Unformatted text preview: The Cyclic Preﬁx of OFDM/DMT – An Analysis Werner Henkel, Georg Taub¨ ck o Telecommunications Research Center (ftw.) A1220 Vienna Henkel,Tauboeck @ftw.at ¨ Per Odling, Per Ola B¨ rjesson, Niklas Petersson o Lund Institute of Technology SE221 00 Lund Per.Odling,POB,Niklas.Petersson @tde.lth.se Albin Johansson Ericsson Telecom AB SE126 25 Stockholm [email protected]
Abstract— We address the impact of a too short cyclic preﬁx on multicarrier systems such as Orthogonal Frequency Division Multiplex (OFDM) and Discrete MultiTone (DMT). The main result is that the intersymbol interference (ISI) and intercarrier interference (ICI) may be spectrally concentrated and analytical expressions showing this are given. A practical implication is, e.g., that the cyclic preﬁx in some xDSL systems can be surprisingly short, as shown in one example of ADSL transmission. Keywords— Intersymbol interference (ISI), intercarrier interference (ICI), multicarrier transmission, cyclic preﬁx, timedomain equalization. We give a mathematical analysis of the ISI and ICI for a system with insufﬁcient length of the cyclic preﬁx for the case when all the tones are used to transmit data in one direction, for instance in simplex communication or for a timedivision duplex (TDD) system. This is done in Section II. In Section III, we present an example where the interference is strikingly concentrated in frequency. Fig. 2. The system under analysis: the receiving end of a multicarrier system II. S IGNAL MODEL AND I NTERFERENCE C ALCULATION The interference that we are studying consists of two parts: the intersymbol interference (ISI) and the intercarrier interference (ICI). We start with analyzing the ISI, which can be derived in a more intuitive fashion. We will also see that the ICI is of very similar structure, and can be described with the same mathematics under our present assumptions. through a chanConsider the transmission of symbols of length . We exnel with the impulse response tend our notations with an index on the input sequence as we need to distinguish between the input corresponding to , which gives rise to the ICI, and the present symbol which causes the ISI1 . The signal the previous data is assumed to be zero mean with a variance of and are assumed to be pairwise uncorrelated. The received signal, which is to be processed by the FFT, This work was partially ﬁnanced by the Telecommunications program of the Swedish National Board for Industrial and Technical Development (NUTEK) ¨ and by the Austrian Kplus program. It was initiated by Per Odling while being with ftw. We do not consider the case where the evaluation frame at the receiver is positioned such that it has post and precursors from both neighboring frames. In a real system, this could be the case. However, this would make the math lengthy and difﬁcult to follow. Thus, in order to outline the fundamental properties, we decided to restrict this presentation to only postcursors from a preceding frame. 0780372573/02/$10.00 2002 IEEE. 221 Fig. 1. Symbol structures F 20 1) ¤ RT SQ V2 `yx H E¢ F 0 2) G) yxW¥( F 20 G) ¤ ¨ ©¦ 20 1)¥( wsuvt F P rp e20 sqfG) i¥Gh(gf01) ¤ 2 0) e 2 ¢F& ¢F & H¢ F 2 d ) bbacb`2 Y$ ) VW2 U$ ) F Va aV X G) 20 I 20 ¦ G) ¨©P §¤ 20 G) ¤ F N this paper, we address the impact of a too short cyclic preﬁx in multicarrier systems [1]. The cyclic preﬁx removes the intersymbol interference (ISI) and intercarrier interference (ICI) [2]. The introduction of the cyclic preﬁx of length , see Fig. 1, gives a constant capacity loss, since the channel does no longer carry data for short periods of time. As such, one would like to minimize the length of the cyclic preﬁx, preferably maintaining performance. Common wisdom is to choose the cyclic preﬁx to be of roughly the same length as the channel (or system) impulse response, thus eliminating ISI and ICI. It is also well established that, if the tail of the impulse response contains only very little energy, it has little impact and can be considered zero allowing a shorter cyclic preﬁx. We show that, furthermore, the ISI and ICI can be spectrally concentrated and sometimes have limited or even zero impact on performance. Although this is not completely unknown among designers of DMT digital subscriber loop (DSL) systems, it has rarely been given a thorough analysis. Our attempt here is to provide an intuitive and immediate understanding of the mechanisms involved. DFT 2C ED) A B $ 2) 01¥( £ 2) I. I NTRODUCTION ¡ 2 ) @ 3 8974© 65 &' ¢ %! $ ¨ ¤ ¢ ! ¡ ¤ £ ¤ ¢ ¨ ¦¤ ©§¥£ (1) ¢ #" ! , i.e., . This is a reasonable assumption. One would certainly not design a DMT (OFDM) system with a frame length shorter than the channel impulse response. While is increased, the upper limit of the inner sum is decreased accordingly. The lower limit is when . If we now reached take the sum over as outer sum, we obtain the upper limit for the inner sum over as and thus, Substituting
impulse responses yields G G G G The signal after the DFT, , then becomes (4) impulse responses H G evaluation frame
Fig. 5. Hypothetic extension ”H” of the guard interval ”G” for computing the ICI Fig. 4. Summation area in (5) Next, we interchange the two sums. Due to the dependence of in the inner sum, we have to investigate the summation over in some more detail. Figure 4 shows the effecthe pair tive pairs that are used in the two sums. There, we assume that Remains now to calculate the ICI term. ICI would not be an issue if the guard interval would be big enough to fake the channel convolution as a cyclic convolution within the frame that is evaluated by the receiver. Thus we just assume an extension of the guard interval that would null the ICI and compute its impact on the evaluation frame. Its negative would then be the ICI 222 ³³ · ´´ ¸ ³´³´ ·¸·¸ ³³ · ´´ ¸ ³´³´ ·¸·¸ ³´ ³´ ³´ ³´ ·¸·¸ ·¸ p u ° °¨ S RT Q R rq`) ¦ 2 rp ±±±µµ ² ² ² ¶ ±² ¶ ±²µ¶ ±±±µµ ² ² ² ¶ ±² ¶ ±²µ¶ ±² ±² ±² ±² µ¶ ±² µ¶ µ¶ e (5) R ¬ F F u ª  x ¦ ® #{z x v `2 ) 2 ¨ ) «¨ p p u® ¨ u x ¯ ¨ x ¬ ¦ ¦ r sp r sp e 2q r`) ® ¨
¦ H 2q r`) With a cyclic preﬁx of length , the residual ISI is determined as rW) 2q
¦ ¨ ¬G bp 2 rqW) ¨©¦ bGp ª ¨ e 2q «©§r`) (3) (10) G 2q rW) bG ! c1 ! p u ¨ F sp r ) a } z{ x y¦ `2 ) 20 $ Gh( vu  w¦
¦p ¨ u ¨ E t F 0G) yW¥( 2 x) rp ¨
¦ a } {z x y¦ `yx v u2  v u2  } z{ x y¦ `yx "n n on '
m wu w svt su F 0 2 x) 1) yW¥( r p t ¨ ¨©¦
¦ w su P t x v u2 0 } {z yw¦ WG) ¤ ¨
¦ 'nSm o Fig. 3. ISI from symbol to symbol j ¢ p 2q r`) ¤ lk Ysj symbol symbol is actually the DFT of the Note that the expression for tail of the impulseresponse. The power spectral density after the FFT due to the intersymbol interference follows to be (8) e evaluation frame x u2 )  ¦ {z yvw¦ WGh( ¦ 2q r`)  ¤ u ¥£ ©u9 a  {z x vw¦ uWGh( r p ¡ 2 ) ¨
¦ p u ¨ F r sp vu ¡¢ z{ x © W2U ) ¨
¦ p u ©u9 ¨ F rs}U ) p2 r sp
¦ ¨ ¨
¦ wsu F (2) In (2), we actually used the concatenated symbols in order to simplify the description. & & d$ ¢ a } {z x y¦ `0 $ v u2  H¢ ¢ f2 $ ) e2 & H X ¢ ¢ &g ¢
) 1¥( 0 e ¢ ¢ H 1)
X 0 H E¢ &¢ 2 Y$ e0 s& ) H E¢ & 0 ¨
¦ ¦ H E¢ ! r2
p ) 0 $ e p u ¨ cG p r sp er`) 2q ¨
¦ 0 e2 rqW) bG p H ! q E¢ where denotes the concatenation of all up to the present symbol. A part of this signal will then be ISI from the previous symbol. Figure 3 illustrates the tails of the impulse response that are not covered by the cyclic preﬁx. Note that in contrast to Fig. 1 not only the guard interval but also its counterpart at the end of the frame is highlighted (hatched) and labeled is with G. The ISI that affects a symbol from a symbol & $ ¢ ¢ X a & dI e¥e¥e¥ ee e¥¥¥ee dd dd dd ¥¥¥dd F 20 G) ¤
H E¢ 0 Wyx V 2 I fg g fg ¥¥¥ ¥f¥¥ ¥¥¥ ¥¥¥fg gf fg fg ¥¥¥ ¥¥¥fg p u ¨ ©7 vt F c1P 0 2 x) 1) yW¥( r p G) e20 ¨
¦ ¨©¦ ¨©¦
u w w
u o t t e p e 2q srW) ¤ es2r`) q e 2 V0 `1) on
n
o
20 1) i hi h¥¥¥ ¥hi¥¥hi hi hi ¥hi¥¥hi cG~p F (6) (7) (9) 0 signal. The hypothetic frame extension is shown in Fig. 5. The negated ICI signal is then similar to (2) and can be written as
mod 0.45 h(n) 0.4 Note that this time, (11) refers to only one input signal block from which the hypothetical guard interval is taken. Correto stay within this frame. In spondingly, we count modulo DFT domain, we obtain mod (12) which corresponds to (4). If we further follow the steps in the derivation of the ISI result down to (10), we see that the only differences are the minus sign and that we count modulo , staying within the same frame. The minus sign disappears with the squaring when computing the power spectral density. If we instead of , does not change the result, either, so consider that we obtain (13) R EFERENCES III. E XAMPLE As a practical example we choose an ADSL transmission with and a guard interval of an FFT length of samples. We select the impulse response of a 4 km long loop of German 0.4 mm cables, which is shown in Fig. 6, cutting noncausal precursors that are due to the underlying cable model. We see that it certainly has a portion exceeding the guard interval. If we compute the ICI and ISI components according to (10) and (13), we see in Fig. 7 that the noise is predominantly disturbing lowfrequency components. It has long been realized that the signaltonoise ratio decreases near DC, which was considered partly to be due to the leakage effect of the DFT which folds highfrequency noise components into the lowfrequency range. The results in here now show that noise due to ISI and ICI is concentrated around DC as well. IV. C ONCLUSIONS We derived closed formulas for the intersymbol and interchannel interference power spectral density and found that they are actually the same. From a typical example we concluded that this noise will be concentrated around DC. [1] Bingham, J.A.C., ‘Multicarrier modulation for data transmission: An idea whose time has come’, IEEE Communications Magazine, vol. 28, no. 5, pp. 5–14, May 1990. [2] Peled, A. and Ruiz, A., ‘Frequency domain data transmission using reduced computational complexity algorithms’, Proc. IEEE ICASSP, pp. 964–967, Denver, Colorado, 1980. [3] Henkel, W., Kessler, Th., ‘Maximizing the Channel Capacity of Multicarrier Transmission by Suitable Adaptation of the TimeDomain Equalizer’, IEEE Transactions on Communications, Vol. 48, No. 12, pp. 20002004, Dec. 2000. 223 ÅÄ¾§ÃÀ É ¾¿ ÂÁ ÅÄ7ÂÃ ¾¿ Æ ¾ ÁÀ ISI and ICI have the same power spectral density. This is an important ﬁrst result that has relevance for practical timedomain equalizer algorithms (see, e.g., [3]). Ë}Ç Èr yÊÇÉ X ¼e ½ ¢ p u ¨ ° ¹ ¦° RT Q X bG y1 ! r sp a R r`) 2q ¡e
¦ ¨ and (14) ! ! vu  } {z xyy¦ S2 Vº2 ! 2 yx
2q rW) 2 yx ¢ X a bG ! ¹ y1 ! fr`) e 2q F 0) 1) ¤ y`h( 2 x) H E¢ ! p u w ©7¨ vt
u F sp r p sp r 01) ¤ 2yW)¥( ) x ¨
¦ ¦ ¹ wGp & ¼e X» 0 ' p u ©7 vt ¨ r sp
F ¨
¦ 0.35 ¨
¦ ! e20 fG)
e 2q rW)
¤F (11) ¹ y1P
¤F 0.3 0.25 0.2 0.15 0.1 0.05 0 0 50 100 150 200 250 sample no. Fig. 6. Impulse response of a 4 km, 0.4 mm loop
H(q) /Hmax [dB]
2 0 −5 −10 −15 −20 −25 −30 −35 −40 0 64 128 192 256 carrier no. Fig. 7. ISI and ICI power spectral density according to (10) and (13) normalized to its maximum ( ) ...
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 Spring '10
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 Telecommunications , spectral density, Impulse response, ICI, cyclic prefix

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