Fundamentals of Digital Quadrature Modulation

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mixed signal Fundamentals of cy) throughout this text. Typically, c is much high-

denoted by the symbol f c (or ω for radian frequen-

g a ua ra ureModulation

er in frequency than the highest frequency compo-nent of the message signal.

The Basics The concept of modulation comes from thetrigonometric identity:

ow a f ormerly analog-onl yapplication, the quadrature

modulator , is finding a homein modern digital CMOS fabrication techniques

By Ken Gentile

he concept of modulation is not at all new, ithas been around since the early days of radio.

Modulation, especially in the context of RFapplications, refers to the mixing of two sinusoidalsignals. One signal is the message signal and con-tains the information to be modulated. It usuallyconsists of a band-limited spectrum of sinusoids(such as music). The other signal is the carrier sig-nal and is generally a pure tone (a sinusoid of asingle frequency). The frequency of the carrier isreferred to as the carrier frequency and will be

If we assume that the message signal is a puretone of frequency, f m, then the message can bemathematically represented as cos(2π f mt). Thesame assumption can be made about the carrier signal, thereby expressing it as cos(2π f ct). The“pure tone” assumption makes the mathematicsmuch more tractable. However, it is important to

keep in mind that the message signal is rarely a pure tone. Typically, it is composed of time variations in

amplitude, frequency, phase, or any combinatiothereof. Even the carrier need not necessarily be a pure sinusoid. Applications exist in which the carri-er signal is a square wave with a fundamental fre-quency, f c. The harmonics of f c inherent in thesquare wave are dealt with by filtering the modu-lated signal.

The mixing process mentioned earlier can bethought of as a multiplication operation.Therefore, the trigonometric identity above ma

be employed to represent the mixing process asfollows:

Figure 1. Functional representation o a single and double side-band modulator and associated frequency spectra

Thus, the mixing of the message and carrier results in a transformation of the frequency of themessage. The message frequency is translated fromits original frequency to two new frequencies — onegreater than the carrier ( f c + f m), and one less thathe carrier ( f c – f m), the upper and lower side bands,respectively. Furthermore, the translated signal

undergoes a 6 dB loss (50 percent reduction) as dic-tated by the factor of ½ appearing on the right-hand side of the equality.

The form of modulation just descr ibed is referred to as “double sideband modulation,”

because the message is translated to a frequencyrange above and below the carrier frequency.Another form of modulation, known as single sideband modulation, can be used to eliminate either the upper or lower sideband. One method of per-forming single sideband modulation is to employ a

40 www.rfdesign.com February 2003



Figure 2. A continuous time representation o a sinusoid

bits carry the magnitude information.Two’s-complement numbers, in whichthe MSB is 0, are positive numbers,while those for which the MSB is 1 arenegative numbers. When the MSB is 0

(positive numbers) the non-MSB bitsquadrature modulator. A quadrature modulator mixes the message with twocarriers. Both carriers operate at thesame frequency, but are shifted inphase by 90 degrees relative to on eanother (hence the “quadrature” term).This simply means that the two carri-ers can be expressed as cos(2π f ct) andsin(2π f ct). The message, too, is modi-fied to consist of two separate signals:the original and a 90 degree phase-shifted version of the original. Th eoriginal is mixed with the cosine com-ponent of the carrier and the phase-

shifted version is mixed with the sinecomponent of the carrier. These twomodifications result in the implemen-tation of the single sideband function.Trigonometrically, this ca n beexpressed as:

⎣⎡cos ( x)cos ( y)⎦⎤ + ⎡⎣sin ( x)sin ( y)⎤⎦

values — 0 or 1. Bits may be concate-nated to form larger numbers just likedecimal digits. For example, a singledigit in the decimal system can take onvalues from 0 to 9, but the number 148uses three digits to represent the sum:8(100) + 4(101) + 1(102) = 8 + 40 + 100 =148. Each digit carries an increasing

power of 10 weighting starting withthe rightmost digit (ones, tens, hun-dreds...). Similarly, binary numbersare made up of bits carrying a power otwo weighting starting with the right-most bit. For example, 10010100 =

0(20

) + 0(21

) + 1(22

) + 0(23

) + 1(24

) +0(25) + 0(26) + 1(27) = 0 + 0 + 4 + 0 + 16+ 0 + 0 + 128 = 148.

The benefit of binary number repre-sentation is not readily apparent. For

instance, binary numbers can becomevery cumbersome to work with.Consider the number 1 million. It ca

be expressed with 7 decimal digits,

are taken as an ordinary binary number. For example, the two’s-complementnumber 0101 is +5 in decimal notation.When the MSB is 1 (negative numbers)the non-MSB bits are first inverted (i.e.,0s become 1s and vice versa) and then 1is added to the result. For example, thetwo’s-complement number 1101 is –3 indecimal notation. The MSB indicates anegative value, while the remaining bits(101) are inverted to yield 010, or 2 indecimal. Then 1 is added to the result tomake 3. So, the end result is –3 as indi-cated by the MSB. Although this may

seem complicated it is readily imple-mented in hardware using fundamentaldigital building blocks.

It should be noted that the digitalimplementation of an analog functionrequires a certain amount of compro-mise. For instance, an analog signal isnot a numeric quantity, but a physicalquantity. A digital signal, on the other

⎡ 1 1 ⎤= cos ( x + )+ cos ( x − ) whereas 20 binary digits are required; hand, is a numeric quantity and serves

⎣⎢ 2

⎡ 1

2

1

⎦⎥

⎤

not very efficient in terms of notation.However, the real beauty of binary

only to model an analog signal.Digital systems rely on a compromise

+ cos ( x − y)− cos ( x + y) numbers comes from the fact that each between absolute numeric accuracy and⎣⎢ 2 2 ⎦⎥ digit can take on only two values. This

is readily modeled by the state of asufficient numeric accuracy. For exam-

ple, the number of amplitude values

Note that the right-hand side of the equation reduces to cos( x – y), the lower sideband, only. In the above equation, x is the carrier and y is the message.Incidentally, changing the sign of theoperator on the left-hand side of theequation results in only the upper side-band appearing on the right hand side.

In figure 1 the functional representa-tion of a single and double sidebanmodulator are shown along with theassociated frequency spectra. However,the message is shown as a band-limitespectrum rather than a pure tone,which better represents a real-worlapplication. Each constituent frequenc

in the message is translated to one or both sides of the carrier, as shown.

Digital Number Representation In addition to an understanding of

modulation basics, it is also helpful tohave at least an elementary under-standing of numeric representation aemployed in the digital world. Thebasic building block of digital numericis the bit, which can only take on two

switch (on or off), which can be electron-ically implemented with a single tran-sistor. Transistors, in turn, can be phys-ically realized by the millions on a sin-gle silicon chip. The ability to place mil-lions of binary switches on a single chipis what gives digital its advantage.Returning to the concept of modula-tion, it was pointed out that modulationis applied in the context of sinusoids.Since a sinusoid is represented by atrigonometric function, it can take on

positive or negative values. So, digitalmodulation will require the ability torepresent negative binary values. Froma notation point of view this is utterly

simple — place a negative sign to theleft of the leftmost bit. However, fromthe point of view of a physical imple-mentation a negative sign does not exist.To tackle the negative number prob-lem, the concept of twos-complement binary representation was developed. Inthis system, the leftmost bit carries the

positive/negative information. The left-most bit is often referred to as the mostsignificant bit or MSB. The remaining

that constitute an analog sinusoidal sig-nal is infinite. That is, we can think oan analog sinusoid as being made up oinfinitely small steps in amplitude. If we elect to make the steps larger, wesacrifice accuracy fo r the luxury of requiring fewer numbers with which torepresent the purely analog waveform.

In effect, we can trade an absolutely pure analog sinusoid for one made up of small (but finite) steps plus some noise(the deviation between each step andthe ideal analog equivalent). The stepsize is directly related to digital resolu-tion. Resolution is the number of bitsused to represent the full range of

amplitude of an analog signal. For example, a 10-bit binary number carepresent an analog signal to an accura-cy of 1 part in 210, or 1/1024 (approxi-mately 0.1 percent).

Sampled Digital Signals Digital modulation is wholly depen-

dent on the fundamental concepts of sampling theory. The subject of sam-

pling is far too broad to be covered here,




Figure 3. A numerically controlled oscillator

but a brief overview is given for the

sake of clarity. Since the topic is modulation, we will

use a sinusoidal signal as a model. A con-tinuous time representation of a sinusoiis shown graphically in figure 2a. At aninstant, “t ,” on the horizontal axis, theamplitude of the sinusoid may be founon the vertical axis. A uniformly sampleversion of the sinusoid is shown in figure 2b . Note that the amplitude is onlyknown at certain discrete points in time (the sampling instants), which are uni-formly distributed in time. Sampling the-ory states that as long as at least twosampling instants occur within a com-

plete cycle of the sampled sinusoid, thenthe sinusoid can be completely recon-structed from the two sample points.

What makes sampling attractive isthat the amplitude of the sample values can be encoded as twos-complement binary numbers. So, from a digital per-spective, if we can generate a proper sequence of numeric values, we can gen-erate a digital sinusoid. But why woulwe want to generate a digital sinusoid ithe first place? Recall that modulatio

requires a carrier signal; i.e., a sinusoid.In the analog world this is implementewith an oscillator circuit that operateat the specified frequency. In the digitaworld, some form of a digital oscillatomust be realized. It turns out that thican be readily accomplished with anumerically controlled oscillator.

Numerically Controlled Oscillator

In its simplest form, a NCO consists of a lookup table made up of sinusoidalsample values, (usually implemented aa read only memory – ROM), a binar counter for addressing the ROM, and clock signal to drive the counter (see fig-ure 3). Successive address locations ithe ROM contain the successive sample

values of the desired sinusoid. As thecounter is clocked, each ne w count

addresses the next ROM location caus-ing the appropriate digital number toappear at the ROM output. The rate atwhich the counter is clocked is the sam-

ple rate of the system. If we examinethe output of the ROM over time, wewould observe a series of numbers thaupdate at the sample rate. The numbers would span a numeric range tha

depends on the bit-width of the ROM

output. Thus, the number of bits in theROM's output word determines the res-olution of the desired digital sinusoid.As an example, if the ROM output were10 bits, then using twos-complementrepresentation would yield a numericrange of –512 to +511 for the amplitudevalues of our sampled sinusoid.

The particular NCO example justdescribed would only be capable of gen-erating a digital sinusoid of one specificfrequency; namely, the sample ratedivided by the number of samplesstored in the ROM (assuming that thesamples stored in ROM span a single

cycle of the sinusoid). A more flexible NCO would use a fairly large ROM(perhaps containing 4,096 samples, or more) and a counter that can count bysome input modulus; that is, count by 1,2, 3, 4, 5, etc. as determined by a “fre-quency control number”. For example, if the sample rate is 10 MHz, the ROM is4,096 words in length, and the frequen-cy control number is 1, then the outputsinusoid would have a frequency of: 10MHz/4096 or 2.44 kHz. If the frequenccontrol number is 5, then the counter umps 5 steps for each input clock peri-

od. This, in turn, causes every 5th ROMaddress to be accessed. The net result ia sinusoid with coarser amplitude steps,

but of greater frequency. Specifically,the frequency of the new sinusoid woul

be: 10 MHz/(4096/5) or 12.21 kHz. Igeneral, this can be expressed as :

s( N /M ) where f s is the system sample

rate, N is the frequency control number,and M is the length of the ROM.

Many variations on this theme can befound in the literature. The main pointis that a NCO serves as a means for gen-erating a digital sinusoid at a specificsample rate but with programmable fre-quency. The frequency is restricted,however, to an integer multiple of the

sample rate divided by the ROM wor length up to a maximum of ½ of thesample rate (the Nyquist constraint).The larger the ROM length (addressrange), the finer the frequency resolu-tion. The more bits in the ROM outputword, the finer the amplitude resolution.

The Digital Quadrature Modulator The foregoing sections provided a

foundation for understanding modula-

Figure 4. The digital quadrature modulator tion, number representation in thedigital domain, and a method for gen-erating a sampled sinusoid. Thesethree concepts are required for a firmunderstanding of digital modulation.The fundamental building blocks of adigital quadrature modulator areessentially the same as those for theanalog single-sideband modulatoshown in figure 1b.

The digital version is shown in figure4 — the main difference being that thetwo multipliers, the adder, and the car-

rier signals are all made of digital build-ing blocks. The model shown in figure 4 can be

readily implemented using digital hard-ware. The digital makeup of the NCOswas discussed earlier. Multipliers anadders, too, are readily designed fromelemental digital building blocks (AND,OR and NOT units). The only real con-straints are the maximum possiblesampling frequency (which is mostldependent on th e semiconductor

process) and power dissipation. There is one fundamental rule that

cannot be overlooked in digital modula-tion. Both the digital carrier signal andthe digital message signal must be sam-

pled at the same rate. In some instan-ces, the message signal consists of a dig-ital signal sampled at a rate less thathe carrier. Such situations require thatthe message signal be digitally up-sam-

pled to match the carrier sample rate.However, this is another topic altogeth-er and is well beyond the scope of thisarticle. Sample rate conversion tech-niques are rigorously covered in theexisting literature.

Returning to figure 4, the digitalquadrature modulator has two mes-sage signal inputs: X and Y . In addi-

tion, two NCOs produce the quadra-ture carrier signals. The same systemclock and frequency control number are provided to both NCOs. However,one NCO has a cosine wave stored inits ROM while the other NCO has asine wave stored in its ROM. The car-rier frequency ( f c) is determined by thefrequency control number.

Typically, the X and Y input signalsare intended to be quadrature compo-




nents. For example, if X were a digital cosine wave of frequency f m, and Y werea digital sine wave of the same frequen-cy, then the output of the quadraturemodulator would be a single-sideband

tone of frequency f c – f m. The Appendixcontains a Mathcad program that pre-cisely models this scenario. For other scenarios, simple modifications can bemade to the program. For example, aupper sideband can be generated bysimply changing the sign of the operator from + to – in the right-hand portion othe “QuadModi” statement.

If a double-sideband signal is desiresimply replace Y in(i) in the right-hand por-tion of the “QuadModi” statement wit

in(i). The NCO ROM parameters, N anD, can also be changed to see the effects oboth frequency and amplitude resolution.

The system sample rate (Fs) and carrier frequency (Fcarrier) can both be changed,as well. Note, however, that the actualcarrier frequency (Fc) may not exactlmatch the value entered for Fcarrier. Thiis due to the fact that the frequency con-trol number (FCN) for the NCO must bean integer value because it controls themodulus fo r a binary counter. This

restriction means that only finite frequen-cy resolution is possible. Specifically, onlthose frequencies that correspond to theFCN values are possible.

About the author Ken 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 Inc. (www.analog.com) where he is responsible for the system level design and analysis of signal synthesis products. His specialtiesare the application of digital signal processing techniques in communicationssystems and analog filter design. Ken has been published in several industrytrade journals and was a primary contributing author to ADI's "A Technical

Tutorial on Digital Signal Synthesis". Gentile ca n be reached at [email protected].

The Results This article has demonstrated the

key elements of digital quadrature mod-ulation. The high speeds available fromtoday's semiconductor processes make itever more practical to implement modu-lation functions in digital rather thananalog technologies.

This trend will likely continue as digi-tal semiconductor technology pushesoperating speeds ever higher. It shoulbe kept in mind that the end result oimplementing analog functions in thedigital domain is a time sequence of digi-tal numbers, a natural consequence othe sampling process. However, thisnumber stream must ultimately be con-

verted to an analog waveform to be oany practical use. Thus, a digital-to-ana-log converter (DAC) must be employeto transform the digital signal into theanalog domain. As such, there is still asignificant role for analog circuitry,especially DACs. As both sample rateand resolution increase, the require-ment for high speed, high resolutionDACs becomes ever more apparent.

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Fundamentals of Digital Quadrature Modulation

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