4.3 Consequences for Bit Error Rate

4.3 Consequences for Bit Error Rate

This section demonstrates a dramatic difference in performance between fading channels and static additive white Gaussian noise (AWGN) channels. We will see that the behaviour is determined almost exclusively by deep fades.

The question: what is the expected bit error rate (BER) if we place a mobile in a randomly selected position? Equivalently, what is the average BER as the mobile moves through the random field? Also, what is the distribution of the errors? To simplify the discussion, assume:

flat fading, so that the channel does not introduce intersymbol interference;

slow fading, so that the primary effect is an SNR that varies slowly compared to the bit rate; i.e., we can ignore the pulse distortion and phase hits that occur in deep fades;

Rayleigh, not Rice, fading.

Clearly, the instantaneous BER depends on the magnitude r of the complex channel gain, which has a Rayleigh distribution . However, it's easier to work with squared amplitude, z = r², or z = |g|², where g is the complex gain, because it has a simple exponential distribution . Denote the instantaneous SNR as g_b=E_b/N_o. where E_b is the instantaneous received energy per bit and N_o is the white noise power spectral density. If we normalize z to have a mean of 1 and denote the expected SNR by G_b, then

As a first step, we'll run a quick simulation to establish an important property of communication in fading channels: errors occur in bursts. In mobile communications, we have long intervals - in fact, most of the time - during which the SNR is very high and there are no errors at all. This good behaviour is punctuated by deep fades, which often produce clusters of several errors. For demonstration purposes, we'll assume differential phase shift keying (DPSK), for which the instantaneous BER is given by

The complex channel gain as a function of time (or position) is produced by the Jakes complex gain generator in Appendix B . Initialize it and choose the mean SNR below:

Initialize the generator:

Choose the mean SNR:

converts from dB

Calculate instantaneous BER P_e(z,G_b) from (4.3.2):

To see other examples, reinitialize the Jakes generator: click the highlighted equation, then press F9.

This is a striking illustration of the burstiness of error patterns in mobile communications.

The fact that errors occur in bursts has a profound effect on the performance of coded systems, whether they are designed for error correction or for error detection and retransmission. However, for many purposes, we are interested in the average BER, and that is what we will now calculate.

To obtain the average BER, we take the expectation of the instantaneous BER P_e(z) with respect to z, calculated as

where p_z(z) is the exponential pdf of z. Because the P_e(z) function is very nonlinear, the average P_{e_av} ¹ Pe(z__av), and we will have to compute (4.3.2) explicitly. Again for simplicity of discussion, assume that we transmit differential PSK and detect it differentially, so that the instantaneous BER is given by (4.3.2). Performing the expectation, we have

which gives

which is asymptotic to

Now compare static channel (4.3.2) and fading channel (4.3.5) BERs with the same average SNR. The static channel BER equals the instantaneous BER with the channel gain set to 1:

Plot parameters:

We see a major difference in behaviour -- at 10^-4, there is roughly 30 dB between the curves! Putting it another way:

On a static channel, a 3 dB increase in SNR squares the BER (e.g., from 10^-3 to 10^-6).

On a fading channel, a 3 dB increase in SNR only halves BER (e.g., from 10^-3 to 0.5x10^-3)

Why is there such a drastic difference between static and fading channels? The reason is straightforward:

P_e(x) is strongly nonlinear, so BER at low SNR is much greater than at high SNR.

When averaging numbers (e.g., BERs) with large dynamic range, such as 10^-6, 10^-7, 10^-5, 10^-6, 10^-2, 10^-5, 10^-6, the sum is determined primarily by the very large values and their probability of occurrence.

So deep fades dominate the behaviour, and the integrands in (4.3.3) and (4.3.4) are significant only for values of z much less than the mean. How likely are those low values? Remember the rule of thumb from Section 4.2 : from the deep fade asymptote for Rayleigh fading, the probability of being below a specific SNR level decreases only inversely with increasing average power. That's why BER decreases as 1/2G_b on Rayleigh channels, instead of exponentially, as it does on static channels.

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