 48 / 54
IDFT-Based Signal Model and its PSD (cont. Example : Virtual roll-o eect by using proper null-subcarrier Very ast roll-o in the PSD may be needed, but designing such a ilter is costly. Because we need a high-rate D/A converter, high order wideband transmit ilter Eicient solution : By nulling subcarriers with indexes PSD S ZZ ( o complex baseband signal Z(t Parameter - =8, L=0, M=0, 9 null-subcarriers - Sample-and-hold in D/A converter (a : No wideband analog iltering (b : Use o an analog ilter to mae p(t the SRRC pulse with =0. Possible to mae a transition band very small D/A Conversion - 49 / 54
IDFT-Based Signal Model and its PSD (cont. D/A Conversion - 50 / 54
IDFT-Based Signal Model and its PSD (cont. In WiBro standard, the signal model is given by NST s( t = Re = NST 0 / C / e jπ ( t t g,0 t T s A rectangular windowing unction without any window overlapping is used. Non-IDFT-based signal model is used. To use IFFT in the transmitter, we need the ollowing or an equivalent : A rectangular windowing sequence without any window overlapping A sample-and-hold D/A converter with inverse-sinc shaped transmit ilter in the requency band or non-null subcarriers D/A Conversion - 5 / 54
IDFT-Based Signal Model and its PSD (cont. W D ( = sin( π ( + L Tc sin( πt sin( π ( + L T πt c c c or large +L and or 0 D/A Conversion - 5 / 54
IDFT-Based Signal Model and its PSD (cont. The transmit ilter must invert the sinc square envelope to equally distribute the power among non-null subcarriers. The transition rom the passband to stopband must lie over the nullsubcarrier requency band. D/A Conversion - 53 / 54
Conclusions D/A conversion is a necessary procedure in modern digital audio systems and in digital communication systems. Y ( P( X ( T c = d Upsampling combined with interpolation is oten used in commercial DACs and in modern radio transceiver. Y ( X ( N d = d IFFT-based OFDM signaling also requires D/A conversion. S P( ZZ ( = α WD( ( + L+ M T 0 = T0 To ease the burden o designing a good interpolation ilter, null subcarriers are used. The interpolation ilter must have the transition rom passband to stopband in the null subcarriers requency band. D/A Conversion - 54 / 54