In

communication theory,

*diversity* means transmitting a signal through multiple

channels that

fades independently, so that the detrimental effects of

fading is minimized. The actual number of independent channels used is called the

*order of diversity*.

We use a flat Rayleigh fading channel as an example. Suppose the modulation scheme used is binary PSK, with coherent demodulation (i.e. carrier phase is known). Without fading effects (i.e., an AWGN (additive white gaussian noise) channel), the bit-error rate P_{b}=Q(sqrt(2γ_{b})), where γ_{b}=E_{b}/N_{0} is the signal-to-noise ratio per bit, E_{b} is the received energy per bit, N_{0} is the one-sided power spectral density of the white gaussian noise, and Q(.) is the Marcum Q function, i.e. Q(x) is the probability that a random variable following standard normal distribution is larger than x.

In a Rayleigh fading channel, the received signal has its amplitude multiplied by α which follows Rayleigh distribution (the carrier phase shift does not matter here), so the conditional bit-error rate for a certain α becomes P_{b}(α)=Q(sqrt(2α^{2} γ_{b})) (now γ_{b} denotes the *average* signal-to-noise ratio), then by averaging over α we obtain the overall bit-error rate P_{b}=∫P_{b}(α) p(α)dα where p(α) is the probability density function of Rayleigh distribution. It can be shown that P_{b} is now roughly proportional to 1/γ_{b}, thus it decreases far more slowly than the AWGN case when γ_{b} increases, or in other words the required signal-to-noise ratio is *much* higher for a reasonably low bit-error rate. Intuitively, it is because a Rayleigh variable has a fair probability of having a very low value, so no matter how large the average received signal-to-noise ratio is, there is always a non-negligible chance that a particular bit gets faded beyond correct detection. The results are roughly the same when noncoherent demodulation is used, and channel coding without interleaving (see below) won't help much either, since if a codeword is received during severe fading, it would be impossible to correct the many resulting errors in *one* codeword, no matter how good the code is.

Utilizing *diversity* is the way to achieve reliable communication in such fading channels. Diversity works by transmitting or receiving the same information (may be coded differently, but they come from the same information) through multiple channels (in an abstract sense) with independent amount of fading, and combine the signal energy received in each channel before detecting the actual bit. The precise bit-error rate performance when utilizing diversity can be calculated with moderate complexity, but we can intuitively analyze it as follows: when a single channel is utilized, the probability that the fading is severe enough to defy detection is roughly proportional to 1/γ_{b}, then when L channels are used, we will fail to detect the transmitted bit correctly only when all these channels go through such severe fading (otherwise the combination algorithm will effectively choose the best received signal and correctly decode the bit), whose probability is roughly 1/γ_{b}^{L}. Much better, eh? With a sufficiently high order of diversity L, the bit-error probability will decrease as rapidly with increasing γ_{b} as the AWGN case, mostly removing the detrimental effects of multipath fading.

There are three basic types of diversity: *time diversity*, *spatial diversity* and *frequency diversity*. These techniques may also be used together.

Time diversity means to repeatedly transmit the same information at different times, with an interval no less than the coherence time of the channel t_{c}, so that each transmission is faded independently (using the terminology above, we regard each transmission as having gone through an abstract "channel"). It is generally implemented by coding followed by interleaving the coded bits, since simple repetition is not as efficient. The interleaving length should be significantly larger than t_{c}, so that each bit in a codeword is faded independently, and diversity of order d_{min} (the minimum Hamming distance of the code) can be achieved. The downside is that interleaving with a large length requires a large buffer, and results in a large delay which may be not tolerable in many real-time applications, such as voice.

Spatial diversity means to use multiple transmitting and receiving antennas, with the space between adjacent antennas no less than a half-wavelength. In this way, the channel between each transmitting antenna and each receiving antenna will have wildly different phase shifts for every path, resulting in essentially independent fading. When multiple transmitting antennas are used, the signal transmitted on each antenna should be coded differently, using space-time coding. When certain criteria are met, diversity of order L=L_{t}L_{r} can be obtained, where L_{t} and L_{r} are the number of transmitting and receiving antennas, respectively. Of course, installing multiple antennas with the required separation increases system cost and size, so it is currently not feasible in your cell phone, although base stations do use multiple antennas.

Frequency diversity means to transmit the signal on multiple frequencies, with the difference between adjacent frequencies no less than the coherent bandwidth f_{c} (equals to the inverse of the delay spread T_{m}), so that each frequency fade independently. Actually, I have not seen any civil system utilizing frequency diversity directly, probably because bandwidth usage is high in this way. However, spread spectrum systems (such as CDMA systems) can be viewed as having some frequency diversity, since spreading usually expands the signal bandwidth beyond the coherence bandwidth of the channel, making the channel frequency-selective, thus decreasing the chance that the signal get faded along the whole bandwidth. The assumption that the signal bandwidth is larger than the coherence bandwidth is equivalent (by inverting both sides) to that the delay spread T_{m} is larger than the chip length T_{c}, so by correlating the received signal with differently delayed versions of the spreading sequence, signal energy of different paths can be separated then combined, therefore realizing the improved diversity. Such a receiver is called a *Rake receiver*.

Diversity may also arise in other cases. For example, many cellular CDMA systems allow for soft handoff, where a mobile station is connected to multiple base stations simultaneously. In this case, on downlink (from base station to mobile station) the multiple base stations can send the same signal to the mobile station, while the latter can use a slightly modified Rake receiver structure to combine these signals. Obviously, the signals from different base stations fade independently, so we can have diversity here, called *macro diversity*.

**Remarks:**

*smartalix* notes that some auto FM tuners use twin-antenna diversity systems to reduce ghosting and multipath.