Article Raised Cosine

5/17/2018 Article Raised Cosine - slidepdf.com

http://slidepdf.com/reader/full/article-raised-cosine 1/6

50 www.rfdesign.com April 200

As digital technology ramps up for this century,an ever-increasing number of RF applications

will involve the transmission of digital data fromone point to another. The general scheme is to con-

vert the data into a suitable baseband signal tha t isthen modulated onto an RF carrier. Some pervasiveexamples include cable modems, mobile phones a ndhigh-definition television (HDTV). In each of thesecases, analog information is converted into digital

form as an ordered set of logical 1’s and 0’s (bits)The task at hand is to transmit these bits betweensource and destinat ion, wheth er by ph one line, coaxial cable, optical fiber or free spa ce.

A brief history lessonIn its simplest form, the transmission of binar

information (i.e., bits) between two points is a simple ta sk. Consider Morse code. The “dots” and “dashes” of Morse code represent a binary form of transmission that has been in us e since the mid-19th century. It found application in the telegraph and shipto-ship light signalin g. In t oday’s environm ent , however, digital transmission has become a much morchallenging proposition.

The main reason is that the number of bits thamust be sent in a given time interval (data rate) icontinually increasing. Unfortunately, the data ratis constr ained by the bandwidth a vailable for

given a pplication. Fu rth ermore, th e presence onoise in a communications system also puts a constraint on the maximum error-free data rate. Thre la t ionship between data ra t e , bandwidth andnoise was qua ntified by Shan non (1948) and m ar kea break thr ough in comm unications theory.

Digital data: Peeling back the layersIn modern data transmission systems, bi ts o

groups of bits (symbols) are t ypically tra nsm itted inthe form of individual pulses of energy. A rectangular pulse is probably the most fundamental. It ieasy to implement in a real-world system because ican be directly compared to opening and closing aswitch, which is synonymous with the concept o

binary in form ation. For exam ple, a “1” bit might bused to turn on an energy source for the duration oone pulse interval (τ seconds), which would producan output level, “A” (see Figure 1a). Alternately, a“0” bit would turn off the energy source, producingan output level of zero during one pulse int erval.

The Fourier t ra nsform of the pu lse yields its spectral characteristics, which is shown in Figure 1bNote tha t a pulse of width τ has the bulk of its energy contained in the main lobe, which spans a onesided bandwidth of 1/ τ Hz. This would imply thathe frequency span of a data transmission channemust be at least 2/ τ Hz wide. More will be said a bouthis later.

Figure 1 shows that a pulse of a given width, τspans a bandwidth t hat is inversely related to τ . If data rate of 1/ τ bits per second is chosen, then eacbit occupies one pulse width (namely, τ seconds)Obviously, if we wish to send bits at a faster ratet h e n t h e v a l u e o f τ m u s t b e m a d e s m a l l e rUnfortuna tely, th is forces the ban dwidth to increasproportionally (see Figure. 1b).

Such data-rate/bandwidth relationships pose problem for band-limited systems. This is mainlybecause most t ransmiss ion systems have banlimitat ions imposed by ei ther the natural bandwidth of the transmission medium (copper wirecoaxial cable, optical fiber) or by governmental oregulatory conditions. Thus, the challenge in dat

The care and feeding

of digital,pulse-shaping filters

m ixed signal

By Ken Gen tile

In the not-too-distant future,data will be the predominant

baggage carried by wirelessand other transm ission system s.

Pulse-shaping filters will

play a critical part in m aintaining signal integrity.




transmission systems is to obtain thehighest possible data r ate in t he band-width allotted with the least numberof errors (preferably none).

Pulse shaping: The detailsBefore delving into the detai ls of

pulse shaping, it is important to under-stand that pulses are sent by the trans-mitter and ultimately detected by thereceiver in any data transmission sys-tem. At th e receiver, th e goal is to sam -ple the received signal at an optimalpoint in the pulse interval to maximizethe probability of an accurate binary

decision. This implies that the funda-mental shapes of the pulses be suchth a t t h ey d o n ot i n t e r fe re w i th o n eanother a t t he optimal sampling point.

There are two cri teria that ensurenoninterference. Criterion one is t hatthe pulse shape exhibits a zero crossingat the sampling point of all pulse inter-vals except i t s own. Oth erwise , theresidual effect of other pulses will intro-duce errors into the decision makingprocess. Criterion two is that the shapeof the pulses be such that the a mplitudedecays rap id ly outs ide of the pulseinterval.

This is important because any reals y s t e m w i l l c o n t a i n t i m i n g j i t t e r ,which mean s tha t th e actual samplingpoint of the receiver will not always beoptimal for each and every pulse. So,even if the pu lse shape pr ovides a zerocrossing at t he optimal sam pling pointof other pulse intervals, timing jitterin the receiver could cause the sam-pling instant to move, thereby missingth e ze ro c ro ss in g p o in t . Th i s , t o o ,

i n t r o d u ces e r r o r i n to t h e d eci s ion -mak ing process. Thus, th e quicker apulse decays outside of its pu lse inter-val, the less likely it is to allow timing

ji t t er t o in t r odu ce er r or s wh en sa m-

pling adjacent pulses. In addit ion tothe noninterference cri teria, th ere ist h e e v e r - p r e s e n t n e e d t o l im i t t h epulse bandwidth, as explained earlier.

The rectangular pulseThe rectangular pulse, by definition,

meets criterion number one because itis zero at all points outside of the pre-sent pulse interval. It clearly cannotcause interference during the samplingtime of other pulses. The trouble withthe rectangular pulse, however, is thatit has significant energy over a fairlylarge bandwidth a s ind ica ted by i t s

Fourier transform (see Figure 1b). Infact, because the spectrum of the pulseis given by the familiar sin(π x)/ π x (sinc)r e s p o n s e , i t s b a n d w i d t h a c t u a l l yextends to infinity. The unbounded fre-quency response of the rec tangularpulse renders it unsuitable for moderntransmission systems. This is wherepulse sh aping filters come int o play.

If the rectangular pulse is not thebest choice for band-limited data trans-mission, then what pulse shape wil ll imit bandwidth, decay quickly, andprovide zero crossings at t he pulse sa m-pling times? The raised cosine pulse,

which is used in a wide variety of mod-ern data t ransmiss ion systems. Themagnitude spectrum , P(ω ), of the raisedcosine pu lse is given by:

(1)

The spectral shape of the raised cosinep u l s e i s s h o w n i n F i g u r e 2 a . T h einverse Fourier transform of P(ω ) yieldsthe time-domain r esponse, p(t ), of theraised cosine pulse (see Figure 2b). This

i s a l s o r e f e r r e d t o a s t h e i m p u l sresponse an d is given by:

(2

Care must be taken when (2) is usedfor calculation because t he den ominatocan go to zero if α t / τ = ±½. Th ereforeany program used to compute p(t ) mustest for the occurrence of α t / τ = ±½Because it can be shown that the limiof p(t ) a s α t / τ approaches ±½ is given b(π /4) sin c(t / τ ), this is the formula to uswhen the special case of α t / τ = ±½ iencountered.

The raised cosine pulseUnl ike the rec tangular pu lse , thraised cosine pulse takes on the shapof a sinc pulse, as indicated by the leftmost term of p(t ). Unfortunately, thname “raised cosine” is misleading. Iactua lly refers t o the pu lse’s frequencspectrum, P(ω ), not to its time domainshape, p (t ). The precise shape of thra ised cosine spectrum is determined bthe parameter, α , where 0 ≤ α ≤ 1.

Specifically, α governs the bandwidthoccupied by the pulse and the rate awhich the tails of the pulse decay. Avalue of α = 0 offers the narrowes

b an d w id th , b u t t h e s l o w es t ra t e odecay in the time domain. When α = 1

p t

t t

t ( )=

−

sincτ

απτ

ατ

cos

12 2

1

01

21

2

( ) ( )=

≤ ≤−( )

( ) ( )= −

−

for

2

P

P

ω τ

ωπ α

τ

ωτ τ

αω

πτ

sin

( )≤ ≤

( )

( ) ( )=

≥(

for1- 1+

3

for1+

π α

τ

ωπ α

τ

ω

ωπ α

P 0

))

τ

Figure 1. A single rectangular pulse and itsFourier transform.

Figure 2. Spectral shape and inverse Fourietransform of the raised cosine pulse.




t h e b a n d w id t h i s 1 / τ , bu t t h e t i m edomain tails decay rapidly. It is inter-esting to note that t he α = 1 case offersa double-sided bandwidth of 2/ τ . Thisexactly matches the bandwidth of the

main lobe of a rectangular pulse, butwith t he a dded benefit of ra pidly decay-in g t ime-d oma in t a i l s . Co n v erse ly ,inverse when α = 0, the bandwidth isreduced to 1/ τ , implying a factor-of-twoincrease in data rat e for t he same ban d-width occupied by a rectangular pulse.However, this comes at the cost of amuch slower ra te of decay in t he ta ils of the pulse. Thus, the parameter α givesthe s ystem designer a tr ade-off betweenincreased data rat e and t ime-domaintail suppression. The latter is of primeimportance for systems with relativelyhigh timing jitter at t he receiver.

Figure 3 shows how a train of raisedcosine pulses interact when the timebetween pulses coincides with the datarate. Note how the zero crossings arecoincident with the pulse centers (thesam pling point) as desired.

I t sh o u ld b e p oin t ed ou t t h a t t h eraised cosine pulse is not a cure-all. Itsapplication is restricted to energy puls-es that are real and even (i.e., symmet-

ric about t = 0). A different form of pulseshaping is required for pulses that arenot real and even. However, regardlessof the necessary pulse shape, once it isexpressible in either the time or fre-

quency domain, the process of designinga p u l s e -s h a p i n g f il t e r r e m a i n s t h esame. In this article, only the raisedcosine pulse shape will be considered.A variant of the raised cosine pulse isoften u sed in modern systems – theroot-raised cosine response. The fre-quency response is expressed simply asthe square root of P(ω ) (and square rootof p(t ) in the time domain). This shapeis used when it is desirable to share thepulse-shaping load between the trans-mitter and receiver.

It’s better in digital

Before the advent of digital fi l terdesign, pulse-shaping filters had to beimplemented as ana log filter designs.Digital filters, however, offer severaladvan tages of ana log designs. They canbe integrated directly on silicon, whichmakes them attractive for system-on-a-chip (SoC) designs. Furthermore, theproblem of component drift due to tem-perature and aging is eliminated. Also,their spectral characteristics are consis-tent and reproducible and do not sufferfrom component tolerance issues.

With the plethora of digital fi l terdesign tools available on the market,

th e designer can design a variety of dig-ital filters with little effort.

Choices, choices, choicesGiven that the pulse shape has been

defined mathematically (such as theraised cosine pulse), the next task is todecide which basic category of digitalfilter to use: finite impulse response(FIR) or infinite impulse response (IIR).

The functional form of FIR and IIRfilters is shown in Figure 4. The funda-mental difference between them is thefact tha t t he IIR conta ins feedback. Thisshould be obvious from t he fact tha t t heb i coefficient’s feedback scaled anddelayed samples of the output y(n ).Hence, the history of the output affectsthe future of the output. This is not truefor the FIR, where y(n) only depends onthe history of the input samples, x(n ).The implication is that the response of an IIR filter to an impu lse (a single non-zero sam ple followed by zero sam ples) isinfinite. That is, the IIR will continue toproduce non-zero output samples longafter t he application of an impulse. Thisis an undesirable consequence for data

pulse transmission (recall the noninterference criteria).

The FIR does not suffer from thiproblem because its architecture doenot contain any feedback elements. Asingle, non-zero impulse at the inpuwill only yield out put sam ples while thimpulse propagates down the dela

stages. Generally, pulse shaping filteremploy FIR designs.

The basic building blocks of a digitafi l ter are adders (⊕), multipliers (⊗)and unit-delays ( D); all of which can breadily implemented in digital formAdders a nd m ultipliers ar e composed ocomb in a t i on a l l o gi c w h i l e t h e u n idelays are composed of latches (whichrequire a clock signal). The basic filtering operation consists of a sequence omu l t i p ly /ad d /d e lay op e ra t i on s t h aoccur each time the delay stages arclocked. This is effectively a convolutionopera tion, which m ay be expressed as:

y(n) = x(n ) * h(n)

In this expression, * is the convolution operat or and should not be taken tmean s imp le mu l t i p l i ca t i o n . ThFourier transform (or z-transform in thcase of digital filters) r eveals tha t filtering is synonymous with convolutionThis is the “secret” of digital filters —by using relatively simple operation(add/multiply/delay), a filtering operat i o n can b e rea l i zed . Th e t r i ck , ocourse, is coming up with the prope

Figure 3. Interaction of raised cosine pulseswhen the time between pulses coincides withthe data rate.

Figure 5. An arbitrary digital filter frequencresponse.

Figure 4. The functional form of FIR and IIRfilters.

Figure 6. Converting an impulse to a raisecosine pulse by filtering.




h (n ) to produce the desired fi l teringoperation (i.e., spectral shaping in thefrequency doma in). This is another topic

altogether, but the many digital filterdesign tools that are available todaymak e t h i s p ro cess ea s i e r t h a n ev e rbefore . Th ese design too ls g ive th edesigner the ability to generate the nec-essary filter coefficients for a desiredfrequency response (or vice versa).

Other variables that enter into thedesign process include determining theoptimal number of filter coefficients andhow much numeric precision (resolu-tion) is required to get the job done.

Resolution refers to the number of bits used to represent the coefficientvalues, as well as the number of bits

used to represent the sample values atany given point in the filter. Resolutionaffects the overall complexity of thedesign because more bits means moredigital ha rdware.

How it comes togetherBefo re p r o ceed in g w i th a d es ig n

example, it should be noted from F igure4b that h (n ) (the impulse response of the filter) is directly determined by thefilter coefficients. This can be used to anadvantage in the filter design process,because an outcome of the FIR filterdesign process is the impulse response,h(n). The h (n ) values can be directlysubstituted for the a i coefficients.

Step oneThe first consideration in FIR filter

design is the sample rate ( f s). This is th erate at which the internal delay stagesare clocked. It tu rns out that the u sefulfrequency response chara cteristic of anyd ig i t a l f i l t e r i s l imi t ed t o ½ f s ( t h eNyquist frequency), not f s as one mightassume. To demonstrate this concept,an arb i t ra ry d ig ita l f i l te r f requency

response is shown in Figure 5. Nowrecall the raised cosine response (seeFigure 2a), which can extend out to a

frequency of 1/ τ (for α = 1). If one wereto operate a digital filter at the datarate (1/ τ ), a problem would surface.

Speci f ica l ly , the f i l te r f requencyresponse is restricted to the Nyquistrate (namely ½τ ). The implication isthat if a digital filter is used for pulseshaping, then it must operate at a sam-ple rate of at least t wice the data rat e tospan the frequency response character-istic of the raised cosine pulse. That is,the filter must oversample the data byat least a factor of two, preferably more.

Step two

The second considera tion in FIR fil-ter design is the number of tap coeffi-cients (the a i values). Typically, this isgoverned by two factors. The first isthe amount of oversampling desired.More oversam pling yields a more a ccu-rate frequency response characteristic.So a designer may elect to oversampleby three, four or more. The second fac-tor is the length of time that the fil-t e r ’s r e sp o n se i s ex p ect ed t o sp an .

Typically, this is determined by thnumber of bi t (or symbol) intervalthat the designer would like the filteres ponse t o occupy.

Remember, an FIR f i l te r impuls

r e s p o n s e l a s t s o n l y a s l on g a s t hnumber of taps. If the filter oversamples by a factor of two and t he desiredimpulse response duration is five bit(or symbols), then 10 t aps a re r equire(2 x 5 = 10). Obviously, a trade-ofexists between the n umber of taps (circ u i t c o m p l e x i t y ) a n d t h e f i l t e r ’response characteristic.

Why it works so wellThe beau ty of the pu lse-sha ping filte

concept is that rectangular pulses canbe used as th e input t o the filter.

Recall th at the basic filtering proces

is synonymous with convolution in thtime domain. Also recall tha t digita l fiters provide a convolution operationF o r e x a m p l e , t h e f i l t e r i m p u l sresponse h (n ) is convolved with thinput samples to yield the output samples. The convolution of a rectangulapulse (more specifically, a unit impulsewith a raised cosine impulse responsresults in a raised cosine pulse at thoutput (see Figure 6). The input to thfilter is a 1 or 0 (scaled to occupy th e fulbit width of the filter’s input word sizeand the output is a raised cosine pulsw i th a l l o f t h e t ime an d f req u en c

domain advantages that such a pulsoffers. All that is required is a digital-toana log converter (DAC) at the output othe filter to convert the digital sampleinto an ana log waveform.

Next, examine a det ailed example oa ra i sed cosine pulse-shaping f i l tedesign. Consider a system in whichdata must be transmitted at a rate o1 Mb/s (i.e., τ = 1 µs). One is also toldthat the t iming j i t te r p resent a t th

Figure 7. The frequency responses of two different versions of the raised cosine pulse-shaping filter.

Figure 8. A block diagram of one potential RF data transmitter design using digital filters.




receiver is not known. Another designconstraint is that the digital circuitryused to construct the digital filter willoperate at a ma ximum rat e of 50 MHz.Additionally, it ha s been given as pa rt

of the design requirement that the fil-ter impulse response span at least fivesymbol periods.

In the absence of specific knowledgeabout timing jitter at the receiver, oneis forced to assu me t he wors t . Thisimplies that a value of α = 1 be used toma ximize th e decay of th e pulse ta ils.

From Figure 2a, it can be seen thatthis corresponds to a single-sided band-width of 1/ τ (1 MHz), which means thatthe medium over which the data aretransmitted must be able to support a2 MHz bandwidth (the double-sidedbandwidth of the raised cosine spec-

trum). If the medium cannot support2 M H z , t h e n o n e m u s t c o n s i d e r ameans of squeezing more bits into thesame bandwidth. This can be done by avariety of modulation schemes (QPSK,16-QAM, etc.). In this example, it isassum ed that a bandwidth of 2 MHz isacceptable.

B e ca u s e i t h a s b e e n d e t e r m i n e dth a t α = 1, it is necessary to operatethe filter at a sample rate of no lessthan twice the data rate (or two sam-ples per symbol). However, to providea more accurate spectral shape, onemay choose to oversample by a factor

of eight (i.e., eight samples/symbol).This means tha t th e digital filter m ustopera te a t a ra te of 8 MHz. This i swell within the specified 50 MHz oper-ating range of the digital circuitry, sothe design is not in jeopar dy.

I t h a s b e e n g iv e n t h a t t h e f i lt e rimpulse response be designed to spanfive symbols, so the filter must containat least 40 ta ps (eight sa mples/symbol xfive symbols = 40 samples). This willprovide the required duration of theimpulse r esponse. However, 41 t aps willbe chosen to avoid the half-symbol delayassociated with an even number of taps .

With α , τ , and the number of tapsdefined, Equation 2 can now be used togenerat e the filter t aps. The value of t isdetermined at increments of 125 ns (thesam pling period of the filter when oper-a t i n g a t 8 MH z) . Th e cen t e r o f t h eimpulse r esponse is given th e value of t = 0. Thus, the first value of t is 20 sam-ples pr ior, or t = -2.5 µs.

How it all came togetherThe author used a PC-based version

of Mathcad to compute p(t) using the

above information (see Appendix). Anysuitable math program will do the job(MatLab, Excel, etc.). It turns out thatbecause p(t ) was computed at the filtersample points, the values of p(t ) corre-

spond one-to-one with h [n], the impulsere sp o n se o f t h e f il t e r . Th e r e su l t s ,rounded t o four decimal places, are list-ed in Table 1.

Note the symmetry of the h (n) valuesabout n = 0. This redundancy can beused to simplify the implementation of the filter har dware. Because t he filter isof the oversampling variety, further

hardware simplification can be gainedby using a polyphase ar chitectur e.

If the filter is designed with floatingpoint multipliers and adders, then thedesign is essent ially done.

In the case of finite arithmetic, 10bits is probably the minimum accept-able resolution to handle tap coeffi-cients that span four decimal places.Formatted as twos-complement num-bers, this will allow a range of –1.000to +0.998046875 with a resolution of 2–1 0 (0.0009765625). This means thatthe multiply and add stages of the fil-ter must be designed to handle 10-bitwords. Also, the coefficients should be

sca l ed b y some f ra c t i on a l v a lu e tavoid overflow conditions in the hardware. A generally accepted scale factois given by:

SF = [∑ h ( k )2]–1/2.

In words, it is the reciprocal of thsquar e root of the s um of the squa re oeach tap value. For the current example, the scale factor is: SF = 0.408249After multiplying the h(n ) by SF, thresulting values are then converted t10-bit words. Figure 7 shows the frequency responses of two versions othe raised cosine pulse-shaping filterOne is a floating-point version of thfilter with the coefficients rounded tfour decimal p laces . The o ther i s scaled, 10-bit, finite-math version o

the filter. Both responses behave welin the passband, but the floating-poinvers ion exhib i t s be t ter ou t -of-banattenuation. Also shown for comparison’s sake is the pa ssban d err or relative to the ideal response (Equa tion 1Note that both responses exhibit lest h a n 0 . 2 d B e r r o r o v e r a r a n g e oabout 90% of the pass band.

The final frontierWith an under sta nding of how to cre

ate digital pulse shaping filters, the RFengineer can tak e on a lar ger role in thdesign of digital transmission systems

This is especially true today with semiconductor manufacturers offering highly integrated, high-speed, mixed-signaICs. Figure 8 shows a detailed blockdiagram of a possible RF da ta tran smitter design. The design is almost completely contained in two ICs (assumina PC or other external device serves athe serial port contr oller).

References

[1] Gibson, J . D., “Pr inciples of Ana loand Digital Communications,” 2nd edPr entice Hall, 1993.

[2 ] I feach o r , E . C . , J e r v i s , B . W“Digital Signal Processing: A PracticaApproach ,” Addisson-WeslePublishing Co., 1996.

[3] The World Book EncyclopediaVolum e 13, World Book, In c., 1994.

n h (n ) n

-20 0 20

-19 -0.0022 19

-18 0.0037 18

-17 0.0031 17

-16 0 16

-15 0.0046 15

-14 0.081 14

-13 0.0072 13

-12 0 12

-11 0.0125 11

-10 0.0234 10

-9 0.0246 9

-8 0 8

-7 0.0624 7

-6 0.1698 6

-5 0.3201 5

-4 0.5 4

-3 0.686 3

-2 0.8488 2

-1 0.9603 1

0 1 0

Table 1. Impulse response values for p(t) of thefilter.



RF Design www.rfdesign.com 61

About the authorKen Gentile graduated with honors from North Carolina State University

where he received a B.S.E.E degree in 1996. Currently, he is a system designengineer at Analog Devices, Greensboro, NC. As a member of the Signal

Synthesis Products group, he is responsible for the system level design and analy-sis of signal synthesis products. His specialties are analog circuit design and theapplication of digital signa l processing techniques in comm unicat ions syst ems. Hecan be r eached a t 336-605-4073 or by e-ma il at ken .gentil e@an alog.com .

Documents

Article Raised Cosine