Shaping Filter Design

7/26/2019 Shaping Filter Design

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As digital technology ramps up for this century,an ever-increasing number of RF applicationswill involve the transmission of digital data fromone point to another. The general scheme is to con-

form as an ordered set of logical 1s and 0s (bits).The ta sk at hand is to transmit these bits betweensource and destina tion, wheth er by phone line, coax-ial cable, optical fiber or free space.

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

information (i.e., bits) between two points is a sim-ple task. C onsider Morse code. The dots a nd d ash -es of Morse code represent a binary form of trans-

mission tha t ha s been in use since the mid-19th cen-tury. It found application in the telegraph and ship-to-ship light signa ling. In t odays environment, h ow-ever, digital transmission has become a much morechallenging proposition.

The main reason is that the number of bits thatmust be sent in a given time interval (data rate) iscontinually increasing. Unfortunately, the da ta rat eis cons t ra ined by the bandwidth ava i lab le for agiven applicat ion. Furthermore, the presence ofnoise in a communications system also puts a con-

straint on the maximum error-free data rate. Ther e l a t io n s h ip b e t w een da t a r a t e , b a n dw id t h a n dnoise wa s qua ntified by Sha nnon (1948) an d ma rkeda brea kthrough in commun ications theory.

Digital data: Peeling back the layersIn modern da t a t ransmiss ion sys tems , b i t s or

groups of bits (symbols) ar e typically tr an smitt ed inthe form of individua l pulses of energy. A rectangu-lar pulse is probably the most fundamental . I t iseasy t o implement in a real-world system because itcan be directly compared to opening and closing aswitch, w hich is synonymous with the concept ofbinary information. For example, a 1 bit might beused to turn on an energy source for th e dura tion ofone pulse interval ( seconds) which would produce

The care and feedingof digital,

pulse-shaping filters

m ixed signal

By Ken Gentile

I n the not-too-distant futur e,

data w il l be the predominant

baggage car r ied by wireless

and other tr ansmission systems.

Pul se-shapi ng fil ters wil l

play a cr it ical par t

in maintain ing signal i ntegrity.


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one pulse interval ( seconds) which would produce

t ransmission systems is to obta in thehighest possible data ra t e in the band-width a l lo t ted wi th the leas t numberof errors (preferably none).

Pulse shaping: The detailsB ef o r e de lv in g in t o t h e de t a i l s o f

pulse shaping, it is important to under-stand t hat pulses are sent by the trans-mit ter and ult imately detected by thereceiver in any data transmission sys-

tem. At the receiver, the goa l is to sam -ple the received s ignal a t an opt imalpoint in the pulse interval to maximizethe probab i l i t y o f an accura te b inary

decision This implies that the funda

i n t r o d u c e s e r r o r i n t o t h e d e c i s i on -making process . Thus, the quicker apulse decay s outside of its pulse inter-val, the less likely it is to allow timingj i t t er to in t roduce errors when sam-pl ing adjacent pulses . In addi t ion tothe noninterference criteria , there ist h e e v e r - p r e s e n t n e e d t o l i m i t t h epulse bandwidth, as explained earlier.

The rectangular pulse

The rectangular pulse, by definition,meets criterion number one because itis zero at all points outside of the pre-sent pulse in terva l . I t c lea r ly cann otcause interference during the samplingtime of other pulses. The trouble withthe rectangular pulse, however, is thati t has s ignif icant energy over a fa ir lyl a r g e b a n d w i d t h a s i n d i ca t e d b y i t sFourier transform (see Figure 1b). Infact, because the spectrum of the pulse

is given by the familiar sin(x)/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 t o infinity. The unbounded fre-qu en c y r es p o n s e o f t h e r ec t a n gu la rpulse renders it unsuitable for modernt ra nsmiss ion sys t ems . This is w herepulse shaping filters come into play.

I f the rec t angula r pulse is not thebest choice for band-limited data trans-miss ion , then wha t pulse shape wi l ll imi t b a n dw id t h , deca y qu ic kly , a n dprovide zero crossings at the pulse sam -pling times? The raised cosine pulse,which is used in a wide variety of mod-er n da t a t r a n s mis s io n s ys t ems . Th emagnitude spectrum P() of the raised

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 s eresponse an d is given by:

(2)

Care must be taken when (2) is usedfor calculat ion because t he denominat or

can go to zero if t/ = . Therefore,any program used to compute p(t) musttest for the occurrence of t/ = .Because it can be shown that the limitof p(t) as t/approaches is given by(/4) s inc(t/), this is the formula to usewhen the specia l case of t/ = isencountered.

The raised cosine pulseU n l ik e t h e r ec t a n gu la r p u l s e, t h e

raised cosine pulse takes on the shapeof a sinc pulse, as indicated by the left-most term of p(t). U nfor tuna tely , thename raised cosine is misleading. Itactually refers to the pulses frequencyspectrum, P(), not to its time domainshape, p(t). The precise shape of thera ised cosine spectrum is determined bythe param eter, , where 0 1.

Specifically, governs the bandw idthoccupied by the pulse and the rate atwhich the ta ils of the pulse decay. Av a lu e o f = 0 of f e r s t h e n a r r o w e s tb a n d w i d t h , b u t t h e s l ow e s t r a t e o fdecay in the time domain. When = 1,

p t

t t

t( )=

sinc

cos

12

2

Figure 1. A single rectangular pulse and itsFourier transform.


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t h e b a n d w i d t h i s 1/, b u t t h e t i m edomain tails decay rapidly. It is inter-esting to note tha t t he = 1 case offersa double-sided bandwidth of 2/. Thisexact ly matches the bandwidth of themain lobe of a rectangular pulse, butwith the added benefit of rapidly decay-

i n g t i m e -d o m a i n t a i l s . C o n v e r s e l y ,inverse when = 0, the bandwidth isreduced to 1/, implying a factor-of-twoincrease in data rat e for the same band-width occupied by a rectangular pulse.However, this comes a t the cost of amuch slower rate of decay in the tails ofthe pulse. Thus, the parameter givesthe system designer a trade-off betweenincreased da t a ra t e and t ime-domaintail suppression. The latter is of primeimportance for systems with relativelyhigh timing jitter a t th e receiver.

Figure 3 shows how a tra in of raisedcosine pulses interact when the t imebetween pulses coincides with the data

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 t ime or fre-quency doma in, the process of designinga p u l s e - s h a p i n g f i l t e r r e m a i n s t h esame. In th is a r t ic le , on ly the r a isedcosine pulse shape will be considered.A variant of the raised cosine pulse iso f ten used in modern sys tems the

root-ra ised cosine r esponse. The fr e-quency response is expressed simply asthe square root of P() (an d squa re rootof p(t) in the time domain). This shapeis used when it is desirable to share thepulse-shaping load between the trans-mitt er and receiver.

Its better in digitalB ef or e t h e a dv en t o f d ig i t a l f i l t e r

design, pulse-shaping filters had to be

implemented as analog f i l ter designs .Digita l f i l ters , however, of fer severaladvantages of analog 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 p lethora o f d ig i t a l f i l t erdesign tools available on the market ,the designer can design a variety of dig-ita l filters wit h little effort.

pulse transmission (recall the noninter-ference criteria).

The FI R does not su f fer f rom t h isproblem because its architecture doesnot contain any feedback elements. Asingle, non-zero impulse at the inputwill only yield output samples while theimp u ls e p r o p a ga t es do w n t h e de l a ystages. Generally, pulse shaping filtersemploy FI R designs.

The basic building blocks of a digitalf i l t er a re adders (), multipliers (),and unit-delays (D); all of which can bereadi ly implemented in dig i t a l form.Adders a nd m ultipliers are composed ofc om b i n a t i o n a l l o g i c w h i l e t h e u n i tdelays are composed of latches (whichrequire a clock signal). The basic filter-ing operation consists of a sequence ofmu l t ip ly/a dd/de la y o p er a t io n s t h a toccur each t ime the delay s t ages a reclocked. This is effectively a convolutionoperat ion, which may be expressed a s:

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

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

Figure 5. An arbitrary digital filter frequencyresponse.


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h(n )to produce the des ired f i l t er ingoperation (i.e. , spectral shaping in thefrequency doma in). This is a nother t opicaltogether, but the many digita l f i l terdes ign too ls tha t a re ava i lab le today

ma k e t h i s p r o c es s ea s ie r t h a n ev erb e f o r e . Th es e des ign t o o l s g iv e t h edesigner the ability to generate the nec-essary filter coefficients for a desiredfrequ ency response (or vice versa ).

Other variables that enter into thedesign process include determining theoptima l number of filter coefficients a ndhow much numeric precision (resolu-tion) is requir ed to get t he job done.

Resolut ion refers to the number of

bits used to represent the coeff icientvalues , as well as the number of b itsused to represent the sample values atany given point in the filter. Resolutiona f fec t s the overa l l complexi ty of the

response is shown in F igure 5 . Nowrecall the raised cosine response (seeFigure 2a), which can extend out to afrequency of 1/ (for = 1). If one wereto opera te a d ig i t a l f i l t er a t the da t a

ra te (1/), a problem would surface.S p e c i fi c a l l y , t h e f i l t e r f r e q u e n c y

response is restr icted to the Nyquis tra te (namely ). The implication isthat if a digital filter is used for pulseshaping, then it must operat e at a sam-ple rate of at least tw ice 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 tw o, preferably more.

Step twoThe second consideration in FIR fil-

ter design is the number of tap coeffi-cients (the a values) Typically this is

Typica l ly , th is i s determined by thenumber o f b i t (or symbol) in terva lsthat the designer would like the filterresponse t o occupy.

R ememb er , a n F I R f i l t e r imp u ls er e s p on s e l a s t s o n l y a s l on g a s t h enumber of taps. If the filter oversam-ples by a factor of two and the desiredimpulse response duration is five bits(or symbols), then 10 taps a re requir ed(2 x 5 = 10). Obvious ly , a t r a de-of f

exists between th e number of ta ps (cir-c 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 sresponse char acteristic.

Why it works so wellThe beauty of the pulse-shaping filter

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

Recall that the basic filtering processis synonymous with convolution in thetime domain. Also recall that digital fil-

ters provide a convolut ion operat ion.F o r e x a m p l e , t h e f i l t e r i m p u l s eresponse h(n) i s convolved wi th theinput samples to yield the output sam-ples. The convolution of a rectangularpulse (more specifically, a unit impulse)with a raised cosine impulse responseresults in a raised cosine pulse at theoutput (see Figure 6). The input to thefilter is a 1 or 0 (scaled t o occupy the fullbit width of the filters input word size)

and the output is a raised cosine pulsew i t h a l l o f t h e t i m e a n d f r e q u en c ydomain a dvantages tha t such a pulseoffers. All tha t is req uired is a digita l-to-an alog converter (DAC) at the output of

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


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receiver is not known. Another designconstra int is that the digita l circuitryused to construct the digital filter willoperate a t a maximum ra te of 50 MHz.Additionally, it has been given as partof the design requirement that the fil-ter impulse response span a t least fivesymbol periods.

In the absence of specific knowledgeabout timing jitter at the receiver, onei s f o r ced t o a s s u me t h e w o r s t . Th is

implies that a value of = 1 be used toma ximize the decay of the pulse ta ils.From Figure 2a, it can be seen that

this corresponds to a single-sided band-width of 1/(1 MHz), which mea ns tha tthe medium over which the da t a a retransmitted must be able to support a2 MHz bandwidth ( the doub le-s idedbandw idth o f the ra ised cos ine spec-trum). I f the medium cannot support2 M H z , t h e n o n e m u s t c o n s i d er a

means of squeezing more bits into thesame bandwidth. This can be done by avariety of modulation schemes (QPSK,16-QAM, etc.) . In this example, i t isassumed tha t a bandwidth of 2 MHz isacceptable.

B e c a u s e i t h a s b e en d e t e r m i n e dt h a t = 1, i t is necessary t o operatethe f i l t er a t a s ample ra te o f no lessthan twice the data ra te (or two sam-ples per symbol). However, to provide

a more accura te spect ra l shape, onemay choose to oversample by a factorof eight (i.e. , eight s a mples /sym bol).This means tha t t he digita l f i l ter musto p er a t e a t a r a t e o f 8 M Hz Th is i s

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 w ith h[n], the impulser es p o n s e o f t h e f i l t e r . Th e r es u l t s ,rounded to four decima l places, are list-ed in Ta ble 1.

Note the symmetry of the h(n)valuesabout n= 0. This redunda ncy can be

used to simplify the implementation ofthe filter ha rdwa re. Because the filter iso f the oversampling var ie ty , fur ther

s c a l e d b y s o m e f r a c t i o n a l v a l u e t oavoid overflow conditions in the hard-wa re. A generally a ccepted scale factoris given by:

SF = [h(k)2]1/2.

In words, it is the reciprocal of thesqua re root of the sum of the squa re ofeach tap va lue. For the current exam-ple, the scale fa ctor is: SF = 0.408249.

After multiplying the h(n) by SF, theresulting values are then converted to10-bit words. Figure 7 shows the fre-quency responses o f two vers ions o fthe raised cosine pulse-shaping filter.One is a floating-point version of thefilter with the coefficients rounded tof o u r dec ima l p l a c es . Th e o t h er i s ascaled, 10-bit , f inite-math version ofthe filter. Both responses behave wellin the passband, but the floating-point

v er s io n ex h ib i t s b e t t e r o u t -o f -b a n dat tenuat ion. Also shown for compari-sons sake is the passband error rela-tive to th e ideal r esponse (Equa tion 1).Note that both responses exhibit lesst h a n 0 . 2 d B e r r o r o v e r a r a n g e o fabout 90%of the passba nd.

The final frontierWith an understanding of how to cre-

ate digital pulse shaping filters, the RF

engineer can ta ke on a la rger role in thedesign of digita l tra nsmission systems.This is especially true today with semi-conductor manufacturers offering high-ly integrated high speed mixed signal

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


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About the authorKen Gent i le gradua ted wi th honors f rom Nor th Caro l ina St a te Univers i t y

where he received a B.S.E.E degree in 1996. Currently, he is a system designen g in eer a t An a lo g Dev ices , G r een s b o r o, N C . A s a m emb er o f t h e S ign a lSynthesis 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 signal processing techniques in communications systems. Hecan be rea ched a t 336-605-4073 or by e-mail a t ken.genti l e@anal og.com.


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Raised Cosine Filter

Raised Cosine Filters

Pulse Shaping

Raised Cosine Filters exist primarily to shape pulses for use in communications systems. Excellent background information on this subject may be found

in Ken Gentile'sarticle, 0402Gentile50.pdf,published by RF Designin April, 2002.


The ideal raised cosine filter frequency response consists of unity gain at low frequencies, a raised cosine function in the middle, and total attenuation at

thigh frequencies. The width of the middle frequencies are defined by the roll off factor constant Alpha, (0


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The ideal raised cosine filter frequency response is shown below:

Ideal Raised Cosine Frequency Response Ideal Raised Cosine Impulse Response (Alpha = 0.35)

A typical raised cosine square wave response is shown below:

http://www.filter-solutions.com/raised.html (2 of 6) [26/12/2003 20:31:32]


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Typical Raised Cosine Square Wave Response, Alpha = 0.35

An FIR Raised cosine filter may be synthesized directly from the impulse response, which is:

Raised Cosine Impulse Response



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Root Raised Cosine Filter

The ideal root raised cosine filter, frequency response consists of unity gain at low frequencies, the square root of raised cosine function in the middle,

and total attenuation at thigh frequencies. The width of the middle frequencies are defined by the roll off factor constant Alpha, (0


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Ideal Root Raised Cosine Frequency Response

An FIR Raised cosine filter may be synthesized directly from the impulse response, which is:

Root Raised Cosine Impulse Response

Data Transmission Filters

Data Transmission Filters are similar to Raised Cosine filters, but are simpler to build in that they do not require a delay equalizer, and are less effective

in removing ISI. The Data Transmission solution offered by Filter Solutions eliminates only postcursor ISI (ISI following the impulse response peak), or

all ISI if the pass band frequency is doubled. Example impulse responses for 7th order filters are shown below.


R i d C i Filt


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7th Order Raised Cosine Filter Impulse Response

Alpha=0.35

7th Order Data Transmission Filter Impulse Response

Alpha=0.35

From inspection, the Data Transmission filter only has one point of significant precursor ISI. If the pass band frequency is doubled, the precursor ISI

disappears or is insignificant. The result is that Data Transmission filters are less effective in removing ISI, or require twice the bandwidth.

Filter Solutions creates Data Transmission filters by removing the delay equalizer from the Raised Cosine filter, and employing numerical methods to

remove the postcursor ISI. This solution is not unique. Other solutions for Data Transmission Filters are known to exist, such as that found in The CRC

Handbook of Electrical Filters. The solutions offered by Filter Solutions has the advantage of offering the user flexibility in designing for accuracy vs.

bandwidth by selecting different Alpha values.

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