1. Advanced Wireless Networks
Lecture 4: Multi-Channel and Multi-Carrier Systems
Multi-Channel Communication in Systems with AWGN
We consider multi-channel signaling over fixed channels that differ only in
attenuation and phase shift. In such a model, the signal waveforms are
generally can be expressed as:
The equivalent baseband signals received from the L channels can be expressed
in the following form: .
The decision variables of the coherent detection can be found via the correlation
metrics:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

2. Multi-Channel Communication in Systems with AWGN (cont.)
In non-coherent detection, no attempt is made to estimate the channel parameters. The demodulator may base its decision either on the sum of the envelopes (envelope decision) or the sum of the squared envelopes (square-law detection). We continue our attention to square-law detection of the received signals of the L channels, which produces the decision variables
An error is occurred if
, or equivalently, if the difference
For non-coherent detection, this difference can be expressed as
with
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

3. Multi-Channel Communication in Systems with AWGN (cont.)
For coherent detection, the difference D can be expressed as
where
Now we will find the probability that the general quadratic form in complex-valued
Gaussian variables is less than zero, i.e.,
This probability is the probability of error of binary multi-channel signaling in AWGN. In coherent PSK, the probability of error takes the simple form:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

4. Multi-Channel Communication in Systems with AWGN (cont.)
If channels are all identical, i.e., for all n, we get:
EL is the total transmitted signal energy for the L signals, that is, above formula shows
that the receiver combines the energy from the L channels in an optimum manner without
any loss during division of the total transmitted signal energy among the L channels.
The same performance is obtained as in the case in which a single waveform having
energy EL.
Here
Other extreme case, is binary DPSK, the probability of error of which can be presented
in the following form:
is transmitted on one channel.
Figure shows the probability from square-law non-coherent combining of the L signals as a function of L for various . As follows from illustrations, it is easily to obtain a form of the curve as a function of , i.e.,
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

5. Characterization of Multipath Channels with Fading
Usually researchers dealt only with classical AWGN channels which are not time-varied. Other situation will occur in ionospheric channels at 3-30 MHz due to scattering from ionosphere, as a time-varying media:
Another example is cellular land communication, stationary and
mobile at 300 MHz-3 GHz due to multipath propagation signals
Base
station
Subscriber
(cellphone)
So, the response of the channel becomes time-dependent and the total signal is now
corrupted not only by Gaussian white noise n(t), but also by multiplicative noise due
to fading, fast and slow:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

6. Characterization of Multipath Channels with Fading (cont.)
Time-invariant (stationary) channel:
The corresponding parameters of fading
obtained empirically are shown in Table:
Channel HF
Ultrasound Underwater
SHF
[Hz]
0.5 5 10
Time-varied (time-dispersive) channel:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

7. Performance of the AWGN Channel without Fading
The received signal in time t, s(t), is then given by
where n(t) is the noise waveform, g(t) is the modulated signal and
A is overall path loss, assumed not to vary in time. The power of
Gaussian noise is:
The signal-to-noise ratio (SNR) at the input of the receiver is then:
or in terms of the corresponding SNR per bit:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

8. Performance of the AWGN Channel without Fading (cont.)
Let us now calculate the bit error rate (BER) performance for binary phase shift
keying (BPSK) signals in AWGN channel. We, first of all, will consider the 2-D
(two-dimensional) case of BPSK signals, where two signals correspond to a
binary 1 and 0. Their complex baseband presentation is
The error rate performance of digital modulation scheme in AWGN channel with
depends on the Euclidean distance d between the transmitted waveforms,
and is determined by the probability of error:
or accounting for Q-function
and for
will get
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

9. Performance of the Channel with Flat Rayleigh Fading
Since the fading varies with time, the SNR at the input of the receiver also varies with time. It is necessary, in contrast with AWGN case, to distinguish between the instantaneous SNR, , and the mean SNR, denoted as .
Then, the signal r(t) can be presented as:
where is the complex fading coefficient at time t.
If the fading is assumed constant over the transmitted symbol duration,
then is also constant over a symbol and SNR is given by:
Then the average SNR for flat fading, having unit variance, equals:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

10. Performance of the Channel with Flat Rayleigh Fading
For Rayleigh distribution the PDF of error equals:
Here
where in slow flat fading channel, where
If so, for the cumulative density function CDF we have:
The average bit error probability for BPSK signal can be defined as:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

11. The average bit error probability for differential phase shift keying:
Performance of the Channel with Flat Rayleigh Fading (cont.)
The average bit error probability for coherent binary frequency shift keying
(FPSK) signal
The average bit error probability for differential phase shift keying:
BER for various modulations vs.
in a flat slow fading channel with respect to
that for AWGN.
and for incoherent binary FPSK
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.