Inter Symbol Interference_ Luan Van

7/31/2019 Inter Symbol Interference_ Luan Van

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Inter Symbol Interference (ISI) v raised cosine filteringInter-symbol interference (ISI) l m t h u qu khng mong mu n c a c h th ng thng tin v tuy n v h u tuy n.

Hnh 1 m t tn hiu pht i (a) v tn hiu nhn c (b)

Trong chu k u tin c a qu trnh truy n d n, cc tn hi u nh n c c xu h ng lm cho di ra v b l n vo nhau. M t xung ng n A short pulse to represent a dot was received as a much-smeared version of the same thing.V n y c lin quan t i c tnh c a mi tr ng truy n d n v kho ng cch

truy n. o m nh ng hi u ng khng mong mu n , cc tr m l p c

thi t l p v cc cch c tnh ton gi m nhi u an d u ISI ny. Hnh 2 ch ra dng d li u 1,0,1,1,0, chu i d li u mong mu n pht i. Chu i d li u ny c d ng xung vung. Cc xung vung c m t tru t ng v mang tnh ch t l thuy t nh ng trong th c t chng r t kh c t o ra. Do chng ta ch t o ra c cc xung vung c hnh dng gi ng nh c ch ra trong ng ng t qung trong hnh sau.Hnh dng v c b n trng gi ng nh 1 xung vung v chng ta c th ni l nh ng chu i d li u c g i i.

u i m c a (1 hm ton h c) hnh dng ny l n lm gi m cc yu c u b ng t n

v c th t o ra c b ng ph n c ng.


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Hnh 2 - Chui 101101 c pht i v ng t qung l hnh dng thc t ca n.Hnh 3 ch ra m i symbol nh n c. Chng ta c th xem mi tr ng truy n d n t o ra 1 ph n trng l p ln nhau. Ph n n ng l ng c a symbols 1 v 2 trng ln symbol 3. M i symbol gy nhi u ln ph n cc symbol khc.

Hnh 3 - Mi symbol c tri theo thi gian trn knh truyn dnHnh 4 ch ra tn hi u t i my thu. N l t ng c a t t c cc symbol b mo. So snh t i ng nt g ch m tn hi u c pht, tn hi u thu c trng khng hon ton r rng. My thu khng phn bi t c tn hi u ny, n ch trng nh ch m nh , gi tr c a bin trong m t kho ng th i gian. r ng symbol 3, gi tr ny ch kho ng gi tr c pht, gi tr t o ra symbol ny th nh y c m h n n nhi u v s th hi n sai v hi n t ng ny l k t qu c a tr nh ng symbol v trng ln nhau smearing.

Hnh 4 Tn hiu nhn c vi tn hiu c pht mc

S tr i v trng ln nhau smearing c a cc symbol nh v y n ng l ng t 1 symbol nh h ng cc symbol ti p theo, theo cch tn hi u thu c c xc su t cao h n di n gi i sai c g i l Inter Symbol Interference or ISI.


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ISI c th gy nn b i nh ng nguyn nhn khc. N c th t o nn b i cc hi u ng l c t ph n c ng ho c nhi u a ng frequency selective fading, t khng

tuy n tnh v t cc hi u ng tch d n. Nhi u h th ng ch ng c nhi u ISI v n g n nh lun lun xu t hi n trong thng tin v tuy n.H th ng thng tin c thi t k cho c h u tuy n v v tuy n lun lun c n thi t k t h p m t vi ph ng php i u khi n n.


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What can you do about ISI?

V n chnh l n ng l ng, ph n chng ta mong mu n h n ch t i nhau, gi m nh h ng t i symbol khc. V th 1 i u n gi n nh t lm gi m ISI, i u

ng ngh a v i vi c lm gi m tn hi u. Ch cho php pht xung ti p theo c a thng tin sau khi tn hi u nh n c gi m ISI, c ch ra trong hnh sau. Th i gian th c hi n lm gi m tn hi u c g i l tr i tr delay spread, ng c l i th i

gian truy n xung nguyn b n original time c a xung c g i l symbol time. N u tr i tr (Xem hnh 1) nh h n ho c cch khc k t qu cn b ng v i th i gian truy n xung nguyn b n sau khi khng cn ISI n a.

Gi m t c truy n bit rate l cch chnh i u khi n ISI trn cc ng truy n d n. Sau s d ng cc chip x l t c cao h n v cho php chng th c hi n x l tn hi u i u khi n ISI v t ng t c truy n d n nhanh chng.

Trong lu n v n ny i v i t c truy n d n cao h n, vi c lm gi m t c d li u l m t v n d dng nh ng l 1 gi i php khng ch p nh n c. Thm vo chng ta c th ki m sot ISI m khng yu c u gi m t c truy n d n?

Ph ng php chnh ki m sot c ISI l hnh d ng xung pht. Hnh d ng xung pht gip ki m sot ISI nh th no? i u b m t l s d ng x l gi i i u ch tn hi u s . Khi cc ph n th i gian xung tn hi u c xc nh gi tr c a tn hi u t i kho ng th i gian nh t nh, i u khng quan tm tn hi u no c hnh dng ra sao tr c khi ho c sau tn hi u . V th n u c cch no chng ta gi cc symbol kh i nhi u sao cho chng khng nh h ng n bin t i slicinginstant, chng ta c th ki m sot c ISI thnh cng. Xem ng bao tn hi u trong hnh 5. Tuy nhin my thu ch xem xt t i m i i m c a th i gian t n t i xung timing pulses (xem tn hi u sau) v ph n cn l i c a s thay i l khng nh h ng. V th t i nh ng i m nh th , chng ta c

th lm gi m nh h ng c a cc symbol li n k , i u yu c u chng ta c n thi t ph i lm gi m nh h ng.

Hnh 5 Chng ta ch quan tm n tn hiu ti thi im ly mu. S dng 1 xung vung

Hy nh ngh a cc i l ng sau Ts = Symbol Time, 1 s trong v d m u sau. (Khng ch p nh n th i gian m. Symbol time c ko di ra c tr c d ng v tr c m)


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Rs, t c truy n symbol l ngh ch o c a Symbol Time, T s. Rs lin quan tr c ti p n b ng thng sao cho t c truy n symbol l n h n, yu c u b ng thng nhi u h n.Rs = 1/TsHnh 6 ch ra 1 xung vung c bin A, truy n trong 1 s. (T -0.5 s n +0.5 s)

Hnh 6 Xung vung trong min thi gianKhi th i gian t n t i symbol l 1 s, t c truy n l 1 symbol / s. p ng t n s c a xung vung ny (theo bi n i Fourier) theo cng th c

Khi Ts = 1 symbol time =1 sec


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Hnh 7 p ng tn s ca xung vung l 1 hm hnh sincTrng hp thng thp th bng thng di.

Trong tr ng h p trn, th i gian t n t i c a symbol l 1 s. Do t c truy n symbol x p x = 1. p ng t n s c a xung vung c d ng c a hm hnh sinc (sin x/x). N c bin

c c i AT s v n i qua tr c honh t i cc i m R s. B ng t n th p c xc nh t i m ban u t i i m 0 u tin, t ng ng v i 2 l n t c truy n symbol ho c l 1 Hz. Tr ng h p b ng thng d i th g p i. Hnh sau ch ra r ng m t vi xung vung v p ng t n s c a chng.


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Hnh 8 Hiu ng thi gian ca xung vung v p ng tn s ca nM t xung h p (xung m g n gi ng v i hm delta) c p ng t n s r ng v n c nhi u a lot of frequency content. A wide one (similar to a flat line if you squint orhave a good imagination) has lesser frequency content v hence its bandwidth issmaller. For each pulse, the bandwidth which we measure is only on the positivehalf andis equal v its symbol rate in Hz.The important thing to note at this point is that a square pulse of symbol rate Rshas abandwidth of Rs Hz (for bandpass signal it is twice that.)A very important relationship:Bandwidth of a square pulse= Rs for lowpass signals,= 2 times Rs for bandpass.The frequency response of the square pulse goes on forever. This is not a goodthing.This would theoretical interfere with others v is not allowed by the FCC.The square pulse has some disadvantages v they are1. The square pulse is difficult to create in time domain because ofrise time v a decay time.2. Its frequency response goes on forever v decays slowly.The second lobe is only 13 dB lower than the first one.3. It is very sensitive to ISI.

If a square pulse gives us a sinc function in the frequency domain, then couldn'twe use asinc function as a pulse shape in time domain v get a brick-wall (square wave)frequency response? We can indeed.


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We can use a pulse that is shaped like a sinc function instead of a square pulsev getthat very nice brick-wall spectrum, with nothing spilling outside the bandwidth.This isgreat.A sequences of bits shaped with sinc pulses may look like this.

Hnh 9 Transmitting a sequence (1011) by shaping the bits as sinc pulsesLet's see what is better about using the sinc pulse as a shape instead of the

square pulse.Now we change the x-axis v go the other way. The sinc pulse becomes the timedomain shape. Its frequency response is the square shape.

Hnh 10 - The effect of sinc pulse v its frequency responseHere is the wonderful part about using the sinc pulse. As opposed to the squarepulse, wesee that using the sinc pulse cuts the bandwidth requirement to one-half! In thefigurebelow, several pulses are shown of symbol rates 1, 2 v 4. In each case, we seethat therequired lowpass bandwidth is one-half of the symbol rate.


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Hnh 11 Pulse width vs. frequency response

The bandwidth achieved by the sinc pulse is called the Nyquist bandwidth inhonor of theman who developed it. It requires only 1/2 Hz per symbol.Can we find something even better? It turns out that we have not been able tofind anyother shape that can improve on this. It l m t ultimate limit for perfect

reconstruction ofthe signal.Wow, two problems gone, this is great! Band-limited spectrum in frequencydomain withno energy going to waste v small total bandwidth requirement. Not so greathowever,because a sinc pulse is actually no more possible to build than is a square pulse.Here is what we are facing1. In time domain a true sinc pulse is of infinite length with tails extending toinfinity so the energy can theoretically continue to add up even after the signalhas

ended. We can only design an approximation to the real sinc pulse of a finitelength. But truncation leads to an imperfect pulse that does not have a true sincpattern v allows ISI to leak in.2. The pulse tails that fall in the adjacent symbols decay at the rate of 1/x so ifthereis some error in timing, this pulse is not very forgiving. It requires near-perfecttiming to achieve decent performance.Nyquist offered ways to build (realizable) shapes that had the same goodqualities asthe sinc pulse v less of the disadvantages.One class of pulses he proposed are called the raised cosine pulses. They are

really amodification of the sinc pulse. Where the sinc pulse has a bandwidth of W, whereWis specified asW = 1/2TsThe raised cosine pulses have an adjustable bandwidth which can be varied fromWto 2W. We want to get as close to W, which is called the Nyquist bandwidth, aspossible with a reasonable amount of power. The factor related the achievedbandwidth to the ideal bandwidth W as


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0

1W

= Wwhere W is Nyquist bandwidth, v W0 is the utilized bandwidth.The factor is called the roll-of factor. It indicates how much bandwidth is beingused over the ideal bandwidth. The smaller this factor, the more efficient thescheme.The percentage over the minimum required W is called the excess bandwidth. Itis100% for roll-off of 1.0 v 50% for roll-off of 50%. The alternate way to expresstheutilized bandwidth issW(1 )R 0 = +The typical roll-off values used for wireless communications range from .2 to .4.Obviously we want to use as small a roll-off as possible, since this gives thesmallestbandwidth.

Here is how the class of raised cosine pulse is defined in time domain.

The first part is the sinc pulse. The second part is a cosine correction applied tothe sincpulse to make it behave better. The sinc pulse insures that the functiontransitions atinteger multiples of symbol rate which makes it easy to extract timinginformation of thesignal. The cosine part works to reduce the excursion in between the samplinginstants.The bandwidth is now adjustable. It can be any where from 1/2 Rs to Rs. It is

greater thanthe Nyquist bandwidth by a factor (1+ ). For = 0, the above equation reducesto thesinc pulse, v for = 1, the equation becomes that of a pure square pulse.Raised Cosine Impulse Response-20 -15 -10 -5 0 5 10 15 20Time, secsAmplitude = .8 = .5 = .1

Sinc pulse


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Hnh 12 Raised cosine impulse response

In frequency domain, the relationship is given by

Why do they call it raised cosine? Because the above response has a cosine

function inehe frequency response looks somewhat like a square pulse as we would expect. Arangethe frequency domain, although other many other trigonometric representationsof thisequation that do not have the cosine-squared term, so it is not always clear whythese arcalled raised cosine.Tof bandwidths are possible depending on the chosen . The bandwidth can be

anywherefrom 1/2 Rs (this term same as W, the Nyquist bandwidth) for the sinc pulse to Rsfor thesquare pulse. The bandwidth utilized is greater than the Nyquist bandwidth by afactor (1+ ). For = 1 the above equation reduces to the sinc pulse, v for = 1 theequationbecomes that of a pure square pulse.

Hnh 13 - The frequency response of the raised cosine pulses of Rs = 1


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xample:ansponder has a bandwidth of 36 MHz. We use QPSK signaling. What dataor unshaped QPSK, which is a square pulse, the bandwidth requirement forlowpass isRs = 1/2 36 MHz = 18 Msps.we use the ideal sinc pulse, we can sendwe shape the pulse with = .3, we can send

Rs = B/(1+

) = 27.7 Mspsor QPSK, with 2 bits per symbol, we get a data rate capability of 27.7*2 = 55.4Mbps.oot-raised Cosine filterEA satellite trrate is possible using a raised cosine signaling with = .3?Fequal to the symbol rate. For passband, the bandwidth requirement is twice that.IfRs = B = 36 Msps

IfB = Rs(1+ )FThis is about standard for most satellite systems.RTo implement the raised cosine response, we split the filtering in two parts tocreate ahees, the whole raised cosine can be applied at once at the transmitter but inpractice it has

Hnh 14 Split filtering of raised cosine response, a root-raised cosine filter at the

he root raised cosine shaping of pulses is also called baseband filtering. Thefrequencymatched set. When we split the raised cosine filtering in two parts, each part iscalled troot-raised cosine. In frequency domain, we take the square root of the frequencyresponse hence the name root-raised cosine.Ybeen found that concatenating two filters each with a root raised cosine response(calledsplit-filtering) works better.


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transmitter v one at the receiver, giving a total response of a raised cosine.Tresponse of the root raised cosine is given byCompare this to the following response for the raised cosine v you see it is just ad theC

square root relation ship. Each of these square root responses are applied in pairsantotal response is that of the raised cosine.Compare the impulse response of the root raised filter to that of the raisedcosine. We donot see much of a difference except that there is a little bit more excursion in therootraisedcosine response. The time domain function is of course NOT the square root. Theroot part applies to frequency domain.

Hnh 15 Impulse response of a. Raised cosine b. Root-raised cosineHnh 16 shows the spectrum of the root raised cosine v the raised cosine. Thebandwidthof the raised cosine filter is specified at the 6 dB point due to the fact that theNyquistfrequency response is reached at 3 dB which is then doubled to give 6 dB forpower. Theroot-raised cosine however is specified at 3 dB point because it is the square root


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Hnh 16 Frequency response of Raised cosine v a Root-raised cosineLooking at the eye diagram at the transmitter v the receiver gives us some

furtherinsight into how the root-raised cosine filtering splits the total filtering.

Hnh 17 Eye diagram of shaped signals, after transmit v receive root raised cosinefiltersRoot raised cosine pulses out of transmitter Root raised cosines pulses at thereceiverThe left side is the signal out of the first root raised cosine filter. The signal doesnot lookthat great. But when it is filtered by the second root raised cosine filter (shown onthe


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right side) the output looks really nice, just as we hoped. As the roll-off factor getslarger,the eye opens up. This says us that if there were no bandwidth limitations, itwould bemuch easier on the receiver if we used a large . But since bandwidth is almostalways alimiting resource, the push is on to make as small as possible.

Design of root-raised cosine filtersThe raised cosine v the root raised cosine filters are both designed as FIR filterswith aspecific number of taps. In time domain the fewer taps mean that the impulseresponsehas been truncated. The response of both the raised v the root raised cosinefilters iseffected by choice of tap length. In general fewer taps give worse rejection. Astapsnumbers are increased, the length of the time domain sequence is increased vthe

rejection increases.

Hnh 18 RRC with taps sizes a. 128 v b. 64.The two versions shown above are for tap length of 128 v 64. The shape for taplength64 is exactly the same as the one for 128 taps; the only difference is that it hasbeentruncated. The effect of this truncation is that in frequency response we get someleakage


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and the spectrum does not drop down as far as one for a longer tap size as shownbelowin Hnh 19. This is true for both raised cosine v root raised cosine filters.

As we can see the rejection increases considerably as number of taps isincreased.However if 3 dB bandwidths used as the discriminator, there is not a large

difference somost RRC filters today use either 48 or 64 tap designs.Figure 19 shows the spectrum of a QPSK signal through the first root-raisedcosine filterof = .3 v various tap lengths.

Hnh 19 Spectrum of a root raised v a raised cosine signal. (a) After the first rootraised cosine at the transmitter, (b) after the second root raised filter at receiver(a) Root raised cosine spectrum

(b) Raised cosine spectrum

Hnh 20 Spectrum of a) root raised v b) raised cosine signal.Sinc equalization of the root raised cosine filterIn the development of raised cosine signaling, we are tacitly assuming that anidealsampler is sampling the signal. We assume that we receive one single pulse for a0 or a 1,which we then shape into root, raised cosine pulse. But in reality the incomingdata maybe analog or digital v may have to be sampled for a variety of reasons such asfor

conversion of sampling rates for modulation.

Hnh 21 - Ideal vs. non-ideal sampling can lead to a sinc biasLet's call the incoming data stream m(t). It has a certain frequency response vhas abandwidth of B. For a perfect reconstruction, we need to sample it at a samplingfrequency of rate 2B or greater. Which means that each pulse should be no morethan1/2B seconds apart, which defines Ts as the pulse time.Ts = 1/fs < 1/2BThe Fourier transform of an ideal impulse train l m t impulse train in the from of

frequency harmonics located at integer multiples of the sampling frequency asshown inFig. 22. So to sample a time domain signal, with a sampling pulse train specifiedby s(t)implies that after sampling we get a harmonically repeating response of themessagestream m(t). Then filter with a lowpass filter of bandwidth B to recover theoriginalsignal.


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But when the sampling pulses have a finite width, they then have their ownfrequencyresponse which gets multiplied by the message signal response.y(t) = m(t) s(t)The Fourier Transform of the two sides gives us the convolution of the FourierTransformof the two signals.

Y( f) = M( f) S( f)The Fourier Transform of a finite width sampling pulses is

The first part is the amplitude at 0; the second part is a sinc function, which is afunctionof the duty cycle of the sampler. The last part just makes the whole thing madediscreteby multiplying it by the delta function. The duty cycle is specified as

The duty cycle states just how wide the sampling pulse is compared to thesymbol time.

A large duty cycle implies a really big (or a square pulse) v a small duty cycleimpliesa very narrow pulse.Using the generic response M(f), for message signal m(t), the convolution of M(f)withthe response of S(f) of sampler impulse train s(t) gives us the following.

The first part again is the amplitude, the second is the sinc function v third termis theresponse of the message repeated at every integer multiple of the samplingfrequency.

The result is that we get a replicating response but its amplitude has beenmultiplied bythe sinc function. The sinc function rolls-off according to the parameter dutycycle. Alarge duty cycle means the sinc function as shown below rolls off faster thanwhen thepulse width is narrow.

Fig. 22 - Sampling with PAM-ike signal leads to a sinc biasThe smaller the duty cycle, the shallower the response since we are approachingthe ideal

case of zero width pulse. The first zero crossing which is important occurs atsamplingtime divided by the duty cycle. In this figure we see that for a duty cycle of .1, vasampling frequency of 1, the first zero crossing is at 10 Hz.What's the significance of all this? Well the closer this number is to the bandwidthof ourlowpass filter, the worse will be the distortion introduced by this flat-top or PAMsampling.


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Hnh 23 Sampling with non-ideal impulses(a) Frequency spectrum of an arbitrary signal which has a lowpass bandwidth of B(b) Frequency spectrum of an ideal impulse train is also an impulse train repeatedatharmonics of fs.(c) Multiplying the two together gives us the replicating spectrum of messagesignal.

This can be filtered to recover the signal.

(d) Now we use a non-ideal sampling pulse train with duty cycle = .3.(e) The Fourier Transform of this sampler is a sinc function with first zero at 3.33Hz.This distorts the spectrum of the message signal. Instead of getting the originalresponse, we get a depressed response.(f) Now if we pre-multiply the signal with the inverse sinc function, then this effectis nulled out and(g) We can recover the message signal despite non-ideal sampling.Once the signal is sampled by a flattop sampler, it can be recovered by using a

low passfilter. But there is some energy loss, particularly in the higher frequencies due tothedownward sloping bias of the sinc faction. This distortion can be significant in twocases,a. when the duty cycle is large, say over .2 v b. when the roll off factor is small.Theeffect of large duty cycle is that there is in-band distortion. However for DutyCycles lessthan .2, the distortion is quite minimal v can be ignored. But anything abovethat needs

to be corrected if we are to make raised cosine pulse perform as it should. Sincethe firstzero crossing occurs at Ts/dc, then a large dc gives us a close-in zero crossing,hencecauses a lot of distortion.The interaction of roll-off factor with this effect is that a smaller roll-off factorresponseis approaching a brick wall spectrum v has large spectrum content in higherfrequencies which are affected more by the distortion. A larger roll factor is lessaffectedbut then it is not so desired either because it needs a larger frequency

bandwidth.The figure below shows the amount of correction (or equalization) required forvariousroll off factor. Any roll-off factor below .4 definitely requires sinc equalization.

Hnh 24 - Sinc equalization affects low roll-off signals a lot moreWe can pre-correct for this distortion by multiplying the desired response by theinversesinc function. See the pre-corrected response as opposed to uncorrectedresponse in Fig


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24. The signal is then sampled v returned to the desired spectrum as shown inHnh 23(f).

The frequency response of a sinc corrected raised cosine spectrum is given bythefollowing equation.

Where the root-raised response at the transmitter v the receiver is given by thefollowing equations. Note that only the transmitter has a sinc correction.Transmit filter response

The receive filter response (note has no sinc correction)

In frequency domain we are transmitting the distorted shape v not the ones that

areshown in solid lines in Hnh 24.There are other ways to mitigate ISI such as with duo-binary or partial responsesignaling. These might be a topic later on.Charan Langton

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Inter Symbol Interference_ Luan Van

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