ECPE 364: view o Small-Carrier Amplitude Modulation his handout is a graphical review o small-carrier amplitude modulation techniques that we studied in class. A Note on Complex Signal Spectra All o the illustrations on this handout will use the ollowing 4 time-domain signals and their equivalent analytical Fourier transorms: x (t) = sn(t) ψ! X () =u( jj) x (t) = sn t ψ! X () =( jj)u( jj) x 3 (t) = sn(t+ )+sn(t )Λ ψ! X 3 () =cos(ß)u jj x 4 (t) = ß J 0(ßt) ψ! X 4 () = u( p j j) 4 hese pairs were computed rom the transorm table at the end o the handout, applying duality whenever necessary. It should be noted that x (t), x (t), x 3 (t), and x 4 (t) all produce real-valued spectra. O course, realistic band-limited signals that carry useul inormation will be complexvalued. he dierence between an ideal, real-valued spectrum and a realistic, complex-valued spectrum is not always apparent rom a two-dimensional plot. Rather, it is best to show the dierence by plotting the signal spectrum on a skewed real and imaginary graph as shown below: Box Spectrum Complex Spectrum he idealized signal spectra are useul or illustrating concepts in AM modulation. All o these concepts work equally well with complex-valued spectra but the operations become much more diicult to visualize. So take everything you see in this handout with a grain o salt! Double Sideband (DSB) Below are the Fourier transorms o cosine and sine waves, which produce spectral lines in the requency domain: Cosine Spectrum Sine Spectrum C - C - C C
he Fourier transorm o the cosine produces even deltas. he Fourier transorm o the sine produces odd deltas that are also imaginary. AM modulation with a sine or cosine shits a band-limited signal up to some carrier requency, c. Consider the case o cosine modulation. Follow the diagram below toseehow a band-limited signal is modulated and demodulated using a cosine carrier wave. he steps o the diagram correspond to the ollowing operations:. Start out with a baseband signal spectrum. his example uses X ().. he signal x (t) is multiplied by cos(ß c t) in the time domain. his has the eect o shiting copies o the baseband spectrum X () up and down the requency axis. he resulting signal x (t)cos(ß c t) is a Double Sideband, Suppressed Carrier (DSB-SC) amplitude modulated signal. 3. o demodulate this signal, the modulated signal is multiplied by cos(ß c t) again in the time domain. he resulting signal, x (t)cos (ß c t), still needs to be passed through an LPF to get rid o the high-requency signal components that exist in the spectrum at c. 4. his is the original recovered spectrum ater sending the signal in Step 3 through the LPF. Cosine Modulation. X ( ). Cosine Modulation C - C 4. Output o ILPF /4 3. Second Cosine Multiply - C - C C /4 C AM modulation may be done with a sine wave carrier as well. Follow the diagram below to see how a band-limited signal is modulated and demodulated using a sine wave. he steps o the diagram correspond to the ollowing operations:. Start out with a baseband signal spectrum. his example uses X ().. he signal x (t) is multiplied by sin(ß c t) in the time domain. his has the eect o shiting copies o the baseband spectrum X () up and down the requency axis. Because the Fourier transorm o a sine wave is odd, imaginary impulses, the requency-shited replicas o our triangle spectrum are also odd, imaginary copies. he resulting signal x (t)sin(ß c t) is a DSB-SC AM modulated signal.
3. o demodulate this signal, the modulated signal is multiplied by sin(ß c t) again in the time domain. Notice how the imaginary copies o the triangle spectrum in Step have lipped back to the real axis. he resulting signal, x (t) sin (ß c t), still needs to be passed through an LPF to get rid o the high-requency signal components that exist in the spectrum at this point. 4. his is the original recovered spectrum ater sending the signal in Step 3 through the LPF. Sine Modulation. X ( ). Sine Modulation - C C 4. Output o ILPF 3. Second Sine Multiply - C C C /4 - C /4 3 Single Sideband (SSB) In truth, we do not require a double sideband modulated signal to recover the original signal. Actually, we can use Single Sideband (SSB) modulation and still recover the exact original signal. his is illustrated by the igure below, which corresponds to the ollowing steps:. Start out with a baseband signal spectrum. his example uses X 3 ().. he signal x 3 (t) ismultiplied by cos(ß c t) in the time domain. Once the copies o X 3 () have been shited up and down the requency axis, one o the sidebands is removed using an near-ideal ilter. his particular illustration uses an IHPF to remove the lower sideband. 3. o demodulate this signal, the modulated signal is multiplied by cos(ß c t) again in the time domain. he resulting signal still needs to be passed through an LPF to get rid o the high-requency signal components. 4. he original spectrum is recovered ater sending the signal in Step 3 through the LPF. SSB AM cuts the amount o transmitted signal bandwidth in hal compared to DSB AM. his additional space in the requency domain may be used or requency division multiplexing (FDM) o other signals. 3
Vestigial Sideband (VSB) modulation is closely related to SSB modulation. For VSB, either the lower or upper sideband o the DSB AM signal is removed using a non-ideal ilter which leaves a small vestige o the spectrum behind. VSB AM has less bandwidth than DSB, but more bandwidth than SSB. Single Sideband (SSB) Modulation. X 3( ). SSB Modulation C - C 4. Output o ILPF /4 3. Second Cosine Multiply /4 C /4 - C C - C 4 Quadrature Amplitude Modulation (QAM) Like SSB, QAM is another way o doubling" the spectral eiciency o signal transmission. Instead o slicing o sidebands, QAM takes advantage o the orthogonality o the in-phase and quadrature channels to transmit two dierent signals simultaneously through the same requency band. he in-phase channel reers to a signal modulated with a cosine. he quadrature channel reers to a signal modulated with a sine. When the in-phase and quadrature channels are demodulated with cosines and sines, respectively, each channel appears invisible to the other. he picture below illustrates two signals (x (t) and x (t)) modulated using QAM. Signal x (t) is modulated on the in-phase channel and signal x (t) is modulated on the quadrature channel. Notice how the cross-channel spectra vanish at baseband in the demodulation steps. 5 Can We Use Both QAM and SSB to Multiplex Signals? I using QAM or SSB doubles the spectral eiciency o DSB-SC AM, we may be tempted to use both techniques together to quadruple the spectral eiciency. Unortunately, using one type o modulation scheme precludes the use o the other. o give anexample, consider the igure at the end o the handout. In this example, we have modulated the single sidebands o 4 signals, x (t), x (t), x 3 (t), and x 4 (t), around a carrier requency o c. wo signals (x (t) and x 3 (t)) are modulated using the in-phase channel, and two signals (x (t) andx 4 (t)) are modulated using the quadrature channel. he 4 signals take up the same bandwidth as a single DSB-SC AM signal. Let us try to extract signal x 3 (t) rom the multiplexed signal. erring to the igure, we ilter out the lower sideband irst, since this just contains x (t) and x (t). hen, we down- 4
Quadrature Amplitude Modulation ) Cosine Modulation C ) - C Sine Modulation C Add In-phase and Quadrature Channels - C C - C Output o In-Phase ILPF Cosine Multiply - C C C - C Output o Quadrature ILPF Sine Multiply C - C C - C 5
convert the spectrum by multiplying by cos(ß c t). Our hope is that, in doing this, we willalso get rid o the quadrature sideband that corresponds to x 4 (t). But that is not what happens. Ater down-shiting the spectrum and iltering out the high-requency components, the sideband corresponding to x 4 (t) still exists in the baseband signal. here is a basic principle here to be learned: removing the sideband rom a DSB-SC AM signal destroys the ability to distinguish between the in-phase and quadrature channels. At this point, we may be tempted to say Look at the igure. I can see the dierence between the S 3 () on the real plane and S 4 () on the imaginary plane. Shouldn't there be a way to separate them?" Here is the problem: this academic example using real-valued spectrum makes the problem o separation appear simpler than it really is. agine i S 3 () ands 4 () were more realistic, complex-valued spectra o the type illustrated on page. he two spectra would be tangled together in such a way that it would be impossible or you (or the receiver) to determine which spectral components had been modulated onto which channel. 6 Appendix Cosine ime Domain Useul Fourier ransorm Pairs Frequency Domain cos(ß 0 t) [i( 0)+i( + 0 )] Even Deltas Sine sin(ß 0 t) j [i( 0) i( + 0 )] Odd Deltas Box - u( jtj) sn() Sinc riangle ( jtj )u( jtj) sn () Squared Sinc Cosine Pulse U-Shape cos jtj ßt u u( jtj) p ( t ) Oset Sincs sn(+ )+sn( )Λ ß J 0(ß ) Bessel 6
Why QAM + SSB Does Not Work ) ) QAM + SSB Modulation o 4 Dierent Signals 3 ) 4 ) - C C Output o ILPF Multiply By Cosine Pass hrough IHPF Beore Downconversion /4 - C /4 C C C - C - C 7