Digital Quadrature Modulation

The concept of modulation is not at all new, it has 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 containsthe information to be modulated. It usually consists of a band-limited spectrum of sinusoids (such as music). The other signal is the carrier signaland is generally a pure tone (a sinusoid of asingle frequency). The frequency of the carrier isreferred to as the carrier frequency and will bedenoted by the symbol fc (or ωc for radian frequency) throughout this text. Typically, fc is much higher in frequency than the highest frequency componentof the message signal.The BasicsThe concept of modulation comes from thetrigonometric identity:If we assume that the message signal is a puretone of frequency, fm, then the message can be mathematically represented as cos(2πfmt). The same assumption can be made about the carriersignal, thereby expressing it as cos(2πfct). The “pure tone” assumption makes the mathematicsmuch more tractable. However, it is important tokeep in mind that the message signal is rarely apure tone.Typically, it is composed of time variations inamplitude, frequency, phase, or any combinationthereof. Even the carrier need not necessarily be apure sinusoid. Applications exist in which the carriersignal is a square wave with a fundamental frequency,fc. The harmonics of fc inherent in the square wave are dealt with by filtering the modulatedsignal.The mixing process mentioned earlier can bethought of as a multiplication operation.Therefore, the trigonometric identity above maybe employed to represent the mixing process asfollows:Thus, the mixing of the message and carrierresults in a transformation of the frequency of themessage. The message frequency is translated fromits original frequency to two new frequencies — onegreater than the carrier (fc + fm), and one less than the carrier (fc – fm), the upper and lower side bands, respectively. Furthermore, the translated signalundergoes a 6 dB loss (50 percent reduction) as dictatedby the factor of ½ appearing on the righthandside of the equality.The form of modulation just described isreferred 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 sidebandmodulation, can be used to eliminate eitherthe upper or lower sideband. One method of performingsingle sideband modulation is to employ a quadrature modulator. A quadraturemodulator mixes the message with twocarriers. Both carriers operate at thesame frequency, but are shifted inphase by 90 degrees relative to oneanother (hence the “quadrature” term).This simply means that the two carrierscan be expressed as cos(2πfct) and sin(2πfct). The message, too, is modified to consist of two separate signals:the original and a 90 degree phaseshiftedversion of the original. Theoriginal is mixed with the cosine componentof the carrier and the phaseshiftedversion is mixed with the sinecomponent of the carrier. These two modifications result in the implementation of the single sideband function.Trigonometrically, this can beexpressed as: