DIGITAL MODULATION: REVEALED
An essay on modulation by Frederick Glenn
Digital modulation schemes are confusing at best. Primarily because there seem to be so many of them. To get an understanding, letís get some terminology straight (read that: understand the acronyms):
AMPS: Advanced Mobile Phone Service. The Cellular analog standard in the US. 832 channels are assigned between 824-849 MHz (Tx) and 869-894 (Rx). Channel spacing is 300 kHz. Modulation used is FM. This is not a digital modulation scheme and class C amplification is used. Except for the cellular configuration of the system, itís pretty straightforward stuff for radio guys to understand.
ADC: (North) American Digital Cellular. The first digital cellular to gain widespread acceptance. Still 832 channels in the same cellular frequency slot but now 3 users per channel. This uses a Time Division channel Multiplexing scheme (TDMA) and p /4-DQPSK modulation. Linear amplification is required. More on this later. The specification that defines this is IS-54.
IS-95: (also known as CDMA North American Digital Cellular). The latest and greatest operational system. Uses a Code Division channel Multiplexing scheme (CDMA) and BPSK/O-QPSK modulation schemes requiring linear amplification. 10 channels of 1250 kHz spacing with 118 users/channel.
GSM: Global System for Mobile communications. Not seen in North America. Widespread acceptance in Europe and elsewhere around the world. Like ADC it uses a Time Division channel Multiplexing scheme but with 124 channels at 200KHz spacing. 8 users per channel. European standards are 890-915 (Tx) and 935-960 MHz (Rx). This uses GMSK modulation. Bandwidth efficiency is not as great, but has constant-envelope modulation characteristics which allow for power efficient class C amplifiers.
The first thing to get straight is the difference between multiplexing (or multiple access) schemes and modulation schemes. TDMA and CDMA are digital multiplexing schemes...used to carry multiple conversations on the same digital data stream. AM, FM, PM, PSK, BPSK, MSK, O-QPSK, p /4-DQPSK, (I could go on!) are modulation methods used to convert the multiplexed data into a form compatible with the communications channel (e.g., radio waves). Also, all standards use FDMA (Frequency Division Multiple Access) which is just changing channel frequencies and repeating all the TDMA/CDMA multiplexing all over again. Except for a scary acronym, FDMA is pretty standard radio stuff and understood by anyone who can tune a car radio. There is also SDMA (Space Division Multiple Access) which just means that if Iím far enough away from you I canít hear your signal and I can use the frequency for myself without interfering with you. Digital Cellular systems use all of these methods in combination.
Now, the object of this game is to cram as many voice or data channels into the least bandwidth with the least cost. Battery drain and efficiency for portables is also a major factor. So we want a multiplexing scheme that can combine as many simultaneous conversations as possible into one data stream and a modulation scheme that occupies minimum bandwidth, reduces battery drain, and uses cheap (read that: class C) amplifiers. Like most engineering problems these are conflicting requirements and engineers have to "squeeze the amoebae" to get an optimum solution. In order to achieve this, 3 areas are played with: audio (compression techniques), digital data (baseband data preprocessing Ė doncha just love it?!) and modulation. All three of these areas are inseparable in that they all effect the common goal. They also depend on 3 theorems first discovered by H. Nyquist (he doesnít seem to have a first name) of Bell Laboratories in the late 1920ís.
The first and most widely known is his sampling theorem. This is the basis for TDMA. It simply says that if you want to transmit analog signals via a data stream you must sample at twice the frequency of the highest frequency component contained in the analog signal. Telephony experience has taught us that f = 3.4 kHz limit to voice signals gives what is known as "toll grade" voice quality. So Mr. Nyquist says that we have to sample at least at a rate of 6.8 kHz. Since this is a theoretical limit and engineers arenít perfect, a sampling rate of 8 kHz has usually been used. The telephony guys have been using a system called PCM (Pulse-Code-Modulation) for decades. (Be careful hereóPCM is not really a modulation scheme as I later define it. Itís really a digitizing scheme. It only takes the analog signal and "modulates" it into a serial data stream.) PCM takes the sampled analog levels and assigns them to one of 256 steps (which can be represented by 8 bits of data). Thus one toll-grade voice channel sampled at 8 kHz to a quantizing accuracy of 8 bits requires a data stream of 64kb/s.
The telephony guys had already played with the audio signal using compression techniques to give more steps at low levels than at high. The 8-bit "word" was determined experimentally to limit quantizing noise and is the basis of our 8/16/32-bit word lengths used in computers today.
Then in the 1970ís they started playing with the data itself. You have to remember that these were the days before microprocessors were prevalent and the hardware was pretty complicated stuff (remember DTL?). They found that if they made an estimate of the next sample based upon previous samples, all they needed to do was subtract this estimate from the actual sampled value to obtain a "predictor error", quantize that and send that along the transmission path instead. The reverse was done at the receiving end. Delta "modulation" was also used along with many other adaptive techniques (this was a science onto itself!). The result was Adaptive Differential Pulse Coded Modulation (ADPCM) which gave "almost" toll-grade voice quality but reduced the data stream to as little as 10Kb/s. Still just a serial data stream incompatible with radio transmission directly (in spite of all this "modulation" going on!).
All they had left was to speed up each voice channel, assign it to a time slot and combine it with other channels (TDMA, remember?) which occupied other time slots. This was the basis for the T1 carrier standard (an old telephony term). The receiving end just did the opposite of all this to recapture the original analog signal.
There were a lot of advantages of digital transmission to the telephone companies. First, digital signals can be decoded and repeated indefinitely without signal to noise impairment. The noise in analog repeating is additive. Secondly, TDMA time slots could be switched by logic circuits or computers without actual troublesome relay contacts. But they still had to modulate it to transmit it over radio.
In radio communications systems, the signal to be transmitted f(t) must be translated to a higher (radio) frequency. Another well known communications theory theorem is known as the frequency-translation theorem. This theorem states that a shift of w o in the frequency domain is equivalent to multiplication by e jw t in the time domain. Since cosw ot = e jw t + e- jw t it follows that multiplication of a signal f(t) by a cosine wave (or sine wave) cos (w o t) translates the frequency spectrum by the amount ± w o (do the math!). This is the process we radio guys define as modulation and is, in fact called the modulation theorem.
This is pretty much where it stood until personal computers, telephone modems, and cellular telephones came around and REALLY BIG BUCKS were being thrown at engineers to save bandwidth both in telephone line (wireline) and radio (now known as "wireless"!) transmission media. So they went back to their textbooks and discovered that this Bell Labs engineer without a first name (H. Nyquist) had a whole lot more to say on the topic of digital transmission way back in 1928!
Nyquistís minimum-bandwidth theorem
"If synchronous impulses, having a transmission rate of fs symbols per second, are applied to an ideal linear-phase brick-wall low-pass channel (in other words, an ideal low-pass filter) having a cut-off frequency of fN = fs / 2 Hz, then the responses to these impulses can be observed without intersymbol interference."
OK, what has that got to do with the price of eggs? Well, it defines a bandwidth required for data transmission at a certain rate if we want to avoid ISI Ė InterSymbol Interference (this guy did this in 1928? Now you know why I own Bell Labs...er... Lucent stock!). If we apply an impulse (remember that an impulse has no width) to an ideal low pass filter we get a distorted pulse output. This distorted pulse has the familiar form of sinx/x . A string of impulses creates the overlapping sin x/x waveforms. What the theorem says is that if we choose our bandwidth carefully, the peak of each pulse will occur only at the zero crossings of the others. In other words, they will not interfere with each other at the times we sample them. Mr. Nyquist says that with a brick wall filter, this occurs if we let fN = fs / 2 Hz. The frequency fN is called the Nyquist frequency.
There are at least two practical problems with the implementation of this theorem. First, our typical data stream is rectangular wave and not impulses. Second, the Brick Wall Filter Co., Inc. is no longer in business. First, thereís a trick we can use to fool the filter into thinking our rectangular pulses are impulses. In an infinite bandwidth medium, the Fourier transform (frequency content) of the impulse function is flat over all frequencies whereas the transform of a rectangular pulse has a sinx/x shape. Since the output Fourier transform of a system is obtained by multiplying the transform of the excitation by the channel transfer function, it is easy to see that if we choose a filter with x/sinx response and multiply it by sinx/x (the transform of the rectangular pulse) we get a system transform = 1 (the same as the impulse functionís transform). With such an equalizer, we can "fake" the system and still get ISI free transmission.
Now what about the filter itself? Mr. Nyquist to the rescue again with yet another theorem:
NYQUISTíS VESTIGIAL SYMMETRY THEOREM
"The addition of a skew-symmetrical, real valued transmittance function Y(w ) to the transmittance of an ideal low pass filter maintains the same zero-crossings of the impulse response."
What this means is that we can add something goofy to the transfer function of the ideal low-pass filter and still keep the same zero crossing points which is the necessary condition for ISI-free transmission. Skew-symmetrical vestigial.... ? All this means is that the desired function has kind of an upside down mirror image symmetry around the frequency fN. . With another mathematical slight of hand we can choose this goofiness such that we end up with a realizable filter. One function that does this is the raised cosine function and is the most commonly used. If you really have to know what that is you can look it up in Feherís book (any of them!). Iím too lazy to type it here. All we need to know here is that contained in this function is a parameter a commonly referred to as the roll-off factor.
So what weíve done to this pulse stream of data is pre-emphasized it with a raised cosine low pass filter and an equalizer of x/sinx form. We can choose the roll-off factor a but there are tradeoffs. For a = 0 we end up with the minimum theoretical bandwidth but an unrealizable filter (remember we havenít modulated any of this yet but keep in mind that the final bandwidth is a function of the modulating signal). For a = 1 the bandwidth is twice the minimum. In both cases the ISI is zero, but jitter approaches ¥ as a approaches zero. Jitter is important because of system timing requirements. The zero crossings of the eye diagram are used to establish system timing and will suffer if a = 0. The compromise value most often used is a = 0.3.
OK, Letís Modulate this stuff
As I defined previously, modulation is just frequency shifting and the simplest thing we can do is multiply the baseband signal we just created by a high frequency carrier and end up with "radio". It can be shown that if the baseband signal has no DC component (we have about as many 1ís as we have 0ís and we level shift so that a 1 is a positive voltage and a 0 is a negative voltage Ė something called NRZ data) then the resulting signal is equivalent to DSB-SC (double sideband, suppressed carrier) and is usually called BPSK just to confuse us. A DC component would show up as a carrier. Well, we all know that DSB-SC carriers redundant information in the two sidebands. Why not SSB-SC? Well, this has to do with coherent Vs. Non-coherent demodulation and Iím not going to get involved with that right now. Besides, thereís a neater trick we can play with all of this.
Letís take the baseband original data (before filtering) and split it into two streams each at half the data rate of the original. If we add a delay to one, then the original data would be transmitted (still digitally) 2 bits at a time, on two different wires. In other words, the odd bits (1,3,5, etc) would be in the first stream (letís call this the I(t) data for In-phase) and the even bits in the Q(t) stream (for quadrature). Weíll set the delay to Ĺ the bit period Ts / 2 . So the Odd and even bits appear at the same time, but in different wires. Now do the filtering as we did before, but on each data stream individually.
Take the I(t) signal and multiply it by cos( w ot ) and multiply the Q(t) signals by sin( w ot) where w o is the desired carrier frequency. Add these two signals together, hook up your antenna, and let the receiver guys try and figure it all out. This technique is called O-QPSK (Offset-Quadrature Phase Shift Keying). If we hadnít shifted the data, it is simply QPSK. QPSK is simply two BPSK systems operating in quadrature. The advantage of offsetting the data is that only one data stream is in transition at a time. This means that the signal envelope never crosses zero as it would with straight QPSK.
It should be noted here that for both QPSK and O-QPSK the data transitions would be instantaneous if the data were not filtered and the resulting RF envelope would be constant. Alas, however, the bandwidth would be infinite and we must bring back the filters to keep the neighbors from complaining. This pre-filtering slows down the data transitions and shows up in envelope variations. Filtered O-QPSK has an envelope variation of 3db (about 30%) whereas filtered QPSK has 100%. Both have a net spectrum efficiency of about 2bits/Hz/s.
Where does p /4 DQPSK come in? If we shift the data by p /4 instead of p /2 as in O-QPSK we get transitions like the following:
QPSK: 0o, ± 90o, ± 180o
O-QPSK: 0o, ± 90o
p /4-DQPSK: 0o, ± 45o, ± 135o
This is really a compromise between QPSK and O-QPSK and was chosen as the US and Japanese standard digital modulation. Envelope variation is larger than O-QPSK and it requires linear amplification to maintain spectrum efficiency. Why was it chosen? Well, remember I said weíd let the receiver boys worry about making sense of all this? It turns out they had something to say about it after all. In the 1980ís it was felt that differential decoding might be necessary for the high-Doppler shifted/faded signals the receiver boys were expecting to get and p /4-QPSK was easy to differentially modulate and demodulate whereas O-QPSK was more difficult. So we got p /4-DQPSK. As it turned out, the fears were largely unfounded and p /4-DQPSK will eventually get the old heave-ho. We could have been using more power efficient O-QPSK instead, but the hindsight is always 20-20 and p /4-DQPSK will be with us for a long time. The newer CDMA systems use O-QPSK however.
What About GMSK?
FSK, or Frequency Shift Keying was one of the earliest techniques used to transmit data. Non-coherent demodulators can be used but they require higher carrier-to noise ratios than coherent detection. If we were to choose our deviation such that
D fpp = 2D f = f2 Ė f1 = 1/(2Tb)
then we could recover the bit timing Tb from the difference f2 Ė f1 and have our coherent demodulation. Then by definition, the modulation index m is
m = D fpp · Tb = Ĺ
and this is called MSK, or Minimum Shift Keying. MSK has a very high bandwidth so filtering techniques need to be used. It can be shown that low pass filtering prior to modulation is equivalent to band pass filtering after modulation. Since baseband low-pass filters are easier to realize (they can be made using DSP techniques) than bandpass RF filters, the low pass filters are always used. Like QPSK, the baseband low-pass filters reduce the occupied bandwidth. Unlike QPSK they do not effect the constant-envelope property of FSK and may be amplified by class C amplifiers. The downside of GMSK? The main lobe of the spectrum is considerably (50%) wider than O-QPSK. Spectrum use is only 1bit/Hz/sec. There ainít no free lunch. Just as a point of interest, the side lobes of an unfiltered MSK signal fall off at a faster rate than the side lobes of an unfiltered QPSK signal. Arenít you glad you know that?
GMSK is the modulation standard of GSM cellular and is a popular modulation for RF modems and the like. Alternative configurations using quadrature modulators are also possible.
So what are we left with? All things in life are compromises, but I hope this little essay helps straighten out some of the acronyms thrown around. This is not an easy subject and this paper is intended to be only the very basic start of a thorough understanding. I skipped over code division multiplexing pretty fast; thatíll be the subject of itís own little essay. One of the main points to grasp is the distinction I have made between multiplexing and modulation methods. Stay tuned because there are many more improvements to come in data compression and modulation techniques. See for example Dr. Kamilo Feherís "IJF" modulation which combines low bandwidth with constant envelope and ISI-free signaling.
FURTHER READING and acknowledgements
Wireless Digital Communications, Dr. Kamilo Feher, Prentice-Hall, 1995
Communications Systems, B.P. Lathi, John Wiley & Sons, 1965
Advanced Digital Communications, Kamilo Feher, Noble House, 1997
The theorems quoted in this essay are quoted from the above textbooks.