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Welcome to the World of CDMA
Optimum Bandwidth for CDMA
Adapted from a presentation by Klein S. Gilhousen
at the International Conference on Personal, Mobile Radio, and
Spread Spectrum Communications
Beijing, China, October 12-14, 1994

The effects of multipath propagation on the CDMA signal are quite different from those of narrowband signals. While increasing spreading bandwidth leads to an asymptotic improvement in Erlang capacity of CDMA per megahertz, multipath propagation phenomena have increasingly detrimental effects. The direct sequence modulation and Rake receiver do mitigate the multipath that leads to narrowband Rayleigh fading. However the complexity of the Rake increases with bandwidth. So what is the "optimum" spreading bandwidth? Why did the designers of IS-95A choose 1.25 MHz?

This question can never be answered precisely because of the varied environments that may be served by the systems in question, and because of the constantly changing technology. It is quite possible that, faced with the same design problem today, that some of the design choices might be different. Some entrepreneurs have, in fact, proposed SSMA schemes with radically larger bandwidths, claiming superior performance.

These pages discuss the effect on system performance, complexity and co-existence with other systems of the choice of spreading bandwidth in a CDMA spread spectrum system. The choice of 1.25 MHz is at least highly plausible, if not provably optimum.

CDMA Capacity and the Law of Large Numbers

In reality, all CDMA systems are also, at least to a certain extent, Frequency Division Multiple Access (FDMA) systems. Usually, a given spectral allocation will be available for provision of a communications service. The system designer has the choice of filling the entire spectral allocation with a single spread spectrum waveform or of dividing the spectral allocation into a number of sub-bands, each of which is filled with a narrower bandwidth spread spectrum waveform. In this latter case, the resulting system is considered to be a hybrid of FDMA and CDMA. IS-95A is such a hybrid system because a number of 1.25 MHz bandwidth channels can be defined and used simultaneously. The radio channel capacity of a CDMA cellular system is defined as the number of subscribers that can access a particular base station of the system simultaneously. Calculation of the number, N, of equal power, fixed rate interfering users in an isolated cell of a CDMA system is generally accepted to be given by
where W/R is the spread bandwidth-to-data rate ratio, sometimes called processing gain, and Eb/(N0+I0)target is the bit energy-to-interference density ratio which the intended receiver's modem needs for an acceptable error performance level (or call quality). This formula follows from elementary dimensional analysis arguments, which can be found in our Capacity pages. Consider a frequency allocation of 10 MHz, used for CDMA by users whose average data rate is 10 kHz. If an energy-to-interference ratio of 6 dB is required, then the number of users that can be served is about 250. This is approximately true whether the signaling uses a spread bandwidth of 10 MHz in a single RF channel or whether the signaling uses a set of 8 RF channels, each of 1.25 MHz in bandwidth. For the narrower bandwidth case, the processing gain factor, W/R, is reduced but the number of RF channels is increased by the same factor, resulting in no net change in capacity in the nominal case.

The capacity of a CDMA system can be readily increased beyond that of (1) by applying additional techniques such as voice activity gating, sectorization and frequency reuse. Voice activity gating reduces the transmission data rate while a subscriber is not talking. Typical conversational speech is active only about 40% of the time, allowing a capacity increase of a factor of about 2.5. Sectorization typically divides the coverage area of a cell into three equal sectors which share the total offered load of the cell, allowing the total capacity of the cell to increase by a factor equal to the number of sectors.

Frequency reuse reduces the capacity of each cell slightly because of the interference from neighboring cells of the system, while multiplying the overall system capacity by the number of cells. The frequency reuse efficiency is about 65% in typical CDMA systems.

These capacity increasing techniques introduce statistical variation into the analysis of capacity. However, if the number of subscribers sharing a CDMA channel is large enough, then the performance will be very close to the average. This is known as a "law of large numbers" effect. Clearly, if the spread bandwidth is substantially reduced, the number of users sharing a given channel will not be "large" and performance will be degraded. Alternatively, as the spreading bandwidth is increased, there are diminishing returns as the capacity approaches its asymptotic maximum. This effect has been studied over a very large range of processing gains from 16 to 1024. In the table below, we show the loss of capacity versus processing gain. The computation is for a probability that the available capacity exceeds instantaneous demand greater than 90% of the time. Note that IS-95A has a processing gain of 128, while a system that filled, say, 10 MHz (the original cellular allocation) would have a processing gain of 1024. Note that the difference in capacity between these two is less than one-half dB. This small improvement in capacity for wider bandwidth can very easily be lost by inefficiencies arising from a variety of sources, as discussed below.
Performance Loss for Finite User Population
Processing Gain (W/R) Efficiency (%) Capacity Loss (dB)
16 63 2.0
32 70 1.5
64 78 1.0
128 83 0.75
256 88 0.55
512 92 0.36
1024 94 0.27
Temporal Dispersion of ChannelIt is well known that the cellular channel is severely impacted by the presence of multipath in the channel. In multipath, the propagating signal is reflected from a number of objects in the physical environment, such as buildings, hills, and vehicles. Multiple replicas of the signal arrive at the receiver after traveling over differing paths, as shown schematically in Figure 1. Each replica has a different phase, attenuation and time delay.

Figure 1. The multipath environment.
The ellipses shown in Figure 1 are the contours where reflectors give a constant propagation delay between the cell-site base station and the mobile. If the channel impulse response is measured at a particular location, it will show multiple delayed components, such as in Figure 2.

Figure 2. The channel impulse response.
Most environments will show an impulse response with a total temporal dispersion of 10 microseconds or less, although dispersions exceeding 20 microseconds are occasionally found.

The extent to which a delayed signal adds coherently depends on the signal bandwidth. An impulse received with an infinite bandwidth totally resolves the multipath, as shown above. Each line represents a single path and does not fade, but it contains only a small fraction of the total signal energy. For a CW signal, all these components add up to a single stationary phasor that changes as the mobile (or the scatterers) move, producing fades. This effect is illustrated below for the narrowband case where all signal components are summed incoherently.

A signal with a finite bandwidth W has a time resolution associated with it
. (2)
A CDMA correlation receiver forms a temporal window such that signals arriving within the window are accepted and signals arriving outside the window are rejected. Some temporal windows are shown schematically in Figure 2. A receiver with only one correlator might accept only a fraction of all the incident signal power in a multipath environment. If multiple correlators are provided, each with different a delay offset, then more of the signal energy can be captured. Furthermore, the correlator outputs can be summed coherently avoiding the fading effect in found in narrowband receivers.
Figure 3. Effect of bandwidth on energy collection.
Multipath delay spread in the outdoor cellular environment can be described statistically, over local regions, as an exponential distribution
where t is the excess delay beyond the delay of the shortest path, and T is the average multipath delay spread. The delay spread T is often used to characterize multipath environments.

Figure 4 shows measured statistics for multipath dispersion in the CDMA test area of San Diego. This data shows the average behavior of the channel, not the impulse response at any particular time or place. This data is typical of many locations (see, for example, Cox). It shows an approximately exponential distribution with a delay spread of about 1.5 microseconds. A spread spectrum bandwidth of 1.25 MHz is well matched to the this channel condition, especially considering that three channel Rake receivers are typically employed in conjunction with IS-95A. It also shows the necessity for increasing the number of Rake receiver elements as the channel bandwidth increases.

Figure 4. Multipath dispersion statistics.
It is of interest to determine the fraction of the total signal energy contained within some time resolution interval, S. For example, suppose that the delay spread T = 1.5 microseconds. Further suppose that the CDMA waveform uses a 1.25 MHz bandwidth signal with about a 1 microsecond temporal resolution, as in IS-95A. A single correlator positioned at delay t, captures an energy equal to

Carrying out the integral, one finds that one correlator, positioned at t=S/2 captures, on average, 49% of the total received energy. Three correlators, integrating from 0 to 3S, can capture up to 86%. The energy losses are, respectively, 3 dB and 0.6 dB relative to a receiver that captures all the energy.

Now compare the 1.25 MHz scheme to a system that uses, for example, the entire 10 MHz original cellular allocation. A 10 MHz bandwidth system will have a temporal resolution of about 0.125 microseconds. A single correlator, optimally located, captures only 8% of the total received energy, for a loss of -11 dB of . This has two implications. First, a large number of correlators would be required to capture a significant fraction of the received energy. Second, even the best correlators will have to cope with very weak signals. This makes a practical Rake receiver very difficult to build. A Rake correlator positioned at 1.5 microseconds offset will capture only 2.9% or -15.3 dB of the total signal. If a spread spectrum receiver fails to capture most of the incident received signal power, then capacity will be lost. For example, in the 10 MHz spread bandwidth case, if only a single correlator is used, there is an 11 dB loss. This dB loss translates directly into capacity. The capacity of this system will be less than 10% of the capacity of a system without multipath. Compare this to the three-correlator 1.25 MHz system, which achieves 87% of the multipath-free capacity.

Practical RAKE Receiver Issues

As discussed above, a Rake receiver is necessary in order to capture a significant fraction of the received signal power of a spread spectrum signal in a multipath environment. A Rake receiver provides other benefits as well. It helps prevent the multipath signal components from combining noncoherently. Noncoherent combining occurs naturally, and unavoidably, in narrowband systems; it is the cause of the familiar Rayleigh fading. Given a finite spread bandwidth, however, one cannot expect the Rake to resolve all the multipath components all the time. This sometimes results in fading in each of the Rake channels (fingers). But to the extent that this residual fading behavior is uncorrelated between fingers, optimal diversity combining still improves performance over a single correlator receiver.

The Forward CDMA Channel is specifically designed to permit coherent combining of the multipath components. It includes an unmodulated pilot signal that gives the receiver a phase reference for demodulating data-bearing channels. The Rake fingers, in effect, remove the random, independent carrier phases from the multipath components. The resultant vector amplitudes then can be added in-phase.

The Reverse CDMA Channel uses non-coherent modulation and has no pilot. Diversity combining must be based on scalar symbol amplitudes and side information from the decoding. There is still diversity gain, however, over a receiver that collects only one component.


Figure 5. Rake receiver suitable for use in a base station receiver.

In the typical Rake, Figure 5, received signals from the antenna array in a multi-sectored site are all downconverted, amplified and converted to digital form in a bank of A/D converters. A selected subset of the sampled antenna signals is presented to each Rake. One Rake is provided for each active traffic channel. A Rake contains, typically, four correlation receivers and a searcher. The searcher also contains one or more correlators. The searcher continuously scans the delay hypothesis window, typically several microseconds, looking for signal energy from the client mobile. Detections are noted, and the correlation receivers are assigned to the strongest four.

Multiple search correlators may be necessary because the signal components being sought are so weak and because the multipath constantly changes due mobile motion. Closed loop power control maintains the received SNR at a very low value for the combined signal. Each of the multipath components has an even lower SNR. Because of the very low SNRs, relatively long time integration is necessary for reliable detection. Multiple correlators, searching simultaneously, thus may be needed for adequate speed.

Another way to look at the bandwidth trade-off for CDMA is to consider the fact that the vector sum of two randomly phased signals is larger than either of the components with probability 65%. This is true for both equal amplitudes and independent Rayleigh-distributed amplitudes. This is important when selecting the spreading bandwidth. Too wide a bandwidth will resolve more multipath components than a more optimal, narrower bandwidth. The resolved components will, with at least this 65% probability, be weaker than the noncoherent vector sum that would be seen by the narrower system. This suggests that there can be too much of a good thing: multipath resolution. In fact, with a 1.25 MHz bandwidth, the three correlator Rake typically used in mobile receivers is already at the point of diminishing returns. The benefit of going beyond three-way diversity is very small, especially when the added diversity is of the same kind as the first three.

Frequency Allocation

One of the more interesting problems with fielding a wide bandwidth system is obtaining an adequate frequency allocation. In the United States, the FCC has allocated a total of 50 MHz in the 800 MHz band for cellular systems. The allocation is split in each service area between two competing carriers, the A (non-wireline) and B (wireline). Initially, the FCC had only allocated 40 MHz, 20 MHz to each carrier and then added 10 MHz later as cellular system proved themselves. This expansion was done in a complicated way, however. Initially, 825-835 MHz was allocated to A and 835-845 MHz was allocated to B for mobile transmit, with corresponding allocations from 870-890 MHz for base transmit. The expansion gave A the bands 824-825 MHz and 845-846.5 MHz; B was given 846.5-849 MHz, with corresponding upper band allocations.
Figure 6. Cellular spectrum allocation.

The problem this causes CDMA systems is apparent. The A' segment will contain at most a single 1.25 MHz bandwidth CDMA channel; the B' segment will contain at most two CDMA channels. Even that is possible only if the two CDMA channels are overlapped slightly, entailing some loss of total capacity. So, if all portions of the cellular spectrum are to be able to be utilized for CDMA, a 1.25 MHz bandwidth is the maximum possible bandwidth for a non-overlay approach. While the situation in the United States is somewhat unique, it is not unusual around the world to find similar situations where very wide bandwidths simply cannot be given exclusive frequency allocations. Typically, existing systems are not very tolerant to a new system sharing the spectrum with them because they were not designed to do so. Hence it is desirable to use relatively narrow bandwidth CDMA in order to minimize the possibility of interference to existing services and to minimize the amount of unusable spectrum allocation.

In the United States there is also a problem of transition. Most cellular carriers will add CDMA service to existing AMPS systems. With IS-95A, the approach to transition is to convert the spectrum in 1.25 MHz segments. The operator would turn off the AMPS radios in about 1.8 MHz (the 1.25 MHz bandwidth of a single CDMA channel, plus guard bands). The CDMA system would then begin operating in this cleared segment. The reduction in capacity of the AMPS system could be handled by increasing the number of AMPS base stations to offset the capacity loss. However if heavy users of the AMPS system can be provided with dual-mode subscriber units before cutover, then the AMPS build-out can be avoided. Distribution of the dual mode units can take place over a prolonged period of time. Initially the dual mode units will operate on the AMPS system. Subsequent to cutover they will seek out the CDMA system, where available, thus relieving the AMPS system of sufficient load to more than compensate for the small reduction in its capacity.


It should be clear that increasing spread bandwidth can only result in increased complexity of the systems. For example, increasing the spread bandwidth a factor of 8 from 1.25 MHz to 10 MHz will require the receiver correlators to operate 8 times faster. This increased speed will require both more silicon area and greater power consumption. In addition, many more correlators are required as shown above in order to adequately process the multipath signals.

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