## Bit Packing

We sometimes estimate the required bandwidth for a binary transmission system by assuming 1 bit/Hz. In most practical cases this is somewhat optimistic. Our goal, remember, is to achieve better than 4.5 bits/Hz.

For digital transmission, phase modulation has a number of excellent attributes. Among these are robustness and improved noise immunity. When we describe ''phase,'' we usually employ a circle to represent 360° of phase. In the case of BPSK (binary phase-shift keying), we describe two phase states. The first phase state we can call a binary 1 and the second, a binary 0. The decision ''distance'' should be as large as possible. In the binary phase case, this distance is 180°. For instance, 0° can be assigned the value of binary

fBy baud rate we mean transitions per second or changes of state per second. Some call this the ''symbol rate,'' but I don't care for that definition. I like to use ''symbol rate'' as the pulse rate at the output of a FEC coder.

Figure 3.2. Conceptual block diagram of a QPSK modulator.

1 and 180° the value of binary 0. However, there is no reason why we cannot assign 45° as the binary 1 and 225° as binary 0, as long as there is 180° distance between the two states. Of course, what we describe here is the familiar BPSK.

Carry this thinking one step further. Instead of two phase states, we'll use four, each separated by 90°. Now assign the binary values to each of the four phase states: 0° = 0,1; 90° = 0,0; 180° = 1,1; and 270° = 1,0. We call this quadrature phase-shift keying or QPSK. A very important point here is that bandwidth is a function of transitions per second or changes of state per second. Of course, in the binary regime, bits per second and transitions per second are the same. Figure 3.2 is a conceptual block diagram of a QPSK (4-PSK) modulator.

With QPSK, we can theoretically achieve bit packing of 2 bits/Hz. Of course, such thinking can be carried yet further as shown in Figure 3.3, which illustrates some signal state diagrams for PSK. In the case of 8-PSK, sometimes called 8-ary PSK, the signal states are separated by 45°, and with 16-ary PSK, by 22.5°. The 8-ary PSK theoretically achieves 3 bits/Hz, and 16-ary PSK achieves 4 bits/Hz.

The generalized name for this type of modulation is M-ary PSK, where the M points are distributed uniformly on a circle. Note that as M increases, the distance between adjacent states decreases. As that distance decreases, with noise and group delay corrupting the received signal, it gets more and more difficult for the receiver demodulator to make a correct decision. System designers, for the PSK case, do not generally let M be greater than 8.

One should also note how the bit packing improves: with QPSK we get 2 bits/Hz (theoretical), 8-ary PSK derives 3 bits/Hz, and 16-ary PSK achieves 4 bits/Hz. That bit-packing value (in bits/Hz) is the square root of the number of states (i.e., 16 = 4).

3.3.2.1 M-QAM, a Hybrid Scheme. QAM stands for quadrature amplitude modulation. It is a hybrid modulation scheme where both amplitude-shift

Figure 3.3. Signal state diagrams for PSK.

keying (ASK) and phase-shift keying are used in conjunction with one another.

One example might be QPSK using four amplitude levels. This is 16-QAM and its state diagram is illustrated in Figure 3.4. The amplitude levels are + 3, +1, — 1, and - 3. 16-QAM provides the same bit packing as 16-ary PSK but is more robust because signal states are derived from two distinct modulation techniques: PSK and ASK. Also, it requires less Eb/N0 for a given error performance than 16-ary PSK.

Figure 3.4. 16-QAM state diagram. I = in-phase, Q = quadrature.

Figure 3.5. Signal state diagrams for 4-, 8-, 16-, 32-, and 64-QAM.

This can be carried yet further to 32-QAM and 64-QAM, shown in Figure 3.5. 32-QAM can be derived from six levels of amplitude modulation by dropping the corner state, achieving bit packing of 5 bits per Hz theoretical (25 = 32).

In a similar fashion there are 128-QAM and 256-QAM; and some systems have been fielded using 512-QAM.

The following is a relationship between the transmitted bit rate (Rb), baud rate (symbol rate) (1 /T), and the value of M:

This formula shows that the bit rate grows linearly with the baud rate (symbol rate) and logarithmically with M.

### 3.3.3 Spectral Efficiency

Let's assume we are assigned a RF bandwidth W. Common bandwidth in the 4-, 6-, 7-, and 11-GHz bands are 20, 30, or 40 MHz. One of these bandwidths may be the value of W.

Let t] be the spectral efficiency; then

Substituting from equation (3.5) we have

Reference 4 states that, in theory, WT can be as low as 1 without adjacent channel interference. Consider now that we will use a raised cosine filter. Using cosine rolloff shaping, this could be achieved by using a rolloff factor of a = 0 or the Nyquist bandwidth. a defines the excess bandwidth, meaning in excess of the Nyquist bandwidth, usually expressed as 1 /T. The Nyquist bandwidth plus the excess bandwidth is commonly expressed as (1 + a)/T. Figure 3.6 shows some typical transmitted spectra for cosine rolloff shaping.

Selecting the value for a is very important. If a = 1, we compromise spectral efficiency because we require double the Nyquist bandwidth. Manufacturing is made more difficult if a = 0, as well as having a system more vulnerable to impairments. In modern digital radios, a is commonly selected as 0.5. It can be shown, with T= fW and a near 0.5, that FCC mask requirements are complied with. The resulting v from (3.7) is f log2 M so

POWER SPECTRUM DENSITY (dB)

POWER SPECTRUM DENSITY (dB)

(Channel Center)

Figure 3.6. Transmitted spectra for typical cosine tions Magazine, Oct. 1986, Figure 5, reprinted with

(Channel Center)

Figure 3.6. Transmitted spectra for typical cosine tions Magazine, Oct. 1986, Figure 5, reprinted with rolloff shaping. (From IEEE Communica-permission; Ref. 4.)

that systems using M = 4, 16, 64, and 256 have spectral efficiencies of 1.5, 3.0, 4.5, and 6.0 bits/Hz respectively.

### 3.3.4 Power Amplifier Distortion

There is a serious distortion consideration for M-QAM waveforms due to the fact that digital microwave transmitter power amplifiers are peak-power-limited devices. These devices become increasingly nonlinear as they approach saturation.

The problem here is the peak power. However, at the receiver we are interested in the average power to achieve a specified bit error rate. Figure 3.7 shows peak instantaneous power relative to QPSK with square pulses as a function of the cosine rolloff factor, a. The figure is for a BER of 1 X 10"6.

For M > 8, as M increases so does the ratio of peak to average power. To accommodate a given M-ary waveform and rolloff factor, the saturation

COSINE ROLLOFF FACTOR, a

Figure 3.7. Peak instantaneous power (relative to QPSK with square pulses) for M-ary waveforms for several values of M. (From IEEE Communications Magazine, Oct. 1986, Figure 8, reprinted with permission; Ref. 4.)

COSINE ROLLOFF FACTOR, a

Figure 3.7. Peak instantaneous power (relative to QPSK with square pulses) for M-ary waveforms for several values of M. (From IEEE Communications Magazine, Oct. 1986, Figure 8, reprinted with permission; Ref. 4.)

power of the transmitter power amplifier must be sufficiently large that the peak power input lies in its linear range.

If power peaks push the transmitter into its nonlinear range, then nonlinear distortion results. It causes spectral spreading of the transmitter output. Predistortion and postdistortion of the transmitter signal help.

+1 0

### Responses

• linda
What is 16QAM diagram?
7 years ago
• regina
What is the circuit diagram of PSK modulation?
5 years ago